path: root/other/burneye2/doc-external/hash/goldreich97tiny.ps

%!PS-Adobe-2.0
%%Creator: dvips 5.519 Copyright 1986, 1993 Radical Eye Software
%%Title: rev.dvi
%%CreationDate: Fri Mar 21 13:57:13 1997
%%Pages: 26
%%PageOrder: Ascend
%%BoundingBox: 0 0 612 792
%%EndComments
%DVIPSCommandLine: dvips -o tiny.ps rev
%DVIPSSource:  TeX output 1997.03.21:1356
%%BeginProcSet: tex.pro
/TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N
/X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72
mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1}
ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale
isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div
hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul
TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get
round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{
statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0
0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{/nn 8 dict N nn
begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X
array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo
setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{/sf 1 N /fntrx
FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]N df-tail}B /E{
pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}
B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup
length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B
/ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type
/stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp
0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2
index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff
ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice
ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]{
ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]}if nn
/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1
sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{cc 1 add D
}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0
moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add
.99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage
userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook
known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X
/IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for
65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0
0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V
{}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7
getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false}
ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley
false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley
scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave
transform round exch round exch itransform moveto rulex 0 rlineto 0
ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta
0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}
B /c{-4 M}B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{
3 M}B /k{4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p
-1 w}B /q{p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{
3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end
%%EndProcSet
TeXDict begin 40258431 52099146 1000 300 300
(/a/oded/PAPERS/AVI/TINY/rev.dvi) @start /Fa 3 113 df<60F0F06004047D8A0B
>1 D<00080000000C0000000C0000000C0000000C0000000C0000000C0000000C000000
0C0000000C0000000C0000000C0000FFFFFF80FFFFFF80000C0000000C0000000C000000
0C0000000C0000000C0000000C0000000C0000000C0000000C0000FFFFFF80FFFFFF8019
1A7E981E>6 D<00000002000000060000000C0000000C00000018000000180000003000
0000300000006000000060000000C0000000C00000018000000180000003000000030000
0006000000060000000C000C000C003C0018004E0018008E003000070030000700600007
0060000380C0000380C00001C1800001C1800000E3000000E30000007600000076000000
3C0000003C000000180000001800001F267E8120>112 D E /Fb
2 49 df<FFFEFFFE0F027D8516>0 D<181818303030606060C0C0050B7E8B09>48
D E /Fc 9 62 df<04081030206040C0C0C0C0C0C0C0C04060203010080406167D8F0B>
40 D<804020301018080C0C0C0C0C0C0C0C0818103020408006167E8F0B>I<00C00000C0
0000C00000C00000C00000C00000C00000C000FFFF80FFFF8000C00000C00000C00000C0
0000C00000C00000C00000C00011127E8D15>43 D<18F818181818181818181818FF080D
7D8C0E>49 D<3E00418080C0C0C000C000C0018003000400084030407F80FF800A0D7E8C
0E>I<3E0041806180018003001E00018000C000C0C0C0C0C041803E000A0D7E8C0E>I<20
803F003C00200020003F00218000C000C0C0C080C041803E000A0D7E8C0E>53
D<0F00118021806000C000DE00E180C0C0C0C0C0C060C021801E000A0D7E8C0E>I<7FFF
00FFFF80000000000000000000000000FFFF807FFF0011087E8815>61
D E /Fd 22 119 df<C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0
021B7A800E>12 D<C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0
C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C00A1B7A8017>I<000600
0C001800300070006000C001C0018003800300070006000E000C001C001C001800380038
0038003000700070007000700070007000E000E000E000E000E000E000E000E000E000E0
00E000E000E000E000E000E000E000E00070007000700070007000700030003800380038
0018001C001C000C000E000600070003000380018001C000C00060007000300018000C00
060F4A788119>16 D<C0006000300018001C000C000600070003000380018001C000C000
E000600070007000300038003800380018001C001C001C001C001C001C000E000E000E00
0E000E000E000E000E000E000E000E000E000E000E000E000E000E000E001C001C001C00
1C001C001C0018003800380038003000700070006000E000C001C0018003800300070006
000C001C00180030006000C0000F4A7F8119>I<0000300000600000C000018000030000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>I<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>I<0000700001F00003C0000780000E00001C000038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>26
D<E00000F000007C00001E00000F000007000003800001C00001C00000E00000E00000E0
0000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E0
0000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E0
0000E00000E00000E00000E000007000007000003800003800001C00000E000007000003
C00000F00000700000E00003C0000700000E00001C0000380000380000700000700000E0
0000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E0
0000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E0
0000E00000E00000E00000E00000E00000E00001C00001C0000380000700000F00001E00
007C0000F00000E0000014637B811F>I<0000180000300000600000E00000C000018000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>32
D<C000006000003000003800001800000C00000E000007000003000003800001800001C0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>I<00000C00001800003800003000006000
00E00000C00001C0000380000380000700000700000E00000E00001C00001C0000380000
380000780000700000F00000F00000E00001E00001E00003C00003C00003C00003C00007
80000780000780000F80000F00000F00000F00001F00001F00001E00001E00001E00003E
00003E00003E00003E00003C00003C00003C00007C00007C00007C00007C00007C00007C
00007C00007C0000780000780000F80000F80000F80000F80000F80000F80000F80000F8
0000F80000F80000F80000F80000F80000F80000F80000F80000F80000164B748024>48
D<C000006000007000003000001800001C00000C00000E00000700000700000380000380
0001C00001C00000E00000E000007000007000007800003800003C00003C00001C00001E
00001E00000F00000F00000F00000F000007800007800007800007C00003C00003C00003
C00003E00003E00001E00001E00001E00001F00001F00001F00001F00000F00000F00000
F00000F80000F80000F80000F80000F80000F80000F80000F800007800007800007C0000
7C00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C0000
7C00007C00007C00007C164B7F8024>I<F80000F80000F80000F80000F80000F80000F8
0000F80000F80000F80000F80000F80000F80000F80000F80000F80000F8000078000078
00007C00007C00007C00007C00007C00007C00007C00007C00003C00003C00003C00003E
00003E00003E00003E00001E00001E00001E00001F00001F00000F00000F00000F00000F
800007800007800007800003C00003C00003C00003C00001E00001E00000E00000F00000
F000007000007800003800003800001C00001C00000E00000E0000070000070000038000
03800001C00000C00000E000006000003000003800001800000C164B748224>64
D<00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C0000
7C00007C00007C00007C00007C00007C0000780000780000F80000F80000F80000F80000
F80000F80000F80000F80000F00000F00000F00001F00001F00001F00001F00001E00001
E00001E00003E00003E00003C00003C00003C00007C0000780000780000780000F00000F
00000F00000F00001E00001E00001C00003C00003C0000380000780000700000700000E0
0000E00001C00001C0000380000380000700000700000E00000C00001C00001800003000
00700000600000C00000164B7F8224>I<FFFFFFFFE0FFFFFFFFF07000001FF078000001
F03C000000781C000000180E0000000C0F000000040700000004038000000203C0000000
01E000000000E0000000007000000000780000000038000000001C000000001E00000000
0F000000000700000000038000000003800000000300000000070000000006000000000C
000000001800000000380000000030000000006000000000C000000001C0000000018000
0002030000000406000000040E0000000C0C00000018180000007830000001F07000001F
F07FFFFFFFF0FFFFFFFFE0272A7E7F2C>80 D<FFFFFFFFFFFFC0FFFFFFFFFFFFE07F0000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>88
D<00000000020000000006000000000C000000000C000000001800000000180000000030
00000000300000000060000000006000000000C000000000C00000000180000000018000
000003000000000300000000060000000006000000000C000000000C0000000018000000
00180000000030000000003000000000600008000060001C0000C0003C0000C000CE0001
80000E000180000E0003000007000300000700060000038006000003800C000001C00C00
0001C018000001E018000000E030000000E0300000007060000000706000000038C00000
0038C00000001D800000001D800000001F000000000F000000000E000000000600000027
327C812A>112 D<000000000200000000060000000006000000000C000000000C000000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>I<000000000200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>115 D<0000018000000180000001800000018000000180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>I<C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0021B64802C>I<
7FFF80FFFF80C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000
C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000
C00000C00000111A64812C>I E /Fe 18 122 df<FFC0FFC0FFC00A037F8B0F>45
D<FFF800FFFF00FFFF80F00FC0F003E0F001E0F000F0F000F0F000F0F000F0F000F0F001
E0F003C0F01F80FFFF00FFFF00FFFF80F007E0F001E0F000F0F00078F00078F00078F000
78F00078F00078F000F0F001F0F007E0FFFFC0FFFF80FFFC0015207B9F1E>66
D<07E03FF87FFC701E401F000F000F000F003F07FF1FFF7E0FF80FF00FF00FF00FF83F7F
FF3FEF1F8F10147E9316>97 D<F00000F00000F00000F00000F00000F00000F00000F000
00F00000F00000F00000F00000F1F000F7FC00FFFE00FC3E00F80F00F00F00F00780F007
80F00780F00780F00780F00780F00780F00F00F00F00F81F00FC3E00FFFC00F7F800F1E0
0011207D9F17>I<00078000078000078000078000078000078000078000078000078000
078000078000078007C7800FF7801FFF803E1F807C0780780780F80780F00780F00780F0
0780F00780F00780F00780F00780780780780F803E1F801FFF800FF78007C78011207E9F
17>100 D<03F0000FFC001FFE003E1F003C0700780700700380FFFF80FFFF80FFFF80F0
0000F00000F000007000007800003C01003E07001FFF0007FE0001F80011147F9314>I<
007E01FE03FE078007000F000F000F000F000F000F000F00FFF0FFF0FFF00F000F000F00
0F000F000F000F000F000F000F000F000F000F000F000F000F000F000F20809F0E>I<F0
F0F0F00000000000000000F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F004207D9F0B
>105 D<F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0F0
04207D9F0B>108 D<F0FC07E0F3FE1FF0F7FF3FF8FE0FF07CF807C03CF807C03CF00780
3CF007803CF007803CF007803CF007803CF007803CF007803CF007803CF007803CF00780
3CF007803CF007803CF007803CF007803C1E147D9327>I<F1F8F3FCF7FEFC1FF80FF80F
F00FF00FF00FF00FF00FF00FF00FF00FF00FF00FF00FF00FF00FF00F10147D9317>I<01
F80007FE001FFF803F0FC03C03C07801E07801E0F000F0F000F0F000F0F000F0F000F0F0
00F07801E07801E03C03C03F0FC01FFF8007FE0001F80014147F9317>I<F1F000F7FC00
FFFE00FC3E00F81F00F00F00F00F80F00780F00780F00780F00780F00780F00780F00F00
F00F00F81F00FC3E00FFFC00F7F800F1E000F00000F00000F00000F00000F00000F00000
F00000F00000F00000111D7D9317>I<F0E0F3E0F7E0FF00FC00FC00F800F800F000F000
F000F000F000F000F000F000F000F000F000F0000B147D9310>114
D<07F01FFC3FFC780C7800780078007C003FC01FF00FF803F8007C003C003CC03CF07CFF
F87FF00FC00E147F9311>I<1E001E001E001E001E001E00FFF0FFF0FFF01E001E001E00
1E001E001E001E001E001E001E001E001E001E001E201FF00FF007C00C1A7F9910>I<78
01E07C03C03E07801E0F000F0F00079E0003FC0003F80001F80000F00001F00001F80003
FC00079E000F0F000E0F001E07803C03C07801E0F801F01414809315>120
D<F003C0F003C07807807807807C07803C0F003C0F001E0F001E1E000E1E000F1C000F1C
00073C0007380003B80003B80003B00001F00001F00000E00000E00001C00001C00001C0
000380000780007F00007E00007C0000121D7F9315>I E /Ff 4
111 df<40E060202040408003087D8209>59 D<040C0000000000705898983030606464
683006127E910B>105 D<0020002000000000000000000000038004C008C008C000C001
8001800180018003000300030003004600CC0078000B1780910D>I<71F09A189C189818
18183030303030323062606460380F0B7E8A13>110 D E /Fg 11
117 df<00F00308060406000C000C000E000F000F8007C003E006F01C70383030307010
6010E010E030E030E020C060E040608031001E000E1A7E9911>14
D<60F0F07010101020204080040B7D830B>59 D<0004000C001800180018003000300030
00600060006000C000C000C00180018001800300030003000600060006000C000C000C00
180018001800300030003000600060006000C000C0000E257E9B13>61
D<01FF80FF0038003800380060003800400070010000700200007004000070080000E020
0000E0400000E0800000E1C00001C5C00001C9C00001D0E00001C0E00003807000038070
0003807000038038000700380007001C0007001C0007001C000E001E00FFE0FF80201A7E
9921>75 D<01FFC000380000380000380000700000700000700000700000E00000E00000
E00000E00001C00001C00001C00001C00003800003800803800803801007001007003007
00200700600E01C0FFFFC0151A7E991A>I<01FC0007F0003C000F00003C000F00003C00
1700005C002E00004E002E00004E004E00004E004E00008E009C00008E011C00008E011C
00008E021C00010E023800010E043800010E083800010E08380002071070000207107000
02072070000207407000040740E000040780E000040780E0000C0700E0001C0601C000FF
861FFC00241A7E9925>I<3FFFFF80380E0180200E0080400E0080401C0080801C008080
1C0080801C00800038000000380000003800000038000000700000007000000070000000
70000000E0000000E0000000E0000000E0000001C0000001C0000001C0000001C0000003
C000007FFE0000191A7F9916>84 D<03000380030000000000000000000000000000003C
004E004E008E008E009C001C001C0038003800390071007100720072003C00091A7E990D
>105 D<0004000E000C000000000000000000000000000000E001300238043804380438
007000700070007000E000E000E000E001C001C001C001C003806380E700C60078000F21
809910>I<383C004CC6008F07008E07008E07008E07001C0E001C0E001C0E001C1C0038
1C40381C40383840383880701900300E0012107E8F17>110 D<01800380038003800380
07000700FFE007000E000E000E000E001C001C001C001C00384038403840388019000E00
0B177F960E>116 D E /Fh 12 117 df<0F00308460C440E8C070C070C06041E03E380E
097E8813>11 D<40E0602020408003077D820A>59 D<004000C001800180018003000300
060006000C000C000C001800180030003000600060006000C000C0000A157E8F0F>61
D<01F8000E0E00180300300100600180C00180C00180C00180C00300C00300C00600600C
003030001FC000110E7E8D16>79 D<0FFF0E06080C08181030006000C0018003040E0418
04300C6018FFF8100E7E8D14>90 D<0F8030C060804000C000C000C04061803E000A097E
880D>99 D<0808000000007098B0303060646870060F7D8E0B>105
D<7800180018001800300030C03360344078007E0063006320C340C1800B0E7E8D10>
107 D<F030303060606060C0C0C0D0D0E0040E7E8D0A>I<73C7009C68809830C0386180
30618030618030631060C32060C1C014097D8819>I<73809C40986030C030C030C03188
619060E00D097D8812>I<0C0C0C18FE1818303030323438070D7E8C0C>116
D E /Fi 11 113 df<FFFFC0FFFFC012027D871A>0 D<40E04003037D880A>I<040E0E1C
1C1C38383070706060C0C0070F7F8F0A>48 D<03FC0FFC1C003000600060006000C000C0
00FFFCFFFCC000C00060006000600030001C000FFC03FC0E147D9016>50
D<0003000300060006000C0018001800300030006000C000C0018001800300030006000C
000C0018001800300060006000C0004000101A7C9300>54 D<FFFFE0FFFFE000C00000C0
0000C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C0
0000C00000C00000C00000400013147D931A>62 D<0000001E0000007E000000FC003000
E000700080007801000078010000780100007C020000BC0200009C0200009E0200009E04
00008F0400010F04000107040001078800020788000203C8000603C8008401F000FC01F0
00F800F000780060001F1880961B>78 D<00E00300060006000600060006000600060006
000600060006001C00F0001C000600060006000600060006000600060006000600060003
0000E00B1D7E9511>102 D<F0001C000600060006000600060006000600060006000600
0600030000E00300060006000600060006000600060006000600060006001C00F0000B1D
7E9511>I<C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0021D
7D950A>106 D<00000040000000C0000001800000018000000300000003000000060000
00060000000C000000180000001800000030000000300000006000000060003000C000D8
00C000180180000C0180000C0300000603000006060000030C0000030C00000198000001
98000000F0000000F0000000600000006000001A1E7F811B>112
D E /Fj 8 56 df<00C00000C00000C00000C00000C00000C00000C00000C00000C000FF
FF80FFFF8000C00000C00000C00000C00000C00000C00000C00000C00000C00011147E8F
17>43 D<0C003C00CC000C000C000C000C000C000C000C000C000C000C000C000C00FF80
09107E8F0F>49 D<1F00618040C08060C0600060006000C00180030006000C0010202020
7FC0FFC00B107F8F0F>I<1F00218060C060C000C0008001800F00008000400060C060C0
60804060801F000B107F8F0F>I<0300030007000F000B001300330023004300C300FFE0
03000300030003001FE00B107F8F0F>I<20803F002C002000200020002F003080204000
6000600060C06080C061801F000B107F8F0F>I<0780184030C060C06000C000CF00F080
E040C060C060C060406060C030801F000B107F8F0F>I<40007FE07FC080808080010002
00040004000C0008000800180018001800180018000B117E900F>I
E /Fk 28 122 df<00600000600000600000F00000F000017800013800013800021C0002
1C00021C00040E00040E00080F000807000807001007801003803803C0FC0FF014147F93
17>3 D<01FC000707001C01C03800E03000607000707000707000707000703000603800
E01800C01800C00C01808C0188840108C603187E03F07E03F07E03F015147F9318>10
D<01020408103020606040C0C0C0C0C0C0C0C0C0C040606020301008040201081E7E950D
>40 D<80402010080C0406060203030303030303030303020606040C0810204080081E7E
950D>I<006000006000006000006000006000006000006000006000006000006000FFFF
F0FFFFF00060000060000060000060000060000060000060000060000060000060001416
7E9119>43 D<0F0030C0606060604020C030C030C030C030C030C030C030C030C0304020
6060606030C00F000C137E9211>48 D<0C001C00EC000C000C000C000C000C000C000C00
0C000C000C000C000C000C000C000C00FFC00A137D9211>I<1F0060C06060F070F03060
3000700070006000C001C00180020004000810101020207FE0FFE00C137E9211>I<0FC0
30707038703870380038003000E00FC0007000380018001C601CF01CF018E03860701FC0
0E137F9211>I<006000E000E00160026006600C600860106020606060C060FFFC006000
6000600060006003FC0E137F9211>I<60607FC07F8044004000400040004F0070C040E0
006000700070E070E070E06040E021C01F000C137E9211>I<07C00C2010702070600060
00C000CF00D0C0E060C020C030C030C03040306020206010C00F000C137E9211>I<4000
7FFC7FF8401080108020004000800100010003000200060006000E000E000E000E000E00
04000E147E9311>I<0FC0107020186018601870183C303F600F800FE031F06078C01CC0
0CC00CC00C601830300FC00E137F9211>I<0F00308060404060C020C030C030C0304030
607030B00F30003000200060E040E08041003E000C137E9211>I<7FFFE0FFFFF0000000
000000000000000000000000000000FFFFF07FFFE0140A7E8B19>61
D<F00030003000300030003000300033E034103808300C30063006300630063006300C38
08343023E00F147F9312>98 D<00780018001800180018001800180F98187820386018C0
18C018C018C018C0186018203810580F9E0F147F9312>100 D<0F80104020206030C010
FFF0C000C000C0006000201018200FC00C0D7F8C0F>I<03C00CE018E018401800180018
00FF00180018001800180018001800180018001800180018007F000B1480930A>I<0F3C
30E62040606060606060204030C02F00600060003FE03FF06018C00CC00CC00C60183030
0FC00F147F8C11>I<F00030003000300030003000300033E03430381830183018301830
1830183018301830183018FC7E0F147F9312>I<F0303030303030303030303030303030
303030FC06147F9309>108 D<0FC0186020106018C00CC00CC00CC00CC00C6018601838
700FC00E0D7F8C11>111 D<F3E034303808300C30063006300630063006300C38083430
33E030003000300030003000FC000F137F8C12>I<3E806180C080C080E0007E003F8003
C080C080C0C0C0E1809F000A0D7F8C0D>115 D<10001000100030007000FF8030003000
3000300030003000300030803080308011000E0009127F910D>I<F87C30183010182018
2018200C400C400680068007800300030002000200E600E400E80070000E137F8C11>
121 D E /Fl 50 123 df<000FF83F00007FFDFFC001F81FE3E003E03F87E007C03F87E0
0F803F07E00F803F03C00F801F00000F801F00000F801F00000F801F00000F801F00000F
801F0000FFFFFFFC00FFFFFFFC000F801F00000F801F00000F801F00000F801F00000F80
1F00000F801F00000F801F00000F801F00000F801F00000F801F00000F801F00000F801F
00000F801F00000F801F00000F801F00000F801F00000F801F00000F801F00007FF0FFF0
007FF0FFF00023237FA221>11 D<387CFEFFFF7F3B03030706060C1C18702008117C8610
>44 D<387CFEFEFE7C3807077C8610>46 D<00180000780001F800FFF800FFF80001F800
01F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F800
01F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F800
7FFFE07FFFE013207C9F1C>49 D<03FC000FFF003C1FC07007E07C07F0FE03F0FE03F8FE
03F8FE01F87C01F83803F80003F80003F00003F00007E00007C0000F80001F00003E0000
380000700000E01801C0180380180700180E00380FFFF01FFFF03FFFF07FFFF0FFFFF0FF
FFF015207D9F1C>I<00FE0007FFC00F07E01E03F03F03F03F81F83F81F83F81F81F03F8
1F03F00003F00003E00007C0001F8001FE0001FF000007C00001F00001F80000FC0000FC
3C00FE7E00FEFF00FEFF00FEFF00FEFF00FC7E01FC7801F81E07F00FFFC001FE0017207E
9F1C>I<0000E00001E00003E00003E00007E0000FE0001FE0001FE00037E00077E000E7
E001C7E00187E00307E00707E00E07E00C07E01807E03807E07007E0E007E0FFFFFEFFFF
FE0007E00007E00007E00007E00007E00007E00007E000FFFE00FFFE17207E9F1C>I<10
00201E01E01FFFC01FFF801FFF001FFE001FF8001BC00018000018000018000018000019
FC001FFF001E0FC01807E01803E00003F00003F00003F80003F83803F87C03F8FE03F8FE
03F8FC03F0FC03F07007E03007C01C1F800FFF0003F80015207D9F1C>I<001F8000FFE0
03F07007C0F00F01F81F01F83E01F83E01F87E00F07C00007C0000FC0800FC7FC0FCFFE0
FD80F0FF00F8FE007CFE007CFC007EFC007EFC007EFC007E7C007E7C007E7C007E3C007C
3E007C1E00F80F00F00783E003FFC000FF0017207E9F1C>I<0000700000000070000000
00F800000000F800000000F800000001FC00000001FC00000003FE00000003FE00000003
FE00000006FF000000067F0000000E7F8000000C3F8000000C3F800000183FC00000181F
C00000381FE00000300FE00000300FE00000600FF000006007F00000E007F80000FFFFF8
0000FFFFF800018001FC00018001FC00038001FE00030000FE00030000FE000600007F00
0600007F00FFE00FFFF8FFE00FFFF825227EA12A>65 D<FFFFFF8000FFFFFFE00007F001
F80007F000FC0007F0007E0007F0007E0007F0007F0007F0007F0007F0007F0007F0007F
0007F0007F0007F0007E0007F000FE0007F000FC0007F003F80007FFFFF00007FFFFF000
07F001FC0007F0007E0007F0003F0007F0003F8007F0001F8007F0001FC007F0001FC007
F0001FC007F0001FC007F0001FC007F0001FC007F0003F8007F0003F8007F0007F0007F0
01FE00FFFFFFF800FFFFFFC00022227EA128>I<0003FE0080001FFF818000FF01E38001
F8003F8003E0001F8007C0000F800F800007801F800007803F000003803F000003807F00
0001807E000001807E00000180FE00000000FE00000000FE00000000FE00000000FE0000
0000FE00000000FE00000000FE000000007E000000007E000001807F000001803F000001
803F000003801F800003000F8000030007C000060003F0000C0001F800380000FF00F000
001FFFC0000003FE000021227DA128>I<FFFFFF8000FFFFFFF00007F003FC0007F0007E
0007F0003F0007F0001F8007F0000FC007F00007E007F00007E007F00007F007F00003F0
07F00003F007F00003F007F00003F807F00003F807F00003F807F00003F807F00003F807
F00003F807F00003F807F00003F807F00003F807F00003F007F00003F007F00003F007F0
0007E007F00007E007F0000FC007F0001F8007F0003F0007F0007E0007F003FC00FFFFFF
F000FFFFFF800025227EA12B>I<FFFFFFFCFFFFFFFC07F000FC07F0003C07F0001C07F0
000C07F0000E07F0000E07F0000607F0180607F0180607F0180607F0180007F0380007F0
780007FFF80007FFF80007F0780007F0380007F0180007F0180007F0180307F0180307F0
000307F0000607F0000607F0000607F0000E07F0000E07F0001E07F0003E07F001FCFFFF
FFFCFFFFFFFC20227EA125>I<0003FE0040001FFFC0C0007F00F1C001F8003FC003F000
0FC007C00007C00FC00003C01F800003C03F000001C03F000001C07F000000C07E000000
C07E000000C0FE00000000FE00000000FE00000000FE00000000FE00000000FE00000000
FE00000000FE000FFFFC7E000FFFFC7F00001FC07F00001FC03F00001FC03F00001FC01F
80001FC00FC0001FC007E0001FC003F0001FC001FC003FC0007F80E7C0001FFFC3C00003
FF00C026227DA12C>71 D<FFFF83FFFEFFFF83FFFE07F0001FC007F0001FC007F0001FC0
07F0001FC007F0001FC007F0001FC007F0001FC007F0001FC007F0001FC007F0001FC007
F0001FC007F0001FC007F0001FC007FFFFFFC007FFFFFFC007F0001FC007F0001FC007F0
001FC007F0001FC007F0001FC007F0001FC007F0001FC007F0001FC007F0001FC007F000
1FC007F0001FC007F0001FC007F0001FC007F0001FC007F0001FC0FFFF83FFFEFFFF83FF
FE27227EA12C>I<FFFFE000FFFFE00007F0000007F0000007F0000007F0000007F00000
07F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0000007F00000
07F0000007F0000007F0000007F0000007F0000007F0001807F0001807F0001807F00018
07F0003807F0003807F0007007F0007007F000F007F001F007F007F0FFFFFFF0FFFFFFF0
1D227EA122>76 D<FFF000000FFFFFF800001FFF07F800001FE006FC000037E006FC0000
37E006FC000037E0067E000067E0067E000067E0063F0000C7E0063F0000C7E0061F8001
87E0061F800187E0060FC00307E0060FC00307E0060FC00307E00607E00607E00607E006
07E00603F00C07E00603F00C07E00601F81807E00601F81807E00601F81807E00600FC30
07E00600FC3007E006007E6007E006007E6007E006003FC007E006003FC007E006001F80
07E006001F8007E006001F8007E006000F0007E0FFF00F00FFFFFFF00600FFFF30227EA1
35>I<0007FC0000003FFF800000FC07E00003F001F80007E000FC000FC0007E001F8000
3F001F80003F003F00001F803F00001F807F00001FC07E00000FC07E00000FC0FE00000F
E0FE00000FE0FE00000FE0FE00000FE0FE00000FE0FE00000FE0FE00000FE0FE00000FE0
FE00000FE07E00000FC07F00001FC07F00001FC03F00001F803F80003F801F80003F000F
C0007E0007E000FC0003F001F80000FC07E000003FFF80000007FC000023227DA12A>79
D<FFFFFF00FFFFFFE007F007F007F001FC07F000FC07F0007E07F0007E07F0007F07F000
7F07F0007F07F0007F07F0007F07F0007E07F0007E07F000FC07F001FC07F007F007FFFF
E007FFFF0007F0000007F0000007F0000007F0000007F0000007F0000007F0000007F000
0007F0000007F0000007F0000007F0000007F00000FFFF8000FFFF800020227EA126>I<
FFFFFE0000FFFFFFC00007F007F00007F001F80007F000FC0007F0007E0007F0007F0007
F0007F0007F0007F0007F0007F0007F0007F0007F0007F0007F0007E0007F000FC0007F0
01F80007F007F00007FFFFC00007FFFF800007F00FE00007F007F00007F003F80007F001
FC0007F001FC0007F001FC0007F001FC0007F001FC0007F001FC0007F001FC0007F001FC
0007F001FC0607F000FE0607F000FF0CFFFF803FF8FFFF800FF027227EA12A>82
D<01FC0407FF8C1F03FC3C007C7C003C78001C78001CF8000CF8000CFC000CFC0000FF00
00FFE0007FFF007FFFC03FFFF01FFFF80FFFFC03FFFE003FFE0003FF00007F00003F0000
3FC0001FC0001FC0001FE0001EE0001EF0003CFC003CFF00F8C7FFE080FF8018227DA11F
>I<7FFFFFFF807FFFFFFF807E03F80F807803F807807003F803806003F80180E003F801
C0E003F801C0C003F800C0C003F800C0C003F800C0C003F800C00003F800000003F80000
0003F800000003F800000003F800000003F800000003F800000003F800000003F8000000
03F800000003F800000003F800000003F800000003F800000003F800000003F800000003
F800000003F800000003F800000003F8000003FFFFF80003FFFFF80022227EA127>I<FF
FF803FFCFFFF803FFC07F000018007F000018007F000018007F000018007F000018007F0
00018007F000018007F000018007F000018007F000018007F000018007F000018007F000
018007F000018007F000018007F000018007F000018007F000018007F000018007F00001
8007F000018007F000018007F000018007F000018003F000030003F800030001F8000600
00FC000E00007E001C00003F80F800000FFFE0000001FF000026227EA12B>I<FFFF0FFF
F01FFEFFFF0FFFF01FFE0FF000FF0000E007F0007F0000C007F0007F0000C003F8007F80
018003F8003F80018003FC003F80038001FC003FC0030001FC003FC0030000FE007FE006
0000FE006FE0060000FF006FE00600007F00C7F00C00007F00C7F00C00007F80C7F81C00
003F8183F81800003F8183F81800001FC383FC3000001FC301FC3000001FE301FC300000
0FE600FE6000000FE600FE6000000FF600FFE0000007FC007FC0000007FC007FC0000003
FC007F80000003F8003F80000003F8003F80000001F0001F00000001F0001F00000000F0
001E00000000E0000E00000000E0000E000037227FA13A>87 D<07FC001FFF803F07C03F
03E03F01E03F01F01E01F00001F00001F0003FF003FDF01FC1F03F01F07E01F0FC01F0FC
01F0FC01F0FC01F07E02F07E0CF81FF87F07E03F18167E951B>97
D<FF000000FF0000001F0000001F0000001F0000001F0000001F0000001F0000001F0000
001F0000001F0000001F0000001F0000001F0FE0001F3FF8001FF07C001F801E001F001F
001F000F801F000F801F000FC01F000FC01F000FC01F000FC01F000FC01F000FC01F000F
C01F000FC01F000F801F001F801F801F001FC03E001EE07C001C3FF800180FC0001A237E
A21F>I<00FF8007FFE00F83F01F03F03E03F07E03F07C01E07C0000FC0000FC0000FC00
00FC0000FC0000FC00007C00007E00007E00003E00301F00600FC0E007FF8000FE001416
7E9519>I<0001FE000001FE0000003E0000003E0000003E0000003E0000003E0000003E
0000003E0000003E0000003E0000003E0000003E0001FC3E0007FFBE000F81FE001F007E
003E003E007E003E007C003E00FC003E00FC003E00FC003E00FC003E00FC003E00FC003E
00FC003E00FC003E007C003E007C003E003E007E001E00FE000F83BE0007FF3FC001FC3F
C01A237EA21F>I<00FE0007FF800F87C01E01E03E01F07C00F07C00F8FC00F8FC00F8FF
FFF8FFFFF8FC0000FC0000FC00007C00007C00007E00003E00181F00300FC07003FFC000
FF0015167E951A>I<003F8000FFC001E3E003C7E007C7E00F87E00F83C00F80000F8000
0F80000F80000F80000F8000FFFC00FFFC000F80000F80000F80000F80000F80000F8000
0F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F8000
7FF8007FF80013237FA211>I<03FC1E0FFF7F1F0F8F3E07CF3C03C07C03E07C03E07C03
E07C03E07C03E03C03C03E07C01F0F801FFF0013FC003000003000003800003FFF801FFF
F00FFFF81FFFFC3800FC70003EF0001EF0001EF0001EF0001E78003C7C007C3F01F80FFF
E001FF0018217E951C>I<FF000000FF0000001F0000001F0000001F0000001F0000001F
0000001F0000001F0000001F0000001F0000001F0000001F0000001F07E0001F1FF8001F
307C001F403C001F803E001F803E001F003E001F003E001F003E001F003E001F003E001F
003E001F003E001F003E001F003E001F003E001F003E001F003E001F003E001F003E00FF
E1FFC0FFE1FFC01A237EA21F>I<1C003F007F007F007F003F001C000000000000000000
000000000000FF00FF001F001F001F001F001F001F001F001F001F001F001F001F001F00
1F001F001F001F001F00FFE0FFE00B247EA310>I<FF000000FF0000001F0000001F0000
001F0000001F0000001F0000001F0000001F0000001F0000001F0000001F0000001F0000
001F00FF801F00FF801F0038001F0060001F01C0001F0380001F0700001F0E00001F1C00
001F7E00001FFF00001FCF00001F0F80001F07C0001F03E0001F01E0001F01F0001F00F8
001F007C001F003C00FFE0FFC0FFE0FFC01A237EA21E>107 D<FF00FF001F001F001F00
1F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F00
1F001F001F001F001F001F001F001F001F001F00FFE0FFE00B237EA210>I<FF07F007F0
00FF1FFC1FFC001F303E303E001F403E403E001F801F801F001F801F801F001F001F001F
001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F
001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F
001F001F001F00FFE0FFE0FFE0FFE0FFE0FFE02B167E9530>I<FF07E000FF1FF8001F30
7C001F403C001F803E001F803E001F003E001F003E001F003E001F003E001F003E001F00
3E001F003E001F003E001F003E001F003E001F003E001F003E001F003E001F003E00FFE1
FFC0FFE1FFC01A167E951F>I<00FE0007FFC00F83E01E00F03E00F87C007C7C007C7C00
7CFC007EFC007EFC007EFC007EFC007EFC007EFC007E7C007C7C007C3E00F81F01F00F83
E007FFC000FE0017167E951C>I<FF0FE000FF3FF8001FF07C001F803E001F001F001F00
1F801F001F801F000FC01F000FC01F000FC01F000FC01F000FC01F000FC01F000FC01F00
0FC01F001F801F001F801F803F001FC03E001FE0FC001F3FF8001F0FC0001F0000001F00
00001F0000001F0000001F0000001F0000001F0000001F000000FFE00000FFE000001A20
7E951F>I<00FE030007FF87000FC1C7001F006F003F003F007E003F007E001F007C001F
00FC001F00FC001F00FC001F00FC001F00FC001F00FC001F00FC001F007E001F007E001F
003E003F001F007F000FC1DF0007FF9F0001FC1F0000001F0000001F0000001F0000001F
0000001F0000001F0000001F0000001F000000FFE00000FFE01B207E951E>I<FE1F00FE
3FC01E67E01EC7E01E87E01E87E01F83C01F00001F00001F00001F00001F00001F00001F
00001F00001F00001F00001F00001F00001F0000FFF000FFF00013167E9517>I<0FF300
3FFF00781F00600700E00300E00300F00300FC00007FE0007FF8003FFE000FFF0001FF00
000F80C00780C00380E00380E00380F00700FC0E00EFFC00C7F00011167E9516>I<0180
000180000180000180000380000380000780000780000F80003F8000FFFF00FFFF000F80
000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F81800F81
800F81800F81800F81800F830007C30003FE0000F80011207F9F16>I<FF01FE00FF01FE
001F003E001F003E001F003E001F003E001F003E001F003E001F003E001F003E001F003E
001F003E001F003E001F003E001F003E001F003E001F003E001F007E001F00FE000F81BE
0007FF3FC001FC3FC01A167E951F>I<FFE01FE0FFE01FE00F8006000F8006000FC00E00
07C00C0007E01C0003E0180003E0180001F0300001F0300000F8600000F86000007CC000
007CC000007FC000003F8000003F8000001F0000001F0000000E0000000E00001B167F95
1E>I<FFE7FF07F8FFE7FF07F81F007800C00F807801800F807C01800F807C018007C07E
030007C0DE030007E0DE070003E0DF060003E18F060001F18F0C0001F38F8C0001FB079C
0000FB07D80000FE03D800007E03F000007E03F000007C01F000003C01E000003800E000
001800C00025167F9528>I<FFE07FC0FFE07FC00F801C0007C0380003E0700003F06000
01F8C00000F98000007F8000003F0000001F0000001F8000003FC0000037C0000063E000
00C1F00001C0F8000380FC0007007E000E003E00FF80FFE0FF80FFE01B167F951E>I<FF
E01FE0FFE01FE00F8006000F8006000FC00E0007C00C0007E01C0003E0180003E0180001
F0300001F0300000F8600000F86000007CC000007CC000007FC000003F8000003F800000
1F0000001F0000000E0000000E0000000C0000000C00000018000078180000FC380000FC
300000FC60000069C000007F8000001F0000001B207F951E>I<7FFFF07FFFF07C03E070
07C0600FC0E01F80C01F00C03E00C07E0000FC0000F80001F00003F03007E03007C0300F
80701F80703F00603E00E07C03E0FFFFE0FFFFE014167E9519>I
E /Fm 46 123 df<03C0000C2080183080301900601900601A00601A00C01C00C0180040
180060380020CD001F0600110D7E8C16>11 D<03C006200C000C000C000C00070007800D
C010C030C06040604060C0C080C0804180410022001C000B147E930F>14
D<0F80180020006000C000FE00C00080008000C000C00061003E00090D7D8C0E>I<1C00
06000300030003800180018001C000C000C000E001E0037006300C30181830186018C00C
C00E0F147E9314>21 D<781818181818303030303060304060C0618063006600D800E000
0D0D7E8C11>23 D<40E04003037D820A>58 D<40E06020202040408003097D820A>I<00
200060006000C000C000C0018001800180030003000300060006000C000C000C00180018
001800300030003000600060006000C000C000C0000B1D7E9511>61
D<C00000F000003800000E000007800001E000007800001C000007000003C00003C00007
00001C0000780001E0000780000E0000380000F00000C0000012147D901A>I<00010000
0300000700000780000B80001B800013800023800023800043800083800083C00101C003
FFC00201C00401C00401C00801C01801E0FE07F815147F9319>65
D<07FFE000E03801C01801C01C01C01C01C01C0380380380700380E003FFC00700E00700
700700300700380E00700E00700E00E00E00E01C0380FFFE0016147F9319>I<07FFE000
E07001C01801C00C01C00C01C00E03800E03800E03800E03800E07001C07001C07001C07
00380E00300E00700E00E00E01C01C0700FFFC0017147F931B>68
D<07FFFC00E01C01C00C01C00C01C00C01C00803810803810003830003FF000702000702
000702080700100E00100E00100E00200E00601C01E0FFFFC016147F9318>I<07FFFC00
E01C01C00C01C00C01C00C01C00803820803820003820003FE0007040007040007040007
00000E00000E00000E00000E00001C0000FFC00016147F9315>I<003F0800C098030070
0600300C0030180030380020700000700000700000E00000E01FF0E001C0E001C0600380
6003803003803007800C0B0007F10015147E931A>I<07FC7FC000E00E0001C01C0001C0
1C0001C01C0001C01C0003803800038038000380380003FFF80007007000070070000700
7000070070000E00E0000E00E0000E00E0000E00E0001C01C000FF8FF8001A147F931B>
I<07FC00E001C001C001C001C0038003800380038007000700070007000E000E000E000E
001C00FF800E147F930F>I<07FC1FC000E0060001C0080001C0100001C0600001C08000
0381000003860000038E0000039E0000076700000787000007038000070380000E01C000
0E01C0000E00E0000E00E0001C00F000FF83FC001A147F931C>75
D<07FE0000E00001C00001C00001C00001C0000380000380000380000380000700000700
000700000700200E00400E00400E00800E01801C0780FFFF0013147F9317>I<07F000FE
00F000F0017001E0017002E0017002E0017004E0027009C0023809C0023811C0023821C0
043823800438438004388380041C8380081D0700081E0700081E0700081C070018180E00
FE187FC01F147F9320>I<07E01FC000E006000170040001700400013804000138040002
1C0800021C0800020E0800020E0800040710000407100004039000040390000801E00008
01E0000800E0000800E00018004000FE0040001A147F931A>I<003F0001C1C003006006
00700C0030180038380038700038700038700038E00070E00070E00070E000E0E000C060
01C07003803806001C1C0007E00015147E9319>I<00F8800305800603000401000C0100
0C01000C00000E00000FE00007F80001FC00001C00000E00000E00400C00400C00400800
601800D020008FC00011147E9314>83 D<FF01F83800603800403800803C01001C01001C
02001C04001C04001C08001E10000E10000E20000E40000E40000E80000F000007000006
000004000015147E9314>86 D<07FC7F8000E01C0000F010000070200000784000003880
0000390000003E0000001C0000001E0000001E0000003F00000067000000C70000018380
00030380000603C0000401C0001C01E000FE07F80019147F931B>88
D<FF80FC1C00301E00400E00C00F01800F010007020007840003880003900003E00001C0
0001C00001C0000380000380000380000380000700003FE00016147F9314>I<07FFF007
00E00C01C00C0380080700080F00100E00001C0000380000700000E00001C00003808007
01000F01000E01001C0200380600701E00FFFC0014147E9317>I<07B00C701070306060
6060606060C0C0C0C8C0C841C862D03C700D0D7E8C12>97 D<07800C4010E031C0600060
006000C000C0004020404021801E000B0D7E8C0F>99 D<007C000C001800180018001800
3007B00C7010703060606060606060C0C0C0C8C0C841C862D03C700E147E9311>I<0038
006C007C004C00C000C000C007F800C00180018001800180018003000300030003000300
0300060006006600E400C80070000E1A7F9310>102 D<3E0006000C000C000C000C0018
0019E01E30183038303030303030306060606460C460C4C0C8C0700E147E9313>104
D<06070600000000384C4C8C98181830326262643808147F930C>I<0060007000600000
000000000000038004C0046008C008C000C000C001800180018001800300030003000300
6600E600CC0078000C1A81930E>I<3E0006000C000C000C000C001800187018B8193832
30340038003E006300631063106310C320C1C00D147E9312>I<7C0C1818181830303030
60606060C0D0D0D0D06006147E930A>I<30F87C00590C86004E0D06009C0E0600980C06
00180C0600180C060030180C0030180C8030181880301818806030190060300E00190D7F
8C1D>I<30F8590C4E0C9C0C980C180C180C30183019303130316032601C100D7F8C15>I<
0C78168C130426062606060606060C0C0C0C0C080C101A2019C018001800300030003000
FC000F13818C11>112 D<0700188019C0318038001E000F0003804180E180C10082007C
000A0D7E8C10>115 D<02000600060006000C00FF800C000C0018001800180018003000
31003100320032001C0009127F910D>I<380C4C0C4C0C8C189818181818183030303230
32307218B40F1C0F0D7F8C14>I<38104C304C108C109810181018103020302030403040
18800F000C0D7F8C11>I<0E3C13CE238E430C43000300030006000608C608E610CA2071
C00F0D7F8C13>120 D<38184C184C188C3098301830183030603060306030E011C00EC0
00C00080E180E30046003C000D137F8C11>I<07100F2010E000400080010002000C0010
2020203840478083000C0D7F8C10>I E /Fn 32 113 df<FFFFFFC0FFFFFFC01A027C8B
23>0 D<70F8F8F87005057C8D0D>I<400004C0000C6000183000301800600C00C0060180
03030001860000CC0000780000300000300000780000CC000186000303000601800C00C0
180060300030600018C0000C40000416187A9623>I<0002000000060000000600000006
000000060000000600000006000000060000000600000006000000060000000600000006
00000006000000060000FFFFFFF0FFFFFFF0000600000006000000060000000600000006
00000006000000060000000600000006000000060000000600000006000000060000FFFF
FFF0FFFFFFF01C207D9E23>6 D<03C00FF01FF83FFC7FFE7FFEFFFFFFFFFFFFFFFFFFFF
FFFF7FFE7FFE3FFC1FF80FF003C010127D9317>15 D<FFFFFFF07FFFFFF0000000000000
00000000000000000000000000000000000000000000FFFFFFF0FFFFFFF0000000000000
000000000000000000000000000000000000000000007FFFFFF0FFFFFFF01C147D9423>
17 D<003FFFC000FFFFC003C00000070000000C00000018000000300000003000000060
00000060000000C0000000C0000000C0000000C0000000C0000000C0000000C000000060
000000600000003000000030000000180000000C0000000700000003C0000000FFFFC000
3FFFC0000000000000000000000000000000000000000000000000000000007FFFFFC07F
FFFFC01A247C9C23>I<000000C0000003C000000F0000003C000000F0000003C0000007
0000001C00000078000001E00000078000001E00000078000000E0000000780000001E00
00000780000001E0000000780000001C0000000700000003C0000000F00000003C000000
0F00000003C0000000C00000000000000000000000000000000000000000000000000000
00007FFFFF80FFFFFFC01A247C9C23>20 D<C0000000F00000003C0000000F00000003C0
000000F0000000380000000E0000000780000001E0000000780000001E00000007800000
01C00000078000001E00000078000001E00000078000000E00000038000000F0000003C0
00000F0000003C00000070000000C0000000000000000000000000000000000000000000
000000000000000000007FFFFF80FFFFFFC01A247C9C23>I<0FC000101FF000103FF800
10703E0030E00F0070C007C0E08001FFC08000FF8080003F0000000000000000000FC000
101FF000103FF80010703E0030E00F0070C007C0E08001FFC08000FF8080003F001C147D
9523>25 D<000000C00C000003C03C000007007000001C01C000007807800000E00E0000
03803800000F00F000003C03C000007007000001C01C000007807800000E00E000003803
800000F00F000000F00F00000038038000000E00E00000078078000001C01C0000007007
0000003C03C000000F00F00000038038000000E00E00000078078000001C01C000000700
70000003C03C000000C00C261E7D992D>28 D<C00C000000F00F00000038038000000E00
E00000078078000001C01C00000070070000003C03C000000F00F00000038038000000E0
0E00000078078000001C01C00000070070000003C03C000003C03C000007007000001C01
C000007807800000E00E000003803800000F00F000003C03C000007007000001C01C0000
07807800000E00E000003803800000F00F000000C00C000000261E7D992D>I<00000004
00000000020000000002000000000100000000008000000000400000000020FFFFFFFFFC
FFFFFFFFFC00000000200000000040000000008000000001000000000200000000020000
0000040026107D922D>33 D<003FF800FFF803C0000700000C0000180000300000300000
600000600000C00000C00000C00000FFFFF8FFFFF8C00000C00000C00000600000600000
3000003000001800000C000007000003C00000FFF8003FF8151C7C981E>50
D<00000C00000C0000180000180000300000300000600000600000C00000C00001800001
80000180000300000300000600000600000C00000C000018000018000030000030000060
0000600000C00000C0000180000180000300000300000600000600000600000C00000C00
00180000180000300000300000600000600000C00000400000162C7AA000>54
D<C0C0C0C0C0C0C0E0E0C0C0C0C0C0C0C003107D9200>I<400001C00003600006600006
60000630000C30000C30000C1800181800181800180FFFF00FFFF00C0030060060060060
0600600300C00300C001818001818001818000C30000C30000C300006600006600006600
003C00003C00003C000018000018001821809F19>I<FFFFC0FFFFC00000C00000C00000
C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C07FFFC07FFF
C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000
C00000C0FFFFC0FFFFC012207D9F19>I<0000FC0007FE001C3E00201E00C01E01801C03
801C07003C0600380E00381C00601C00003C000038000038000078000078000070000070
0000F00000F00000F00000F00000F00000F00000F80018F800307800707C00C07E00803F
83001FFC0007F0001721809F18>67 D<00001F8000007F80000087C0000307C0000603C0
000E03C0001C0380001C0300003800000038000000780000007000000070000000F00000
00E0000000E0000001E0000001E0000001C0000001C0000003C000000380000003800000
0780000007000000070000000E0000600E0000E01FF001C01FFE018030FFC300601FFC00
C003F8001B217F9F1F>76 D<0000000000F80000000003F00000000007F00000C0000FF0
0001C0000E000003E00018000003E00010000003E00010000003E00020000007F0002000
0007F00020000004F00020000004F80040000004F8004000000478004000000878008000
00087C00800000087C00800000083C00800000103E01000000103E01000000101E010000
00101F01000000201F02000000200F02000000200F82000000400F820000004007C40000
004007C40000008003C40000008003E40000010003EC0000630001F80000FE0001F80000
FE0000F00000FC00006000007800000000002D2581A225>78 D<003FFFE00001FFFFF800
071E01FE00081E007F00181E001F00701E000F00701E000F00C01E000F00001C000E0000
1C000E00003C001E00003C001C00003C0018000038003000003800600000780080000078
0700000070FC00000071F8000000F0FC000000E07C000000E03E000001E03E000001C01F
000001C01F000003C00F001003800F806003800F80E0078007C180070007E300060003FE
000C0001F00024207F9E27>82 D<00007F000001FFC000060FE0000C03E0003801E00030
01E0007001E000F0018000F0000000F0000000F8000000FC0000007E0000003F0000001F
C000000FF0000003F8000000FC0000007E0000003F000C001F0018000F0070000F007000
0F00F0000E00F0000E00F0001C00F8001800FC0030007E0060003F8180001FFE000007F8
00001B217F9F1C>I<400002C00006C00006C00006C00006C00006C00006C00006C00006
C00006C00006C00006C00006C00006C00006C00006C00006C00006C00006C00006C00006
60000C60000C3000181C00700F01E003FF8000FE00171C7D9A1E>91
D<00FE0003FF800F01E01C007030001860000C60000CC00006C00006C00006C00006C000
06C00006C00006C00006C00006C00006C00006C00006C00006C00006C00006C00006C000
06C00006C00006C00006400002171C7D9A1E>I<001000003800003800006C00006C0000
6C0000C60000C6000183000183000183000301800301800600C00600C00600C00C00600C
006018003018003018003030001830001860000C60000C60000CC00006C00002171C7D9A
1E>94 D<000F0038006000E001C001C001C001C001C001C001C001C001C001C001C001C0
01C001C001C0038007001E00F8001E000700038001C001C001C001C001C001C001C001C0
01C001C001C001C001C001C001C000E000600038000F102D7DA117>102
D<F8001E000700038001C001C001C001C001C001C001C001C001C001C001C001C001C001
C001C000E000600038000F0038006000E001C001C001C001C001C001C001C001C001C001
C001C001C001C001C001C0038007001E00F800102D7DA117>I<C0C0C0C0C0C0C0C0C0C0
C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C002
2D7BA10D>106 D<4020C030C030C030C030C030C030C030C030C030C030C030C030C030
C030C030C030C030C030C030C030C030C030C030C030C030C030C030C030C030C030C030
C030C030C030C030C030C030C030C030C030C030C030C030C03040200C2E7BA117>I<C0
00C000C0006000600060003000300030001800180018000C000C000C0006000600060003
000300030001800180018000C000C000C000600060006000300030003000180018001800
0C000C000C000600060006000300030001102D7DA117>110 D<00000000080000000018
000000003000000000300000000060000000006000000000C000000000C0000000018000
0000018000000003000000000300000000060000000006000000000C000000000C000000
00180000000018000000003000000000300000000060000000006000000000C000060000
C0001E000180002F000180004F000300008780030000078006000003C006000003C00C00
0003C00C000001E018000001E018000000F030000000F030000000786000000078600000
003CC00000003CC00000001F800000001F800000000F000000000F000000000600000000
06000000252E7E8126>112 D E /Fo 61 123 df<007C0001C3000701810E01C11E00C1
1C00E23C00E27800E27800E47800E4F000E8F000F0F000F0F000E0F000E07000E07003E0
30046118383207C01C18147E931D>11 D<0000F800030600040600080300100300200300
400700400700800700800601000E01000C0107F80104700207D802001C02001C02001E04
001E04001E04001E04001E08003C08003C08003C0800781800701400F01400E01201C021
8700207C0020000020000040000040000040000040000080000080000080000018297F9F
1A>I<001E0000610000C08001808001800003800003000003800003800003C00001F000
00F800007C0001FC00070E000E0E001E06001C06003C0600780600780600780600F00400
F00400F00C00F00800F008007018007010003020001840000F800011207E9F14>14
D<007E01C007000E001C003C003800780078007FF0F000F000F000700070007000300018
000C1807E00F147E9312>I<000800000800000800000FC00018200027C0004000008000
0100000300000600000400000C0000080000180000100000300000300000600000600000
600000600000600000E00000E00000E00000E000006000007000007C00003F00001FE000
07F80001FC00003C00000C00000C00000C00000C0003180000F00013297E9F14>I<1E07
C023186023A03043C0304380384380388700700700700700700700700E00E00E00E00E00
E00E00E01C01C01C01C01C01C01C01C03803801803800003800003800007000007000007
00000700000E00000E00000E00000C00151E7E9317>I<07000001C00000E00000E00000
F000007000007000007800003800003800003C00001C00001C00001E00000E00000E0000
0F00000700000F0000378000638000C38001C3C00381C00701C00E01E01C00E03800E070
00F0F00070E00070C0003815207D9F1B>21 D<01801801C01C0380380380380380380380
380700700700700700700700700E00E00E00E00E00E00E00E11E01C21E01C21E03C21E05
C43F08C439F078380000380000700000700000700000700000E00000E00000E00000C000
00181E7F931B>I<0F00307F00380E00700E00700E00700E00E01C00E01C01C01C01C01C
0380380300380600380C0038180070300070600070C000730000FC0000F0000015147E93
16>I<001000001000001000001FC000782000EFC001C00003C0000780000780000F0000
0F00000F00000F00000F0000070000077E0003810006FE000C0000180000100000300000
600000600000600000E00000E00000E000007000007C00003F00001FE00007F80000FC00
003E00000E00000600020600010C0000F80013297F9F14>I<000F800038C000606000C0
7001C0700380780380780700780700780700780E00F00E00F00E00F00E01E01C01C01C01
C01E03801E0700390C0038F000380000380000700000700000700000700000E00000E000
00E00000C00000151E7F9318>26 D<04000180080003C0100003E0100001E0200000E020
0000E02000004040040040400C0040400C0040800C008080080080C0080180C0180300C0
380600E07C0E00FFEFFC007FCFF8003F87F0001E03C0001B1480931C>33
D<70F8F8F87005057C840D>58 D<70F8FCFC74040404080810102040060E7C840D>I<00
0001C00000078000001E00000078000001E00000078000000E00000038000000F0000003
C000000F0000003C000000F0000000F00000003C0000000F00000003C0000000F0000000
380000000E0000000780000001E0000000780000001E0000000780000001C01A1A7C9723
>I<000100030003000600060006000C000C000C00180018001800300030003000600060
006000C000C000C00180018001800300030003000600060006000C000C000C0018001800
1800300030003000600060006000C000C000C000102D7DA117>I<E0000000780000001E
0000000780000001E0000000780000001C0000000700000003C0000000F00000003C0000
000F00000003C0000003C000000F0000003C000000F0000003C00000070000001C000000
78000001E00000078000001E00000078000000E00000001A1A7C9723>I<000002000000
060000000E0000000E0000001E0000001F0000002F0000002F0000004F0000008F000000
8F0000010F0000010F0000020F0000040F0000040F0000080F8000080780001007800020
078000200780007FFF800040078000800780018007800100078002000780020007C00400
03C00C0003C01E0007C0FF807FFC1E207E9F22>65 D<00FFFFE0000F0078000F003C000F
001C000F001E001E001E001E001E001E001E001E001E003C003C003C003C003C0078003C
00F0007803C0007FFF80007803C0007801E000F000F000F000F000F000F000F0007001E0
00F001E000F001E000F001E000E003C001E003C003C003C0038003C00F0007801E00FFFF
F0001F1F7E9E22>I<00FFFFE000000F007800000F001C00000F000E00000F000700001E
000700001E000380001E000380001E000380003C000380003C000380003C000380003C00
0380007800078000780007800078000780007800078000F0000F0000F0000F0000F0000E
0000F0001E0001E0001C0001E0003C0001E000380001E000700003C000E00003C001C000
03C003800003C007000007803C0000FFFFF00000211F7E9E26>68
D<00FFFFFF000F000E000F0006000F0002000F0002001E0002001E0002001E0002001E00
02003C0404003C0400003C0400003C0C0000781800007FF800007818000078180000F010
0000F0100000F0100000F0000401E0000801E0000801E0001001E0001003C0002003C000
6003C0004003C001C0078007C0FFFFFF80201F7E9E22>I<00FFFFFF000F000E000F0006
000F0002000F0002001E0002001E0002001E0002001E0002003C0004003C0400003C0400
003C04000078080000781800007FF8000078180000F0100000F0100000F0100000F01000
01E0000001E0000001E0000001E0000003C0000003C0000003C0000003C0000007C00000
FFFE0000201F7E9E1D>I<00007E0100038183000E00460038002E0070001E00E0000E01
C0000C0380000C0700000C0F00000C1E0000081E0000083C0000003C0000007800000078
0000007800000078000000F0000000F0007FFCF00001E0F00001E0F00003C0700003C070
0003C0700003C038000780380007801C000F800C000B80060033000380C100007F000020
217E9F24>I<00FFF9FFF0000F801F00000F001E00000F001E00000F001E00001E003C00
001E003C00001E003C00001E003C00003C007800003C007800003C007800003C00780000
7800F000007FFFF000007800F000007800F00000F001E00000F001E00000F001E00000F0
01E00001E003C00001E003C00001E003C00001E003C00003C007800003C007800003C007
800003C007800007C00F8000FFF8FFF800241F7E9E26>I<00FFFC000F80000F00000F00
000F00001E00001E00001E00001E00003C00003C00003C00003C00007800007800007800
00780000F00000F00000F00000F00001E00001E00001E00001E00003C00003C00003C000
03C00007C000FFFC00161F7F9E14>I<00FFF80FF8000F8003E0000F000380000F000200
000F000400001E000800001E002000001E004000001E008000003C010000003C04000000
3C080000003C180000007838000000787C000000793C0000007A3C000000F41E000000F8
1E000000F01E000000F00F000001E00F000001E00F000001E007800001E007800003C007
800003C003C00003C003C00003C003C00007C003E000FFFC3FFC00251F7E9E27>75
D<00FFFC00000F8000000F0000000F0000000F0000001E0000001E0000001E0000001E00
00003C0000003C0000003C0000003C00000078000000780000007800000078000000F000
0000F0000000F0000000F0004001E0008001E0008001E0018001E0010003C0030003C003
0003C0060003C00E0007803C00FFFFFC001A1F7E9E1F>I<00FF00001FF0000F00003F00
000B80003E00000B80005E00000B80005E0000138000BC00001380013C00001380013C00
001380023C000023800278000023800478000023800878000021C00878000041C010F000
0041C020F0000041C020F0000041C040F0000081C041E0000081C081E0000081C101E000
0081C101E0000100E203C0000100E203C0000100E403C0000100E803C0000200E8078000
0200F00780000200F00780000600E00780000F00C00F8000FFE0C1FFF8002C1F7E9E2C>
I<00FF803FF0000F800780000F800200000BC00200000BC002000013C004000011E00400
0011E004000011E004000020F008000020F008000020F808000020780800004078100000
403C100000403C100000403C100000801E200000801E200000801E200000800F20000100
0F400001000F4000010007C000010007C000020007800002000380000200038000060003
80000F00010000FFE0010000241F7E9E25>I<0001FC0000070700001C01C0003000E000
E0006001C000700380007007800038070000380E0000381E0000381C0000383C0000383C
00003878000078780000787800007878000078F00000F0F00000F0F00000E0F00001E0F0
0001C0F00003C0700003807000070078000F0038001E0038003C001C0070000E00E00007
83800001FC00001D217E9F23>I<00FFFFC0000F0070000F0038000F001C000F001E001E
001E001E001E001E001E001E001E003C003C003C003C003C0078003C0070007800E00078
0380007FFE000078000000F0000000F0000000F0000000F0000001E0000001E0000001E0
000001E0000003C0000003C0000003C0000003C0000007C00000FFFC00001F1F7E9E1D>
I<0007E0800018118000300B000060070000C0070001C003000180020003800200038002
0003800200038000000380000003C0000003F8000003FF800001FFC00000FFE000003FF0
000003F0000000F0000000700000007000000070002000700020007000200060006000E0
006000C0006001C00070018000E8030000C60E000081F8000019217D9F1C>83
D<7FFC1FF807C003C00780010007800100078001000F0002000F0002000F0002000F0002
001E0004001E0004001E0004001E0004003C0008003C0008003C0008003C000800780010
00780010007800100078001000F0002000F0002000F0002000F0004000F0004000700080
007001000030020000380400000C18000007E000001D207C9E1F>85
D<FFF801FF0F8000780F0000600F0000400F800040078000800780008007800100078002
0007800200078004000780080007C0080003C0100003C0100003C0200003C0400003C040
0003C0800003C1800003C1000003E2000001E2000001E4000001E8000001E8000001F000
0001F0000001E0000001C0000000C000000080000020207E9E1B>I<00FFF83FF8000FC0
0F80000F80060000078004000007C008000003C010000003C020000003E040000001E080
000001F100000000F300000000F600000000FC0000000078000000007C000000007C0000
00007C00000000BE000000011E000000021E000000061F0000000C0F000000080F800000
100780000020078000004007C000008003C000010003E000030003E0000F0007E000FFE0
1FFE00251F7F9E26>88 D<FFF801FF0F8000780F8000600780004007C0008007C0018003
C0010003E0020003E0040001E0080001F0180000F0100000F0200000F840000078800000
7D0000007D0000003E0000003C0000003C00000038000000780000007800000078000000
70000000F0000000F0000000F0000000F0000001E000003FFF0000201F7F9E1A>I<007F
FFF800FC00F000E001E000C003C0018007800100078003000F0002001E0002003C000400
78000000F8000000F0000001E0000003C00000078000000F0000000F0000001E0000003C
00000078008000F0008001F0010001E0010003C00300078002000F0006001E0004003E00
0C003C003C007800F800FFFFF8001D1F7D9E1F>I<00F1800389C00707800E03801C0380
3C0380380700780700780700780700F00E00F00E00F00E00F00E10F01C20F01C20703C20
705C40308C400F078014147E9318>97 D<007C01C207010E0F1E0F1C0E3C047800780078
00F000F000F000F000F00070017002300418380FC010147E9314>99
D<0000780003F80000700000700000700000700000E00000E00000E00000E00001C00001
C000F1C00389C00707800E03801C03803C0380380700780700780700780700F00E00F00E
00F00E00F00E10F01C20F01C20703C20705C40308C400F078015207E9F18>I<007C01C2
07010E011C013C013802780C7BF07C00F000F000F000F0007000700170023004183807C0
10147E9315>I<00007C0000CE00019E00039E00030C000700000700000700000700000E
00000E00000E0000FFF0000E00000E00001C00001C00001C00001C00001C000038000038
0000380000380000380000700000700000700000700000700000E00000E00000E00000E0
0000C00001C000318000798000F300006200003C000017297E9F16>I<001E3000713800
E0F001C0700380700780700700E00F00E00F00E00F00E01E01C01E01C01E01C01E01C01E
03801E03800E07800E0B8006170001E700000700000700000E00000E00300E00781C00F0
38006070003FC000151D809316>I<01E0000FE00001C00001C00001C00001C000038000
038000038000038000070000070000071F000761800E80C00F00C00E00E00E00E01C01C0
1C01C01C01C01C01C0380380380380380380380704700708700E08700E10700610E00620
6003C016207E9F1A>I<00E001E001E000C000000000000000000000000000000E001300
23804380438043808700070007000E000E001C001C001C20384038403840388019000E00
0B1F7E9E10>I<0000C00001E00001E00001C00000000000000000000000000000000000
00000000001E00006300004380008380010380010380020700000700000700000700000E
00000E00000E00000E00001C00001C00001C00001C000038000038000038000038000070
0000700030700078E000F1C0006380003E00001328819E13>I<01E0000FE00001C00001
C00001C00001C0000380000380000380000380000700000700000701E00706100E08700E
10F00E20F00E40601C80001D00001E00001FC000387000383800383800381C2070384070
3840703840701880E01880600F0014207E9F18>I<03C01FC00380038003800380070007
00070007000E000E000E000E001C001C001C001C00380038003800380070007000700071
00E200E200E200E200640038000A207E9F0E>I<1E07C07C00231861860023A032030043
C03403004380380380438038038087007007000700700700070070070007007007000E00
E00E000E00E00E000E00E00E000E00E01C101C01C01C201C01C038201C01C038401C01C0
184038038018801801800F0024147E9328>I<1E07802318C023A06043C0704380704380
708700E00700E00700E00700E00E01C00E01C00E01C00E03821C03841C07041C07081C03
083803101801E017147E931B>I<007C0001C3000301800E01C01E01C01C01E03C01E078
01E07801E07801E0F003C0F003C0F003C0F00780F00700700F00700E0030180018700007
C00013147E9316>I<03C1E004621804741C08781C08701E08701E10E01E00E01E00E01E
00E01E01C03C01C03C01C03C01C0380380780380700380E003C1C0072380071E00070000
0700000E00000E00000E00000E00001C00001C0000FFC000171D819317>I<00F0400388
C00705800E03801C03803C0380380700780700780700780700F00E00F00E00F00E00F00E
00F01C00F01C00703C00705C0030B8000F38000038000038000070000070000070000070
0000E00000E0000FFE00121D7E9314>I<1E1E0023210023C38043C78043878043830087
00000700000700000700000E00000E00000E00000E00001C00001C00001C00001C000038
000018000011147E9315>I<007C018203010603060706060E00078007F803FC01FE001F
00077007F006F006E004400820301FC010147E9315>I<00C000E001C001C001C001C003
800380FFF8038007000700070007000E000E000E000E001C001C001C001C103820382038
20384018800F000D1C7F9B10>I<0F00601180702180E021C0E041C0E04380E08381C007
01C00701C00701C00E03800E03800E03800E03840E07080C07080C07080E0F1006131003
E1E016147E931A>I<0F01801183C02183E021C1E041C0E0438060838040070040070040
0700400E00800E00800E00800E01000E01000C02000E04000E040006180001E00013147E
9316>I<03C1C00C62201034701038F02038F020386040700000700000700000700000E0
0000E00000E00000E02061C040F1C040F1C080E2C080446300383C0014147E931A>120
D<0F00601180702180E021C0E041C0E04380E08381C00701C00701C00701C00E03800E03
800E03800E03800E07000C07000C07000E0F00061E0003EE00000E00000E00001C007818
0078380070700060600021C0001F0000141D7E9316>I<01E02003F04007F8C00C1F8008
010000020000040000080000100000600000C0000100000200000400800801001003003F
060061FC0040F80080700013147E9315>I E /Fp 61 123 df<00003F03E00000C38670
0001878CF00003879CF00003031860000700380000070038000007003800000E00380000
0E007000000E007000000E00700000FFFFFF80001C007000001C00E000001C00E000001C
00E000001C00E000003800E000003801C000003801C000003801C000003801C000007001
C0000070038000007003800000700380000070038000006003800000E007000000E00700
0000E007000000E007000000C006000001C00E000001C00E000031860C0000798F180000
F31E100000620C6000003C07C000002429829F1C>11 D<00003FE00000E0100001803800
0380780003007800070030000700000007000000070000000E0000000E0000000E000000
FFFFE0000E00E0001C01C0001C01C0001C01C0001C01C0001C0380003803800038038000
3803800038070000380700007007000070071000700E2000700E2000700E2000E00E2000
E0064000E0038000E0000000C0000001C0000001C000003180000079800000F300000062
0000003C0000001D29829F1A>I<00003FC0FF800000E0E38040000181E600E0000381EC
01E0000300DC01E00007001C00C0000700180000000700380000000E00380000000E0038
0000000E00380000000E0070000000FFFFFFFF80001C00700380001C00700700001C0070
0700001C00700700001C00E00700001C00E00E00003800E00E00003800E00E00003800E0
0E00003801C01C00003801C01C00007001C01C00007001C01C40007001C0388000700380
388000700380388000E00380388000E00380190000E003000E0000E00700000000C00700
000001C00600000001C00600000031860E000000798F0C000000F31E18000000620C3000
00003C07C00000002B29829F28>14 D<0E1F3F3F1D0102020404081020C0080E779F0E>
39 D<000100020004000800100020006000C0018001800300070006000E000C001C0018
003800380030007000700060006000E000E000C000C000C000C000C000C000C000C000C0
00C000C000C000C0004000600060002000100010000800102E79A113>I<001000000800
000400000600000200000300000300000300000100000180000180000180000180000180
000180000180000380000380000380000300000300000300000700000700000600000600
000E00000C00000C00001C0000180000380000300000700000600000E00000C000018000
0100000300000600000C0000180000300000600000800000112E80A113>I<1C3C3C3C3C
040408081020204080060E7D840E>44 D<7FF0FFE07FE00C037D8A10>I<70F8F8F0E005
057B840E>I<000F800030E000E07001C0700380300380380700380F00780F00780E0078
1E00781E00703C00F03C00F03C00F03C00F07801E07801E07801E07801C07003C0F003C0
F00380F00780F00700700700700E00701C003038001870000FC000151F7C9D17>48
D<000200020006000E003C00DC031C001C0038003800380038007000700070007000E000
E000E000E001C001C001C001C003800380038003800780FFF80F1E7B9D17>I<001F0000
61800080E00100E00200700220700420700410700820F00820F00820F00840E00881E007
03C0000380000700000C0000180000600000800003000004000008004010004010008020
01807E030047FF0041FE0080FC00807800141F7C9D17>I<001F800060E0008070010030
0200380420380420380410380420700460700380600000E00001C000030000FE00001C00
000600000700000780000780000780300780780780780780F00F00800F00401E00401C00
40380020E0001F8000151F7C9D17>I<0000600000E00000E00000E00001C00001C00001
C0000380000380000300000700000700000600000E00000C000018000018000030000030
0000630000C700008700010700030700060E00040E00080E003F8E00607C00801FC0001C
00001C0000380000380000380000380000700000700000600013277E9D17>I<00C06000
FFC001FF8001FE00010000010000020000020000020000020000040000047800058C0006
06000C0700080700000780000780000780000780000F00700F00F00F00F00E00E01E0080
1C0080380080300040600061C0001F0000131F7B9D17>I<001F000061800080C0010060
0300600600600600600600600E00C00F00800F818007C30007E40003F80001F80003FC00
047E00183F00300F00200700600700C00300C00300C00300800600800600C00C00C00800
4030003060001F8000131F7B9D17>56 D<001F0000718000C0C00180C00380E00700E00F
00E00F01E01E01E01E01E01E01E01E01C01C03C01C03C01C03C01C07C01C0F800C0F8006
378003C700000F00000E00000E00001C00601C00F03800F07000E0600080C0004380003E
0000131F7B9D17>I<070F1F1F0E0000000000000000000070F8F8F0E008147B930E>I<01
C003C007C007C0038000000000000000000000000000000000000000001C003C003C003C
003C000400040008000800100020002000400080000A1D7D930E>I<0000020000000600
0000060000000E0000001E0000001E0000003F0000002F0000004F0000004F0000008F00
00010F0000010F0000020F0000020F0000040F00000C0F0000080F0000100F0000100F00
00200F80003FFF800040078000C007800080078001000780010007800200078002000780
060007801E000F80FF807FF81D207E9F22>65 D<01FFFFC0001E00F0001E0078001E0038
001E003C003C003C003C003C003C003C003C003C0078007800780078007800F0007801E0
00F0078000FFFE0000F00F8000F003C001E001C001E001E001E001E001E001E003C001E0
03C001E003C001E003C001C0078003C00780078007800F0007801E000F007800FFFFE000
1E1F7D9E20>I<0000FE0200078186001C004C0038003C0060003C00C0001C01C0001803
800018070000180F0000181E0000101E0000103C0000003C000000780000007800000078
00000078000000F0000000F0000000F0000000F0000000F0000080700000807000008070
0001003800010038000200180004000C001800060020000381C00000FE00001F217A9F21
>I<01FFFFFE001E001C001E000C001E0004001E0004003C0004003C0004003C0004003C
00040078080800780800007808000078180000F0300000FFF00000F0300000F0300001E0
200001E0200001E0200001E0001003C0002003C0002003C0004003C00040078000800780
018007800100078007000F001F00FFFFFE001F1F7D9E1F>69 D<01FFFFFC001E0038001E
0018001E0008001E0008003C0008003C0008003C0008003C000800780010007808000078
08000078080000F0100000F0300000FFF00000F0300001E0200001E0200001E0200001E0
200003C0000003C0000003C0000003C00000078000000780000007800000078000000F80
0000FFF800001E1F7D9E1E>I<01FFF0001F00001E00001E00001E00003C00003C00003C
00003C0000780000780000780000780000F00000F00000F00000F00001E00001E00001E0
0001E00003C00003C00003C00003C0000780000780000780000780000F8000FFF800141F
7D9E12>73 D<001FFF0000F80000F00000F00000F00001E00001E00001E00001E00003C0
0003C00003C00003C0000780000780000780000780000F00000F00000F00000F00001E00
001E00301E00781E00F83C00F83C00F0780080700040E00021C0001F000018207D9E18>
I<01FFF800001F0000001E0000001E0000001E0000003C0000003C0000003C0000003C00
000078000000780000007800000078000000F0000000F0000000F0000000F0000001E000
0001E0000001E0000001E0008003C0010003C0010003C0030003C0020007800600078006
0007800C0007801C000F007800FFFFF800191F7D9E1D>76 D<01FE00007FC0001E0000FC
00001E0000F80000170001780000170001780000270002F00000270004F00000270004F0
0000270008F00000470009E00000470011E00000470021E00000470021E00000870043C0
0000838043C00000838083C00000838083C0000103810780000103820780000103820780
000103840780000203840F00000203880F00000203900F00000203900F00000401E01E00
000401E01E00000401C01E00000C01801E00001C01803E0000FF8103FFC0002A1F7D9E29
>I<01FF007FE0001F000F00001F0004000017800400001780040000278008000023C008
000023C008000023C008000041E010000041E010000041F010000040F010000080F02000
00807820000080782000008078200001003C400001003C400001003C400001001E400002
001E800002001E800002000F800002000F800004000F0000040007000004000700000C00
0700001C00020000FF80020000231F7D9E22>I<0001FC0000070700001C01C0003000E0
00E0006001C000700380007007800038070000380E0000381E0000381C0000383C000038
3C00003878000078780000787800007878000078F00000F0F00000F0F00000E0F00001E0
F00001C0F00003C0700003807000070078000F0038001E0038003C001C0070000E00E000
0783800001FC00001D217A9F23>I<01FFFF80001E00E0001E0070001E0038001E003C00
3C003C003C003C003C003C003C003C0078007800780078007800F0007800E000F003C000
F00F0000FFFC0000F0000001E0000001E0000001E0000001E0000003C0000003C0000003
C0000003C00000078000000780000007800000078000000F800000FFF000001E1F7D9E1F
>I<01FFFF00001E03C0001E00E0001E0070001E0078003C0078003C0078003C0078003C
0078007800F0007800F0007801E0007801C000F0070000F01E0000FFF00000F0380001E0
1C0001E01E0001E00E0001E00F0003C01E0003C01E0003C01E0003C01E0007803C000780
3C0807803C0807803C100F801C10FFF00C20000007C01D207D9E21>82
D<0007E040001C18C0003005800060038000C0038001C001800180010003800100038001
00038001000380000003C0000003C0000003F8000001FF800001FFE000007FF000001FF0
000001F80000007800000078000000380000003800200038002000380020003000600070
00600060006000E0007000C000E8038000C606000081F800001A217D9F1A>I<0FFFFFF0
1E0780E0180780201007802020078020200F0020600F0020400F0020400F0020801E0040
001E0000001E0000001E0000003C0000003C0000003C0000003C00000078000000780000
007800000078000000F0000000F0000000F0000000F0000001E0000001E0000001E00000
01E0000003E00000FFFF00001C1F789E21>I<FFF1FFC0FF801F003E001C001F003C0018
000F003C0010000F003C0010000F003C0020000F003C0020000F003E0040000F003E00C0
000F005E0080000F005E0100000F009E0100000F009E0200000F011E0200000F021E0400
000F021E0400000F041E0800000F041E0800000F081E1000000F081E2000000F101E2000
000F101E4000000F201E4000000F601E8000000FC01E80000007801F00000007801F0000
0007001E00000007001E00000006000C0000000600080000000400080000002920779E2D
>87 D<00F1800389C00707800E03801C03803C0380380700780700780700780700F00E00
F00E00F00E00F00E20F01C40F01C40703C40705C40308C800F070013147C9317>97
D<07803F8007000700070007000E000E000E000E001C001C001CF01D0C3A0E3C0E380F38
0F700F700F700F700FE01EE01EE01EE01CE03CE038607060E031C01F0010207B9F15>I<
007E0001C1000300800E07801E07801C07003C0200780000780000780000F00000F00000
F00000F00000F0000070010070020030040018380007C00011147C9315>I<0000780003
F80000700000700000700000700000E00000E00000E00000E00001C00001C000F1C00389
C00707800E03801C03803C0380380700780700780700780700F00E00F00E00F00E00F00E
20F01C40F01C40703C40705C40308C800F070015207C9F17>I<007C01C207010E011C01
3C013802780C7BF07C00F000F000F000F0007000700170023804183807C010147C9315>
I<00007800019C00033C00033C000718000700000700000E00000E00000E00000E00000E
0001FFE0001C00001C00001C00001C000038000038000038000038000038000070000070
0000700000700000700000700000E00000E00000E00000E00000C00001C00001C0000180
003180007B0000F300006600003C00001629829F0E>I<003C6000E27001C1E00380E007
00E00F00E00E01C01E01C01E01C01E01C03C03803C03803C03803C03803C07003C07001C
0F001C17000C2E0003CE00000E00000E00001C00001C00301C00783800F0700060E0003F
8000141D7E9315>I<01E0000FE00001C00001C00001C00001C000038000038000038000
038000070000070000071E000763000E81800F01C00E01C00E01C01C03801C03801C0380
1C0380380700380700380700380E10700E20700C20701C20700C40E00CC060070014207D
9F17>I<00C001E001E001C000000000000000000000000000000E003300230043804300
470087000E000E000E001C001C001C003840388030807080310033001C000B1F7C9E0E>
I<0001800003C00003C0000380000000000000000000000000000000000000000000003C
00004600008700008700010700010700020E00000E00000E00000E00001C00001C00001C
00001C0000380000380000380000380000700000700000700000700000E00000E00030E0
0079C000F180006300003C00001228829E0E>I<01E0000FE00001C00001C00001C00001
C0000380000380000380000380000700000700000703C00704200E08E00E11E00E21E00E
40C01C80001D00001E00001FC00038E00038700038700038384070708070708070708070
3100E03100601E0013207D9F15>I<03C01FC0038003800380038007000700070007000E
000E000E000E001C001C001C001C0038003800380038007000700070007100E200E200E2
00E200640038000A207C9F0C>I<1C0F80F0002630C318004740640C004780680E004700
700E004700700E008E00E01C000E00E01C000E00E01C000E00E01C001C01C038001C01C0
38001C01C038001C01C0708038038071003803806100380380E100380380620070070066
00300300380021147C9325>I<1C0F802630C04740604780604700704700708E00E00E00
E00E00E00E00E01C01C01C01C01C01C01C03843803883803083807083803107003303001
C016147C931A>I<007C0001C3000301800E01C01E01C01C01E03C01E07801E07801E078
01E0F003C0F003C0F003C0F00780F00700700F00700E0030180018700007C00013147C93
17>I<01C1E002621804741C04781C04701E04701E08E01E00E01E00E01E00E01E01C03C
01C03C01C03C01C0380380780380700380E003C1C0072380071E000700000700000E0000
0E00000E00000E00001C00001C0000FFC000171D809317>I<00F0400388C00705800E03
801C03803C0380380700780700780700780700F00E00F00E00F00E00F00E00F01C00F01C
00703C00705C0030B8000F380000380000380000700000700000700000700000E00000E0
000FFE00121D7C9315>I<1C1E002661004783804787804707804703008E00000E00000E
00000E00001C00001C00001C00001C000038000038000038000038000070000030000011
147C9313>I<00FC030206010C030C070C060C000F800FF007F803FC003E000E700EF00C
F00CE008401020601F8010147D9313>I<018001C0038003800380038007000700FFF007
000E000E000E000E001C001C001C001C003800380038003820704070407080708031001E
000C1C7C9B0F>I<0E00C03300E02301C04381C04301C04701C08703800E03800E03800E
03801C07001C07001C07001C07101C0E20180E20180E201C1E200C264007C38014147C93
18>I<0E03803307802307C04383C04301C04700C08700800E00800E00800E00801C0100
1C01001C01001C02001C02001C04001C04001C08000E300003C00012147C9315>I<0E00
C1C03300E3C02301C3E04381C1E04301C0E04701C060870380400E0380400E0380400E03
80401C0700801C0700801C0700801C0701001C0701001C0602001C0F02000C0F04000E13
080003E1F0001B147C931E>I<0383800CC4401068E01071E02071E02070C040E00000E0
0000E00000E00001C00001C00001C00001C040638080F38080F38100E5810084C6007878
0013147D9315>I<0E00C03300E02301C04381C04301C04701C08703800E03800E03800E
03801C07001C07001C07001C07001C0E00180E00180E001C1E000C3C0007DC00001C0000
1C00003800F03800F07000E06000C0C0004380003E0000131D7C9316>I<01C04003E080
07F1800C1F00080200000400000800001000002000004000008000010000020000040100
0802001002003E0C0063FC0041F80080E00012147D9313>I E /Fq
40 122 df<FFFFFFFFFFFFFFFFFFFFFFFF10067F9016>45 D<000E00001E00007E0007FE
00FFFE00FFFE00F8FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE
0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE
0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE007FFFFE7FFFFE7FFFFE1727
7BA622>49 D<00FF800007FFF0000FFFFC001E03FE003800FF807C003F80FE003FC0FF00
1FC0FF001FE0FF000FE0FF000FE07E000FE03C001FE000001FE000001FC000001FC00000
3F8000003F0000007E000000FC000000F8000001F0000003E00000078000000F0000001E
0000003C00E0007000E000E000E001C001C0038001C0060001C00FFFFFC01FFFFFC03FFF
FFC07FFFFFC0FFFFFF80FFFFFF80FFFFFF801B277DA622>I<007F800003FFF00007FFFC
000F80FE001F007F003F807F003F803F803F803F803F803F801F803F801F003F8000007F
0000007F0000007E000000FC000001F8000007F00000FFC00000FFC0000001F80000007E
0000003F0000003F8000001FC000001FC000001FE000001FE03C001FE07E001FE0FF001F
E0FF001FE0FF001FC0FF003FC0FE003F807C007F003F00FE001FFFFC0007FFF00000FF80
001B277DA622>I<00000E0000001E0000003E0000007E000000FE000000FE000001FE00
0003FE0000077E00000E7E00000E7E00001C7E0000387E0000707E0000E07E0000E07E00
01C07E0003807E0007007E000E007E000E007E001C007E0038007E0070007E00E0007E00
FFFFFFF8FFFFFFF8FFFFFFF80000FE000000FE000000FE000000FE000000FE000000FE00
0000FE000000FE00007FFFF8007FFFF8007FFFF81D277EA622>I<180003001F801F001F
FFFE001FFFFC001FFFF8001FFFF0001FFFC0001FFF00001C0000001C0000001C0000001C
0000001C0000001C0000001C0000001C7FC0001DFFF8001F80FC001E003F0008003F0000
001F8000001FC000001FC000001FE000001FE018001FE07C001FE0FE001FE0FE001FE0FE
001FE0FE001FC0FC001FC078003F8078003F803C007F001F01FE000FFFFC0003FFF00000
FF80001B277DA622>I<0007F800003FFE0000FFFF0001FC078003F00FC007C01FC00F80
1FC01F801FC01F001FC03F000F803F0000007E0000007E0000007E000000FE020000FE1F
F000FE3FFC00FE603E00FE801F00FF801F80FF000FC0FF000FC0FE000FE0FE000FE0FE00
0FE0FE000FE07E000FE07E000FE07E000FE07E000FE03E000FE03F000FC01F000FC01F00
1F800F801F0007E07E0003FFFC0001FFF800003FC0001B277DA622>I<00000780000000
000780000000000FC0000000000FC0000000000FC0000000001FE0000000001FE0000000
003FF0000000003FF0000000003FF00000000077F80000000077F800000000F7FC000000
00E3FC00000000E3FC00000001C1FE00000001C1FE00000003C1FF0000000380FF000000
0380FF00000007007F80000007007F8000000F007FC000000E003FC000000E003FC00000
1C001FE000001C001FE000003FFFFFF000003FFFFFF000003FFFFFF00000700007F80000
700007F80000F00007FC0000E00003FC0000E00003FC0001C00001FE0001C00001FE0003
C00001FF00FFFE003FFFFCFFFE003FFFFCFFFE003FFFFC2E297EA833>65
D<FFFFFFFFE0FFFFFFFFE0FFFFFFFFE003FC001FE003FC0007F003FC0001F003FC0001F0
03FC0000F003FC00007003FC00007003FC00007003FC01C07803FC01C03803FC01C03803
FC01C03803FC03C00003FC03C00003FC0FC00003FFFFC00003FFFFC00003FFFFC00003FC
0FC00003FC03C00003FC03C00003FC01C00E03FC01C00E03FC01C00E03FC01C01C03FC00
001C03FC00001C03FC00001C03FC00003C03FC00003803FC00007803FC0000F803FC0001
F803FC0003F803FC001FF8FFFFFFFFF0FFFFFFFFF0FFFFFFFFF027297EA82C>69
D<FFFFFFFFC0FFFFFFFFC0FFFFFFFFC003FC003FC003FC000FE003FC0003E003FC0001E0
03FC0001E003FC0000E003FC0000E003FC0000E003FC0000F003FC01C07003FC01C07003
FC01C07003FC01C00003FC03C00003FC03C00003FC0FC00003FFFFC00003FFFFC00003FF
FFC00003FC0FC00003FC03C00003FC03C00003FC01C00003FC01C00003FC01C00003FC01
C00003FC00000003FC00000003FC00000003FC00000003FC00000003FC00000003FC0000
0003FC00000003FC000000FFFFFC0000FFFFFC0000FFFFFC000024297EA82A>I<FFFFF0
0FFFFFFFFFF00FFFFFFFFFF00FFFFF03FC00003FC003FC00003FC003FC00003FC003FC00
003FC003FC00003FC003FC00003FC003FC00003FC003FC00003FC003FC00003FC003FC00
003FC003FC00003FC003FC00003FC003FC00003FC003FC00003FC003FC00003FC003FFFF
FFFFC003FFFFFFFFC003FFFFFFFFC003FC00003FC003FC00003FC003FC00003FC003FC00
003FC003FC00003FC003FC00003FC003FC00003FC003FC00003FC003FC00003FC003FC00
003FC003FC00003FC003FC00003FC003FC00003FC003FC00003FC003FC00003FC003FC00
003FC003FC00003FC0FFFFF00FFFFFFFFFF00FFFFFFFFFF00FFFFF30297EA835>72
D<FFFFFCFFFFFCFFFFFC01FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE
0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE
0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE0001FE
0001FE0001FE00FFFFFCFFFFFCFFFFFC16297FA819>I<FFFFFC0000FFFFFC0000FFFFFC
000003FC00000003FC00000003FC00000003FC00000003FC00000003FC00000003FC0000
0003FC00000003FC00000003FC00000003FC00000003FC00000003FC00000003FC000000
03FC00000003FC00000003FC00000003FC00000003FC00000003FC00000003FC00000003
FC0001C003FC0001C003FC0001C003FC0001C003FC0003C003FC00038003FC00038003FC
00078003FC00078003FC000F8003FC000F8003FC001F8003FC007F8003FC01FF00FFFFFF
FF00FFFFFFFF00FFFFFFFF0022297EA828>76 D<FFFE0000003FFF80FFFE0000003FFF80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>I<FFFC00007F
FFFFFE00007FFFFFFF00007FFF03FF800001C003FFC00001C003BFE00001C0039FE00001
C0039FF00001C0038FF80001C00387FC0001C00383FE0001C00381FF0001C00380FF8001
C003807F8001C003807FC001C003803FE001C003801FF001C003800FF801C0038007FC01
C0038003FC01C0038003FE01C0038001FF01C0038000FF81C00380007FC1C00380003FE1
C00380001FF1C00380000FF1C00380000FF9C003800007FDC003800003FFC003800001FF
C003800000FFC0038000007FC0038000007FC0038000003FC0038000001FC0038000000F
C00380000007C0FFFE000003C0FFFE000001C0FFFE000001C030297EA835>I<FFFFFFF8
00FFFFFFFF00FFFFFFFFC003FC003FE003FC0007F003FC0003F803FC0003FC03FC0001FC
03FC0001FE03FC0001FE03FC0001FE03FC0001FE03FC0001FE03FC0001FE03FC0001FE03
FC0001FC03FC0003FC03FC0003F803FC0007F003FC003FE003FFFFFF8003FFFFFE0003FC
00000003FC00000003FC00000003FC00000003FC00000003FC00000003FC00000003FC00
000003FC00000003FC00000003FC00000003FC00000003FC00000003FC00000003FC0000
0003FC000000FFFFF00000FFFFF00000FFFFF0000027297EA82E>80
D<FFFFFFE00000FFFFFFFE0000FFFFFFFF800003FC003FE00003FC000FF00003FC0007F8
0003FC0003FC0003FC0001FC0003FC0001FE0003FC0001FE0003FC0001FE0003FC0001FE
0003FC0001FE0003FC0001FE0003FC0001FC0003FC0003F80003FC0007F80003FC000FE0
0003FC003FC00003FFFFFE000003FFFFFE000003FC00FF800003FC003FC00003FC001FE0
0003FC000FF00003FC0007F80003FC0007F80003FC0007F80003FC0007F80003FC0007F8
0003FC0007F80003FC0007F80003FC0007F80003FC0007F80003FC0007F80E03FC0007F8
0E03FC0003F80E03FC0001FC1CFFFFF000FE1CFFFFF0007FF8FFFFF0000FE02F297EA832
>82 D<00FF00C003FFE1C00FFFF9C01F80FFC03F003FC03E000FC07C0007C07C0007C0FC
0003C0FC0003C0FC0001C0FE0001C0FE0001C0FF000000FFC000007FFC00007FFFE0003F
FFF8001FFFFE001FFFFF0007FFFF8003FFFFC000FFFFC0000FFFE000007FE000001FF000
000FF0000007F0E00003F0E00003F0E00003F0E00003F0F00003E0F00003E0F80007E0FC
0007C0FF000F80FFE01F80E3FFFF00E1FFFC00C01FF0001C297CA825>I<7FFFFFFFFF80
7FFFFFFFFF807FFFFFFFFF807F807F807F807C007F800F8078007F80078078007F800780
70007F800380F0007F8003C0F0007F8003C0E0007F8001C0E0007F8001C0E0007F8001C0
E0007F8001C0E0007F8001C000007F80000000007F80000000007F80000000007F800000
00007F80000000007F80000000007F80000000007F80000000007F80000000007F800000
00007F80000000007F80000000007F80000000007F80000000007F80000000007F800000
00007F80000000007F80000000007F80000000007F80000000007F80000000007F800000
00FFFFFFC00000FFFFFFC00000FFFFFFC0002A287EA72F>I<03FF80000FFFF0001F01FC
003F80FE003F807F003F803F003F803F801F003F8000003F8000003F8000003F8000003F
80003FFF8001FC3F800FE03F801F803F803F003F807E003F80FC003F80FC003F80FC003F
80FC003F80FC005F807E00DF803F839FFC1FFE0FFC03F803FC1E1B7E9A21>97
D<003FF00001FFFC0003F03E000FC07F001F807F003F007F003F007F007F003E007E0000
007E000000FE000000FE000000FE000000FE000000FE000000FE000000FE0000007E0000
007E0000007F0000003F0003803F8003801F8007000FE00E0003F83C0001FFF800003FC0
00191B7E9A1E>99 D<00007FF000007FF000007FF0000007F0000007F0000007F0000007
F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0000007F0003F87
F001FFF7F007F03FF00FC00FF01F8007F03F0007F03F0007F07E0007F07E0007F07E0007
F0FE0007F0FE0007F0FE0007F0FE0007F0FE0007F0FE0007F0FE0007F0FE0007F07E0007
F07E0007F03F0007F03F0007F01F800FF00FC01FF007E07FFF01FFE7FF007F87FF202A7E
A925>I<003FC00001FFF00003E07C000F803E001F801F001F001F003F000F807E000F80
7E000FC07E000FC0FE0007C0FE0007C0FFFFFFC0FFFFFFC0FE000000FE000000FE000000
7E0000007E0000007F0000003F0001C01F0001C00F80038007C0070003F01E0000FFFC00
003FE0001A1B7E9A1F>I<0007F8003FFC007E3E01FC7F03F87F03F07F07F07F07F03E07
F00007F00007F00007F00007F00007F00007F000FFFFC0FFFFC0FFFFC007F00007F00007
F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007
F00007F00007F00007F00007F00007F00007F0007FFF807FFF807FFF80182A7EA915>I<
007F80F001FFE3F807C0FE1C0F807C7C1F003E7C1F003E103F003F003F003F003F003F00
3F003F003F003F003F003F001F003E001F003E000F807C0007C0F80005FFE0000C7F8000
180000001C0000001C0000001E0000001FFFF8001FFFFF000FFFFFC007FFFFE003FFFFF0
0FFFFFF03E0007F07C0001F8F80000F8F80000F8F80000F8F80000F87C0001F07C0001F0
3F0007E00FC01F8007FFFF00007FF0001E287E9A22>I<FFE00000FFE00000FFE000000F
E000000FE000000FE000000FE000000FE000000FE000000FE000000FE000000FE000000F
E000000FE000000FE000000FE07E000FE1FF800FE30FC00FE40FE00FE807E00FF807F00F
F007F00FF007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F00F
E007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F0FF
FE3FFFFFFE3FFFFFFE3FFF202A7DA925>I<07000FC01FE03FE03FE03FE01FE00FC00700
0000000000000000000000000000FFE0FFE0FFE00FE00FE00FE00FE00FE00FE00FE00FE0
0FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE0FFFEFFFEFFFE0F2B7EAA
12>I<FFE00000FFE00000FFE000000FE000000FE000000FE000000FE000000FE000000F
E000000FE000000FE000000FE000000FE000000FE000000FE000000FE03FF80FE03FF80F
E03FF80FE007000FE00E000FE03C000FE078000FE0F0000FE1E0000FE3C0000FE780000F
EFC0000FFFE0000FFFE0000FF7F0000FE3F8000FC1FC000FC1FC000FC0FE000FC07F000F
C07F000FC03F800FC01FC00FC01FC0FFFC7FFCFFFC7FFCFFFC7FFC1E2A7EA923>107
D<FFE0FFE0FFE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00F
E00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00F
E00FE00FE00FE0FFFEFFFEFFFE0F2A7EA912>I<FFC07F001FC000FFC1FFC07FF000FFC3
07E0C1F8000FC407F101FC000FC803F200FC000FD803FE00FE000FD003FC00FE000FD003
FC00FE000FE003F800FE000FE003F800FE000FE003F800FE000FE003F800FE000FE003F8
00FE000FE003F800FE000FE003F800FE000FE003F800FE000FE003F800FE000FE003F800
FE000FE003F800FE000FE003F800FE000FE003F800FE000FE003F800FE000FE003F800FE
000FE003F800FE00FFFE3FFF8FFFE0FFFE3FFF8FFFE0FFFE3FFF8FFFE0331B7D9A38>I<
FFC07E00FFC1FF80FFC30FC00FC40FE00FC807E00FD807F00FD007F00FD007F00FE007F0
0FE007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F0
0FE007F00FE007F00FE007F00FE007F00FE007F00FE007F0FFFE3FFFFFFE3FFFFFFE3FFF
201B7D9A25>I<003FE00001FFFC0003F07E000FC01F801F800FC03F0007E03F0007E07E
0003F07E0003F07E0003F0FE0003F8FE0003F8FE0003F8FE0003F8FE0003F8FE0003F8FE
0003F8FE0003F87E0003F07E0003F03F0007E03F0007E01F800FC00FC01F8007F07F0001
FFFC00003FE0001D1B7E9A22>I<FFE1FE00FFE7FF80FFFE0FE00FF803F00FF001F80FE0
01FC0FE000FC0FE000FE0FE000FE0FE0007F0FE0007F0FE0007F0FE0007F0FE0007F0FE0
007F0FE0007F0FE0007F0FE0007E0FE000FE0FE000FE0FE000FC0FE001FC0FF001F80FF8
03F00FFC0FE00FEFFF800FE1FC000FE000000FE000000FE000000FE000000FE000000FE0
00000FE000000FE000000FE00000FFFE0000FFFE0000FFFE000020277E9A25>I<FFC3E0
FFC7F8FFCC7C0FD8FE0FD0FE0FD0FE0FF0FE0FE07C0FE0000FE0000FE0000FE0000FE000
0FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE000FFFF00
FFFF00FFFF00171B7E9A1B>114 D<03FE300FFFF03E03F07800F07000F0F00070F00070
F80070FE0000FFE0007FFF007FFFC03FFFE01FFFF007FFF800FFF80007FC0000FCE0007C
E0003CF0003CF00038F80038FC0070FF01E0E7FFC0C1FF00161B7E9A1B>I<0070000070
0000700000700000F00000F00000F00001F00003F00003F00007F0001FFFE0FFFFE0FFFF
E007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F0
0007F00007F07007F07007F07007F07007F07007F07007F07003F0E001F8C000FFC0003F
0014267FA51A>I<FFE07FF0FFE07FF0FFE07FF00FE007F00FE007F00FE007F00FE007F0
0FE007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F0
0FE007F00FE007F00FE007F00FE007F00FE007F00FE00FF00FE00FF007E017F003F067FF
01FFC7FF007F87FF201B7D9A25>I<FFFC7FFC1FFCFFFC7FFC1FFCFFFC7FFC1FFC0FE00F
E001C007F007E0038007F007E0038007F807F0078003F807F0070003F807F8070001FC0F
F80E0001FC0FF80E0001FE1FFC1E0000FE1CFC1C0000FE1CFE1C0000FF387E3C00007F38
7E3800007F787F3800003FF03F7000003FF03F7000003FE01FF000001FE01FE000001FE0
1FE000000FC00FC000000FC00FC000000FC00FC0000007800780000007800780002E1B7F
9A31>119 D<FFFC1FFEFFFC1FFEFFFC1FFE07F0078003F8070001FC0F0001FE1E0000FE
3C00007F7800003FF800003FF000001FE000000FE0000007F0000007F800000FF800001F
FC00003DFE000038FF0000787F0000F03F8001E03FC003C01FE003800FE0FFF03FFFFFF0
3FFFFFF03FFF201B7F9A23>I<FFFE07FFFFFE07FFFFFE07FF07F000E007F000E007F801
E003F801C003F801C001FC038001FC038001FE078000FE070000FF0F00007F0E00007F0E
00003F9C00003F9C00003FFC00001FF800001FF800000FF000000FF0000007F0000007E0
000007E0000003C0000003C000000380000003800000078000380700007C070000FE0E00
00FE0E0000FE1C0000FE3800007C7000003FE000000F80000020277F9A23>I
E /Fr 21 122 df<007C0001860002030004038004838008838010838010838010838011
07001207000C0E00001C000030000060000180000200000C00001001002001003C060067
FE00C1FC0080F00011187D9714>50 D<001F000060800180800303800603800E00001C00
0018000038000039F000721800740C00780E00700E00F00E00E00E00E00E00E00E00E01C
00E01C0060380060700030C0001F800011187C9714>54 D<0000200000600000600000E0
0001E00001E0000270000270000470000870000870001070001070002070002070004070
00807000FFF00100380100380200380400380400380C00381C0038FF01FF181A7E991D>
65 D<000F8200706200C01603801E07000C0E000C1C000C18000C380008300008700000
700000E00000E00000E00000E00000E00020E00020E00020E00040600040600080300100
1006000C180003E000171A7A991B>67 D<03F8001FC00078003C000078003C000078005C
0000B800B80000B800B800009C013800009C013800011C027000011C027000011C047000
011C087000021C08E000021C10E000021C10E000021C20E000041C41C000041C41C00004
1C81C000041C81C000080F038000080F038000080E038000180C038000380C070000FF08
3FF000221A7D9922>77 D<003F1000609001807001007003002006002006002006002006
000007000007C00003F80001FE00007F00000F8000038000018000018020018020018060
0300600300600600700C00C8180087E000141A7D9916>83 D<3FFFFC381C0C201C04401C
0440380480380480380480380400700000700000700000700000E00000E00000E00000E0
0001C00001C00001C00001C000038000038000038000038000078000FFF800161A79991B
>I<01E006181C08380870087010FFE0E000E000E000E000E0086010602030C01F000D10
7C8F12>101 D<000700001980001B80003B000030000030000070000070000070000070
0007FF0000E00000E00000E00000E00000E00001C00001C00001C00001C00001C0000380
00038000038000038000038000070000070000070000660000E40000CC00007000001121
81990C>I<00F300038B800607000E07000C07001C0700380E00380E00380E00380E0030
1C00301C00301C00183C0018780007B800003800003800007000607000E0E000C1C0007F
000011177E8F12>I<1F80000380000380000380000700000700000700000700000E0000
0E00000E7C000F86001E07001E07001C07001C0700380E00380E00380E00381C00701C80
701C80703880703900E01900600E00111A7E9914>I<030706000000000000384C4E8E9C
9C1C3838707272E2E4643808197C980C>I<307C1E00598663009E0783809E0703809C07
03809C070380380E0700380E0700380E0700380E0E00701C0E40701C0E40701C1C40701C
1C80E0380C80601807001A107C8F1F>109 D<307C005986009E07009E07009C07009C07
00380E00380E00380E00381C00701C80701C80703880703900E01900600E0011107C8F16
>I<01F006180C0C180E300E700E600EE00EE00EE00CE01CE018E030606030C01F000F10
7C8F14>I<030F000590C009E0C009C06009C06009C0600380E00380E00380E00380E007
01C00701800703800703000E8E000E78000E00000E00001C00001C00001C00001C0000FF
00001317808F14>I<30F059189E389C189C009C00380038003800380070007000700070
00E00060000D107C8F10>114 D<03E004300830187018601C001F801FC00FE000E00060
E060E06080C041803E000C107D8F10>I<06000E000E000E000E001C001C00FFC01C0038
003800380038007000700070007000E100E100E100E200640038000A177C960D>I<3806
4C074E0E8E0E9C0E9C0E1C1C381C381C381C7039703970393079389A0F0C10107C8F15>
I<38064C074E0E8E0E9C0E9C0E1C1C381C381C381C703870387038307838F00F70007000
6060E0E1C0C18047003C0010177C8F13>121 D E /Fs 64 124 df<00FC000182000703
000607000E02000E00000E00000E00000E00000E0000FFFF000E07000E07000E07000E07
000E07000E07000E07000E07000E07000E07000E07000E07000E07000E07007F0FE0131A
809915>12 D<007E1F8001C170400703C060060380E00E0380400E0380000E0380000E03
80000E0380000E038000FFFFFFE00E0380E00E0380E00E0380E00E0380E00E0380E00E03
80E00E0380E00E0380E00E0380E00E0380E00E0380E00E0380E00E0380E00E0380E07F8F
E3FC1E1A809920>14 D<60F0F868080808101020C0050B7D990B>39
D<00800100020004000C00080018003000300030006000600060006000E000E000E000E0
00E000E000E000E000E000E0006000600060006000300030003000180008000C00040002
000100008009267D9B0F>I<8000400020001000180008000C0006000600060003000300
030003000380038003800380038003800380038003800380030003000300030006000600
06000C0008001800100020004000800009267E9B0F>I<60F0F07010101020204080040B
7D830B>44 D<FFC0FFC00A0280880D>I<60F0F06004047D830B>I<078018603030303060
186018E01CE01CE01CE01CE01CE01CE01CE01CE01CE01CE01CE01C601860187038303018
6007800E187E9713>48 D<03000700FF0007000700070007000700070007000700070007
000700070007000700070007000700070007000700FFF00C187D9713>I<0F8010602030
4038803CC01CE01C401C003C003800380070006000C00180010002000404080410043008
3FF87FF8FFF80E187E9713>I<0F8010E02070607870382038007800700070006000C00F
8000E000700038003C003CE03CE03CC03C4038407030E00F800E187E9713>I<00300030
007000F000F001700370027004700C7008701070307020704070C070FFFF007000700070
00700070007007FF10187F9713>I<30183FF03FE03FC02000200020002000200027C038
60203000380018001C001C401CE01CE01C80184038403030E00F800E187E9713>I<01E0
06100C1818383038300070006000E000E7C0E860F030F018E018E01CE01CE01C601C601C
701830183030186007C00E187E9713>I<078018603030201860186018601870103C303E
600F8007C019F030F86038401CC00CC00CC00CC00C6008201018600FC00E187E9713>56
D<07801860303070306018E018E018E01CE01CE01C601C603C303C185C0F9C001C001800
18003870307060604021801F000E187E9713>I<FFFFFF80FFFFFF800000000000000000
00000000000000000000000000000000FFFFFF80FFFFFF80190A7E8D1E>61
D<000C0000000C0000000C0000001E0000001E0000003F00000027000000270000004380
0000438000004380000081C0000081C0000081C0000100E0000100E00001FFE000020070
000200700006007800040038000400380008001C0008001C001C001E00FF00FFC01A1A7F
991D>65 D<FFFF000E01C00E00E00E00700E00780E00780E00780E00780E00780E00F00E
00E00E03C00FFF800E01E00E00700E00780E003C0E003C0E003C0E003C0E003C0E00380E
00780E00F00E01E0FFFF80161A7E991B>I<003F0201C0C603002E0E001E1C000E1C0006
380006780002700002700002F00000F00000F00000F00000F00000F00000700002700002
7800023800041C00041C00080E000803003001C0C0003F00171A7E991C>I<FFFFE00E00
E00E00600E00200E00300E00100E00100E00100E04000E04000E04000E0C000FFC000E0C
000E04000E04000E04000E00000E00000E00000E00000E00000E00000E00000E0000FFF0
00141A7E9919>70 D<003F020001C0C60003002E000E001E001C000E001C000600380006
00780002007000020070000200F0000000F0000000F0000000F0000000F0000000F001FF
C070000E0070000E0078000E0038000E001C000E001C000E000E000E000300160001C066
00003F82001A1A7E991E>I<FFE7FF0E00700E00700E00700E00700E00700E00700E0070
0E00700E00700E00700E00700FFFF00E00700E00700E00700E00700E00700E00700E0070
0E00700E00700E00700E00700E0070FFE7FF181A7E991D>I<FFE00E000E000E000E000E
000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E
000E00FFE00B1A7F990E>I<1FFC00E000E000E000E000E000E000E000E000E000E000E0
00E000E000E000E000E000E000E000E040E0E0E0E0E041C061801E000E1A7D9914>I<FF
0003FC0F0003C00F0003C00B8005C00B8005C00B8005C009C009C009C009C009C009C008
E011C008E011C008E011C0087021C0087021C0083841C0083841C0083841C0081C81C008
1C81C0081C81C0080F01C0080F01C0080F01C0080601C01C0601C0FF861FFC1E1A7E9923
>77 D<FE01FF0F00380F00100B80100B801009C01008E01008E010087010087010083810
081C10081C10080E10080E100807100803900803900801D00801D00800F0080070080070
0800301C0030FF8010181A7E991D>I<007F000001C1C000070070000E0038001C001C00
3C001E0038000E0078000F0070000700F0000780F0000780F0000780F0000780F0000780
F0000780F0000780F000078078000F0078000F0038000E003C001E001C001C000E003800
0700700001C1C000007F0000191A7E991E>I<007F000001C1C000070070000E0038001C
001C003C001E0038000E0078000F0070000700F0000780F0000780F0000780F0000780F0
000780F0000780F0000780F00007807000070078000F0038000E003C1C1E001C221C000E
4138000741F00001E1C000007F80800000C0800000C0800000E18000007F0000007F0000
003E0000001C0019217E991E>81 D<FFFC00000E0780000E01C0000E00E0000E00F0000E
00F0000E00F0000E00F0000E00F0000E00E0000E01C0000E0780000FFC00000E0600000E
0300000E0180000E01C0000E01C0000E01C0000E01E0000E01E0000E01E0000E01E0800E
00F0800E007100FFE03E00191A7E991C>I<0FC21836200E6006C006C002C002C002E000
70007E003FE01FF807FC003E000E00070003800380038003C002C006E004D81887E0101A
7E9915>I<7FFFFF00701C0700401C0100401C0100C01C0180801C0080801C0080801C00
80001C0000001C0000001C0000001C0000001C0000001C0000001C0000001C0000001C00
00001C0000001C0000001C0000001C0000001C0000001C0000001C0000001C000003FFE0
00191A7F991C>I<FFE1FF0E00380E00100E00100E00100E00100E00100E00100E00100E
00100E00100E00100E00100E00100E00100E00100E00100E00100E00100E001006002007
002003004001804000C180003E00181A7E991D>I<FF83FF0FF03C007801C01C00780080
1C007800800E007801000E007801000E009C010007009C020007009C020007010E020007
010E020003810E04000382070400038207040001C207080001C403880001C403880000E4
03900000E403900000E801D000007801E000007801E000007000E000007000E000003000
C0000020004000241A7F9927>87 D<FEFEC0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0
C0C0C0C0C0C0C0C0C0C0C0C0C0C0FEFE07257D9B0B>91 D<FEFE06060606060606060606
0606060606060606060606060606060606060606060606FEFE0725809B0B>93
D<3F8070C070E020700070007007F01C7030707070E070E071E071E0F171FB1E3C10107E
8F13>97 D<FC00001C00001C00001C00001C00001C00001C00001C00001C00001C00001C
F8001F0E001E07001C03801C01801C01C01C01C01C01C01C01C01C01C01C01C01C03801C
03001E07001B0C0010F000121A7F9915>I<07F80C1C381C30087000E000E000E000E000
E000E0007000300438080C1807E00E107F8F11>I<007E00000E00000E00000E00000E00
000E00000E00000E00000E00000E0003CE000C3E00380E00300E00700E00E00E00E00E00
E00E00E00E00E00E00E00E00600E00700E00381E001C2E0007CFC0121A7F9915>I<07C0
1C3030187018600CE00CFFFCE000E000E000E0006000300438080C1807E00E107F8F11>
I<01F0031807380E100E000E000E000E000E000E00FFC00E000E000E000E000E000E000E
000E000E000E000E000E000E000E007FE00D1A80990C>I<0FCE18733030703870387038
7038303018602FC02000600070003FF03FFC1FFE600FC003C003C003C0036006381C07E0
10187F8F13>I<FC00001C00001C00001C00001C00001C00001C00001C00001C00001C00
001CF8001D0C001E0E001E0E001C0E001C0E001C0E001C0E001C0E001C0E001C0E001C0E
001C0E001C0E001C0E00FF9FC0121A7F9915>I<18003C003C0018000000000000000000
00000000FC001C001C001C001C001C001C001C001C001C001C001C001C001C001C00FF80
091A80990A>I<018003C003C001800000000000000000000000000FC001C001C001C001
C001C001C001C001C001C001C001C001C001C001C001C001C001C001C041C0E180E3007E
000A2182990C>I<FC00001C00001C00001C00001C00001C00001C00001C00001C00001C
00001C3F801C1E001C18001C10001C20001C40001DC0001FE0001CE0001C70001C78001C
38001C1C001C1E001C1F00FF3FC0121A7F9914>I<FC001C001C001C001C001C001C001C
001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C00FF
80091A80990A>I<FC7C1F001D8E63801E0781C01E0781C01C0701C01C0701C01C0701C0
1C0701C01C0701C01C0701C01C0701C01C0701C01C0701C01C0701C01C0701C0FF9FE7F8
1D107F8F20>I<FCF8001D0C001E0E001E0E001C0E001C0E001C0E001C0E001C0E001C0E
001C0E001C0E001C0E001C0E001C0E00FF9FC012107F8F15>I<07E01C38300C700E6006
E007E007E007E007E007E0076006700E381C1C3807E010107F8F13>I<FCF8001F0E001E
07001C03801C03801C01C01C01C01C01C01C01C01C01C01C01C01C03801C03001E07001F
0C001CF0001C00001C00001C00001C00001C00001C0000FF800012177F8F15>I<03C200
0C2600381E00300E00700E00E00E00E00E00E00E00E00E00E00E00E00E00700E00700E00
381E001C2E0007CE00000E00000E00000E00000E00000E00000E00007FC012177F8F14>
I<FCE01D701E701E201C001C001C001C001C001C001C001C001C001C001C00FFC00C107F
8F0F>I<1F2060E04020C020C020F0007F003FC01FE000F080708030C030C020F0408F80
0C107F8F0F>I<0400040004000C000C001C003C00FFC01C001C001C001C001C001C001C
001C001C201C201C201C201C200E4003800B177F960F>I<FC7E001C0E001C0E001C0E00
1C0E001C0E001C0E001C0E001C0E001C0E001C0E001C0E001C0E001C1E000C2E0007CFC0
12107F8F15>I<FF1F803C06001C04001C04001E0C000E08000E08000710000710000790
0003A00003A00001C00001C00001C00000800011107F8F14>I<FF3F9F803C0E0700380E
06001C1604001C1704001E170C000E2308000E2388000F239800074190000741D00003C1
E0000380E0000380E0000180C0000100400019107F8F1C>I<FF3F803C1C001C18000E10
0007200007600003C00001C00001E00003E000027000043800083800181C00381E00FC3F
C012107F8F14>I<FF1F803C06001C04001C04001E0C000E08000E080007100007100007
900003A00003A00001C00001C00001C000008000008000010000010000E10000E20000E4
000078000011177F8F14>I<7FF86070407040E041C041C00380070007000E081C081C08
381070107030FFF00D107F8F11>I<FFFFC01201808913>I E /Ft
4 123 df<FFFFC0FFFFC012027D8618>0 D<0C000C008C40EDC07F800C007F80EDC08C40
0C000C000A0B7D8B10>3 D<1818181818FFFF1818181818181818181818181818180816
7D900E>121 D<1818181818FF18181818180018181818FFFF1818181808167D900E>I
E /Fu 93 128 df<00008000000001C000000001C000000003E000000003E000000005F0
00000004F000000008F80000000878000000107C000000103C000000203E000000201E00
0000401F000000400F000000800F80000080078000010007C000010003C000020003E000
020001E000040001F000040000F000080000F80008000078001000007C001000003C0020
00003E002000001E007FFFFFFF007FFFFFFF00FFFFFFFF8021207E9F26>1
D<001F800000F0F00001C0380007801E000F000F000E0007001E0007803C0003C03C0003
C07C0003E07C0003E0780001E0F80001F0F84021F0F84021F0F87FE1F0F87FE1F0F87FE1
F0F84021F0F84021F0F80001F0780001E0780001E07C0003E03C0003C03C0003C01E0007
800E0007000F000F0007801E0001C0380000F0F000001F80001C217D9F23>I<00020000
00070000000700000007000000070000000F8000000F8000000F80000013C0000013C000
0013C0000021E0000021E0000021E0000040F0000040F0000040F0000080780000807800
0080780001003C0001003C0001003C0002003E0002001E0002001E0004001F0004000F00
04000F000C000F801E000F80FF80FFF81D207F9F20>I<003FC00000E0700003801C0007
000E000F000F001E0007803E0007C03C0003C07C0003E07C0003E07C0003E07C0003E07C
0003E07C0003E07C0003E03C0003C03E0007C01E0007801E0007800E0007000F000F0007
000E0003000C0003801C0001801800818018108080101040C03020404020207FC03FE03F
C03FC03FC03FC01C207E9F21>10 D<001F83E000F06E3001C078780380F8780300F03007
007000070070000700700007007000070070000700700007007000FFFFFF800700700007
007000070070000700700007007000070070000700700007007000070070000700700007
007000070070000700700007007000070070000700700007007000070070007FE3FF001D
20809F1B>I<003F0000E0C001C0C00381E00701E00701E0070000070000070000070000
070000070000FFFFE00700E00700E00700E00700E00700E00700E00700E00700E00700E0
0700E00700E00700E00700E00700E00700E00700E00700E00700E07FC3FE1720809F19>
I<003FE000E0E001C1E00381E00700E00700E00700E00700E00700E00700E00700E00700
E0FFFFE00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700
E00700E00700E00700E00700E00700E00700E00700E07FE7FE1720809F19>I<001F81F8
0000F04F040001C07C06000380F80F000300F00F000700F00F0007007000000700700000
0700700000070070000007007000000700700000FFFFFFFF000700700700070070070007
007007000700700700070070070007007007000700700700070070070007007007000700
700700070070070007007007000700700700070070070007007007000700700700070070
070007007007007FE3FE3FF02420809F26>I<07070F1C383060C00808779F17>19
D<C0C061803F001E000C000A057A9C17>I<FFFF10017D9A17>22
D<7038F87CFC7EFC7E743A0402040204020804080410081008201040200F0E7E9F17>34
D<0078000000840000018400000302000007020000070200000702000007020000070400
000704000007080000070800000310000003A00FFC03C003E0038001C001C0008001C001
0003E0010004E0020008F00200187004003078080070380800701C1000F01E1000F00E20
00F0074000F003C0087003C0087801C010380670301C18386007E00F801E227EA023>38
D<70F8FCFC74040404080810102040060E7C9F0D>I<0020004000800100020006000C00
0C00180018003000300030007000600060006000E000E000E000E000E000E000E000E000
E000E000E000E0006000600060007000300030003000180018000C000C00060002000100
0080004000200B2E7DA112>I<800040002000100008000C000600060003000300018001
80018001C000C000C000C000E000E000E000E000E000E000E000E000E000E000E000E000
C000C000C001C001800180018003000300060006000C00080010002000400080000B2E7D
A112>I<0006000000060000000600000006000000060000000600000006000000060000
00060000000600000006000000060000000600000006000000060000FFFFFFF0FFFFFFF0
000600000006000000060000000600000006000000060000000600000006000000060000
0006000000060000000600000006000000060000000600001C207D9A23>43
D<70F8FCFC74040404080810102040060E7C840D>I<FFC0FFC00A027F8A0F>I<70F8F8F8
7005057C840D>I<000100030003000600060006000C000C000C00180018001800300030
003000600060006000C000C000C00180018001800300030003000600060006000C000C00
0C00180018001800300030003000600060006000C000C000C000102D7DA117>I<03F000
0E1C001C0E00180600380700700380700380700380700380F003C0F003C0F003C0F003C0
F003C0F003C0F003C0F003C0F003C0F003C0F003C0F003C0F003C0700380700380700380
7807803807001806001C0E000E1C0003F000121F7E9D17>I<018003800F80F380038003
800380038003800380038003800380038003800380038003800380038003800380038003
80038003800380038007C0FFFE0F1E7C9D17>I<03F0000C1C00100E0020070040078080
0780F007C0F803C0F803C0F803C02007C00007C0000780000780000F00000E00001C0000
380000700000600000C0000180000300000600400C00401800401000803FFF807FFF80FF
FF80121E7E9D17>I<03F0000C1C00100E00200F00780F80780780780780380F80000F80
000F00000F00000E00001C0000380003F000003C00000E00000F000007800007800007C0
2007C0F807C0F807C0F807C0F00780400780400F00200E001C3C0003F000121F7E9D17>
I<000600000600000E00000E00001E00002E00002E00004E00008E00008E00010E00020E
00020E00040E00080E00080E00100E00200E00200E00400E00C00E00FFFFF0000E00000E
00000E00000E00000E00000E00000E0000FFE0141E7F9D17>I<1803001FFE001FFC001F
F8001FE00010000010000010000010000010000010000011F000161C00180E0010070010
07800003800003800003C00003C00003C07003C0F003C0F003C0E0038040038040070020
0600100E000C380003E000121F7E9D17>I<007C000182000701000E03800C07801C0780
380300380000780000700000700000F1F000F21C00F40600F80700F80380F80380F003C0
F003C0F003C0F003C0F003C07003C07003C07003803803803807001807000C0E00061C00
01F000121F7E9D17>I<4000007FFFC07FFF807FFF804001008002008002008004000008
0000080000100000200000200000400000400000C00000C00001C0000180000380000380
00038000038000078000078000078000078000078000078000078000030000121F7D9D17
>I<03F0000C0C001006003003002001806001806001806001807001807803003E03003F
06001FC8000FF00003F80007FC000C7E00103F00300F806003804001C0C001C0C000C0C0
00C0C000C0C000806001802001001002000C0C0003F000121F7E9D17>I<03F0000E1800
1C0C00380600380700700700700380F00380F00380F003C0F003C0F003C0F003C0F003C0
7007C07007C03807C0180BC00E13C003E3C0000380000380000380000700300700780600
780E00700C002018001070000FC000121F7E9D17>I<70F8F8F870000000000000000000
0070F8F8F87005147C930D>I<70F8F8F8700000000000000000000070F0F8F878080808
101010202040051D7C930D>I<7FFFFFE0FFFFFFF0000000000000000000000000000000
0000000000000000000000000000000000FFFFFFF07FFFFFE01C0C7D9023>61
D<0FC0307040384038E03CF03CF03C603C0038007000E000C00180018001000300020002
0002000200020002000000000000000000000007000F800F800F8007000E207D9F15>63
D<000100000003800000038000000380000007C0000007C0000007C0000009E0000009E0
000009E0000010F0000010F0000010F00000207800002078000020780000403C0000403C
0000403C0000801E0000801E0000FFFE0001000F0001000F0001000F0002000780020007
8002000780040003C00E0003C01F0007E0FFC03FFE1F207F9F22>65
D<FFFFE0000F80380007801E0007801F0007800F0007800F8007800F8007800F8007800F
8007800F8007800F0007801F0007801E0007803C0007FFF00007803C0007801E0007800F
0007800F8007800780078007C0078007C0078007C0078007C0078007C00780078007800F
8007800F0007801F000F803C00FFFFF0001A1F7E9E20>I<000FC040007030C001C009C0
038005C0070003C00E0001C01E0000C01C0000C03C0000C07C0000407C00004078000040
F8000000F8000000F8000000F8000000F8000000F8000000F8000000F8000000F8000000
780000007C0000407C0000403C0000401C0000401E0000800E0000800700010003800200
01C0040000703800000FC0001A217D9F21>I<FFFFE0000F803C0007801E000780070007
800380078003C0078001E0078001E0078001F0078000F0078000F0078000F8078000F807
8000F8078000F8078000F8078000F8078000F8078000F8078000F8078000F0078000F007
8000F0078001E0078001E0078003C0078003800780070007800E000F803C00FFFFE0001D
1F7E9E23>I<FFFFFF000F800F0007800300078003000780010007800180078000800780
008007800080078080800780800007808000078080000781800007FF8000078180000780
800007808000078080000780800007800020078000200780002007800040078000400780
0040078000C0078000C0078001800F800F80FFFFFF801B1F7E9E1F>I<FFFFFF000F800F
000780030007800300078001000780018007800080078000800780008007800080078080
000780800007808000078080000781800007FF8000078180000780800007808000078080
000780800007800000078000000780000007800000078000000780000007800000078000
000FC00000FFFE0000191F7E9E1E>I<000FE0200078186000E004E0038002E0070001E0
0F0000E01E0000601E0000603C0000603C0000207C00002078000020F8000000F8000000
F8000000F8000000F8000000F8000000F8000000F8007FFCF80003E0780001E07C0001E0
3C0001E03C0001E01E0001E01E0001E00F0001E0070001E0038002E000E0046000781820
000FE0001E217D9F24>I<FFF8FFF80F800F8007800F0007800F0007800F0007800F0007
800F0007800F0007800F0007800F0007800F0007800F0007800F0007800F0007FFFF0007
800F0007800F0007800F0007800F0007800F0007800F0007800F0007800F0007800F0007
800F0007800F0007800F0007800F0007800F000F800F80FFF8FFF81D1F7E9E22>I<FFFC
0FC007800780078007800780078007800780078007800780078007800780078007800780
07800780078007800780078007800780078007800FC0FFFC0E1F7F9E10>I<0FFFC0007C
00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C
00003C00003C00003C00003C00003C00003C00003C00003C00003C00203C00F83C00F83C
00F83C00F0380040780040700030E0000F800012207E9E17>I<FFFC0FFC0FC003E00780
018007800100078002000780040007800800078010000780200007804000078080000781
00000783000007878000078F80000793C0000791E00007A1E00007C0F0000780F0000780
780007803C0007803C0007801E0007801E0007800F000780078007800780078007C00FC0
07E0FFFC3FFC1E1F7E9E23>I<FFFE000FC0000780000780000780000780000780000780
000780000780000780000780000780000780000780000780000780000780000780000780
0007800207800207800207800207800607800407800407800C07801C0F807CFFFFFC171F
7E9E1C>I<FF80001FF80F80001F800780001F0005C0002F0005C0002F0005C0002F0004
E0004F0004E0004F000470008F000470008F000470008F000438010F000438010F000438
010F00041C020F00041C020F00041C020F00040E040F00040E040F00040E040F00040708
0F000407080F000407080F000403900F000403900F000401E00F000401E00F000401E00F
000E00C00F001F00C01F80FFE0C1FFF8251F7E9E2A>I<FF803FF807C007C007C0038005
E0010005E0010004F001000478010004780100043C0100043C0100041E0100040F010004
0F010004078100040781000403C1000401E1000401E1000400F1000400F1000400790004
003D0004003D0004001F0004001F0004000F0004000700040007000E0003001F000300FF
E001001D1F7E9E22>I<001F800000F0F00001C0380007801E000F000F000E0007001E00
07803C0003C03C0003C07C0003E0780001E0780001E0F80001F0F80001F0F80001F0F800
01F0F80001F0F80001F0F80001F0F80001F0F80001F0780001E07C0003E07C0003E03C00
03C03C0003C01E0007800E0007000F000F0007801E0001C0380000F0F000001F80001C21
7D9F23>I<FFFFE0000F80780007801C0007801E0007800F0007800F8007800F8007800F
8007800F8007800F8007800F8007800F0007801E0007801C000780780007FFE000078000
000780000007800000078000000780000007800000078000000780000007800000078000
000780000007800000078000000FC00000FFFC0000191F7E9E1F>I<FFFF80000F80F000
0780780007803C0007801E0007801E0007801F0007801F0007801F0007801F0007801E00
07801E0007803C00078078000780F00007FF80000781C0000780E0000780F00007807000
07807800078078000780780007807C0007807C0007807C0007807C0407807E0407803E04
0FC01E08FFFC0F10000003E01E207E9E21>82 D<07E0800C198010078030038060018060
0180E00180E00080E00080E00080F00000F000007800007F00003FF0001FFC000FFE0003
FF00001F800007800003C00003C00001C08001C08001C08001C08001C0C00180C00380E0
0300F00600CE0C0081F80012217D9F19>I<7FFFFFE0780F01E0600F0060400F0020400F
0020C00F0030800F0010800F0010800F0010800F0010000F0000000F0000000F0000000F
0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F
0000000F0000000F0000000F0000000F0000000F0000000F0000001F800007FFFE001C1F
7E9E21>I<FFFC3FF80FC007C00780038007800100078001000780010007800100078001
000780010007800100078001000780010007800100078001000780010007800100078001
000780010007800100078001000780010007800100078001000780010003800200038002
0001C0020001C0040000E008000070180000382000000FC0001D207E9E22>I<FFF003FE
1F8000F80F0000600F800060078000400780004003C0008003C0008003C0008001E00100
01E0010001F0010000F0020000F0020000F806000078040000780400003C0800003C0800
003C0800001E1000001E1000001F3000000F2000000F20000007C0000007C0000007C000
000380000003800000038000000100001F207F9E22>I<FFF07FF81FF01F800FC007C00F
00078003800F00078001000F0007C00100078007C00200078007C00200078007C0020003
C009E0040003C009E0040003C009E0040003E010F00C0001E010F0080001E010F0080001
F02078080000F02078100000F02078100000F0403C10000078403C20000078403C200000
78C03E2000003C801E4000003C801E4000003C801E4000001F000F8000001F000F800000
1F000F8000001E00078000000E00070000000E00070000000C000300000004000200002C
207F9E2F>I<FFF003FF1F8000F80F8000600780004007C0004003E0008001E0008001F0
010000F0030000F80200007C0400003C0400003E0800001E0800001F1000000FB0000007
A0000007C0000003C0000003C0000003C0000003C0000003C0000003C0000003C0000003
C0000003C0000003C0000003C0000007C000007FFE00201F7F9E22>89
D<7FFFF87C00F87000F06001E04001E0C003C0C003C0800780800F80800F00001E00001E
00003C00003C0000780000F80000F00001E00001E00003C00403C0040780040F80040F00
0C1E000C1E00083C00183C0018780038F801F8FFFFF8161F7D9E1C>I<FEFEC0C0C0C0C0
C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0
FEFE072D7CA10D>I<080410082010201040204020804080408040B85CFC7EFC7E7C3E38
1C0F0E7B9F17>I<FEFE0606060606060606060606060606060606060606060606060606
060606060606060606060606060606FEFE072D7FA10D>I<0C001E0033006180C0C08040
0A067A9E17>I<081020204040808080B8FCFC7C38060E7D9F0D>96
D<1FE000303000781800781C00300E00000E00000E00000E0000FE00078E001E0E00380E
00780E00F00E10F00E10F00E10F01E10781E103867200F83C014147E9317>I<0E0000FE
00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E3E000E
C3800F01C00F00E00E00E00E00700E00700E00780E00780E00780E00780E00780E00780E
00700E00700E00E00F00E00D01C00CC300083E0015207F9F19>I<03F80E0C1C1E381E38
0C70007000F000F000F000F000F000F00070007000380138011C020E0C03F010147E9314
>I<000380003F8000038000038000038000038000038000038000038000038000038000
038003E380061B801C0780380380380380700380700380F00380F00380F00380F00380F0
0380F003807003807003803803803807801C07800E1B8003E3F815207E9F19>I<03F000
0E1C001C0E00380700380700700700700380F00380F00380FFFF80F00000F00000F00000
7000007000003800801800800C010007060001F80011147F9314>I<007C00C6018F038F
07060700070007000700070007000700FFF0070007000700070007000700070007000700
0700070007000700070007000700070007007FF01020809F0E>I<0000E003E3300E3C30
1C1C30380E00780F00780F00780F00780F00780F00380E001C1C001E380033E000200000
2000003000003000003FFE001FFF800FFFC03001E0600070C00030C00030C00030C00030
6000603000C01C038003FC00141F7F9417>I<0E0000FE00000E00000E00000E00000E00
000E00000E00000E00000E00000E00000E00000E3E000E43000E81800F01C00F01C00E01
C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01
C00E01C0FFE7FC16207F9F19>I<1C001E003E001E001C00000000000000000000000000
0E007E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E00
0E00FFC00A1F809E0C>I<00E001F001F001F000E0000000000000000000000000007007
F000F0007000700070007000700070007000700070007000700070007000700070007000
7000700070007000706070F060F0C061803F000C28829E0E>I<0E0000FE00000E00000E
00000E00000E00000E00000E00000E00000E00000E00000E00000E0FF00E03C00E03000E
02000E04000E08000E10000E30000E70000EF8000F38000E1C000E1E000E0E000E07000E
07800E03800E03C00E03E0FFCFF815207F9F18>I<0E00FE000E000E000E000E000E000E
000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E
000E000E000E000E000E00FFE00B20809F0C>I<0E1F01F000FE618618000E81C81C000F
00F00E000F00F00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00
E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E0
0E000E00E00E00FFE7FE7FE023147F9326>I<0E3E00FE43000E81800F01C00F01C00E01
C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01
C00E01C0FFE7FC16147F9319>I<01F800070E001C03803801C03801C07000E07000E0F0
00F0F000F0F000F0F000F0F000F0F000F07000E07000E03801C03801C01C0380070E0001
F80014147F9317>I<0E3E00FEC3800F01C00F00E00E00E00E00F00E00700E00780E0078
0E00780E00780E00780E00780E00700E00F00E00E00F01E00F01C00EC3000E3E000E0000
0E00000E00000E00000E00000E00000E00000E0000FFE000151D7F9319>I<03E0800619
801C05803C0780380380780380700380F00380F00380F00380F00380F00380F003807003
807803803803803807801C0B800E138003E3800003800003800003800003800003800003
80000380000380003FF8151D7E9318>I<0E78FE8C0F1E0F1E0F0C0E000E000E000E000E
000E000E000E000E000E000E000E000E000E00FFE00F147F9312>I<1F9030704030C010
C010C010E00078007F803FE00FF00070803880188018C018C018E030D0608F800D147E93
12>I<020002000200060006000E000E003E00FFF80E000E000E000E000E000E000E000E
000E000E000E000E080E080E080E080E080610031001E00D1C7F9B12>I<0E01C0FE1FC0
0E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C0
0E01C00E01C00E03C00603C0030DC001F1FC16147F9319>I<FF83F81E01E01C00C00E00
800E00800E008007010007010003820003820003820001C40001C40001EC0000E80000E8
0000700000700000700000200015147F9318>I<FF9FE1FC3C0780701C0300601C038020
0E0380400E0380400E03C0400707C0800704C0800704E080038861000388710003C87300
01D0320001D03A0000F03C0000E01C0000E01C0000601800004008001E147F9321>I<7F
C3FC0F01E00701C007018003810001C20000E40000EC00007800003800003C00007C0000
4E000087000107000303800201C00601E01E01E0FF07FE1714809318>I<FF83F81E01E0
1C00C00E00800E00800E008007010007010003820003820003820001C40001C40001EC00
00E80000E800007000007000007000002000002000004000004000004000F08000F08000
F100006200003C0000151D7F9318>I<3FFF380E200E201C40384078407000E001E001C0
0380078007010E011E011C0338027006700EFFFE10147F9314>I<FFFFFC1601808C17>I<
1C043F1863F080E00E047C9D17>126 D<30307878F87C787830300E057C9E17>I
E /Fv 46 122 df<000FE000007FF80000F81C0001E07C0003E07C0007C07C0007C07C00
07C0380007C0000007C0000007C0000007C1FE00FFFFFE00FFFFFE0007C03E0007C03E00
07C03E0007C03E0007C03E0007C03E0007C03E0007C03E0007C03E0007C03E0007C03E00
07C03E0007C03E0007C03E0007C03E0007C03E003FF9FFC03FF9FFC01A20809F1D>12
D<387CFEFEFE7C3807077C860F>46 D<00E00001E0000FE000FFE000F3E00003E00003E0
0003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E0
0003E00003E00003E00003E00003E00003E00003E00003E000FFFF80FFFF80111D7C9C1A
>49 D<07F0001FFE00383F007C1F80FE0FC0FE0FC0FE0FE0FE07E07C07E03807E0000FE0
000FC0000FC0001F80001F00003E0000780000F00000E00001C0000380600700600E0060
1C00E01FFFC03FFFC07FFFC0FFFFC0FFFFC0131D7D9C1A>I<01FC0007FF000E0F801E0F
C03F07E03F07E03F07E03F07E01E0FC0000FC0000F80001F0001FC0001FC00000F800007
C00003E00003F00003F83803F87C03F8FE03F8FE03F8FE03F0FC03F07807E03C0FC01FFF
8003FC00151D7E9C1A>I<0001C00003C00007C00007C0000FC0001FC0003BC00073C000
63C000C3C00183C00383C00703C00E03C00C03C01803C03803C07003C0E003C0FFFFFEFF
FFFE0007C00007C00007C00007C00007C00007C000FFFE00FFFE171D7F9C1A>I<380380
3FFF803FFF003FFE003FFC003FF0003F800030000030000030000030000033F80037FE00
3C1F00380F801007C00007C00007E00007E07807E0FC07E0FC07E0FC07E0FC07C0780FC0
600F80381F001FFC0007F000131D7D9C1A>I<003F0001FFC007E0E00F81E01F03F01E03
F03E03F07C03F07C01E07C0000FC1000FCFF00FDFFC0FD03E0FE01F0FE01F0FC01F8FC01
F8FC01F8FC01F87C01F87C01F87C01F83C01F03E01F01E03E00F07C007FF8001FE00151D
7E9C1A>I<6000007FFFF87FFFF87FFFF07FFFE07FFFC0E00180C00300C00300C0060000
0C0000180000380000380000780000700000F00000F00001F00001F00001F00001F00003
F00003F00003F00003F00003F00003F00001E00000C000151E7D9D1A>I<01FC0007FF00
0F07801E03C01C01E03C01E03C01E03E01E03F01E03FC3C01FE3801FFF000FFE0007FF80
07FFC01FFFE03C3FF0780FF07803F8F001F8F000F8F00078F00078F000707800707C00E0
3E03C00FFF8003FC00151D7E9C1A>I<387CFEFEFE7C38000000000000387CFEFEFE7C38
07147C930F>58 D<0007FC02003FFF0E00FE03DE03F000FE07E0003E0FC0001E1F80001E
3F00000E3F00000E7F0000067E0000067E000006FE000000FE000000FE000000FE000000
FE000000FE000000FE0000007E0000007E0000067F0000063F0000063F00000C1F80000C
0FC0001807E0003803F0007000FE01C0003FFF800007FC001F1F7D9E26>67
D<FFFFFE0000FFFFFFC00007E007F00007E001F80007E000FC0007E0007E0007E0003F00
07E0003F0007E0001F8007E0001F8007E0001F8007E0001FC007E0001FC007E0001FC007
E0001FC007E0001FC007E0001FC007E0001FC007E0001FC007E0001FC007E0001F8007E0
001F8007E0001F8007E0003F0007E0003F0007E0007E0007E000FC0007E001F80007E007
F000FFFFFFC000FFFFFE0000221F7E9E28>I<FFFFFFE0FFFFFFE007E007E007E001E007
E000E007E0006007E0007007E0003007E0003007E0603007E0603007E0600007E0E00007
E1E00007FFE00007FFE00007E1E00007E0E00007E0600007E0600C07E0600C07E0000C07
E0001807E0001807E0001807E0003807E0007807E000F807E003F0FFFFFFF0FFFFFFF01E
1F7E9E22>I<FFFF0FFFF0FFFF0FFFF007E0007E0007E0007E0007E0007E0007E0007E00
07E0007E0007E0007E0007E0007E0007E0007E0007E0007E0007E0007E0007E0007E0007
E0007E0007FFFFFE0007FFFFFE0007E0007E0007E0007E0007E0007E0007E0007E0007E0
007E0007E0007E0007E0007E0007E0007E0007E0007E0007E0007E0007E0007E0007E000
7E0007E0007E00FFFF0FFFF0FFFF0FFFF0241F7E9E29>72 D<FFFFFFFF07E007E007E007
E007E007E007E007E007E007E007E007E007E007E007E007E007E007E007E007E007E007
E007E007E007E007E007E0FFFFFFFF101F7E9E14>I<FFFF00FFE0FFFF00FFE007E0001E
0007E000180007E000300007E000600007E001C00007E003800007E006000007E00C0000
07E018000007E030000007E0F0000007E1F8000007E3F8000007E6FC000007EC7E000007
F87F000007F03F000007E01F800007E00FC00007E00FE00007E007E00007E003F00007E0
01F80007E000FC0007E000FC0007E0007E0007E0007F00FFFF03FFF0FFFF03FFF0241F7E
9E29>75 D<FFFF8000FFFF800007E0000007E0000007E0000007E0000007E0000007E000
0007E0000007E0000007E0000007E0000007E0000007E0000007E0000007E0000007E000
0007E0000007E0000007E000C007E000C007E000C007E001C007E001C007E001C007E003
8007E0038007E00F8007E01F80FFFFFF80FFFFFF801A1F7E9E1F>I<FFE000003FF8FFF0
00007FF807F000007F0006F80000DF0006F80000DF0006F80000DF00067C00019F00067C
00019F00063E00031F00063E00031F00061F00061F00061F00061F00060F800C1F00060F
800C1F000607C0181F000607C0181F000607C0181F000603E0301F000603E0301F000601
F0601F000601F0601F000600F8C01F000600F8C01F0006007D801F0006007D801F000600
3F001F0006003F001F0006003F001F0006001E001F00FFF01E03FFF8FFF00C03FFF82D1F
7E9E32>I<001FF80000FFFF0001F81F8007E007E00FC003F01F8001F81F0000F83F0000
FC7F0000FE7E00007E7E00007EFE00007FFE00007FFE00007FFE00007FFE00007FFE0000
7FFE00007FFE00007FFE00007F7E00007E7F0000FE7F0000FE3F0000FC3F8001FC1F8001
F80FC003F007E007E001F81F8000FFFF00001FF800201F7D9E27>79
D<FFFFFE00FFFFFF8007E00FE007E003F007E001F807E001F807E001FC07E001FC07E001
FC07E001FC07E001FC07E001F807E001F807E003F007E00FE007FFFF8007FFFE0007E000
0007E0000007E0000007E0000007E0000007E0000007E0000007E0000007E0000007E000
0007E0000007E00000FFFF0000FFFF00001E1F7E9E24>I<FFFFF80000FFFFFF000007E0
1FC00007E007E00007E003F00007E003F00007E003F80007E003F80007E003F80007E003
F80007E003F00007E003F00007E007E00007E01FC00007FFFF000007FFFC000007E03E00
0007E01F000007E00F800007E00F800007E00FC00007E00FC00007E00FC00007E00FE000
07E00FE00007E00FE00007E00FE03007E007F03007E003F860FFFF01FFC0FFFF007F8024
1F7E9E27>82 D<03FC080FFF381E03F83800F8700078700038F00038F00018F00018F800
00FC00007FC0007FFE003FFF801FFFE00FFFF007FFF000FFF80007F80000FC00007C0000
3CC0003CC0003CC0003CE00038E00078F80070FE01E0E7FFC081FF00161F7D9E1D>I<7F
FFFFFC7FFFFFFC7C07E07C7007E01C6007E00C6007E00CE007E00EC007E006C007E006C0
07E006C007E0060007E0000007E0000007E0000007E0000007E0000007E0000007E00000
07E0000007E0000007E0000007E0000007E0000007E0000007E0000007E0000007E00000
07E00003FFFFC003FFFFC01F1E7E9D24>I<07FC001FFF003F0F803F07C03F03E03F03E0
0C03E00003E0007FE007FBE01F03E03C03E07C03E0F803E0F803E0F803E0FC05E07E0DE0
3FF8FE0FE07E17147F9319>97 D<01FE0007FF801F0FC03E0FC03E0FC07C0FC07C0300FC
0000FC0000FC0000FC0000FC0000FC00007C00007E00003E00603F00C01F81C007FF0001
FC0013147E9317>99 D<0007F80007F80000F80000F80000F80000F80000F80000F80000
F80000F80000F80000F801F8F80FFEF81F83F83E01F87E00F87C00F87C00F8FC00F8FC00
F8FC00F8FC00F8FC00F8FC00F87C00F87C00F87E00F83E01F81F07F80FFEFF03F8FF1820
7E9F1D>I<01FE0007FF800F83C01E01E03E00F07C00F07C00F8FC00F8FFFFF8FFFFF8FC
0000FC0000FC00007C00007C00003E00181E00180F807007FFE000FF8015147F9318>I<
001F8000FFC001F3E003E7E003C7E007C7E007C3C007C00007C00007C00007C00007C000
FFFC00FFFC0007C00007C00007C00007C00007C00007C00007C00007C00007C00007C000
07C00007C00007C00007C00007C00007C0003FFC003FFC0013207F9F10>I<01FC3C07FF
FE0F079E1E03DE3E03E03E03E03E03E03E03E03E03E01E03C00F07800FFF0009FC001800
001800001C00001FFF800FFFF007FFF81FFFFC3C007C70003EF0001EF0001EF0001E7800
3C78003C3F01F80FFFE001FF00171E7F931A>I<FF0000FF00001F00001F00001F00001F
00001F00001F00001F00001F00001F00001F00001F0FC01F3FE01F61F01FC0F81F80F81F
00F81F00F81F00F81F00F81F00F81F00F81F00F81F00F81F00F81F00F81F00F81F00F81F
00F8FFE3FFFFE3FF18207D9F1D>I<1C003E003F007F003F003E001C0000000000000000
0000000000FF00FF001F001F001F001F001F001F001F001F001F001F001F001F001F001F
001F001F00FFE0FFE00B217EA00E>I<FF0000FF00001F00001F00001F00001F00001F00
001F00001F00001F00001F00001F00001F01FE1F01FE1F00F01F00C01F03801F07001F0C
001F18001F7C001FFC001F9E001F0F001E0F801E07C01E03C01E01E01E01F01E00F8FFC3
FFFFC3FF18207E9F1C>107 D<FF00FF001F001F001F001F001F001F001F001F001F001F
001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F
00FFE0FFE00B207E9F0E>I<FE0FE03F80FE1FF07FC01E70F9C3E01E407D01F01E807E01
F01F807E01F01F007C01F01F007C01F01F007C01F01F007C01F01F007C01F01F007C01F0
1F007C01F01F007C01F01F007C01F01F007C01F01F007C01F01F007C01F0FFE3FF8FFEFF
E3FF8FFE27147D932C>I<FE0FC0FE3FE01E61F01EC0F81E80F81F00F81F00F81F00F81F
00F81F00F81F00F81F00F81F00F81F00F81F00F81F00F81F00F81F00F8FFE3FFFFE3FF18
147D931D>I<01FF0007FFC01F83F03E00F83E00F87C007C7C007CFC007EFC007EFC007E
FC007EFC007EFC007E7C007C7C007C3E00F83E00F81F83F007FFC001FF0017147F931A>
I<FF1FC0FF7FF01FE1F81F80FC1F007E1F007E1F003E1F003F1F003F1F003F1F003F1F00
3F1F003F1F003E1F007E1F007C1F80FC1FC1F81F7FE01F1F801F00001F00001F00001F00
001F00001F00001F0000FFE000FFE000181D7E931D>I<FE3E00FE7F801ECFC01E8FC01E
8FC01F8FC01F03001F00001F00001F00001F00001F00001F00001F00001F00001F00001F
00001F0000FFF000FFF00012147E9316>114 D<0FE63FFE701E600EE006E006F800FFC0
7FF83FFC1FFE03FE001FC007C007E007F006F81EFFFCC7F010147E9315>I<0180018001
8003800380038007800F803F80FFFCFFFC0F800F800F800F800F800F800F800F800F800F
800F860F860F860F860F8607CC03F801F00F1D7F9C14>I<FF07F8FF07F81F00F81F00F8
1F00F81F00F81F00F81F00F81F00F81F00F81F00F81F00F81F00F81F00F81F00F81F01F8
1F01F80F06F807FCFF03F8FF18147D931D>I<FFE07F80FFE07F801F001C000F8018000F
80180007C0300007C0300003E0600003E0600001F0C00001F0C00001F9C00000F9800000
FF8000007F0000007F0000003E0000003E0000001C0000001C000019147F931C>I<FFE7
FE1FE0FFE7FE1FE01F00F003001F00F803000F80F806000F80F8060007C1BC0C0007C1BC
0C0007C1BE0C0003E31E180003E31E180001F60F300001F60F300001F60FB00000FC07E0
0000FC07E000007803C000007803C000007803C000003001800023147F9326>I<FFE1FF
00FFE1FF000F80700007C0E00007E0C00003E1800001F3800000FF0000007E0000003E00
00003F0000007F8000006F800000C7C0000183E0000381F0000701F8000E00FC00FF81FF
80FF81FF8019147F931C>I<FFE07F80FFE07F801F001C000F8018000F80180007C03000
07C0300003E0600003E0600001F0C00001F0C00001F9C00000F9800000FF8000007F0000
007F0000003E0000003E0000001C0000001C0000001800000018000078300000FC300000
FC600000C0E00000E1C000007F8000001E000000191D7F931C>I
E /Fw 11 121 df<03CC063C0C3C181C3838303870387038E070E070E070E070E0E2C0E2
C0E261E462643C380F127B9115>97 D<01F007080C08181C3838300070007000E000E000
E000E000E000E008E010602030C01F000E127B9113>99 D<01E007100C10180838107010
70607F80E000E000E000E000E000E0086010602030C01F000D127B9113>101
D<00F3018F030F06070E0E0C0E1C0E1C0E381C381C381C381C383830383038187818F00F
700070007000E000E0C0C0E1C0C3007E00101A7D9113>103 D<01800380010000000000
000000000000000000001C002600470047008E008E000E001C001C001C00380038007100
71007100720072003C00091C7C9B0D>105 D<3C1E0780266318C04683A0E04703C0E08E
0380E08E0380E00E0380E00E0380E01C0701C01C0701C01C0701C01C070380380E038838
0E0388380E0708380E0710701C0320300C01C01D127C9122>109
D<3C3C002646004687004707008E07008E07000E07000E07001C0E001C0E001C0E001C1C
00381C40381C40383840383880701900300E0012127C9117>I<01E007180C0C180C380C
300E700E700EE01CE01CE01CE018E038E030E06060C031801E000F127B9115>I<3C3C26
C2468747078E068E000E000E001C001C001C001C0038003800380038007000300010127C
9112>114 D<00C001C001C001C00380038003800380FFE00700070007000E000E000E00
0E001C001C001C001C00384038403840388019000E000B1A7D990E>116
D<070E0019910010E38020E38041C30041C00001C00001C0000380000380000380000380
00070200670200E70400CB04008B080070F00011127D9113>120
D E /Fx 36 124 df<007E1F0001C1B1800303E3C00703C3C00E03C1800E01C0000E01C0
000E01C0000E01C0000E01C0000E01C000FFFFFC000E01C0000E01C0000E01C0000E01C0
000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0
000E01C0000E01C0000E01C0007F87FC001A1D809C18>11 D<007E0001C1800301800703
C00E03C00E01800E00000E00000E00000E00000E0000FFFFC00E01C00E01C00E01C00E01
C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01
C07F87F8151D809C17>I<004000800100020006000C000C001800180030003000700060
0060006000E000E000E000E000E000E000E000E000E000E000E000E00060006000600070
0030003000180018000C000C00060002000100008000400A2A7D9E10>40
D<800040002000100018000C000C000600060003000300038001800180018001C001C001
C001C001C001C001C001C001C001C001C001C0018001800180038003000300060006000C
000C00180010002000400080000A2A7E9E10>I<60F0F0701010101020204080040C7C83
0C>44 D<FFE0FFE00B0280890E>I<60F0F06004047C830C>I<7FFFFFC0700F01C0600F00
C0400F0040400F0040C00F0020800F0020800F0020800F0020000F0000000F0000000F00
00000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F00
00000F0000000F0000000F0000000F0000000F0000001F800003FFFC001B1C7F9B1E>84
D<FFF07FC00F000E000F0004000F0004000F0004000F0004000F0004000F0004000F0004
000F0004000F0004000F0004000F0004000F0004000F0004000F0004000F0004000F0004
000F0004000F0004000F0004000F0004000700080007800800038010000180100000C020
000070C000001F00001A1D7E9B1F>I<FFE0FFE0FF1F001F003C1E001E00180F001F0010
0F001F00100F001F001007801F00200780278020078027802003C027804003C043C04003
C043C04003E043C04001E081E08001E081E08001E081E08000F100F10000F100F10000F1
00F100007900FA00007A007A00007A007A00003E007C00003C003C00003C003C00003C00
3C00001800180000180018000018001800281D7F9B2B>87 D<1FC000307000783800781C
00301C00001C00001C0001FC000F1C00381C00701C00601C00E01C40E01C40E01C40603C
40304E801F870012127E9115>97 D<FC00001C00001C00001C00001C00001C00001C0000
1C00001C00001C00001C00001C7C001D86001E03001C01801C01C01C00C01C00E01C00E0
1C00E01C00E01C00E01C00E01C00C01C01C01C01801E030019060010F800131D7F9C17>
I<07E00C301878307870306000E000E000E000E000E000E00060007004300418080C3007
C00E127E9112>I<003F0000070000070000070000070000070000070000070000070000
070000070003E7000C1700180F00300700700700600700E00700E00700E00700E00700E0
0700E00700600700700700300700180F000C370007C7E0131D7E9C17>I<03E00C301818
300C700E6006E006FFFEE000E000E000E00060007002300218040C1803E00F127F9112>
I<00F8018C071E061E0E0C0E000E000E000E000E000E00FFE00E000E000E000E000E000E
000E000E000E000E000E000E000E000E000E000E007FE00F1D809C0D>I<00038003C4C0
0C38C01C3880181800381C00381C00381C00381C001818001C38000C300013C000100000
3000001800001FF8001FFF001FFF803003806001C0C000C0C000C0C000C0600180300300
1C0E0007F800121C7F9215>I<FC00001C00001C00001C00001C00001C00001C00001C00
001C00001C00001C00001C7C001C87001D03001E03801C03801C03801C03801C03801C03
801C03801C03801C03801C03801C03801C03801C03801C0380FF9FF0141D7F9C17>I<18
003C003C0018000000000000000000000000000000FC001C001C001C001C001C001C001C
001C001C001C001C001C001C001C001C001C00FF80091D7F9C0C>I<FC00001C00001C00
001C00001C00001C00001C00001C00001C00001C00001C00001C3FC01C0F001C0C001C08
001C10001C20001C40001CE0001DE0001E70001C78001C38001C3C001C1C001C0E001C0F
001C0F80FF9FE0131D7F9C16>107 D<FC001C001C001C001C001C001C001C001C001C00
1C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C00
FF80091D7F9C0C>I<FC7E07E0001C838838001D019018001E01E01C001C01C01C001C01
C01C001C01C01C001C01C01C001C01C01C001C01C01C001C01C01C001C01C01C001C01C0
1C001C01C01C001C01C01C001C01C01C001C01C01C00FF8FF8FF8021127F9124>I<FC7C
001C87001D03001E03801C03801C03801C03801C03801C03801C03801C03801C03801C03
801C03801C03801C03801C0380FF9FF014127F9117>I<03F0000E1C0018060030030070
0380600180E001C0E001C0E001C0E001C0E001C0E001C06001807003803003001806000E
1C0003F00012127F9115>I<FC7C001D86001E03001C01801C01C01C00C01C00E01C00E0
1C00E01C00E01C00E01C00E01C01C01C01C01C01801E03001D06001CF8001C00001C0000
1C00001C00001C00001C00001C0000FF8000131A7F9117>I<03C1000C3300180B00300F
00700700700700E00700E00700E00700E00700E00700E00700600700700700300F00180F
000C370007C700000700000700000700000700000700000700000700003FE0131A7E9116
>I<FCE01D301E781E781C301C001C001C001C001C001C001C001C001C001C001C001C00
FFC00D127F9110>I<1F9030704030C010C010E010F8007F803FE00FF000F880388018C0
18C018E010D0608FC00D127F9110>I<04000400040004000C000C001C003C00FFE01C00
1C001C001C001C001C001C001C001C001C101C101C101C101C100C100E2003C00C1A7F99
10>I<FC1F801C03801C03801C03801C03801C03801C03801C03801C03801C03801C0380
1C03801C03801C03801C07800C07800E1B8003E3F014127F9117>I<FF07E03C03801C01
001C01000E02000E020007040007040007040003880003880003D80001D00001D00000E0
0000E00000E00000400013127F9116>I<FF3FCFE03C0F03801C0701801C0701001C0B01
000E0B82000E0B82000E1182000711C4000711C4000720C40003A0E80003A0E80003C068
0001C0700001C0700001803000008020001B127F911E>I<7F8FF00F03800F0300070200
03840001C80001D80000F00000700000780000F800009C00010E00020E00060700040380
1E07C0FF0FF81512809116>I<FF07E03C03801C01001C01000E02000E02000704000704
0007040003880003880003D80001D00001D00000E00000E00000E0000040000040000080
00008000F08000F10000F300006600003C0000131A7F9116>I<7FFC70386038407040F0
40E041C003C0038007000F040E041C043C0C380870087038FFF80E127F9112>I<FFFFF0
1401808B15>I E /Fy 7 117 df<00038000000380000007C0000007C0000007C000000F
E000000FE000001FF000001BF000001BF0000031F8000031F8000061FC000060FC0000E0
FE0000C07E0000C07E0001803F0001FFFF0003FFFF8003001F8003001F8006000FC00600
0FC00E000FE00C0007E0FFC07FFEFFC07FFE1F1C7E9B24>65 D<0FF8001C1E003E0F803E
07803E07C01C07C00007C0007FC007E7C01F07C03C07C07C07C0F807C0F807C0F807C078
0BC03E13F80FE1F815127F9117>97 D<FF0000FF00001F00001F00001F00001F00001F00
001F00001F00001F00001F00001F3F801FE1E01F80701F00781F003C1F003C1F003E1F00
3E1F003E1F003E1F003E1F003E1F003C1F003C1F00781F80701EC1E01C3F00171D7F9C1B
>I<03FC000E0E001C1F003C1F00781F00780E00F80000F80000F80000F80000F80000F8
00007800007801803C01801C03000E0E0003F80011127E9115>I<FE3E00FE47001E8F80
1E8F801E8F801F07001F00001F00001F00001F00001F00001F00001F00001F00001F0000
1F0000FFF000FFF00011127F9114>114 D<1FD830786018E018E018F000FF807FE07FF0
1FF807FC007CC01CC01CE01CE018F830CFC00E127E9113>I<0300030003000300070007
000F000F003FFCFFFC1F001F001F001F001F001F001F001F001F001F0C1F0C1F0C1F0C0F
08079803F00E1A7F9913>I E /Fz 2 123 df<06000600060006000600060006000600FF
F0FFF0060006000600060006000600060006000600060006000600060006000600060006
00060006000C1D7E9611>121 D<060006000600060006000600FFF0FFF0060006000600
0600060006000000060006000600060006000600FFF0FFF0060006000600060006000600
0C1D7E9611>I E /FA 41 123 df<70F8FCFC7404040404080810102040060F7C840E>
44 D<70F8F8F87005057C840E>46 D<008003800F80F380038003800380038003800380
038003800380038003800380038003800380038003800380038003800380038003800380
03800380038007C0FFFE0F217CA018>49 D<03F0000C1C001007002007804003C04003C0
8003E0F003E0F801E0F801E0F801E02003E00003E00003C00003C0000780000700000E00
001C0000180000300000600000C000018000010000020020040020080020180060300040
3FFFC07FFFC0FFFFC013217EA018>I<4000006000007FFFE07FFFC07FFFC0400080C001
008001008002008002000004000008000008000010000030000020000060000060000060
0000E00000C00000C00001C00001C00001C00001C00003C00003C00003C00003C00003C0
0003C00003C00003C00001800013237DA118>55 D<01F000060C000C0600180700380380
700380700380F001C0F001C0F001C0F001E0F001E0F001E0F001E0F001E07001E07003E0
3803E01805E00C05E00619E003E1E00001C00001C00001C0000380000380300300780700
780600700C002018001030000FC00013227EA018>57 D<00018000000180000001800000
03C0000003C0000003C0000005E0000005E000000DF0000008F0000008F0000010F80000
1078000010780000203C0000203C0000203C0000401E0000401E0000401E0000800F0000
800F0000FFFF000100078001000780030007C0020003C0020003C0040003E0040001E004
0001E00C0000F00C0000F03E0001F8FF800FFF20237EA225>65 D<0007E0100038183000
E0063001C00170038000F0070000F00E0000701E0000701C0000303C0000303C0000307C
0000107800001078000010F8000000F8000000F8000000F8000000F8000000F8000000F8
000000F800000078000000780000107C0000103C0000103C0000101C0000201E0000200E
000040070000400380008001C0010000E0020000381C000007E0001C247DA223>67
D<FFFFF0000F801E0007800700078003C0078001C0078000E0078000F007800078078000
780780007C0780003C0780003C0780003C0780003E0780003E0780003E0780003E078000
3E0780003E0780003E0780003E0780003E0780003C0780003C0780007C07800078078000
78078000F0078000E0078001E0078003C0078007000F801E00FFFFF8001F227EA125>I<
0007F008003C0C1800E0021801C001B8038000F8070000780F0000381E0000381E000018
3C0000183C0000187C0000087800000878000008F8000000F8000000F8000000F8000000
F8000000F8000000F8000000F8001FFF780000F8780000787C0000783C0000783C000078
1E0000781E0000780F00007807000078038000B801C000B800E00318003C0C080007F000
20247DA226>71 D<FFFC3FFF0FC003F0078001E0078001E0078001E0078001E0078001E0
078001E0078001E0078001E0078001E0078001E0078001E0078001E0078001E0078001E0
07FFFFE0078001E0078001E0078001E0078001E0078001E0078001E0078001E0078001E0
078001E0078001E0078001E0078001E0078001E0078001E0078001E00FC003F0FFFC3FFF
20227EA125>I<FFFC0FC007800780078007800780078007800780078007800780078007
80078007800780078007800780078007800780078007800780078007800780078007800F
C0FFFC0E227EA112>I<03FFF0001F00000F00000F00000F00000F00000F00000F00000F
00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F
00000F00000F00000F00000F00000F00700F00F80F00F80F00F80E00F01E00401C002038
0018700007C00014237EA119>I<FFC00003FF0FC00003F007C00003E005E00005E005E0
0005E004F00009E004F00009E004F00009E004780011E004780011E004780011E0043C00
21E0043C0021E0043C0021E0041E0041E0041E0041E0040F0081E0040F0081E0040F0081
E004078101E004078101E004078101E00403C201E00403C201E00401E401E00401E401E0
0401E401E00400F801E00400F801E00400F801E004007001E00E007001E01F007003F0FF
E0203FFF28227EA12D>77 D<000FE00000783C0000E00E0003C00780078003C00F0001E0
0E0000E01E0000F03C0000783C0000787C00007C7C00007C7800003C7800003CF800003E
F800003EF800003EF800003EF800003EF800003EF800003EF800003EF800003E7800003C
7C00007C7C00007C3C0000783E0000F81E0000F00F0001E00F0001E0078003C003C00780
00E00E0000783C00000FE0001F247DA226>79 D<FFFFE000000F803C000007800E000007
80078000078007C000078003C000078003E000078003E000078003E000078003E0000780
03E000078003C000078007C000078007800007800E000007803C000007FFE00000078070
0000078038000007801C000007801E000007800E000007800F000007800F000007800F00
0007800F000007800F800007800F800007800F800007800F808007800FC080078007C080
0FC003C100FFFC01E2000000007C0021237EA124>82 D<03F0200C0C601802603001E070
00E0600060E00060E00060E00020E00020E00020F00000F000007800007F00003FF0001F
FE000FFF0003FF80003FC00007E00001E00000F00000F000007080007080007080007080
0070C00060C00060E000C0F000C0C80180C6070081FC0014247DA21B>I<FFFC07FF0FC0
00F807800070078000200780002007800020078000200780002007800020078000200780
002007800020078000200780002007800020078000200780002007800020078000200780
00200780002007800020078000200780002007800020078000200380004003C0004003C0
004001C0008000E000800060010000300600001C08000003F00020237EA125>85
D<FFF03FFC03FE1F8007E000F80F0003C000700F0003C000200F0003C00020078001E000
40078001E00040078001E0004003C002F0008003C002F0008003C002F0008001E0047801
0001E00478010001E00478010000F0083C020000F0083C020000F0083C020000F8183E06
000078101E04000078101E0400007C101E0400003C200F0800003C200F0800003C200F08
00001E40079000001E40079000001E40079000000F8003E000000F8003E000000F8003E0
0000070001C00000070001C00000070001C0000003000180000002000080002F237FA132
>87 D<0FE0001838003C0C003C0E0018070000070000070000070000FF0007C7001E0700
3C0700780700700700F00708F00708F00708F00F087817083C23900FC1E015157E9418>
97 D<0E0000FE00001E00000E00000E00000E00000E00000E00000E00000E00000E0000
0E00000E00000E00000E1F000E61C00E80600F00300E00380E003C0E001C0E001E0E001E
0E001E0E001E0E001E0E001E0E001E0E001C0E003C0E00380F00700C80600C41C0083F00
17237FA21B>I<01FE000703000C07801C0780380300780000700000F00000F00000F000
00F00000F00000F00000F000007000007800403800401C00800C010007060001F8001215
7E9416>I<0000E0000FE00001E00000E00000E00000E00000E00000E00000E00000E000
00E00000E00000E00000E001F8E00704E00C02E01C01E03800E07800E07000E0F000E0F0
00E0F000E0F000E0F000E0F000E0F000E07000E07800E03800E01801E00C02E0070CF001
F0FE17237EA21B>I<01FC000707000C03801C01C03801C07801E07000E0F000E0FFFFE0
F00000F00000F00000F00000F000007000007800203800201C00400E008007030000FC00
13157F9416>I<003C00C6018F038F030F070007000700070007000700070007000700FF
F80700070007000700070007000700070007000700070007000700070007000700070007
0007807FF8102380A20F>I<00007001F198071E180E0E181C07001C07003C07803C0780
3C07803C07801C07001C07000E0E000F1C0019F0001000001000001800001800001FFE00
0FFFC00FFFE03800F0600030400018C00018C00018C000186000306000303800E00E0380
03FE0015217F9518>I<0E0000FE00001E00000E00000E00000E00000E00000E00000E00
000E00000E00000E00000E00000E00000E1F800E60C00E80E00F00700F00700E00700E00
700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00
700E0070FFE7FF18237FA21B>I<1C001E003E001E001C00000000000000000000000000
000000000E00FE001E000E000E000E000E000E000E000E000E000E000E000E000E000E00
0E000E000E000E00FFC00A227FA10E>I<0E00FE001E000E000E000E000E000E000E000E
000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E
000E000E000E000E000E000E00FFE00B237FA20E>108 D<0E1FC07F00FE60E183801E80
7201C00F003C00E00F003C00E00E003800E00E003800E00E003800E00E003800E00E0038
00E00E003800E00E003800E00E003800E00E003800E00E003800E00E003800E00E003800
E00E003800E00E003800E00E003800E0FFE3FF8FFE27157F942A>I<0E1F80FE60C01E80
E00F00700F00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E00
700E00700E00700E00700E00700E0070FFE7FF18157F941B>I<01FC000707000C018018
00C03800E0700070700070F00078F00078F00078F00078F00078F00078F0007870007078
00F03800E01C01C00E038007070001FC0015157F9418>I<0E1F00FE61C00E80600F0070
0E00380E003C0E001C0E001E0E001E0E001E0E001E0E001E0E001E0E001E0E003C0E003C
0E00380F00700E80E00E41C00E3F000E00000E00000E00000E00000E00000E00000E0000
0E00000E0000FFE000171F7F941B>I<0E3CFE461E8F0F0F0F060F000E000E000E000E00
0E000E000E000E000E000E000E000E000E000F00FFF010157F9413>114
D<0F8830786018C018C008C008E008F0007F803FE00FF001F8003C801C800C800CC00CC0
08E018D0308FC00E157E9413>I<02000200020002000600060006000E001E003E00FFF8
0E000E000E000E000E000E000E000E000E000E000E000E040E040E040E040E040E040708
030801F00E1F7F9E13>I<0E0070FE07F01E00F00E00700E00700E00700E00700E00700E
00700E00700E00700E00700E00700E00700E00700E00700E00F00E00F006017003827800
FC7F18157F941B>I<FFC1FE1E00780E00300E00200E0020070040070040038080038080
03808001C10001C10000E20000E20000E200007400007400003800003800003800001000
17157F941A>I<FF8FF8FF1E01E03C1C01C0180E01C0180E01E0100E01E0100702602007
0270200702702003843040038438400384384001C8188001C81C8001C81C8000F00D0000
F00F0000F00F0000600600006006000060060020157F9423>I<FFC1FE1E00780E00300E
00200E002007004007004003808003808003808001C10001C10000E20000E20000E20000
7400007400003800003800003800001000001000002000002000002000004000F04000F0
8000F180004300003C0000171F7F941A>121 D<3FFFC0380380300780200700600E0040
1C00403C0040380000700000E00001E00001C0000380400700400F00400E00C01C008038
0080780180700780FFFF8012157F9416>I E /FB 1 4 df<00C00000C00000C00000C000
00C000C0C0C0F0C3C038C7000EDC0003F00000C00003F0000EDC0038C700F0C3C0C0C0C0
00C00000C00000C00000C00000C00012157D9619>3 D E /FC 31
124 df<0001FE01F8000701C704001C002E0E007800FC1F00F001F81F00E001F81F01E0
01F00E03C000F00003C000F00003C000F00003C000F00003C000F00003C000F00003C000
F00003C000F00003C000F00003C000F00003C000F00003C000F000FFFFFFFFF0FFFFFFFF
F003C000F00003C000F00003C000F00003C000F00003C000F00003C000F00003C000F000
03C000F00003C000F00003C000F00003C000F00003C000F00003C000F00003C000F00003
C000F00003C000F00003C000F00003C000F00003C000F00003C000F00003C000F00003C0
00F00003C000F00003C000F00003C000F00003C000F00007E001F8007FFE1FFFC07FFE1F
FFC028327FB126>11 D<78FCFCFCFC780000000000000000000000000000000000000078
FCFCFCFC78061F7A9E12>58 D<0000030000000000030000000000030000000000078000
0000000780000000000FC0000000000FC0000000000FC00000000017E00000000013E000
00000013E00000000023F00000000021F00000000021F00000000040F80000000040F800
00000040F800000000807C00000000807C00000001807E00000001003E00000001003E00
000002003F00000002001F00000002001F00000004000F80000004000F80000004000F80
0000080007C00000080007C00000180007E000001FFFFFE000001FFFFFE00000200003F0
0000200001F00000200001F00000400001F80000400000F80000400000F800008000007C
00008000007C00008000007C00010000003E00010000003E00030000003F00030000001F
00070000001F001F8000003F80FFE00003FFFCFFE00003FFFC2E327EB132>65
D<FFFFFFFFC0FFFFFFFFC007E0001FC003E00003C003E00001E003E00000E003E0000060
03E000006003E000002003E000002003E000002003E000002003E000001003E000001003
E000001003E000801003E000800003E000800003E000800003E000800003E001800003E0
01800003E007800003FFFF800003FFFF800003E007800003E001800003E001800003E000
800003E000800003E000800003E000800003E000800003E000000003E000000003E00000
0003E000000003E000000003E000000003E000000003E000000003E000000003E0000000
03E000000003E000000003E000000007F0000000FFFFC00000FFFFC0000024317CB02B>
70 D<FFFF807FFFC0FFFF807FFFC007F00003F80003E00001F00003E00001F00003E000
01F00003E00001F00003E00001F00003E00001F00003E00001F00003E00001F00003E000
01F00003E00001F00003E00001F00003E00001F00003E00001F00003E00001F00003E000
01F00003E00001F00003E00001F00003E00001F00003E00001F00003E00001F00003FFFF
FFF00003FFFFFFF00003E00001F00003E00001F00003E00001F00003E00001F00003E000
01F00003E00001F00003E00001F00003E00001F00003E00001F00003E00001F00003E000
01F00003E00001F00003E00001F00003E00001F00003E00001F00003E00001F00003E000
01F00003E00001F00003E00001F00003E00001F00003E00001F00007F00003F800FFFF80
7FFFC0FFFF807FFFC02A317CB032>72 D<FFFFFFC000FFFFFFF80007E0007E0003E0001F
0003E000078003E00003C003E00001E003E00001F003E00001F003E00000F003E00000F8
03E00000F803E00000F803E00000F803E00000F803E00000F803E00000F003E00001F003
E00001E003E00003E003E00003C003E000078003E0001F0003E0007C0003FFFFF00003E0
00000003E000000003E000000003E000000003E000000003E000000003E000000003E000
000003E000000003E000000003E000000003E000000003E000000003E000000003E00000
0003E000000003E000000003E000000003E000000003E000000003E000000007F0000000
FFFF800000FFFF80000025317CB02D>80 D<00003FC000000001C03800000007000E0000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>I<FFFFFF000000FFFFFFF0000007E001FC000003E0003E00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>I<007F8020
01FFE02007C078600F001C601E0006E03C0003E0380001E0780000E0700000E070000060
F0000060F0000060F0000020F0000020F0000020F8000020F80000007C0000007E000000
3F0000003FC000001FF800000FFF800007FFF80003FFFC0000FFFF00000FFF800000FFC0
00001FE0000007E0000003F0000001F0000000F0000000F8000000F88000007880000078
800000788000007880000078C0000078C0000070E00000F0E00000E0F00000E0F80001C0
EC000380C7000700C1F01E00807FFC00800FF0001D337CB125>I<7FFFFFFFFFE07FFFFF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>I<00FE00000303C0000C00E00010007000100038003C003C003E001C003E001E00
3E001E0008001E0000001E0000001E0000001E00000FFE0000FC1E0003E01E000F801E00
1F001E003E001E003C001E007C001E00F8001E04F8001E04F8001E04F8003E04F8003E04
78003E047C005E043E008F080F0307F003FC03E01E1F7D9E21>97
D<003F8000E0600380180700040F00041E001E1C003E3C003E7C003E7C0008780000F800
00F80000F80000F80000F80000F80000F80000F80000F800007800007C00007C00003C00
011E00011E00020F000207000403801800E060003F80181F7D9E1D>99
D<000001E000003FE000003FE0000003E0000001E0000001E0000001E0000001E0000001
E0000001E0000001E0000001E0000001E0000001E0000001E0000001E0000001E0000001
E0000001E0001F81E000F061E001C019E0078005E00F0003E00E0003E01E0001E03C0001
E03C0001E07C0001E0780001E0F80001E0F80001E0F80001E0F80001E0F80001E0F80001
E0F80001E0F80001E0F80001E0780001E0780001E03C0001E03C0001E01C0001E01E0003
E00E0005E0070009E0038011F000E061FF003F81FF20327DB125>I<003F800000E0E000
0380380007003C000E001E001E001E001C000F003C000F007C000F0078000F8078000780
F8000780F8000780FFFFFF80F8000000F8000000F8000000F8000000F8000000F8000000
780000007C0000003C0000003C0000801E0000800E0001000F0002000780020001C00C00
00F03000001FC000191F7E9E1D>I<0007E0001C1000383800707C00E07C01E07C01C038
03C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C000
FFFFC0FFFFC003C00003C00003C00003C00003C00003C00003C00003C00003C00003C000
03C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C000
03C00003C00003C00003C00007E0007FFF007FFF0016327FB114>I<000000F0007F0308
01C1C41C0380E81C070070080F0078001E003C001E003C003E003E003E003E003E003E00
3E003E003E003E003E003E001E003C001E003C000F007800070070000780E00009C1C000
087F000018000000180000001800000018000000180000001C0000000E0000000FFFF800
07FFFF0003FFFF800E000FC0180001E0300000F070000070E0000038E0000038E0000038
E0000038E00000387000007070000070380000E01C0001C00700070001C01C00003FE000
1E2F7E9F21>I<0780000000FF80000000FF800000000F80000000078000000007800000
000780000000078000000007800000000780000000078000000007800000000780000000
0780000000078000000007800000000780000000078000000007800000000780FE000007
83078000078C03C000079001E00007A001E00007A000F00007C000F00007C000F0000780
00F000078000F000078000F000078000F000078000F000078000F000078000F000078000
F000078000F000078000F000078000F000078000F000078000F000078000F000078000F0
00078000F000078000F000078000F000078000F000078000F0000FC001F800FFFC1FFF80
FFFC1FFF8021327EB125>I<07000F801F801F800F800700000000000000000000000000
000000000000000000000780FF80FF800F80078007800780078007800780078007800780
0780078007800780078007800780078007800780078007800780078007800FC0FFF8FFF8
0D307EAF12>I<0780FF80FF800F80078007800780078007800780078007800780078007
800780078007800780078007800780078007800780078007800780078007800780078007
80078007800780078007800780078007800780078007800780078007800FC0FFFCFFFC0E
327EB112>108 D<0780FE001FC000FF83078060F000FF8C03C18078000F9001E2003C00
07A001E4003C0007A000F4001E0007C000F8001E0007C000F8001E00078000F0001E0007
8000F0001E00078000F0001E00078000F0001E00078000F0001E00078000F0001E000780
00F0001E00078000F0001E00078000F0001E00078000F0001E00078000F0001E00078000
F0001E00078000F0001E00078000F0001E00078000F0001E00078000F0001E00078000F0
001E00078000F0001E00078000F0001E00078000F0001E000FC001F8003F00FFFC1FFF83
FFF0FFFC1FFF83FFF0341F7E9E38>I<0780FE0000FF83078000FF8C03C0000F9001E000
07A001E00007A000F00007C000F00007C000F000078000F000078000F000078000F00007
8000F000078000F000078000F000078000F000078000F000078000F000078000F0000780
00F000078000F000078000F000078000F000078000F000078000F000078000F000078000
F000078000F000078000F0000FC001F800FFFC1FFF80FFFC1FFF80211F7E9E25>I<001F
C00000F0780001C01C00070007000F0007801E0003C01C0001C03C0001E03C0001E07800
00F0780000F0780000F0F80000F8F80000F8F80000F8F80000F8F80000F8F80000F8F800
00F8F80000F8780000F07C0001F03C0001E03C0001E01E0003C01E0003C00F0007800780
0F0001C01C0000F07800001FC0001D1F7E9E21>I<0781FC00FF860700FF8803C00F9001
E007A000F007C00078078000780780003C0780003C0780003E0780001E0780001F078000
1F0780001F0780001F0780001F0780001F0780001F0780001F0780001F0780003E078000
3E0780003C0780007C0780007807C000F007A000F007A001E00798038007860F000781F8
000780000007800000078000000780000007800000078000000780000007800000078000
0007800000078000000FC00000FFFC0000FFFC0000202D7E9E25>I<0783E0FF8C18FF90
7C0F907C07A07C07C03807C00007C00007C0000780000780000780000780000780000780
000780000780000780000780000780000780000780000780000780000780000780000780
000780000FC000FFFE00FFFE00161F7E9E19>114 D<01FC100E03301800F03000706000
30E00030E00010E00010E00010F00010F800007E00003FF0001FFF000FFFC003FFE0003F
F00001F80000F880003C80003C80001CC0001CC0001CE0001CE00018F00038F00030CC00
60C301C080FE00161F7E9E1A>I<00400000400000400000400000400000C00000C00000
C00001C00001C00003C00007C0000FC0001FFFE0FFFFE003C00003C00003C00003C00003
C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003
C01003C01003C01003C01003C01003C01003C01003C01001C02001E02000E0400078C000
1F00142C7FAB19>I<078000F000FF801FF000FF801FF0000F8001F000078000F0000780
00F000078000F000078000F000078000F000078000F000078000F000078000F000078000
F000078000F000078000F000078000F000078000F000078000F000078000F000078000F0
00078000F000078000F000078000F000078001F000078001F000078001F000038002F000
03C004F00001C008F800007030FF80001FC0FF80211F7E9E25>I<FFF07FF80FFCFFF07F
F80FFC0FC007C003F00F8003C001C0078003C00080078003C0008003C003E0010003C003
E0010003C007E0010001E004F0020001E004F0020001E00870020000F00878040000F008
78040000F0103804000078103C08000078103C08000078201C0800003C201E1000003C20
1E1000003C400E1000001E400F2000001E400F2000001E80072000000F8007C000000F80
07C000000F0003C0000007000380000007000380000006000180000002000100002E1F7F
9E30>119 D<FFF801FF80FFF801FF800FC0007C00078000380007C000300003C0002000
03C000200001E000400001E000400001F000400000F000800000F0008000007801000000
78010000007C010000003C020000003C020000001E040000001E040000001F040000000F
080000000F080000000790000000079000000007D000000003E000000003E000000001C0
00000001C000000001C00000000080000000008000000001000000000100000000010000
000002000000000200000000040000007004000000F80C000000F808000000F810000000
703000000030400000001F80000000212D7F9E23>121 D<3FFFFF3E001E38001E30003C
2000782000786000F04001E04001E04003C0400780000780000F00001E00001E00003C00
00780000780000F00101E00101E00103C0010780010780030F00021E00021E00063C0006
78000E78007EFFFFFE181F7E9E1D>I<FFFFFFFF802101809321>I
E end
%%EndProlog
%%BeginSetup
%%Feature: *Resolution 300dpi
TeXDict begin

%%EndSetup
%%Page: 0 1
0 0 bop 218 233 a FC(Tin)n(y)21 b(F)-6 b(amilies)21 b(of)g(F)-6
b(unctions)21 b(with)h(Random)e(Prop)r(erties:)436 333
y(A)h(Qualit)n(y{Size)g(T)-6 b(rade{o\013)22 b(for)f(Hashing)1515
307 y FB(\003)412 465 y FA(Oded)16 b(Goldreic)o(h)748
447 y Fz(y)223 529 y FA(Departmen)o(t)f(of)i(Computer)e(Science)382
593 y(and)i(Applied)e(Mathematics)256 657 y(W)l(eizmann)f(Institute)h
(of)i(Science)412 721 y(Reho)o(v)o(ot,)f(Israel.)1238
465 y(Avi)f(Wigderson)1556 447 y Fz(z)1068 529 y FA(Institute)g(for)i
(Computer)e(Science)1199 593 y(Hebrew)h(Univ)o(ersit)o(y)1279
657 y(Giv)m(at)h(Ram)1212 721 y(Jerusalem,)d(Israel.)811
825 y(Marc)o(h)i(21,)g(1997)884 1047 y Fy(Abstract)176
1128 y Fx(W)m(e)d(presen)o(t)i(three)g(explicit)e(constructions)i(of)e
(hash)h(functions,)f(whic)o(h)g(exhibit)h(a)f(trade-o\013)h(b)q(et)o(w)
o(een)114 1183 y(the)e(size)g(of)f(the)h(family)d(\(and)i(hence)i(the)f
(n)o(um)o(b)q(er)f(of)g(random)f(bits)h(needed)i(to)f(generate)h(a)e
(mem)o(b)q(er)f(of)h(the)114 1237 y(family\),)e(and)j(the)g(qualit)o(y)
f(\(or)i(error)g(parameter\))f(of)f(the)i(pseudo-random)e(prop)q(ert)o
(y)i(it)f(ac)o(hiev)o(es.)18 b(Unlik)o(e)114 1292 y(previous)g
(constructions,)i(most)c(notably)h(univ)o(ersal)h(hashing,)g(the)h
(size)f(of)f(our)h(families)e(is)h(essen)o(tially)114
1347 y(indep)q(enden)o(t)e(of)e(the)i(size)f(of)g(the)g(domain)e(on)h
(whic)o(h)h(the)h(functions)f(op)q(erate.)176 1402 y(The)e(\014rst)g
(construction)g(is)f(for)g(the)h Fw(mixing)g Fx(prop)q(ert)o(y)g({)f
(mapping)e(a)i(prop)q(ortional)g(part)g(of)g(an)o(y)g(subset)114
1457 y(of)j(the)j(domain)c(to)i(an)o(y)g(other)h(subset.)24
b(The)16 b(other)g(t)o(w)o(o)f(are)h(for)f(the)h Fw(extr)n(action)32
b Fx(prop)q(ert)o(y)16 b({)g(mapping)114 1511 y(an)o(y)d(subset)j(of)e
(the)h(domain)d(almost)g(uniformly)g(in)o(to)h(a)i(range)f(smaller)f
(than)h(it.)19 b(The)c(second)g(and)f(third)114 1566
y(constructions)h(handle)f(\(resp)q(ectiv)o(ely\))i(the)e(extreme)g
(situations)g(when)g(the)h(range)f(is)g(v)o(ery)g(large)g(or)g(v)o(ery)
114 1621 y(small.)176 1676 y(W)m(e)i(pro)o(vide)f(lo)o(w)o(er)h(b)q
(ounds)g(sho)o(wing)g(that)g(our)g(constructions)h(are)g(nearly)f
(optimal,)d(and)j(men)o(tion)114 1731 y(some)d(applications)f(of)i(the)
g(new)h(constructions.)0 2191 y Fv(Keyw)o(ords:)43 b
Fu(Randomness)11 b(and)f(Computation,)h(Randomness)f(Extractors,)f
(Sampling)j(Algorithms,)f(Random-)0 2254 y(Lo)q(oking)j(F)l(unctions,)g
(Expander)g(Graphs,)f(Raman)o(ujan)h(Graphs,)f(Univ)o(ersal)h(Hashing,)
g(Small-Biased)i(Prob-)0 2316 y(abilit)o(y)g(Spaces,)g(Lindsey's)g
(Lemma.)p 0 2380 780 2 v 51 2407 a Ft(\003)69 2423 y
Fs(An)d(extended)h(abstract)f(of)g(this)g(pap)q(er)h(has)f(app)q(eared)
h(in)g(the)f Fr(26th)f(A)o(CM)i(Symp)n(osium)d(on)i(The)n(ory)g(of)g
(Computing)d Fs(\(STOC)0 2473 y(94\))j(held)h(in)g(Mon)o(treal,)g(Queb)
q(ec,)f(Canada,)h(Ma)o(y)f(23-25,)g(1994.)53 2505 y Ft(y)69
2521 y Fs(Researc)o(h)h(w)o(as)f(supp)q(orted)h(in)g(part)f(b)o(y)g
(gran)o(t)g(No.)f(92-00226)i(from)f(the)g(United)h(States)f({)g(Israel)
g(Binational)j(Science)e(F)m(oun-)0 2572 y(dation)g(\(BSF\),)f
(Jerusalem,)h(Israel.)53 2604 y Ft(z)69 2620 y Fs(Researc)o(h)d(w)o(as)
g(supp)q(orted)h(in)f(part)g(b)o(y)f(the)h(W)m(olfson)h(Researc)o(h)f
(Aw)o(ards,)g(administered)i(b)o(y)d(the)h(Israel)g(Academ)o(y)g(of)f
(Sciences)0 2670 y(and)k(Humanities.)964 2795 y Fu(0)p
eop
%%Page: 1 2
1 1 bop 0 42 a Fq(1)67 b(In)n(tro)r(duction)0 149 y Fu(In)12
b(1979,)f(Carter)f(and)i(W)l(egman)f(in)o(tro)q(duced)h(the)g(notion)f
(of)g(univ)o(ersal)i(hashing)f(functions)g([9)o(].)18
b(Though)12 b(these)0 211 y(functions)k(w)o(ere)e(in)o(tro)q(duced)i
(with)f(data)f(storage)g(application)i(in)g(mind,)f(they)g(found)g(man)
o(y)f(applications)j(to)0 273 y(complexit)o(y)h(theory)f([30)o(,)g(32,)
g(34)o(,)g(18,)g(17)o(,)g(21)o(,)g(22,)g(19)o(,)g(20,)g(27)o(,)g(28,)g
(36)o(].)26 b(This)18 b(wide)g(range)f(of)g(applications)0
335 y(o)o(wns)f(its)g(existence)i(to)e(t)o(w)o(o)f(related)i(`random')e
(prop)q(erties)i(of)f(these)h(succinct)h(and)e(e\016cien)o(tly)i
(computable)0 397 y(functions:)j(the)15 b Fp(extr)n(action)g
Fu(and)g(the)h Fp(mixing)e Fu(prop)q(erties.)71 459 y(F)l(or)i(a)g
(family)i Fo(F)23 b Fu(of)16 b(functions,)i(eac)o(h)e(mapping)i
Fo(n)p Fu(-bit)g(strings)e(to)g Fo(m)p Fu(-bit)i(strings,)e(the)h
Fp(extr)n(action)g Fu(prop-)0 521 y(ert)o(y)e(asserts)f(the)i(follo)o
(wing.)21 b(Ev)o(ery)15 b(subset)g(of)g Fo(K)e Fn(\001)d
Fu(2)950 505 y Fm(m)997 521 y Fu(strings)15 b(in)h(the)g(domain)g
Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1551 505 y Fm(n)1571
521 y Fu(,)15 b(is)h(mapp)q(ed)g(almost)0 583 y(uniformly)21
b(to)e(the)h(range)g Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)597
567 y Fm(m)627 583 y Fu(,)20 b(b)o(y)g(all)h(but)f(a)g(small)h
(fraction)e(of)h(the)g(functions)h(in)g(the)f(family)l(.)35
b(The)0 645 y(parameter)14 b Fo(K)h(>)e Fu(1)i(determines)h(the)e
(qualit)o(y)i(of)e(the)h(appro)o(ximation)g(to)f(the)g(uniform)i
(distribution)g(and)f(the)0 708 y(fraction)i(of)f(bad)h(functions)h(in)
g Fo(F)23 b Fu(\(i.e.)i(those)17 b(that)f(don't)g(ac)o(hiev)o(e)i(this)
f(appro)o(ximation\).)25 b(The)17 b(extraction)0 770
y(prop)q(ert)o(y)12 b(is)h(the)g(heart)f(of)g(the)h(Lefto)o(v)o(er)f
(Hash)g(Lemma)h([21)o(])f(and)h(its)f(precursors,)h(whic)o(h)g(w)o(ere)
f(k)o(ey)h(to)f(n)o(umer-)0 832 y(ous)18 b(results,)g(e.g.)28
b(in)19 b(sa)o(ving)e(randomness)h([22)o(],)g(w)o(eak)g(random)f
(sources)h([36)o(],)g(pseudorandom)g(generators)0 894
y([17)o(,)g(21)o(])g(and)h(in)o(teractiv)o(e)f(pro)q(ofs)g([18)o(].)29
b(\(Alternativ)o(e)19 b(function)g(families)h(with)e(extraction)g(prop)
q(ert)o(y)g(w)o(ere)0 956 y(previously)f(constructed)e(in)h([29)o(],)e
(with)i(a)f(v)m(ariet)o(y)g(of)g(other)g(applications.\))71
1018 y(The)i Fp(mixing)g Fu(prop)q(ert)o(y)g(is)h(meaningful)h(also)f
(in)g(case)f Fo(m)g Fu(=)f Fo(n)p Fu(,)i(and)g(in)g(fact)f(it)h(is)g
(commonly)g(used)g(with)0 1080 y(this)f(c)o(hoice.)26
b(Hence,)18 b(w)o(e)f(assume)g(for)f(simplicit)o(y)j(that)e
Fo(m)e Fu(=)h Fo(n)p Fu(.)25 b(Lo)q(osely)18 b(sp)q(eaking,)g(the)f
(mixing)h(prop)q(ert)o(y)0 1142 y(asserts)h(that,)g(for)g(all)h(but)f
(a)g(small)i(fraction)e(of)g(the)g(functions)h Fo(f)25
b Fu(in)20 b(the)f(family)i Fo(F)6 b Fu(,)20 b(the)g(mem)o(b)q(ership)g
(in)0 1204 y Fo(A)11 b Fn(\002)g Fo(B)18 b Fu(of)e(a)g(pair)g(\()p
Fo(r)o(;)8 b(f)d Fu(\()p Fo(r)q Fu(\)\))13 b(with)j Fo(r)h
Fu(b)q(eing)h(a)d(random)h(elemen)o(t)h(from)e(the)h(domain,)g(is)h
(essen)o(tially)g(the)f(same)0 1266 y(as)d(that)g(of)g(a)h(random)f
(pair)h(\()p Fo(r)o(;)8 b(s)p Fu(\))j(of)j(elemen)o(ts.)20
b(The)13 b(prime)i(use)f(of)f(the)g(mixing)i(prop)q(ert)o(y)e(is)h(in)h
(the)f(logspace)0 1329 y(pseudorandom)h(generators)g(of)f(Nisan)i([27)o
(,)f(28)o(].)71 1391 y(In)22 b(the)f(de\014nitions)j(ab)q(o)o(v)o(e,)e
(there)g(is)g(an)f(error)g(parameter)g Fo(\017)h Fu(\(e.g.)38
b(the)21 b(fraction)h(of)f(bad)h(functions,)0 1453 y(the)e(distance)g
(from)f(the)h(uniform)f(distribution)j(etc.\),)d(whic)o(h)i(determines)
f(the)g(qualit)o(y)g(of)f(the)h(mixing)g(or)0 1515 y(extraction)13
b(ac)o(hiev)o(ed)h(b)o(y)e(the)h(family)h Fo(F)6 b Fu(.)20
b(All)14 b(the)f(applications)h(men)o(tioned)g(ab)q(o)o(v)o(e)f(tak)o
(e)f Fo(F)19 b Fu(to)13 b(b)q(e)g(a)g(univ)o(ersal)0
1577 y(family)e(of)g(hash)f(functions.)20 b(This)11 b(family)g(ac)o
(hiev)o(es)g(the)g(b)q(est)g(p)q(ossible)i(qualit)o(y)e(parameter:)17
b Fo(\017)11 b Fu(is)g(exp)q(onen)o(tially)0 1639 y(small)k(in)g
Fo(m)p Fu(.)20 b(Ho)o(w)o(ev)o(er,)13 b(while)j(small)f(enough)g(for)e
(these)i(applications,)h(a)e(univ)o(ersal)h(family)g(has)g(to)e(b)q(e)i
(large:)0 1701 y(exp)q(onen)o(tial)i(in)f Fo(n)p Fu(.)71
1763 y(But)d(in)i(some)e(applications)j(w)o(e)d(ma)o(y)g(b)q(e)i(con)o
(ten)o(t)e(with)h(a)f(larger)h Fo(\017)g Fu(\(i.e.)19
b(lo)o(w)o(er)14 b(qualit)o(y\),)g(sa)o(y)f(constan)o(t)f(or)0
1825 y(1)p Fo(=pol)q(y)r Fu(\()p Fo(n)p Fu(\).)17 b(Can)c(w)o(e)g(use)g
(m)o(uc)o(h)h(smaller)f(families)i Fo(F)k Fu(in)14 b(this)g(case)f(and)
g(ac)o(hiev)o(e)h(similar)g(random)f(prop)q(erties?)0
1887 y(A)20 b(straigh)o(tforw)o(ard)e(coun)o(ting)i(argumen)o(t)g(sho)o
(ws)f(that)g(there)h(exist)h(families)g Fo(F)27 b Fu(of)19
b(size)i Fo(pol)q(y)r Fu(\(1)p Fo(=\017)p Fu(\))e(\(resp.,)0
1950 y Fo(pol)q(y)r Fu(\()p Fo(n=\017)p Fu(\)\))i(ac)o(hieving)i(the)f
(mixing)h(\(resp.,)g(extraction\))f(prop)q(erties)h(with)f(qualit)o(y)h
Fo(\017)p Fu(.)41 b(Note)21 b(that)h(these)0 2012 y(b)q(ounds)16
b(dep)q(end)h(essen)o(tially)g(only)e(on)g(the)h(qualit)o(y)g
(required,)f(and)h(not)f(on)g(the)g(size)h(of)f(the)g(domain.)0
2139 y Fl(1.1)56 b(Our)18 b(Results)0 2231 y Fu(The)g(main)h(con)o
(tribution)g(of)f(this)g(pap)q(er)h(is)g(in)g(presen)o(ting)f(explicit)
j(constructions)d(of)g(suc)o(h)g(families,)i(th)o(us)0
2293 y(yielding)f(a)e(trade-o\013)f(b)q(et)o(w)o(een)h(the)g(size)h(of)
e(the)h(family)h(and)f(the)g(desired)h(qualit)o(y)l(.)26
b(The)17 b(\014rst)g(construction)0 2355 y(is)i(for)e(mixing,)j(where)e
(w)o(e)g(obtain)g(a)g(complete)h(trade-o\013.)27 b(The)19
b(second)f(and)h(third)f(constructions)h(are)f(for)0
2417 y(extraction,)c(where)h(w)o(e)f(\(resp)q(ectiv)o(ely\))i(handle)g
(t)o(w)o(o)d(extreme)h(cases:)20 b(when)15 b Fo(n)9 b
Fn(\000)g Fo(m)k Fn(\034)g Fo(n)i Fu(and)f(when)h Fo(m)e
Fn(\034)g Fo(n)p Fu(.)0 2479 y(Our)22 b(constructions)f(are)g(relativ)o
(ely)i(simple.)40 b(The)22 b(\014rst)f(t)o(w)o(o)f(of)h(them)g(com)o
(bine)h(univ)o(ersal)h(hashing)f(and)0 2541 y(expander)c(graphs.)25
b(\(It)17 b(is)h(in)o(teresting)g(to)e(note)h(that)g(despite)h(the)f
(similarit)o(y)i(in)f(these)f(t)o(w)o(o)f(constructions,)0
2603 y(the)h(pro)q(ofs)g(are)g(completely)i(di\013eren)o(t\).)26
b(The)18 b(third)g(construction)g(uses)f(small-bias)i(probabilit)o(y)g
(spaces)e(of)0 2665 y(small)j(size,)g(and)f(its)g(analysis)g(utilizes)i
(a)e(new)g(generalization)h(of)e(Lindsey's)i(Lemma.)31
b(W)l(e)19 b(pro)o(vide)g(lo)o(w)o(er)964 2795 y(1)p
eop
%%Page: 2 3
2 2 bop 0 42 a Fu(b)q(ounds)19 b(to)e(sho)o(w)h(that)f(the)h(\014rst)f
(construction)i(is)f(nearly)h(optimal,)f(and)h(the)f(third)g(is)h
(nearly)f(optimal)h(for)0 104 y Fo(m)i Fu(=)g Fo(O)q
Fu(\(log)8 b Fo(n)p Fu(\).)35 b(By)20 b(nearly)h(optimal)f(here)h(w)o
(e)f(mean)g(that)f(the)i(n)o(um)o(b)q(er)f(of)g(bits)g(needed)i(to)d
(describ)q(e)j(a)0 166 y(mem)o(b)q(er)15 b(of)g(the)g(family)h(in)g
(our)f(constructions)h(is)f(within)i(a)e(constan)o(t)f(factor)g(of)h
(the)g(lo)o(w)o(er)g(b)q(ound.)71 228 y(Using)g(the)f(\014rst)g
(construction)h(w)o(e)f(reduce)h(the)f(randomness)h(complexit)o(y)g(of)
f(t)o(w)o(o)f(generic)i(pro)q(cedures)g(as)0 290 y(follo)o(ws:)56
389 y(1.)22 b(F)l(or)14 b(sampling)i(pro)q(cedures,)f(whic)o(h)h(use)f
(an)f(asymptotically)i(optimal)f(n)o(um)o(b)q(er)g(of)g(sample)g(p)q
(oin)o(ts,)g(the)114 451 y(amoun)o(t)j(of)g(randomness)h(required)h(to)
e(generate)h(the)g(sample)g(p)q(oin)o(ts)g(is)h(reduced)g(b)o(y)e(a)h
Fp(factor)g Fu(of)g(2,)114 514 y(yielding)e(an)e(optimal)h(result)g
(up-to)f(a)g(small)h Fp(additive)g(term)p Fu(;)f(and)56
613 y(2.)22 b(The)14 b(randomness)h(complexit)o(y)g(of)f(Nisan's)h
(\\generalized)h(logspace")f(generator)e([27)o(],)h(is)h(reduced)h(b)o
(y)e(a)114 675 y(logarithmic)i(factor.)0 775 y(The)f(second)h
(construction)f(implies)j(a)d(randomness{e\016cien)o(t)g(lefto)o(v)o
(er)g(hash)g(lemma,)g(whic)o(h)h(is)g(particularly)0
837 y(app)q(ealing)k(in)g(case)e Fo(n)13 b Fn(\000)f
Fo(m)19 b Fn(\034)f Fo(n)p Fu(.)30 b(The)19 b(third)g(construction)g
(turned)g(out)f(to)g(b)q(e)h(the)g(main)g(tec)o(hnical)h(to)q(ol)0
899 y(in)d(the)e(recen)o(t)h(adv)m(ances)h(on)e(constructing)h(nearly)h
(optimal)f Fp(extr)n(actors)1284 882 y Fk(1)1318 899
y Fu(for)g(an)o(y)f Fo(m)f Fu(=)f(\002\()p Fo(n)p Fu(\),)j(on)g(whic)o
(h)g(w)o(e)0 961 y(elab)q(orate)f(b)q(elo)o(w.)0 1088
y Fl(1.2)56 b(Previous,)17 b(Concurren)n(t)i(and)g(Subsequen)n(t)g(W)-5
b(ork)0 1180 y Fu(Despite)17 b(the)f(general)h(in)o(terest)f(in)h
(reducing)h(the)e(size)h(of)f(sample)h(spaces)g(ac)o(hieving)g(v)m
(arious)g(random)f(prop-)0 1242 y(erties,)k(v)o(ery)f(little)h(w)o(as)f
(done)g(for)g(the)g(prop)q(erties)h(pro)o(vided)g(b)o(y)f(univ)o(ersal)
h(hashing.)32 b(The)20 b(only)f(previous)0 1304 y(result)e(ac)o
(hieving)g(suc)o(h)g(a)f(qualit)o(y-size)i(trade-o\013)d(is)i(b)o(y)f
(Nisan)h(and)g(Zuc)o(k)o(erman)f([29)o(].)23 b(They)16
b(deal)h(with)g(the)0 1366 y(extraction)i(problem)g(in)h(the)e
(di\016cult)j(range)d Fo(m)h Fu(=)f(\002\()p Fo(n)p Fu(\))h(\(whic)o(h)
g(w)o(e)g(cannot)f(handle\),)j(via)e(an)f(ingenious)0
1428 y(construction,)13 b(follo)o(wing)h(earlier)g(w)o(ork)e(of)g(Zuc)o
(k)o(erman)h([36)o(].)18 b(In)c(addition,)g(they)f(applied)i(their)e
(extractors)f(to)0 1490 y(sho)o(w)i(that)g(p)q(oly)q(\()p
Fo(S)s Fu(\))f(man)o(y)h(random)g(bits)h(add)g(no)f(p)q(o)o(w)o(er)g
(at)g(all)h(to)f(space)q(\()p Fo(S)s Fu(\))f(T)l(uring)i(mac)o(hines.)
20 b(\(Actually)l(,)0 1552 y(they)15 b(sho)o(w)o(ed)f(ho)o(w)g(to)g
(sim)o(ulate)i(p)q(oly)q(\()p Fo(S)s Fu(\))e(man)o(y)g(random)g(bits,)h
(in)h(space\()p Fo(S)s Fu(\))e(computations)g(b)o(y)h(using)g
Fo(O)q Fu(\()p Fo(S)s Fu(\))0 1615 y(man)o(y)g(random)g(coins.\))71
1677 y(Sriniv)m(asan)22 b(and)g(Zuc)o(k)o(erman)e([33])g(ha)o(v)o(e)h
(indep)q(enden)o(tly)j(disco)o(v)o(ered)e(a)f(construction)g(similar)i
(to)d(our)0 1739 y(third)e(construction.)25 b(Their)18
b(construction)f(is)h(di\013eren)o(t)f(and)g(its)g(analysis)h(is)f
(simpler)i(than)e(the)g(analysis)g(of)0 1801 y(our)g(construction.)27
b(F)l(urthermore,)16 b(they)i(ha)o(v)o(e)f(used)h(suc)o(h)f(a)g
(construction)h(as)f(the)g(main)h(tec)o(hnical)h(to)q(ol)e(in)0
1863 y(reducing)f(the)g(size)g(of)f(extractors,)e(for)i(the)g(range)g
Fo(m)d Fu(=)h(\002\()p Fo(n)p Fu(\).)71 1925 y(Subsequen)o(tly)l(,)i
(Zuc)o(k)o(erman)e([37)o(],)g(using)i(ideas)f(from)f([35)o(,)g(33],)g
(obtained)h(nearly)h(optimal)f(results)g(for)f(the)0
1987 y(extraction)h(problem)g(in)h(the)f(range)f Fo(m)g
Fu(=)g(\002\()p Fo(n)p Fu(\).)19 b(This)c(construction)f(has)f(n)o
(umerous)h(applications)i(whic)o(h)e(w)o(e)0 2049 y(shall)i(not)f(elab)
q(orate)g(here.)71 2111 y(W)l(e)h(stress)f(that)g(although)h(all)h(the)
f(ab)q(o)o(v)o(e)g(results)g(impro)o(v)o(e)g(on)g(our)f(second)i
(construction)f(in)h(case)f Fo(m)e Fu(=)0 2174 y(\002\()p
Fo(n)p Fu(\),)f(our)f(second)h(construction)g(is)g(b)q(etter)g(in)g
(case)g Fo(n)5 b Fn(\000)g Fo(m)13 b Fn(\034)g Fo(n)g
Fu(\(sp)q(eci\014cally)l(,)i(in)f(case)e Fo(n)5 b Fn(\000)g
Fo(m)13 b Fn(\024)g Fo(O)q Fu(\(log)8 b(1)p Fo(=\017)p
Fu(\)\).)0 2301 y Fl(1.3)56 b(Con)n(v)n(en)n(tions)0
2392 y Fu(Most)15 b(probabilistic)k(expressions)f(refer)e(to)g
(explicitly)j(de\014ned)f(random)e(v)m(ariables)i(whic)o(h)f(are)f(t)o
(ypically)i(de-)0 2455 y(noted)d Fo(X)q(;)8 b(Y)s(;)g(Z)q(;)g(U)320
2462 y Fm(n)355 2455 y Fu(and)15 b Fo(\020)s Fu(.)20
b(In)c(this)g(case)f(w)o(e)g(write)457 2562 y(Prob\(Bo)q(olean)g
(expression)i(in)f(these)f(random)g(v)m(ariables)q(\))p
0 2604 780 2 v 52 2635 a Fj(1)69 2650 y Fs(An)e(extractor)h(is)f(a)g
(family)h(of)f(functions)i(ha)o(ving)f(the)g(extraction)g(prop)q(ert)o
(y)m(.)964 2795 y Fu(2)p eop
%%Page: 3 4
3 3 bop 0 42 a Fu(and)18 b(it)f(is)h(understo)q(o)q(d)g(that)f(the)g
(probabilit)o(y)i(space)f(is)g(the)f(one)h(used)g(in)g(the)f
(de\014nition)j(of)d(these)g(random)0 104 y(v)m(ariables.)j(In)14
b(a)e(few)h(cases,)g(the)f(probabilistic)k(expression)d(also)g(in)o(v)o
(olv)o(es)g(a)g(uniformly)g(selected)h(ob)s(ject,)f(suc)o(h)0
166 y(as)i Fo(f)j Fn(2)13 b Fo(F)6 b Fu(.)20 b(In)c(suc)o(h)f(a)g(case)
g(w)o(e)g(write)331 257 y(Prob)427 264 y Fm(f)s Fi(2)p
Fm(F)497 257 y Fu(\(Bo)q(olean)g(expression)h(in)g(these)g(random)f(v)m
(ariables)h(and)g(in)g Fo(f)5 b Fu(\))0 382 y Fl(1.4)56
b(Organization)0 474 y Fu(W)l(e)14 b(start)e(b)o(y)i(recalling)i(the)d
(tec)o(hnical)j(to)q(ols)d(used)i(in)f(our)g(pro)q(ofs.)19
b(The)14 b(follo)o(wing)g(three)g(sections)g(\(Sec.)g(3{5\))0
536 y(are)h(dev)o(oted)g(to)f(the)i(corresp)q(onding)g(three)f
(constructions)g(men)o(tioned)h(ab)q(o)o(v)o(e.)j(Eac)o(h)c(section)h
(starts)e(with)h(a)0 598 y(brief)j(in)o(tuitiv)o(e)i(summary)d(of)g
(the)h(results)g(obtained.)28 b(Next,)18 b(comes)g(a)g(formal)f
(statemen)o(t)g(of)g(the)h(result,)g(a)0 660 y(description)e(of)d(the)h
(construction)h(whic)o(h)f(ac)o(hiev)o(es)h(it)f(and)h(an)f(analysis)g
(of)g(this)g(construction.)20 b(W)l(e)14 b(conclude)0
722 y(eac)o(h)19 b(section)h(with)f(a)g(relev)m(an)o(t)h(lo)o(w)o(er)f
(b)q(ound.)33 b(In)19 b(Section)h(6)f(w)o(e)g(presen)o(t)g(the)h(new)f
(sampling)h(pro)q(cedure)0 784 y(men)o(tioned)c(as)f(an)g(application)i
(ab)q(o)o(v)o(e.)0 933 y Fq(2)67 b(T)-6 b(ec)n(hnical)24
b(T)-6 b(o)r(ols)0 1040 y Fu(Univ)o(ersal)23 b(Hashing)g(and)f
(Expanders)h(are)e(used)i(in)g(our)f(\014rst)g(t)o(w)o(o)f
(constructions,)i(whereas)f(Small)i(Bias)0 1102 y(Probabilit)o(y)16
b(Spaces)g(are)f(used)h(in)g(the)f(third.)20 b(Expanders)c(are)f(also)g
(used)h(in)g(Section)g(6.)0 1227 y Fl(2.1)56 b(Univ)n(ersal)17
b(Hashing)0 1318 y Fu(Lo)q(osely)d(sp)q(eaking,)g(univ)o(ersal)g
(families)g(of)f(hashing)h(functions)f(consist)h(of)e(functions)i(op)q
(erating)f(on)g(the)g(same)0 1380 y(domain-range)j(pair)h(so)e(that)h
(a)f(function)i(uniformly)g(selected)g(in)g(the)f(family)h(maps)f(eac)o
(h)g(pair)g(of)g(p)q(oin)o(ts)g(in)0 1442 y(a)e(pairwise)h(indep)q
(enden)o(t)i(and)d(uniform)h(manner.)k(Sp)q(eci\014cally)m(,)e(a)c
(family)l(,)i Fo(H)1375 1449 y Fm(n;m)1437 1442 y Fu(,)f(of)g
(functions)h(from)e Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1929
1426 y Fm(n)0 1504 y Fu(to)15 b Fn(f)p Fu(0)p Fo(;)8
b Fu(1)p Fn(g)169 1488 y Fm(m)198 1504 y Fu(,)15 b(is)h(called)g
Fp(universal)f Fu(if)h(for)e(ev)o(ery)h Fo(x)e Fn(6)p
Fu(=)g Fo(y)h Fn(2)f(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1104
1488 y Fm(n)1140 1504 y Fu(and)16 b Fo(\013;)8 b(\014)14
b Fn(2)f(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1475 1488 y
Fm(m)1519 1504 y Fu(it)16 b(holds)581 1596 y(Prob)678
1603 y Fm(h)p Fi(2)p Fm(H)746 1607 y Fh(n;m)804 1596
y Fu(\()p Fo(h)p Fu(\()p Fo(x)p Fu(\))5 b(=)g Fo(\013)10
b Fn(^)h Fo(h)p Fu(\()p Fo(y)r Fu(\))5 b(=)g Fo(\014)r
Fu(\))12 b(=)h(2)1295 1577 y Fi(\000)p Fk(2)p Fm(m)0
1688 y Fu(where)i(the)h(probabilit)o(y)g(is)g(tak)o(en)f(o)o(v)o(er)f
(all)i(c)o(hoices)g(of)f Fo(h)e Fn(2)5 b Fo(H)1089 1695
y Fm(n;m)1166 1688 y Fu(with)15 b(uniform)h(probabilit)o(y)g
(distribution.)71 1750 y(Sev)o(eral)i(e\016cien)o(t)g(families)h(of)f
(univ)o(ersal)h(hashing)f(functions)g(are)g(kno)o(wn)f([9].)27
b(The)18 b(functions)g(in)h(these)0 1812 y(families)g(can)f(b)q(e)h
(describ)q(ed)g(using)g Fo(O)q Fu(\()p Fo(n)12 b Fu(+)g
Fo(m)p Fu(\))17 b(bits)i(and)f(p)q(osses)g(an)f(e\016cien)o(t)i
(\(e.g.,)e(p)q(olynomial-time)j(and)0 1874 y(ev)o(en)15
b(logspace\))g(ev)m(aluating)h(algorithms.)j(F)l(or)14
b(example,)i(linear)g(transformations)d(with)i(T)l(o)q(eplitz)1748
1858 y Fk(2)1783 1874 y Fu(matrices)0 1936 y(require)h(only)g
Fo(n)10 b Fu(+)h(2)p Fo(m)e Fn(\000)i Fu(1)k(bits.)20
b(The)15 b(t)o(w)o(o)f(main)i(facts)f(w)o(e)f(will)j(use)f(ab)q(out)f
(univ)o(ersal)h(hash)f(families)i(are:)0 2059 y Fv(P)o(airwise)h(Indep)
q(endence.)45 b Fu(The)16 b(set)g(of)f(random)h(v)m(ariables)h
Fn(f)p Fo(h)p Fu(\()p Fo(x)p Fu(\))p Fn(j)p Fo(x)c Fn(2)h(f)p
Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1480 2043 y Fm(n)1500 2059
y Fn(g)16 b Fu(de\014ned)h(b)o(y)f(a)f(random)0 2122
y Fo(h)e Fn(2)g Fo(H)18 b Fu(are)d(pairwise)h(indep)q(enden)o(t)i(and)d
(uniformly)i(distributed)f(in)g Fn(f)p Fu(0)p Fo(;)8
b Fu(1)p Fn(g)1343 2105 y Fm(m)1373 2122 y Fu(.)0 2245
y Fv(Lefto)o(v)o(er)18 b(Hash)g(Lemma.)45 b Fu(This)17
b(fundamen)o(tal)g(lemma)f(of)g([21)o(])g(asserts)f(that)h(a)g(random)g
(hash)g(function)0 2307 y(from)c(a)g(univ)o(ersal)i(family)g(will)g
(smo)q(oth)e(min-en)o(trop)o(y)h Fo(k)h Fu(whenev)o(er)f(the)g(range)f
(parameter)g Fo(m)g Fu(is)i(smaller)f(than)0 2369 y Fo(k)q
Fu(.)20 b(More)14 b(precisely)0 2462 y Fv(Lemma)j(2.1)23
b Fu(\(Lefto)o(v)o(er)c(Hash)g(Lemma)h([21)o(]\):)28
b Fp(L)n(et)20 b Fo(X)k Fp(b)n(e)c(any)g(r)n(andom)g(variable)h(on)f
Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1713 2446 y Fm(n)1754
2462 y Fp(with)21 b(min-)0 2524 y(entr)n(opy)c Fo(k)i
Fu(\(i.e.,)d(Prob\()p Fo(X)10 b Fu(=)d Fo(x)p Fu(\))15
b Fn(\024)h Fu(2)650 2508 y Fi(\000)p Fm(k)712 2524 y
Fu(for)g(all)i Fo(x)p Fu('s\))p Fp(.)24 b(Then)16 b(the)i(distribution)
g Fu(\()p Fo(h;)8 b(h)p Fu(\()p Fo(X)t Fu(\)\))p Fp(,)15
b(with)j Fo(h)g Fp(chosen)f(at)0 2586 y(r)n(andom)g(fr)n(om)f
Fo(H)311 2593 y Fm(n;m)373 2586 y Fp(,)g(has)g Fu(\(norm-1\))f
Fp(distanc)n(e)h Fu(2)876 2570 y Fi(\000)p Fk(\()p Fm(k)q
Fi(\000)p Fm(m)p Fk(\))p Fm(=)p Fk(3)1053 2586 y Fp(fr)n(om)g(the)h
(uniform)g(distribution.)p 0 2624 780 2 v 52 2654 a Fj(2)69
2670 y Fs(A)c(T)m(o)q(eplitz)h(matrix,)g Fg(T)i Fs(=)11
b(\()p Fg(t)504 2674 y Ff(i;j)540 2670 y Fs(\),)i(satis\014es)h
Fg(t)734 2674 y Ff(i;j)781 2670 y Fs(=)d Fg(t)836 2674
y Ff(i)p Fj(+1)p Ff(;j)q Fj(+1)950 2670 y Fs(,)h(for)h(all)h
Fg(i;)6 b(j)r Fs(.)964 2795 y Fu(3)p eop
%%Page: 4 5
4 4 bop 0 42 a Fl(2.2)56 b(Expanders)0 133 y Fu(An)13
b(\()p Fo(N)r(;)8 b(d;)g(\025)p Fu(\))p Fe(-expander)j
Fu(is)j(a)f Fo(d)p Fu(-regular)g(graph)f(with)i Fo(N)j
Fu(v)o(ertices)c(so)g(that)f(the)h(absolute)h(v)m(alue)g(of)e(all)i
(eigen)o(v)m(al-)0 195 y(ues)i(\(except)f(the)h(biggest)f(one\))g(of)g
(its)h(adjacency)g(matrix)f(is)h(b)q(ounded)g(b)o(y)g
Fo(\025)p Fu(.)k(A)15 b(\()p Fo(d;)8 b(\025)p Fu(\))p
Fe(-fam)n(ily)k Fu(is)k(an)f(in\014nite)0 257 y(sequence)i(of)e(graphs)
h(so)f(that)g(the)h Fo(n)650 241 y Fk(th)699 257 y Fu(graph)g(is)g(a)f
(\(2)954 241 y Fm(n)976 257 y Fo(;)8 b(d;)g(\025)p Fu(\)-expander.)20
b(W)l(e)c(sa)o(y)f(that)g(suc)o(h)h(a)f(family)i(is)f
Fp(ef-)0 319 y(\014ciently)e(c)n(onstructible)j Fu(if)e(there)f(exists)
h(a)f(log-space)h(algorithm)f(whic)o(h)h(giv)o(en)g(a)f(v)o(ertex,)g
Fo(v)r Fu(,)g(in)h(the)f(expander)0 381 y(and)i(an)f(index)i
Fo(i)c Fn(2)h Fu([)p Fo(d)p Fu(])410 356 y Fk(def)415
381 y Fu(=)19 b Fn(f)p Fu(1)p Fo(;)8 b(:::;)g(d)p Fn(g)p
Fu(,)k(returns)j(the)h Fo(i)920 365 y Fk(th)969 381 y
Fu(neigh)o(b)q(or)g(of)f Fo(v)r Fu(.)21 b(W)l(e)16 b(\014rst)f(recall)i
(that)e(for)g Fo(d)e Fu(=)h(16)h(and)0 443 y(some)g Fo(\025)d(<)h
Fu(16,)h(e\016cien)o(tly)j(constructible)f(\(16)p Fo(;)8
b(\025)p Fu(\)-families)15 b(do)g(exist)g(\(cf.,)g([16)o(]\))1410
427 y Fk(3)1428 443 y Fu(.)71 506 y(In)20 b(our)g(applications)h(w)o(e)
f(use)g(\(parameterized\))g(expanders)h(satisfying)1397
488 y Fm(\025)p 1397 495 20 2 v 1398 521 a(d)1442 506
y Fo(<)g(\013)f Fu(and)h Fo(d)f Fu(=)h(p)q(oly)q(\(1)p
Fo(=\013)p Fu(\),)0 568 y(where)c Fo(\013)h Fu(is)g(an)f
(application-sp)q(eci\014c)k(parameter.)k(Suc)o(h)18
b(\(parameterized\))f(expanders)h(are)e(also)i(e\016cien)o(tly)0
630 y(constructible.)40 b(F)l(or)20 b(example,)k(w)o(e)d(ma)o(y)f
(obtain)i(them)f(b)o(y)h(taking)f(paths)g(of)g(length)h
Fo(O)q Fu(\(log\(1)p Fo(=\013)p Fu(\))f(on)g(an)0 692
y(expander)e(as)f(ab)q(o)o(v)o(e.)30 b(Sp)q(eci\014cally)l(,)22
b(giv)o(en)d(a)g(parameter)e Fo(\013)i(>)f Fu(0,)h(w)o(e)f(obtain)h(an)
g(e\016cien)o(tly)h(constructible)0 754 y(\()p Fo(D)q(;)8
b Fu(\003\)-family)20 b(satisfying)502 736 y Fk(\003)p
500 743 29 2 v 500 770 a Fm(D)555 754 y Fo(<)j(\013)f
Fu(and)f Fo(D)i Fu(=)g(p)q(oly)r(\(1)p Fo(=\013)p Fu(\))d(as)h(follo)o
(ws.)38 b(W)l(e)21 b(start)f(with)i(a)f(constructible)0
823 y(\(16)p Fo(;)8 b(\025)p Fu(\)-family)l(,)k(set)i
Fo(k)397 798 y Fk(def)402 823 y Fu(=)k(log)514 834 y
Fk(16)p Fm(=\025)585 823 y Fu(\(1)p Fo(=\013)p Fu(\))12
b(=)h Fo(O)q Fu(\(log)8 b(1)p Fo(=\013)p Fu(\))13 b(and)h(consider)h
(the)f(paths)f(of)h(length)g Fo(k)h Fu(in)g(eac)o(h)f(graph.)0
890 y(This)i(yields)h(a)d(constructible)j(\(16)604 874
y Fm(k)624 890 y Fo(;)8 b(\025)672 874 y Fm(k)691 890
y Fu(\)-family)l(,)15 b(and)h(b)q(oth)1079 872 y Fm(\025)1099
860 y Fh(k)p 1072 879 52 2 v 1072 906 a Fk(16)1106 898
y Fh(k)1141 890 y Fo(<)d(\013)i Fu(and)h(16)1368 874
y Fm(k)1400 890 y Fu(=)d(p)q(oly)r(\(1)p Fo(=\013)p Fu(\))h(indeed)j
(hold.)71 952 y(T)l(o)j(obtain)h(the)f(b)q(est)h(constan)o(ts)f(in)h
(Sections)g(3,)f(4)g(and)h(6.2,)f(one)h(ma)o(y)f(use)h(e\016cien)o(tly)
h(constructible)0 1014 y(Raman)o(ujan)15 b(Graphs)g([24)o(].)21
b(F)l(urthermore,)14 b(using)j(Raman)o(ujan)e(Graphs)g(is)h(essen)o
(tial)g(for)f(our)g(pro)q(of)h(of)f(The-)0 1077 y(orem)i(6.5.)24
b(Raman)o(ujan)17 b(Graphs)g(satisfy)g Fo(\025)e Fn(\024)h
Fu(2)879 1038 y Fn(p)p 917 1038 24 2 v 39 x Fo(d)h Fu(and)g(so,)g
(setting)g Fo(d)e Fu(=)i(4)p Fo(=\013)p Fu(,)f(w)o(e)h(obtain)1683
1059 y Fm(\025)p 1683 1066 20 2 v 1684 1092 a(d)1723
1077 y Fo(<)f(\013)p Fu(,)i(where)0 1139 y Fo(\013)h
Fu(is)h(an)f(application-sp)q(eci\014c)k(parameter.)30
b(Here)20 b(some)e(minor)i(tec)o(hnicalities)h(arise)f(as)e(these)i
(graphs)e(are)0 1201 y(giv)o(en)d(only)g(for)f(certain)h(degrees)g(and)
f(certain)h(sizes.)21 b(Sp)q(eci\014cally)l(,)d(they)c(can)h(b)q(e)g
(e\016cien)o(tly)h(constructed)f(for)5 1245 y Fk(1)p
5 1252 17 2 v 5 1279 a(2)37 1263 y Fn(\001)c Fo(q)83
1246 y Fm(k)114 1263 y Fn(\001)g Fu(\()p Fo(q)178 1246
y Fk(2)p Fm(k)225 1263 y Fn(\000)h Fu(1\))k(v)o(ertices,)g(where)h
Fo(q)g Fn(\021)e Fo(d)10 b Fn(\000)i Fu(1)i Fn(\021)h
Fu(1)d(mo)q(d)h(4)j(and)h Fo(d)11 b Fn(\000)g Fu(1)16
b(b)q(eing)i(a)e(quadratic)h(residue)g(mo)q(dulo)0 1325
y Fo(q)j Fu(\(cf.,)d([3)o(,)i(Sec.)f(I)q(I]\).)g(Still,)i(\014xing)f
Fo(d)f Fu(and)g Fo(\017;)8 b(N)d Fu(,)18 b(w)o(e)f(ma)o(y)h(\014nd)g(a)
g Fo(q)i Fu(satisfying)f(the)f(ab)q(o)o(v)o(e)f(conditions)j(with)5
1369 y Fk(1)p 5 1376 V 5 1403 a(2)39 1387 y Fn(\001)11
b Fo(q)85 1371 y Fm(k)118 1387 y Fn(\001)h Fu(\()p Fo(q)183
1371 y Fk(2)p Fm(k)231 1387 y Fn(\000)h Fu(1\))k Fn(2)h
Fu([\(1)11 b Fn(\000)h Fo(\017)p Fu(\))h Fn(\001)e Fo(N)r(;)d(N)d
Fu(],)17 b(in)i(time)g(p)q(olynomial)g(in)h(1)p Fo(=\017)p
Fu(.)28 b(This)19 b(de\014nes)g(a)f(Raman)o(ujan)g(Graph)0
1449 y(whic)o(h)e(is)g(adequate)f(for)g(all)h(our)f(applications)h
(\(and)f(sp)q(eci\014cally)m(,)i(for)e(the)g(pro)q(of)g(of)g(Theorem)g
(6.5\).)0 1575 y Fv(The)21 b(Expander)f(Mixing)h(Lemma.)45
b Fu(The)18 b(follo)o(wing)h(lemma)f(is)g(folklore)h(and)f(has)f(app)q
(eared)i(in)f(man)o(y)0 1637 y(pap)q(ers.)32 b(Lo)q(osely)19
b(sp)q(eaking,)i(the)e(lemma)g(asserts)f(that)g(expander)i(graphs)f
(\(for)f(whic)o(h)h Fo(d)g Fn(\035)g Fo(\025)p Fu(\))f(ha)o(v)o(e)h
(the)0 1699 y(prop)q(ert)o(y)h(that)f(the)h(fraction)g(of)g(edges)g(b)q
(et)o(w)o(een)g(t)o(w)o(o)f(large)h(sets)g(of)g(v)o(ertices)g(appro)o
(ximately)g(equals)h(the)0 1761 y(pro)q(duct)16 b(of)e(the)i(densities)
g(of)f(these)h(sets.)j(This)d(prop)q(ert)o(y)f(is)h(called)g
Fp(mixing)p Fu(.)0 1873 y Fv(Lemma)h(2.2)23 b Fu(\(Expander)16
b(Mixing)g(Lemma\):)21 b Fp(L)n(et)16 b Fo(G)d Fu(=)h(\()p
Fo(V)s(;)8 b(E)s Fu(\))13 b Fp(b)n(e)k(an)f(exp)n(ander)h(gr)n(aph)g
(of)g(de)n(gr)n(e)n(e)e Fo(d)i Fp(and)f Fo(\025)0 1935
y Fp(b)n(e)h(an)h(upp)n(er)g(b)n(ound)g(on)g(the)g(absolute)g(value)f
(of)h(al)r(l)g(eigenvalues,)f(save)g(the)h(biggest)f(one,)h(of)g(the)g
(adjac)n(ency)0 1997 y(matrix)f(of)f(the)h(gr)n(aph.)k(Then)16
b(for)g(every)h(two)f(subsets,)g Fo(A;)8 b(B)14 b Fn(\022)f
Fo(V)d Fp(,)16 b(it)g(holds)508 2065 y Fd(\014)508 2090
y(\014)508 2115 y(\014)508 2140 y(\014)527 2096 y Fn(j)p
Fu(\()p Fo(A)10 b Fn(\002)g Fo(B)r Fu(\))h Fn(\\)f Fo(E)s
Fn(j)p 527 2116 274 2 v 633 2158 a(j)p Fo(E)s Fn(j)816
2127 y(\000)868 2096 y(j)p Fo(A)p Fn(j)p 866 2116 62
2 v 866 2158 a(j)p Fo(V)g Fn(j)943 2127 y(\001)971 2096
y(j)p Fo(B)r Fn(j)p 971 2116 63 2 v 971 2158 a(j)p Fo(V)f
Fn(j)1038 2065 y Fd(\014)1038 2090 y(\014)1038 2115 y(\014)1038
2140 y(\014)1064 2127 y Fn(\024)1117 2096 y Fo(\025)1144
2059 y Fd(p)p 1185 2059 155 2 v 1185 2096 a Fn(j)p Fo(A)p
Fn(j)h(\001)g(j)p Fo(B)r Fn(j)p 1117 2116 223 2 v 1169
2158 a Fo(d)g Fn(\001)g(j)p Fo(V)f Fn(j)1357 2127 y Fo(<)1410
2096 y(\025)p 1410 2116 27 2 v 1411 2158 a(d)0 2346 y
Fu(The)15 b(lemma)h(\(and)f(a)g(pro)q(of)t(\))f(app)q(ears)h(as)g
(Corollary)g(2.5)f(in)i([6,)e(Chap.)h(9].)p 0 2388 780
2 v 52 2415 a Fj(3)69 2431 y Fs(The)e(are)g(minor)h(tec)o(hnicalitie)q
(s)h(whic)o(h)f(can)g(b)q(e)f(easily)h(\014xed.)k(Firstly)m(,)c(the)f
(Gab)q(er{Galil)j(expanders)f(are)e(de\014ned)h(\(only\))g(for)0
2481 y(graph)g(sizes)g(whic)o(h)f(are)g(p)q(erfect)g(squares)h([16].)j
(This)c(su\016ces)h(for)f(ev)o(en)g Fg(n)p Fs('s.)k(F)m(or)12
b(o)q(dd)i Fg(n)p Fs('s,)e(w)o(e)h(ma)o(y)g(use)g(a)g(trivial)i(mo)q
(di\014cation,)0 2531 y(suc)o(h)g(as)f(taking)i(t)o(w)o(o)e(copies)h
(of)f(the)g(graph)h(of)f(size)h(2)804 2516 y Ff(n)p Ft(\000)p
Fj(1)879 2531 y Fs(and)g(connecting)h(eac)o(h)f(pair)g(of)f(corresp)q
(onding)j(v)o(ertices.)22 b(Finally)m(,)16 b(w)o(e)0
2582 y(add)e(m)o(ultiple)h(edges)f(so)f(that)g(the)g(degree)h(b)q
(ecomes)g(16,)e(rather)i(than)f(b)q(eing)i(14)e(for)g(ev)o(en)g
Fg(n)p Fs('s)g(and)h(15)f(for)g(o)q(dd)g Fg(n)p Fs('s.)964
2795 y Fu(4)p eop
%%Page: 5 6
5 5 bop 0 42 a Fv(The)15 b(Expander)f(Smo)q(othing)i(Lemma.)45
b Fu(Random)13 b(w)o(alks)f(on)h(expander)g(graphs)f(are)h(kno)o(wn)f
(to)g(increase)0 104 y(the)j(en)o(trop)o(y)f(of)h(a)f(distribution)j(v)
o(ery)d(fast.)19 b(That)c(is,)g(if)g(one)g(starts)f(with)h(some)f
(\(non-uniform\))h(distribution)0 166 y(on)i(the)f(v)o(ertices)i(of)e
(the)h(expander)g(and)g(tak)o(es)f(a)g(short)g(random)h(w)o(alk)f(then)
h(one)g(arriv)o(es)g(at)f(a)g(distribution)0 228 y(whic)o(h)j(is)g
(closer)g(to)e(uniform.)30 b(The)18 b(follo)o(wing)h(lemma)g(refers)f
(to)f(the)i(e\013ect)f(of)f(a)h(single)i(random)e(step.)29
b(It)0 290 y(follo)o(ws)18 b(easily)h(b)o(y)e(the)h(standard)g(tec)o
(hniques)h(of)e(dealing)i(with)f(random)g(w)o(alks)f(on)h(expander)h
(graphs)e(\(cf.,)0 352 y([1)o(,)e(6]\).)0 436 y Fv(Lemma)i(2.3)23
b Fu(\(Expander)12 b(Smo)q(othing)g(Lemma\):)18 b Fp(L)n(et)13
b Fo(G)f Fu(=)h(\()p Fo(V)s(;)8 b(E)s Fu(\))p Fp(,)i
Fo(d)j Fp(and)g Fo(\025)g Fp(b)n(e)f(as)h(in)g(the)g(pr)n(evious)g
(lemma.)0 498 y(L)n(et)i Fo(X)k Fp(b)n(e)c(a)h(r)n(andom)g(variable,)f
(distribute)n(d)h(over)g Fo(V)9 b Fp(,)16 b(so)g(that)g
Fu(Prob\()p Fo(X)7 b Fu(=)e Fo(v)r Fu(\))13 b Fn(\024)1431
480 y Fm(K)p 1423 487 47 2 v 1423 514 a Fi(j)p Fm(V)6
b Fi(j)1474 498 y Fp(,)16 b(for)g(every)f Fo(v)7 b Fn(2)e
Fo(V)10 b Fp(,)16 b(and)g Fo(Y)0 560 y Fp(denote)g(the)h(vertex)f(r)n
(e)n(ache)n(d)g(fr)n(om)g Fo(X)k Fp(by)c(fol)r(lowing)g(a)g(uniformly)h
(chosen)e(e)n(dge.)21 b(Then)601 616 y Fd(X)597 706 y
Fm(v)q Fi(2)p Fm(V)672 596 y Fd(\014)672 621 y(\014)672
646 y(\014)672 670 y(\014)686 657 y Fu(Prob)o(\()p Fo(Y)16
b Fu(=)5 b Fo(v)r Fu(\))k Fn(\000)1003 626 y Fu(1)p 984
646 62 2 v 984 688 a Fn(j)p Fo(V)g Fn(j)1051 596 y Fd(\014)1051
621 y(\014)1051 646 y(\014)1051 670 y(\014)1077 657 y
Fn(\024)1130 626 y Fo(\025)p 1130 646 27 2 v 1131 688
a(d)1172 657 y Fn(\001)1194 618 y(p)p 1232 618 121 2
v 39 x Fo(K)k Fn(\000)e Fu(1)0 824 y Fv(Pro)q(of:)24
b Fu(Let)18 b Fo(N)308 799 y Fk(def)314 824 y Fu(=)k
Fn(j)p Fo(V)9 b Fn(j)p Fu(,)17 b(and)h(let)g Fo(x)g Fu(denote)g(the)f
Fo(N)5 b Fu(-dimensional)20 b(probabilit)o(y)e(v)o(ector)f(de\014ned)i
(b)o(y)f Fo(X)j Fu(\(i.e.,)0 886 y Fo(x)26 893 y Fm(i)56
861 y Fk(def)61 886 y Fu(=)h(Prob\()p Fo(X)11 b Fu(=)e
Fo(i)p Fu(\)\).)25 b(Let)18 b Fo(A)f Fu(denote)h(the)g(Mark)o(o)o(v)d
(pro)q(cess)j(de\014ned)h(b)o(y)e(tra)o(v)o(ersing)f(a)h(uniformly)i
(selected)0 949 y(edge)14 b(in)g Fo(G)p Fu(;)g(namely)l(,)g(the)g
(matrix)f Fo(A)h Fu(is)g(the)f(adjacency)h(matrix)g(of)f(the)h(graph)f
Fo(G)p Fu(,)g(normalized)i(b)o(y)e(division)j(b)o(y)0
1011 y Fo(d)p Fu(.)j(Denote)12 b(the)h(eigen)o(v)m(alues)i(of)d
Fo(A)h Fu(b)o(y)f Fo(\025)701 1018 y Fk(1)720 1011 y
Fo(;)c(:::;)g(\025)827 1018 y Fm(N)856 1011 y Fu(,)13
b(and)g(note)f(that)g Fo(\025)1189 1018 y Fk(1)1220 1011
y Fu(=)h(1)f(and)h Fn(j)p Fo(\025)1429 1018 y Fm(i)1442
1011 y Fn(j)g(\024)1521 993 y Fm(\025)p 1521 1000 20
2 v 1522 1026 a(d)1545 1011 y Fu(,)g(for)f(ev)o(ery)h
Fo(i)f(>)h Fu(1.)19 b(W)l(e)0 1073 y(consider)13 b(the)g(orthogonal)e
(eigen)o(v)o(ector)i(basis,)g Fo(e)851 1080 y Fk(1)870
1073 y Fo(;)8 b(:::;)g(e)972 1080 y Fm(N)1000 1073 y
Fu(,)13 b(where)g Fo(e)1176 1080 y Fm(i)1190 1073 y Fo(e)1211
1056 y Fi(>)1211 1084 y Fm(i)1251 1073 y Fu(=)1311 1055
y Fk(1)p 1304 1062 30 2 v 1304 1088 a Fm(N)1351 1073
y Fu(for)f(eac)o(h)g Fo(i)p Fu(,)g Fo(e)1579 1080 y Fk(1)1611
1073 y Fu(=)h(\()1688 1055 y Fk(1)p 1682 1062 V 1682
1088 a Fm(N)1716 1073 y Fo(;)8 b(:::;)1806 1055 y Fk(1)p
1799 1062 V 1799 1088 a Fm(N)1834 1073 y Fu(\),)k(and)0
1135 y(write)k(eac)o(h)g(v)o(ector)g(as)g(a)f(linear)j(com)o(bination)e
(of)g(the)g(v)o(ectors)g(in)h(this)f(basis.)23 b(Denote)16
b(b)o(y)g Fo(c)1644 1142 y Fm(i)1674 1135 y Fu(the)g(co)q(e\016cien)o
(t)0 1197 y(of)f Fo(x)g Fu(in)h(the)f(direction)i(of)d
Fo(e)486 1204 y Fm(i)500 1197 y Fu(.)20 b(W)l(e)c(start)e(b)o(y)h(b)q
(ounding)984 1165 y Fd(P)1028 1208 y Fm(i)1050 1197 y
Fo(c)1070 1180 y Fk(2)1070 1208 y Fm(i)1103 1197 y Fu(as)g(follo)o(ws)
605 1254 y Fd(X)629 1342 y Fm(i)672 1294 y Fo(c)692 1275
y Fk(2)692 1305 y Fm(i)721 1294 y Fn(\001)758 1263 y
Fu(1)p 748 1284 42 2 v 748 1325 a Fo(N)836 1294 y Fu(=)42
b(\()931 1254 y Fd(X)955 1342 y Fm(i)998 1294 y Fo(c)1018
1301 y Fm(i)1032 1294 y Fo(e)1053 1275 y Fi(>)1053 1305
y Fm(i)1081 1294 y Fu(\))10 b Fn(\001)g Fu(\()1150 1254
y Fd(X)1173 1342 y Fm(i)1217 1294 y Fo(c)1237 1301 y
Fm(i)1250 1294 y Fo(e)1271 1275 y Fi(>)1271 1305 y Fm(i)1300
1294 y Fu(\))1318 1275 y Fi(>)836 1402 y Fu(=)42 b Fo(x)10
b Fn(\001)g Fo(x)998 1383 y Fi(>)836 1476 y Fu(=)913
1436 y Fd(X)937 1524 y Fm(i)981 1476 y Fo(x)1007 1458
y Fk(2)1007 1488 y Fm(i)836 1615 y Fn(\024)918 1585 y
Fo(N)p 918 1605 V 918 1647 a(K)975 1615 y Fn(\001)998
1556 y Fd(\022)1033 1585 y Fo(K)p 1033 1605 V 1033 1647
a(N)1080 1556 y Fd(\023)1111 1564 y Fk(2)0 1774 y Fu(getting)158
1742 y Fd(P)202 1785 y Fm(i)223 1774 y Fo(c)243 1757
y Fk(2)243 1785 y Fm(i)281 1774 y Fn(\024)20 b Fo(K)s
Fu(.)32 b(It)19 b(is)h(also)f(easy)g(to)g(see)h(that)e
Fo(c)985 1781 y Fk(1)1023 1774 y Fu(=)i(1.)32 b(W)l(e)19
b(no)o(w)g(consider)h(the)g(di\013erences)g(v)o(ector,)0
1836 y(denoted)c Fo(z)r Fu(,)f(represen)o(ting)g(the)h(deviation)g(of)f
(the)g(random)g(v)m(ariable)h Fo(Y)26 b Fu(from)14 b(the)i(uniform)f
(distribution.)716 1919 y Fo(z)739 1901 y Fi(>)809 1894
y Fk(def)814 1919 y Fu(=)47 b Fo(Ax)956 1901 y Fi(>)994
1919 y Fn(\000)10 b Fo(e)1060 1901 y Fi(>)1060 1931 y
Fk(1)814 1994 y Fu(=)47 b Fo(A)p Fu(\()948 1954 y Fd(X)972
2042 y Fm(i)1015 1994 y Fo(c)1035 2001 y Fm(i)1049 1994
y Fo(e)1070 2001 y Fm(i)1084 1994 y Fu(\))1102 1975 y
Fi(>)1139 1994 y Fn(\000)11 b Fo(e)1206 1975 y Fi(>)1206
2005 y Fk(1)814 2107 y Fu(=)896 2067 y Fd(X)899 2155
y Fm(i>)p Fk(1)963 2107 y Fo(\025)990 2114 y Fm(i)1004
2107 y Fo(c)1024 2114 y Fm(i)1037 2107 y Fo(e)1058 2089
y Fi(>)1058 2119 y Fm(i)0 2219 y Fu(Recall)18 b(that)d(the)h(lemma)g
(claims)h(an)e(upp)q(er)i(b)q(ound)g(on)f(the)g(norm-1)f(of)g
Fo(z)r Fu(.)22 b(Instead,)16 b(w)o(e)g(start)e(b)o(y)i(pro)o(viding)0
2281 y(a)f(b)q(ound)h(on)f(its)h(norm-2:)667 2324 y Fd(X)691
2413 y Fm(i)734 2365 y Fo(z)757 2346 y Fk(2)755 2376
y Fm(i)818 2365 y Fu(=)894 2324 y Fd(X)897 2413 y Fm(i>)p
Fk(1)962 2365 y Fo(\025)989 2346 y Fk(2)989 2376 y Fm(i)1007
2365 y Fo(c)1027 2346 y Fk(2)1027 2376 y Fm(i)1045 2365
y Fo(e)1066 2372 y Fm(i)1080 2365 y Fo(e)1101 2346 y
Fi(>)1101 2376 y Fm(i)818 2506 y Fn(\024)909 2476 y Fu(1)p
899 2496 V 899 2538 a Fo(N)956 2506 y Fn(\001)979 2447
y Fd(\022)1014 2476 y Fo(\025)p 1014 2496 27 2 v 1015
2538 a(d)1046 2447 y Fd(\023)1076 2454 y Fk(2)1102 2466
y Fd(X)1105 2554 y Fm(i>)p Fk(1)1170 2506 y Fo(c)1190
2488 y Fk(2)1190 2518 y Fm(i)818 2648 y Fn(\024)909 2617
y Fu(1)p 899 2638 42 2 v 899 2679 a Fo(N)956 2648 y Fn(\001)979
2589 y Fd(\022)1014 2617 y Fo(\025)p 1014 2638 27 2 v
1015 2679 a(d)1046 2589 y Fd(\023)1076 2596 y Fk(2)1105
2648 y Fn(\001)10 b Fu(\()p Fo(K)i Fn(\000)f Fu(1\))964
2795 y(5)p eop
%%Page: 6 7
6 6 bop 0 42 a Fu(Maximizing)16 b(the)g(sum)f(of)g(the)g
Fn(j)p Fo(z)586 49 y Fm(i)600 42 y Fn(j)p Fu('s,)f(sub)s(ject)h(to)f
(the)h(ab)q(o)o(v)o(e)g(b)q(ound,)h(the)f(lemma)h(follo)o(ws.)p
1685 48 34 40 v 0 179 a Fl(2.3)56 b(Small)17 b(Probabilit)n(y)g(Spaces)
i(with)g(the)f(Small)f(Bias)h(Prop)r(ert)n(y)0 271 y
Fu(The)h(follo)o(wing)g(de\014nition)i(of)d(small-bias)i(sample)f
(spaces)g(implies)h(the)f(informal)g(de\014nition)i(in)e(Section)h(5.)0
333 y(Both)e(de\014nitions)h(are)f(legitimate)h(generalizations)g(of)f
(the)g(de\014nition)i(of)d(small-biased)j(sample)e(spaces)h(for)0
395 y(the)c(binary)h(case)f(\(and)g(indeed)i(they)f(are)e(equiv)m(alen)
o(t)j(for)e Fo(p)d Fu(=)h(2\).)0 507 y Fv(De\014nition)19
b(2.4)k Fp(L)n(et)13 b Fo(t)h Fp(b)n(e)g(an)f(inte)n(ger,)h
Fo(p)g Fp(b)n(e)f(a)h(prime)g(and)g Fo(!)i Fp(b)n(e)e(a)g
Fo(p)1218 490 y Fk(th)1265 507 y Fp(r)n(o)n(ot)g(of)g(unity)g(\(in)f
(the)h(c)n(omplex)g(\014eld\).)0 569 y(A)19 b(set)g Fo(S)h
Fn(\022)e Fo(GF)6 b Fu(\()p Fo(p)p Fu(\))358 552 y Fm(t)392
569 y Fp(is)18 b(said)i(to)f(have)h Fo(\017)f Fe(bias)h
Fu(\(sample)f(space)f(for)g Fo(GF)6 b Fu(\()p Fo(p)p
Fu(\))1337 552 y Fm(t)1351 569 y Fu(\))19 b Fp(if,)h(for)f(every)g
Fo(t)p Fp(-long)h(se)n(quenc)n(e)0 631 y Fu(\()p Fo(a)42
638 y Fk(1)60 631 y Fo(;)8 b(:::;)g(a)165 638 y Fm(t)177
631 y Fu(\))15 b Fp(of)f(elements)g(in)g Fo(GF)6 b Fu(\()p
Fo(p)p Fu(\))p Fp(,)15 b(so)f(that)h(not)g(al)r(l)g Fo(a)974
638 y Fm(i)987 631 y Fp('s)g(ar)n(e)f(zer)n(o,)h(the)g(exp)n(e)n
(ctation)f(of)h(\(the)f(magnitude)i(of)5 b(\))0 697 y
Fo(!)30 649 y Fd(P)74 659 y Fh(t)74 693 y(i)p Fc(=1)128
677 y Fm(a)146 681 y Fh(i)159 677 y Fm(s)175 681 y Fh(i)190
697 y Fp(,)16 b(taken)g(over)h(al)r(l)f Fu(\()p Fo(s)547
704 y Fk(1)565 697 y Fo(;)8 b(:::;)g(s)667 704 y Fm(t)680
697 y Fu(\))k Fn(2)h Fo(S)19 b Fp(with)d(uniform)h(distribution,)f(is)g
(b)n(ounde)n(d)g(ab)n(ove)g(by)g Fo(\017)p Fp(.)22 b(That)16
b(is,)645 761 y Fd(\015)645 786 y(\015)645 811 y(\015)668
810 y Fu(Exp)748 820 y Fk(\()p Fm(s)777 824 y Fc(1)793
820 y Fm(;:::;s)859 824 y Fh(t)871 820 y Fk(\))p Fi(2)p
Fm(S)938 763 y Fd(\020)963 810 y Fo(!)993 761 y Fd(P)1037
771 y Fh(t)1037 805 y(i)p Fc(=1)1091 789 y Fm(a)1109
793 y Fh(i)1122 789 y Fm(s)1138 793 y Fh(i)1153 763 y
Fd(\021)1178 761 y(\015)1178 786 y(\015)1178 811 y(\015)1226
810 y Fn(\024)26 b Fo(\017)0 921 y Fu(The)20 b(follo)o(wing)g(theorem,)
g(due)g(to)f(G.)g(Ev)o(en)g([14)o(],)h(is)g(obtained)g(b)o(y)g
(generalizing)h(a)e(construction)h(of)f(Alon)0 984 y(et.)i(al.)h([4)o
(].)39 b(Sp)q(eci\014cally)l(,)26 b(Ev)o(en)c(generalizes)h(the)f(LFSR)
g(construction)g(b)o(y)g(considering)h(sequences)g(o)o(v)o(er)0
1046 y Fo(GF)6 b Fu(\()p Fo(p)p Fu(\))15 b(\(rather)f(than)h(o)o(v)o
(er)f Fo(GF)6 b Fu(\(2\)\).)0 1158 y Fv(Theorem)17 b(2.5)22
b Fu([14,)d(15)o(]:)28 b Fp(F)m(or)20 b(every)g(inte)n(ger)f
Fo(t)p Fp(,)j(prime)e Fo(p)g Fp(and)h Fo(\017)f(>)g Fu(0)p
Fp(,)h(ther)n(e)f(exists)f(an)h(e\016ciently)f(c)n(on-)0
1220 y(structible)d Fo(\017)p Fp(-bias)h(sample)f(sp)n(ac)n(e)f(for)i
Fo(GF)6 b Fu(\()p Fo(p)p Fu(\))797 1203 y Fm(t)827 1220
y Fp(of)16 b(size)g Fu(\(2)p Fo(t=\017)p Fu(\))1085 1203
y Fk(2)1103 1220 y Fp(.)0 1370 y Fq(3)67 b(Tin)n(y)24
b(F)-6 b(amilies)24 b(of)d(F)-6 b(unctions)24 b(with)g(Mixing)g(Prop)r
(erties)0 1478 y Fu(Recall)16 b(that)e(a)h(function)g
Fo(f)20 b Fu(is)15 b(mixing)h(for)e(subsets)g Fo(A)h
Fu(and)g Fo(B)i Fu(of)d(the)h(domain)g(if)g(mem)o(b)q(ership)h(in)f
Fo(A)9 b Fn(\002)h Fo(B)17 b Fu(of)d(a)0 1540 y(pair)f(\()p
Fo(r)o(;)8 b(f)d Fu(\()p Fo(r)q Fu(\)\),)11 b(with)i
Fo(r)h Fu(b)q(eing)g(a)f(random)g(elemen)o(t)h(in)g(the)f(domain,)h(o)q
(ccurs)f(roughly)g(as)g(often)g(as)g(it)g(w)o(ould)h(for)0
1602 y(a)f(random)g(pair)g(\()p Fo(r)o(;)8 b(s)p Fu(\))k(of)g(elemen)o
(ts.)20 b(The)14 b(main)f(result)h(of)f(this)h(section)f(is)h(the)g
(explicit)h(construction)e(of)g(an)g Fo(\017)p Fu(-)0
1664 y(mixing)g(family)g(of)e(size)i(p)q(oly\(1)p Fo(=\017)p
Fu(\).)19 b(Here)12 b Fo(\017)g Fu(stands)g(b)q(oth)g(for)f(distance)i
(from)e(truly)h(random)g(b)q(eha)o(vior,)g(as)g(w)o(ell)0
1726 y(as)j(the)h(fraction)f(of)g(bad)h(functions)g(whic)o(h)g(do)g
(not)f(ac)o(hiev)o(e)h(this)g(distance.)21 b(W)l(e)16
b(state)e(the)i(precise)h(theorem,)0 1788 y(then)d(describ)q(e)i(the)e
(construction.)20 b(W)l(e)14 b(pro)o(v)o(e)f(that)g(our)h(family)h(has)
f(optimal)g(size)h(up)f(to)g(a)f(p)q(olynomial,)j(and)0
1850 y(presen)o(t)f(an)h(application:)22 b(sa)o(ving)15
b(randomness)g(in)h(the)g(generalized)h(logspace)f(mo)q(del)g(of)f([27)
o(].)20 b(W)l(e)15 b(conclude)0 1912 y(with)h(a)e(di\013eren)o(t)i(p)q
(ersp)q(ectiv)o(e)h(of)d(this)i(result,)f(adv)o(o)q(cated)g(b)o(y)g
(Linial.)0 2040 y Fl(3.1)56 b(Main)18 b(result)0 2131
y Fv(Theorem)f(3.1)22 b Fu(\(The)c(Mixing)g(F)l(amily\):)25
b Fp(L)n(et)18 b Fo(n)g Fp(b)n(e)g(an)g(inte)n(ger)f(and)i
Fo(\017)e(>)f Fu(2)1375 2115 y Fi(\000)p Fm(n=O)q Fk(\(1\))1509
2131 y Fp(.)27 b(Then,)18 b(ther)n(e)g(exists)g(a)0 2193
y(family)e(of)h(functions,)e(e)n(ach)h(mapping)h Fn(f)p
Fu(0)p Fo(;)8 b Fu(1)p Fn(g)802 2177 y Fm(n)839 2193
y Fp(to)16 b(itself,)g(satisfying)f(the)h(fol)r(lowing)g(pr)n(op)n
(erties.)68 2293 y Fn(\017)23 b Fu(succinctness:)i Fp(the)18
b(family)g(c)n(ontains)e(a)i(p)n(olynomial)g(in)1123
2275 y Fk(1)p 1123 2282 17 2 v 1124 2308 a Fm(\017)1162
2293 y Fp(numb)n(er)g(of)g(functions,)f(and)h(e)n(ach)f(function)114
2355 y(is)e(r)n(epr)n(esente)n(d)g(by)i(a)f(unique)g(string)g(of)g
(length)g Fo(l)q Fu(\()p Fo(\017)p Fu(\))c(=)h Fo(O)q
Fu(\(log)1215 2337 y Fk(1)p 1215 2344 V 1216 2371 a Fm(\017)1236
2355 y Fu(\))p Fp(.)68 2454 y Fn(\017)23 b Fu(e\016cien)o(t)11
b(ev)m(aluation:)19 b Fp(Ther)n(e)12 b(exists)f(a)h(lo)n(gsp)n(ac)n(e)f
(algorithm)h(that,)h(on)f(input)g(a)g(description)g(of)g(a)g(function)
114 2516 y Fo(f)21 b Fp(and)16 b(a)h(string)e Fo(x)p
Fp(,)i(r)n(eturns)e Fo(f)5 b Fu(\()p Fo(x)p Fu(\))p Fp(.)964
2795 y Fu(6)p eop
%%Page: 7 8
7 7 bop 68 42 a Fn(\017)23 b Fu(mixing)18 b(prop)q(ert)o(y:)24
b Fp(F)m(or)17 b(every)h(two)h(subsets)e Fo(A;)8 b(B)18
b Fn(\022)e(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1195 25 y
Fm(n)1215 42 y Fp(,)19 b(al)r(l)f(but)g(at)g(most)h(an)e
Fo(\017)i Fp(fr)n(action)f(of)g(the)114 104 y(functions)d
Fo(f)22 b Fp(in)15 b(the)i(family)f(satisfy)606 202 y
Fn(j)p Fu(Prob)o(\()p Fo(U)764 209 y Fm(n)791 202 y Fn(2)5
b Fo(A)11 b Fn(^)f Fo(f)5 b Fu(\()p Fo(U)987 209 y Fm(n)1010
202 y Fu(\))g Fn(2)g Fo(B)r Fu(\))10 b Fn(\000)h Fo(\032)p
Fu(\()p Fo(A)p Fu(\))p Fo(\032)p Fu(\()p Fo(B)r Fu(\))p
Fn(j)g(\024)i Fo(\017)114 301 y Fp(wher)n(e)19 b Fo(\032)p
Fu(\()p Fo(S)s Fu(\))354 276 y Fk(def)359 301 y Fu(=)423
280 y Fi(j)p Fm(S)r Fi(j)p 423 291 42 2 v 425 317 a Fk(2)442
309 y Fh(n)490 301 y Fp(denotes)g(the)h(density)f(of)h(the)g(set)g
Fo(S)i Fp(and)e Fo(U)1279 308 y Fm(n)1321 301 y Fp(is)f(a)h(r)n(andom)g
(variable)g(uniformly)114 363 y(distribute)n(d)c(over)h
Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)550 347 y Fm(n)570
363 y Fp(.)0 465 y Fu(Using)19 b(Raman)o(ujan)f(Graphs)g(as)g
(expanders)h(and)g(T)l(o)q(eplitz)h(matrices)e(as)g(Univ)o(ersal)i
(Hashing,)f(in)g(the)g(con-)0 527 y(struction)c(b)q(elo)o(w,)h(one)f
(ma)o(y)g(obtain)g Fo(l)q Fu(\()p Fo(\017)p Fu(\))d(=)h(7)8
b(log)875 538 y Fk(2)893 527 y Fu(\(1)p Fo(=\017)p Fu(\))i(+)g
Fo(O)q Fu(\(1\).)0 653 y Fl(3.2)56 b(The)18 b(Construction)0
745 y Fu(The)g(construction)g(mak)o(e)f(used)h(of)f(t)o(w)o(o)f(basic)j
(to)q(ols)e(whic)o(h)h(are)g(frequen)o(tly)g(used)g(for)f(sa)o(ving)g
(randomness:)0 807 y(univ)o(ersal)12 b(hashing)g(functions)f(and)g
(expander)h(graphs)e(\(see)h(Section)h(2\).)18 b(W)l(e)11
b(start)e(b)o(y)i(setting)g(the)g(parameters)0 869 y(for)k(the)g
(expander)h(graph)f(and)g(the)g(univ)o(ersal)i(hashing)e(family)h(to)f
(b)q(e)h(used.)0 993 y Fv(The)g(expander)f(graph.)45
b Fu(W)l(e)13 b(use)h(an)g(e\016cien)o(tly)g(constructible)h(expander)f
(graph,)f(denoted)h Fo(G)p Fu(,)g(of)f(degree)0 1055
y Fo(d)p Fu(,)23 b(second)f(eigen)o(v)m(alue)i Fo(\025)p
Fu(,)f(and)f(v)o(ertex)g(set)f Fn(f)p Fu(0)p Fo(;)8 b
Fu(1)p Fn(g)930 1039 y Fm(n)951 1055 y Fu(,)23 b(so)f(that)1160
1037 y Fm(\025)p 1160 1044 20 2 v 1161 1071 a(d)1208
1055 y Fn(\024)1274 1037 y Fm(\017)p 1273 1044 17 2 v
1273 1071 a Fk(2)1316 1055 y Fu(and)g Fo(d)i Fu(=)g(p)q(oly)q(\(1)p
Fo(=\017)p Fu(\).)40 b(F)l(or)21 b(ev)o(ery)0 1124 y
Fo(i)16 b Fn(2)g Fu([)p Fo(d)p Fu(])144 1099 y Fk(def)149
1124 y Fu(=)21 b Fn(f)p Fu(1)p Fo(;)8 b Fu(2)p Fo(:::;)g(d)p
Fn(g)14 b Fu(and)j Fo(v)h Fn(2)f(f)p Fu(0)p Fo(;)8 b
Fu(1)p Fn(g)706 1108 y Fm(n)726 1124 y Fu(,)18 b(denote)f(b)o(y)h
Fo(g)993 1131 y Fm(i)1006 1124 y Fu(\()p Fo(v)r Fu(\))f(the)g(v)o
(ertex)g(reac)o(hed)h(b)o(y)f(mo)o(ving)g(along)h(the)f
Fo(i)1917 1108 y Fk(th)0 1187 y Fu(edge)f(of)e(the)i(v)o(ertex)e
Fo(v)r Fu(.)0 1311 y Fv(The)k(univ)o(ersal)g(hashing)h(family)l(.)45
b Fu(W)l(e)16 b(consider)h(a)e(univ)o(ersal)i(family)l(,)f(denoted)h
Fo(H)t Fu(,)d(of)i(hash)g(functions,)0 1373 y(eac)o(h)j(mapping)g
Fo(l)329 1348 y Fk(def)334 1373 y Fu(=)k(3)8 b(log)481
1384 y Fk(2)500 1373 y Fu(\(2)p Fo(=\017)p Fu(\)-bit)19
b(long)f(strings)h(to)f([)p Fo(d)p Fu(])f(\(where)i([)p
Fo(d)p Fu(])e(=)h Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1453
1356 y Fm(m)1483 1373 y Fu(,)19 b(for)f(some)g Fo(m)p
Fu(,)h(as)f Fo(d)g Fu(is)h(a)0 1435 y(p)q(o)o(w)o(er)e(of)f(2\).)25
b(Namely)l(,)18 b(a)f Fp(uniformly)h(chosen)f(function)g
Fo(h)f Fn(2)g Fo(H)k Fu(maps)d(eac)o(h)g(string)g Fo(\013)f
Fn(2)g(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1730 1419 y Fm(l)1758
1435 y Fu(uniformly)0 1497 y(in)o(to)15 b([)p Fo(d)p
Fu(])f(so)h(that)g(ev)o(ery)g(t)o(w)o(o)e(strings)j(are)e(mapp)q(ed)i
(in)g(an)g(indep)q(enden)o(t)h(manner.)0 1621 y Fv(Our)22
b(construction.)46 b Fu(W)l(e)20 b(no)o(w)e(de\014ne)j(the)e(functions)
h(in)h(our)e(family)l(,)i(denoted)e Fo(F)6 b Fu(.)33
b(F)l(or)19 b(eac)o(h)g(hashing)0 1684 y(function)d Fo(h)d
Fn(2)g Fo(H)t Fu(,)h(w)o(e)h(in)o(tro)q(duce)h(a)f(function)h
Fo(f)i Fn(2)13 b Fo(F)21 b Fu(de\014ned)c(b)o(y)797 1782
y Fo(f)5 b Fu(\()p Fo(v)r Fu(\))895 1757 y Fk(def)901
1782 y Fu(=)18 b Fo(g)976 1789 y Fm(h)p Fk(\(lsb)q(\()p
Fm(v)q Fk(\))1094 1782 y Fu(\()p Fo(v)r Fu(\))0 1881
y(where)f(lsb)q(\()p Fo(v)r Fu(\))e(returns)h(the)h Fo(l)g
Fu(least)f(signi\014can)o(t)i(bits)f(of)f Fo(v)g Fn(2)f(f)p
Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1197 1865 y Fm(n)1218 1881
y Fu(.)23 b(Namely)l(,)18 b Fo(f)5 b Fu(\()p Fo(v)r Fu(\))15
b(is)i(the)g(v)o(ertex)f(reac)o(hed)0 1943 y(from)f Fo(v)i
Fu(b)o(y)e(follo)o(wing)i(the)e Fo(i)497 1927 y Fk(th)546
1943 y Fu(edge)g(of)h Fo(v)r Fu(,)e(where)i Fo(i)f Fu(is)h(the)g(image)
f(of)h(the)f Fo(l)h Fu(least)g(signi\014can)o(t)g(bits)g(of)f
Fo(v)i Fu(under)0 2005 y(the)d(function)h Fo(h)p Fu(.)20
b(\(W)l(e)14 b(remark)f(that)g(our)h(c)o(hoice)h(of)f(using)h(the)f
Fo(l)g Fu(least)g(signi\014can)o(t)h(bits)g(is)g(arbitrary)e(and)h(an)o
(y)0 2068 y(other)h(e\016cien)o(t)h(partition)f(of)g
Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)641 2051 y Fm(n)677
2068 y Fu(in)o(to)15 b(2)792 2051 y Fm(l)820 2068 y Fu(parts,)f(of)h
(appro)o(ximately)g(the)g(same)g(size,)h(will)h(do.\))0
2194 y Fl(3.3)56 b(Analysis)0 2285 y Fu(The)19 b(tec)o(hnical)g(to)q
(ols)g(used)g(in)g(our)f(analysis)h(are)f(the)h(Expander)f(Mixing)i
(Lemma)e(\(Lemma)g(2.2\))f(and)i(the)0 2347 y(pairwise)d(indep)q
(endence)j(of)14 b(images)i(under)g(Univ)o(ersal)g(Hashing)g
(functions.)71 2409 y(Clearly)l(,)j(the)g(family)g Fo(F)25
b Fu(satis\014es)18 b(the)h(succinctness)h(and)e(e\016ciency)i
(requiremen)o(ts)f(\(of)e(Theorem)i(3.1\).)0 2471 y(W)l(e)d(no)o(w)f
(turn)g(to)g(pro)o(v)o(e)g(that)g(it)h(satis\014es)g(the)f(mixing)i
(prop)q(ert)o(y)l(.)k(T)l(o)o(w)o(ards)14 b(this)i(end)g(w)o(e)g(\014x)
f(t)o(w)o(o)g(arbitrary)0 2534 y(sets)g Fo(A)g Fu(and)h
Fo(B)r Fu(.)k(W)l(e)15 b(\014rst)g(observ)o(e)g(that)g(b)o(y)g(the)g
(Expander)h(Mixing)g(Lemma,)f(it)g(holds)h(that)411 2587
y Fd(\014)411 2612 y(\014)411 2637 y(\014)411 2662 y(\014)429
2617 y Fn(jf)p Fu(\()p Fo(x;)8 b(i)p Fu(\))j(:)h Fo(x)5
b Fn(2)g Fo(A)10 b Fn(^)h Fo(g)773 2624 y Fm(i)786 2617
y Fu(\()p Fo(x)p Fu(\))5 b Fn(2)g Fo(B)r Fn(gj)p 429
2638 531 2 v 644 2679 a Fo(d)10 b Fn(\001)f Fu(2)723
2666 y Fm(n)975 2648 y Fn(\000)i Fo(\032)p Fu(\()p Fo(A)p
Fu(\))p Fo(\032)p Fu(\()p Fo(B)r Fu(\))1211 2587 y Fd(\014)1211
2612 y(\014)1211 2637 y(\014)1211 2662 y(\014)1248 2648
y Fo(<)1314 2617 y(\025)p 1314 2638 27 2 v 1315 2679
a(d)1371 2648 y Fn(\024)1438 2617 y Fo(\017)p 1436 2638
23 2 v 1436 2679 a Fu(2)1892 2648 y(\(1\))964 2795 y(7)p
eop
%%Page: 8 9
8 8 bop 0 42 a Fu(Let)583 144 y Fo(e)604 151 y Fm(i)618
144 y Fu(\()p Fo(A;)8 b(B)r Fu(\))785 119 y Fk(def)791
144 y Fu(=)47 b(2)896 125 y Fi(\000)p Fm(n)954 144 y
Fn(\001)10 b(jf)p Fo(x)i Fn(2)5 b Fo(A)13 b Fu(:)f Fo(g)1180
151 y Fm(i)1194 144 y Fu(\()p Fo(x)p Fu(\))5 b Fn(2)g
Fo(B)r Fn(gj)524 b Fu(\(2\))0 246 y(Th)o(us,)15 b(Eq.)f(\(1\))h(can)g
(b)q(e)h(rewritten)f(as)594 305 y Fd(\014)594 330 y(\014)594
355 y(\014)594 380 y(\014)594 405 y(\014)614 348 y Fu(1)p
613 368 24 2 v 613 410 a Fo(d)652 379 y Fn(\001)696 326
y Fm(d)675 338 y Fd(X)678 427 y Fm(i)p Fk(=1)742 379
y Fo(e)763 386 y Fm(i)777 379 y Fu(\()p Fo(A;)8 b(B)r
Fu(\))i Fn(\000)g Fo(\032)p Fu(\()p Fo(A)p Fu(\))p Fo(\032)p
Fu(\()p Fo(B)r Fu(\))1149 305 y Fd(\014)1149 330 y(\014)1149
355 y(\014)1149 380 y(\014)1149 405 y(\014)1187 379 y
Fn(\024)1255 348 y Fo(\017)p 1252 368 23 2 v 1252 410
a Fu(2)1892 379 y(\(3\))0 534 y Fv(Ov)o(erview.)44 b
Fu(Eq.)16 b(\(3\))f(states)h(that)669 517 y Fk(1)p 669
524 18 2 v 669 550 a Fm(d)699 502 y Fd(P)742 546 y Fm(i)764
534 y Fo(e)785 541 y Fm(i)799 534 y Fu(\()p Fo(A;)8 b(B)r
Fu(\))16 b(is)h(a)f(go)q(o)q(d)g(appro)o(ximation)h(of)f
Fo(\032)p Fu(\()p Fo(A)p Fu(\))p Fo(\032)p Fu(\()p Fo(B)r
Fu(\).)22 b(If,)17 b(for)f(most)0 597 y Fo(i)h Fn(2)h
Fu([)p Fo(d)p Fu(],)f(eac)o(h)i Fo(e)288 604 y Fm(i)302
597 y Fu(\()p Fo(A;)8 b(B)r Fu(\))17 b(w)o(ere)h(a)g(go)q(o)q(d)g
(appro)o(ximation)g(to)f Fo(\032)p Fu(\()p Fo(A)p Fu(\))p
Fo(\032)p Fu(\()p Fo(B)r Fu(\))g(then)h(w)o(e)g(w)o(ould)h(b)q(e)g
(done.)29 b(But,)18 b(w)o(e)0 659 y(don't)e(kno)o(w)h(whether)f(this)i
(prop)q(ert)o(y)e(holds.)25 b(Instead,)18 b(w)o(e)e(partition)h
Fo(A)g Fu(in)o(to)g(a)f(small)i(n)o(um)o(b)q(er)f(of)f(subsets,)0
721 y(denoted)f Fo(A)205 728 y Fm(\013)229 721 y Fu('s,)f(asso)q(ciate)
g(a)h(random)f Fo(i)697 728 y Fm(\013)733 721 y Fn(2)f
Fu([)p Fo(d)p Fu(])g(with)i(eac)o(h)g(suc)o(h)g Fo(A)1180
728 y Fm(\013)1219 721 y Fu(and)f(consider)i(ho)o(w)e(w)o(ell)1669
689 y Fd(P)1713 732 y Fm(\013)1744 721 y Fo(e)1765 728
y Fm(i)1777 732 y Fh(\013)1800 721 y Fu(\()p Fo(A)1852
728 y Fm(\013)1875 721 y Fo(;)8 b(B)r Fu(\))0 783 y(appro)o(ximates)283
751 y Fd(P)326 794 y Fm(\013)358 783 y Fo(\032)p Fu(\()p
Fo(A)434 790 y Fm(\013)456 783 y Fu(\))p Fo(\032)p Fu(\()p
Fo(B)r Fu(\))15 b(=)h Fo(\032)p Fu(\()p Fo(A)p Fu(\))p
Fo(\032)p Fu(\()p Fo(B)r Fu(\).)24 b(W)l(e)17 b(sho)o(w)g(that)f(when)i
(the)f Fo(i)1375 790 y Fm(\013)1398 783 y Fu('s)g(are)f(uniformly)i
(distributed)0 845 y(in)e(a)g(pairwise)g(indep)q(enden)o(t)i(manner,)e
(as)f(is)h(the)g(case)f(when)h(setting)g Fo(i)1267 852
y Fm(\013)1303 845 y Fu(=)e Fo(h)p Fu(\()p Fo(\013)p
Fu(\))h(for)g(one)h(uniformly)h(c)o(hosen)0 907 y Fo(h)c
Fn(2)g Fo(H)t Fu(,)h(the)h(appro)o(ximation)g(is)h(go)q(o)q(d)f(with)h
(high)g(probabilit)o(y)l(.)71 992 y(Returning)h(to)f(the)g(formal)g
(pro)q(of,)f(w)o(e)h(consider)h(a)f(partition)h(of)f
Fo(A)g Fu(in)o(to)g Fo(L)1416 967 y Fk(def)1421 992 y
Fu(=)k(2)1499 976 y Fm(l)1528 992 y Fu(subsets)c(so)g(that)g
Fo(A)1877 999 y Fm(\013)1915 992 y Fu(=)0 1054 y Fn(f)p
Fo(x)h Fn(2)g Fo(A)g Fu(:)f(lsb)q(\()p Fo(x)p Fu(\))9
b(=)g Fo(\013)p Fn(g)p Fu(,)18 b(for)f(ev)o(ery)h Fo(\013)f
Fn(2)g(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)848 1038 y Fm(l)859
1054 y Fu(.)28 b(W)l(e)18 b(de\014ne)g Fo(L)g Fu(random)f(v)m
(ariables,)j Fo(\020)1557 1061 y Fk(0)1574 1053 y Fh(l)1587
1054 y Fo(;)8 b(:::;)g(\020)1688 1061 y Fk(1)1705 1053
y Fh(l)1716 1054 y Fu(,)18 b(so)f(that)g Fo(\020)1926
1061 y Fm(\013)0 1116 y Fu(represen)o(ts)h(the)g(densit)o(y)g(of)g(the)
g(set)g Fn(f)p Fo(x)e Fn(2)10 b Fo(A)806 1123 y Fm(\013)847
1116 y Fu(:)17 b Fo(g)899 1123 y Fm(h)p Fk(\()p Fm(\013)p
Fk(\))968 1116 y Fu(\()p Fo(x)p Fu(\))f Fn(2)i Fo(B)r
Fn(g)g Fu(\(in)g Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1357
1100 y Fm(n)1378 1116 y Fu(\).)28 b(Note)17 b(that)g(the)h
Fo(\020)1751 1123 y Fm(\013)1775 1116 y Fu('s)f(are)h(all)0
1179 y(determined)e(b)o(y)f(the)g(c)o(hoice)g(of)f Fo(h)p
Fu(,)h(and)g(th)o(us)g(the)f(probabilit)o(y)i(space)f(is)h(uniform)f(o)
o(v)o(er)f(all)h(c)o(hoices)h(of)e Fo(h)f Fn(2)g Fo(H)t
Fu(.)0 1241 y(Alternativ)o(ely)l(,)j(since)h(lsb\()p
Fo(x)p Fu(\))12 b(=)h Fo(\013)j Fu(for)e(ev)o(ery)h Fo(x)e
Fn(2)g Fo(A)923 1248 y Fm(\013)947 1241 y Fu(,)h(w)o(e)h(can)h(write)
597 1343 y Fo(\020)617 1350 y Fm(\013)653 1343 y Fu(=)d(2)724
1324 y Fi(\000)p Fm(n)782 1343 y Fn(\001)d(jf)p Fo(x)i
Fn(2)5 b Fo(A)948 1350 y Fm(\013)984 1343 y Fu(:)13 b
Fo(g)1032 1350 y Fm(h)p Fk(\(lsb)q(\()p Fm(x)p Fk(\)\))1164
1343 y Fu(\()p Fo(x)p Fu(\))f Fn(2)h Fo(B)r Fn(gj)0 1445
y Fu(Observ)o(e)k(that)f(the)h(set)g Fn(f)p Fo(x)e Fn(2)8
b Fo(A)15 b Fu(:)g Fo(g)630 1452 y Fm(h)p Fk(\(lsb)q(\()p
Fm(x)p Fk(\)\))763 1445 y Fu(\()p Fo(x)p Fu(\))g Fn(2)g
Fo(B)r Fn(g)j Fu(can)f(b)q(e)g(written)g(as)1339 1412
y Fm(:)1328 1445 y Fn([)1359 1452 y Fm(\013)1398 1445
y Fn(f)p Fo(x)e Fn(2)8 b Fo(A)1534 1452 y Fm(\013)1573
1445 y Fu(:)15 b Fo(g)1623 1452 y Fm(h)p Fk(\(lsb)q(\()p
Fm(x)p Fk(\)\))1756 1445 y Fu(\()p Fo(x)p Fu(\))f Fn(2)i
Fo(B)r Fn(g)p Fu(.)0 1507 y(Th)o(us,)131 1475 y Fd(P)175
1519 y Fm(\013)206 1507 y Fo(\020)226 1514 y Fm(\013)267
1507 y Fu(=)h(2)342 1491 y Fi(\000)p Fm(n)402 1507 y
Fn(\001)12 b(jf)p Fo(x)k Fn(2)10 b Fo(A)17 b Fu(:)f Fo(g)647
1514 y Fm(h)p Fk(\(lsb)q(\()p Fm(x)p Fk(\)\))780 1507
y Fu(\()p Fo(x)p Fu(\))g Fn(2)i Fo(B)r Fn(gj)g Fu(and)g(the)f(mixing)i
(prop)q(ert)o(y)f(\(to)f(b)q(e)h(pro)o(v)o(en\))f(can)h(b)q(e)0
1569 y(rephrased)e(as)e(asserting)509 1709 y(Prob)613
1624 y Fd(0)613 1699 y(@)650 1622 y(\014)650 1647 y(\014)650
1672 y(\014)650 1697 y(\014)650 1722 y(\014)650 1747
y(\014)700 1668 y(X)663 1760 y Fm(\013)p Fi(2f)p Fk(0)p
Fm(;)p Fk(1)p Fi(g)785 1752 y Fh(l)804 1709 y Fo(\020)824
1716 y Fm(\013)858 1709 y Fn(\000)c Fo(\032)p Fu(\()p
Fo(A)p Fu(\))f Fn(\001)h Fo(\032)p Fu(\()p Fo(B)r Fu(\))1125
1622 y Fd(\014)1125 1647 y(\014)1125 1672 y(\014)1125
1697 y(\014)1125 1722 y(\014)1125 1747 y(\014)1151 1709
y Fo(>)j(\017)1217 1624 y Fd(1)1217 1699 y(A)1279 1709
y Fn(\024)26 b Fo(\017)534 b Fu(\(4\))0 1850 y(where)16
b(the)g(probabilit)o(y)h(space)g(is)f(o)o(v)o(er)f(all)i(p)q(ossible)h
(c)o(hoices)f(of)e Fo(h)f Fn(2)h Fo(H)k Fu(\(or,)c(equiv)m(alen)o(tly)j
(of)e Fo(f)j Fn(2)14 b Fo(F)6 b Fu(\).)22 b(This)0 1912
y(claim)e(is)g(v)o(ery)f(app)q(ealing)i(since)g(eac)o(h)e
Fo(\020)731 1919 y Fm(\013)774 1912 y Fu(is)h(exp)q(ected)g(to)f(appro)
o(ximate)g Fo(\032)p Fu(\()p Fo(A)1420 1919 y Fm(\013)1443
1912 y Fu(\))12 b Fn(\001)h Fo(\032)p Fu(\()p Fo(B)r
Fu(\).)32 b(Indeed,)21 b(Eq.)e(\(4\))0 1974 y(follo)o(ws)c(b)o(y)g(com)
o(bining)i(the)e(t)o(w)o(o)f(items)h(of)g(the)g(next)h(lemma.)0
2080 y Fv(Lemma)h(3.2)23 b Fp(L)n(et)15 b(the)i Fo(\020)449
2087 y Fm(\013)472 2080 y Fp('s)f(b)n(e)g(as)g(ab)n(ove)g(and)g
Fo(I)883 2055 y Fk(def)888 2080 y Fu(=)i Fn(f)p Fu(0)p
Fo(;)8 b Fu(1)p Fn(g)1054 2063 y Fm(l)1065 2080 y Fp(.)21
b(Then)726 2136 y Fd(\014)726 2160 y(\014)726 2185 y(\014)726
2210 y(\014)726 2235 y(\014)740 2169 y(X)739 2258 y Fm(\013)p
Fi(2)p Fm(I)808 2209 y Fu(Exp\()p Fo(\020)926 2216 y
Fm(\013)949 2209 y Fu(\))10 b Fn(\000)g Fo(\032)p Fu(\()p
Fo(A)p Fu(\))s Fn(\001)s Fo(\032)p Fu(\()p Fo(B)r Fu(\))1231
2136 y Fd(\014)1231 2160 y(\014)1231 2185 y(\014)1231
2210 y(\014)1231 2235 y(\014)1284 2209 y Fo(<)1368 2178
y(\017)p 1366 2199 23 2 v 1366 2240 a Fu(2)1892 2209
y(\(5\))556 2354 y(Prob)661 2282 y Fd( )694 2281 y(\014)694
2306 y(\014)694 2331 y(\014)694 2356 y(\014)694 2380
y(\014)708 2314 y(X)707 2403 y Fm(\013)p Fi(2)p Fm(I)776
2354 y Fo(\020)796 2361 y Fm(\013)830 2354 y Fn(\000)875
2314 y Fd(X)875 2403 y Fm(\013)p Fi(2)p Fm(I)943 2354
y Fu(Exp)q(\()p Fo(\020)1062 2361 y Fm(\013)1085 2354
y Fu(\))1103 2281 y Fd(\014)1103 2306 y(\014)1103 2331
y(\014)1103 2356 y(\014)1103 2380 y(\014)1129 2354 y
Fo(>)1184 2324 y(\017)p 1182 2344 V 1182 2386 a Fu(2)1210
2282 y Fd(!)1284 2354 y Fo(<)42 b(\017)513 b Fu(\(6\))0
2491 y Fv(Pro)q(of:)22 b Fu(Using)16 b(the)h(fact)e(that)h
Fo(h)p Fu(\()p Fo(\013)p Fu(\))g(is)g(uniformly)i(distributed)f(on)f([)
p Fo(d)p Fu(])f(and)i(recalling)h(the)e(de\014nition)i(of)e
Fo(\020)1914 2498 y Fm(\013)1937 2491 y Fu(,)0 2553 y(w)o(e)f(ha)o(v)o
(e)570 2634 y(Exp\()p Fo(\020)688 2641 y Fm(\013)711
2634 y Fu(\))25 b(=)820 2604 y(1)p 820 2624 24 2 v 820
2666 a Fo(d)858 2634 y Fn(\001)902 2582 y Fm(d)881 2594
y Fd(X)884 2682 y Fm(i)p Fk(=1)954 2604 y Fn(jf)p Fo(x)11
b Fn(2)i Fo(A)1104 2611 y Fm(\013)1141 2604 y Fu(:)f
Fo(g)1188 2611 y Fm(i)1202 2604 y Fu(\()p Fo(x)p Fu(\))5
b Fn(2)g Fo(B)r Fn(gj)p 954 2624 422 2 v 1142 2666 a
Fu(2)1165 2653 y Fm(n)964 2795 y Fu(8)p eop
%%Page: 9 10
9 9 bop 0 42 a Fu(Using)16 b Fo(A)c Fu(=)220 8 y Fm(:)209
42 y Fn([)240 49 y Fm(\013)276 42 y Fo(A)310 49 y Fm(\013)349
42 y Fu(and)k(Eq.)e(\(2\),)g(w)o(e)h(ha)o(v)o(e)462 117
y Fd(X)462 206 y Fm(\013)p Fi(2)p Fm(I)530 158 y Fu(Exp)q(\()p
Fo(\020)649 165 y Fm(\013)672 158 y Fu(\))41 b(=)813
127 y(1)p 813 147 24 2 v 813 189 a Fo(d)852 158 y Fn(\001)896
105 y Fm(d)874 117 y Fd(X)877 206 y Fm(i)p Fk(=1)947
95 y Fd(P)991 138 y Fm(\013)p Fi(2)p Fm(I)1061 127 y
Fn(jf)p Fo(x)12 b Fn(2)h Fo(A)1212 134 y Fm(\013)1248
127 y Fu(:)g Fo(g)1296 134 y Fm(i)1309 127 y Fu(\()p
Fo(x)p Fu(\))5 b Fn(2)g Fo(B)r Fn(gj)p 947 147 537 2
v 1192 189 a Fu(2)1215 176 y Fm(n)731 306 y Fu(=)813
276 y(1)p 813 296 24 2 v 813 338 a Fo(d)852 306 y Fn(\001)896
254 y Fm(d)874 266 y Fd(X)877 354 y Fm(i)p Fk(=1)942
306 y Fo(e)963 313 y Fm(i)977 306 y Fu(\()p Fo(A;)j(B)r
Fu(\))0 420 y(Using)16 b(Eq.)e(\(3\),)g(w)o(e)h(establish)i(Eq.)d
(\(5\).)71 482 y(Next)20 b(w)o(e)f(use)i(Cheb)o(yshev)g(Inequalit)o(y)g
(to)f(pro)o(v)o(e)f(Eq.)h(\(6\):)29 b(here)20 b(w)o(e)g(use)h(the)f
(fact)f(that)h(the)g Fo(\020)1815 489 y Fm(\013)1839
482 y Fu('s)f(are)0 545 y(pairwise)d(indep)q(enden)o(t)i(\(since)e(the)
f Fo(h)p Fu(\()p Fo(\013)p Fu(\)'s)g(are)f(pairwise)j(indep)q(enden)o
(t\).)423 657 y(Prob)527 585 y Fd( )560 583 y(\014)560
608 y(\014)560 633 y(\014)560 658 y(\014)560 683 y(\014)574
617 y(X)574 706 y Fm(\013)p Fi(2)p Fm(I)642 657 y Fo(\020)662
664 y Fm(\013)696 657 y Fn(\000)742 617 y Fd(X)742 706
y Fm(\013)p Fi(2)p Fm(I)810 657 y Fu(Exp\()p Fo(\020)928
664 y Fm(\013)951 657 y Fu(\))969 583 y Fd(\014)969 608
y(\014)969 633 y(\014)969 658 y(\014)969 683 y(\014)996
657 y Fo(>)1051 626 y(\017)p 1048 647 23 2 v 1048 688
a Fu(2)1076 585 y Fd(!)1151 657 y Fo(<)1232 626 y Fu(V)l(ar\()1321
594 y Fd(P)1365 638 y Fm(\013)1396 626 y Fo(\020)1416
633 y Fm(\013)1440 626 y Fu(\))p 1232 647 225 2 v 1286
688 a(\()p Fo(\017=)p Fu(2\))1386 675 y Fk(2)1151 795
y Fn(\024)1232 764 y Fu(4)10 b Fn(\001)1288 732 y Fd(P)1332
775 y Fm(\013)1363 764 y Fu(Exp\()p Fo(\020)1484 747
y Fk(2)1481 775 y Fm(\013)1504 764 y Fu(\))p 1232 784
290 2 v 1359 826 a Fo(\017)1377 813 y Fk(2)0 890 y Fu(Using)129
858 y Fd(P)172 902 y Fm(\013)204 890 y Fu(Exp\()p Fo(\020)325
874 y Fk(2)322 901 y Fm(\013)345 890 y Fu(\))k Fn(\024)426
858 y Fd(P)470 902 y Fm(\013)501 890 y Fo(\032)p Fu(\()p
Fo(A)577 897 y Fm(\013)600 890 y Fu(\))618 874 y Fk(2)650
890 y Fn(\024)h Fo(L)10 b Fn(\001)h Fu(\()790 872 y Fk(1)p
787 879 23 2 v 787 906 a Fm(L)815 890 y Fu(\))833 874
y Fk(2)851 890 y Fu(,)16 b(and)g(the)h(de\014nition)g(of)f
Fo(L)g Fu(\(=)e(8)p Fo(=\017)1482 874 y Fk(3)1501 890
y Fu(\),)h(w)o(e)h(upp)q(er)h(b)q(ound)g(the)0 952 y(ab)q(o)o(v)o(e)e
(b)o(y)204 934 y Fk(4)p 197 941 31 2 v 197 968 a Fm(\017)211
960 y Fc(2)242 952 y Fn(\001)273 934 y Fk(1)p 270 941
23 2 v 270 968 a Fm(L)310 952 y Fu(=)e Fo(\017=)p Fu(2,)i(and)g(so)g
(Eq.)g(\(6\))f(follo)o(ws.)p 966 958 34 40 v 0 1086 a
Fl(3.4)56 b(Lo)n(w)n(er)19 b(Bound)0 1178 y Fv(Theorem)e(3.3)22
b Fp(A)17 b(family)f(with)h(mixing)e(pr)n(op)n(erty)i(of)g(ac)n(cur)n
(acy)f Fo(\017)p Fp(,)h(must)f(have)h(size)e(at)i(le)n(ast)1676
1130 y Fd(q)p 1717 1130 27 2 v 1722 1160 a Fk(4)p 1722
1167 17 2 v 1723 1193 a Fm(\017)1744 1178 y Fp(.)0 1264
y Fv(Pro)q(of:)22 b Fu(Otherwise,)17 b(let)g Fo(F)22
b Fu(=)15 b Fn(f)p Fo(f)606 1271 y Fm(i)634 1264 y Fu(:)f(1)7
b Fn(\024)g Fo(i)g Fn(\024)g Fo(t)p Fn(g)18 b Fu(b)q(e)f(a)f(family)h
(of)f(functions)h(o)o(v)o(er)f Fn(f)p Fu(0)p Fo(;)8 b
Fu(1)p Fn(g)1559 1247 y Fm(n)1580 1264 y Fu(,)16 b(con)o(tradicting)h
(the)0 1326 y(claim.)k(W)l(e)16 b(construct)f(a)g(graph)g(with)h(v)o
(ertex)f(set)g Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1011
1309 y Fm(n)1047 1326 y Fu(and)16 b(edges)f(set)g Fn(f)p
Fu(\()p Fo(x;)8 b(f)d Fu(\()p Fo(x)p Fu(\)\))11 b(:)i
Fo(x)5 b Fn(2)g(f)p Fu(0)p Fo(;)j Fu(1)p Fn(g)1739 1309
y Fm(n)1770 1326 y Fn(^)j Fo(f)f Fn(2)5 b Fo(F)h Fn(g)p
Fu(.)0 1388 y(Clearly)l(,)20 b(the)f(graph)f(has)h(an)g(indep)q(enden)o
(t)i(set)d(of)h(size)g Fo(N)q(=t)p Fu(,)g(where)g Fo(N)1321
1363 y Fk(def)1327 1388 y Fu(=)24 b(2)1409 1371 y Fm(n)1431
1388 y Fu(.)30 b(Consequen)o(tly)l(,)20 b(there)f(are)0
1450 y(t)o(w)o(o)13 b(sets,)h Fo(A)h Fu(and)g Fo(B)r
Fu(,)g(eac)o(h)f(of)g(cardinalit)o(y)i Fo(N)q(=)p Fu(2)p
Fo(t)p Fu(,)e(so)g(that)g(there)h(exists)g(no)f(function)i
Fo(f)h Fn(2)c Fo(F)21 b Fu(for)14 b(whic)o(h)i(b)q(oth)0
1512 y Fo(x)d Fn(2)g Fo(A)i Fu(and)g Fo(f)5 b Fu(\()p
Fo(x)p Fu(\))13 b Fn(2)g Fo(B)r Fu(.)20 b(On)c(the)f(other)g(hand,)h
Fo(\032)p Fu(\()p Fo(A)p Fu(\))9 b Fn(\001)h Fo(\032)p
Fu(\()p Fo(B)r Fu(\))i(=)h(\(1)p Fo(=)p Fu(2)p Fo(t)p
Fu(\))1236 1496 y Fk(2)1253 1512 y Fu(,)i(and)h(the)f(theorem)g(follo)o
(ws)g(\(ev)o(en)h(for)0 1574 y(the)f(sp)q(ecial)i(case)e(of)g(hitting)h
Fo(A)10 b Fn(\002)h Fo(B)18 b Fu(when)d Fo(\032)p Fu(\()p
Fo(A)p Fu(\))10 b Fn(\001)f Fo(\032)p Fu(\()p Fo(B)r
Fu(\))j Fn(\025)h Fo(\017)p Fu(\).)p 1184 1580 34 40
v 0 1708 a Fl(3.5)56 b(Application)18 b(to)g(Generalized)f(Random)h
(Logspace)0 1800 y Fu(In)g([27)o(],)f(Nisan)g(considered)i(the)e
(problem)h(of)f(sa)o(ving)g(randomness)g(in)h(a)f(con)o(text)f(in)i
(whic)o(h)g Fo(m)f Fu(randomized)0 1862 y(algorithms)11
b(are)h(executed)g(and)g(their)g(output)f(is)h(fed)g(to)f(an)g
Fo(s)p Fu(-space)i(b)q(ounded)g(mac)o(hine)f(whic)o(h)g(then)g(pro)q
(duces)0 1924 y(a)j(\014nal)h(Bo)q(olean)h(output.)j(\(Actually)l(,)c
(the)g(problem)g(is)g(not)f(a\013ected)h(if)g(the)f Fo(s)p
Fu(-space)h(mac)o(hine)h(is)f(allo)o(w)o(ed)g(to)0 1986
y(ha)o(v)o(e)g(output)g(of)g(length)h(b)q(ounded)h(b)o(y)f
Fo(O)q Fu(\()p Fo(s)p Fu(\).\))23 b(F)l(or)15 b(simplicit)o(y)l(,)k
(assume)d(that)g(eac)o(h)h(of)f(the)g(algorithms)h(uses)0
2048 y Fo(n)f Fu(coin)g(\015ips.)22 b(The)16 b(ob)o(vious)g(w)o(a)o(y)e
(of)i(running)g(the)g(en)o(tire)g(pro)q(cedure)h(requires)f
Fo(m)10 b Fn(\001)g Fo(n)16 b Fu(coin)g(\015ips.)22 b(In)17
b(case)e(w)o(e)0 2110 y(are)h(willing)i(to)d(tolerate)g(an)h
Fo(\017)g Fu(additiv)o(e)h(error/deviation)f(in)h(the)f(\014nal)g
(output,)g(more)f(randomness-e\016cien)o(t)0 2172 y(solutions)20
b(are)f(p)q(ossible.)33 b(In)20 b(particular,)g(Nisan)g(sho)o(w)o(ed)f
([27)o(])f(that)h(the)g(randomness)g(complexit)o(y)h(can)f(b)q(e)0
2235 y(decreased)d(to)655 2297 y Fo(O)q Fu(\(max)o Fn(f)p
Fo(n;)8 b(s)i Fu(+)g(log)q(\()p Fo(m=\017)p Fu(\))p Fn(g)f(\001)h
Fu(log)e Fo(m)p Fu(\))0 2375 y(Replacing)16 b(the)e(univ)o(ersal)h
(hash)f(functions)h(used)f(in)h([27)o(])e(b)o(y)h(our)g(family)g(of)g
(mixing)h(functions,)f(w)o(e)g(note)g(that)0 2437 y Fp(the)j(ab)n(ove)f
(pr)n(oblem)g(c)n(an)f(b)n(e)h(solve)n(d)g(with)g(r)n(andomness)g(c)n
(omplexity)685 2522 y Fo(n)10 b Fu(+)h Fo(O)q Fu(\(\()p
Fo(s)e Fu(+)i(log\()p Fo(m=\017)p Fu(\)\))f Fn(\001)f
Fu(log)g Fo(m)p Fu(\))0 2608 y(W)l(e)16 b(remark)f(that)h(in)g(man)o(y)
g(applications)h Fo(n)e Fn(\035)f Fo(s)d Fu(+)g(log\()p
Fo(m=\017)p Fu(\).)22 b(F)l(or)15 b(these)h(cases,)g(our)g(impro)o(v)o
(emen)o(t)f(yields)0 2670 y(a)g(logarithmic)h(reduction)g(in)g(the)f
(randomness)g(complexit)o(y)l(.)964 2795 y(9)p eop
%%Page: 10 11
10 10 bop 0 42 a Fv(Remark:)44 b Fu(Theorem)22 b(6.2)f(follo)o(ws)h(as)
g(a)g(sp)q(ecial)i(case)e(of)f(the)h(ab)q(o)o(v)o(e)g(\(alas)g(with)g
(a)g(more)g(complicated)0 104 y(construction\).)0 231
y Fl(3.6)56 b(A)19 b(Di\013eren)n(t)e(P)n(ersp)r(ectiv)n(e)0
323 y Fu(The)k(mixing)h(prop)q(ert)o(y)f(of)f(families)i(of)f
(functions)h(should)f(not)g(b)q(e)h(confused)f(with)g(the)g(mixing)h
(prop)q(ert)o(y)0 385 y(of)g(graphs.)40 b(Y)l(et,)23
b(the)g(t)o(w)o(o)d(are)i(related)h(as)e(w)o(e)h(shall)h(see)g(b)q(elo)
o(w.)41 b(W)l(e)22 b(sa)o(y)f(that)h(a)g(graph)f(has)h(a)g(go)q(o)q(d)0
447 y(mixing)15 b(prop)q(ert)o(y)e(if)h(for)e(ev)o(ery)i(t)o(w)o(o)e
(subsets)h(of)g(v)o(ertices)h(the)g(fraction)f(of)g(edges)h(connecting)
g(these)g(subsets)f(is)0 509 y(appro)o(ximately)i(equal)g(to)f(the)h
(pro)q(duct)g(of)f(the)h(densities)h(of)e(these)h(subsets.)20
b(Clearly)l(,)15 b(a)g(family)g(of)f(functions)0 571
y(o)o(v)o(er)h Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)211
555 y Fm(n)231 571 y Fu(,)15 b(with)h(go)q(o)q(d)g(mixing,)g(induces)h
(a)e(regular)h(m)o(ulti-graph)1226 555 y Fk(4)1261 571
y Fu(with)f(go)q(o)q(d)h(mixing.)22 b(The)15 b(con)o(v)o(erse)g(is)0
633 y(not)d(ob)o(vious.)20 b(Sp)q(eci\014cally)l(,)c(it)d(w)o(as)f(not)
g(ev)o(en)h(kno)o(wn)f(whether)h(the)g(edges)g(of)f(some)h(small)g
(degree)g(graph)g(with)0 695 y(go)q(o)q(d)i(mixing)h(prop)q(ert)o(y)f
(\(e.g.,)f(an)h(expander\))h(can)f(b)q(e)h(so)f(colored)h(that)e(they)i
(induce)h(a)d(family)i(of)f(functions)0 757 y(with)h(a)e(go)q(o)q(d)i
(mixing)g(prop)q(ert)o(y)l(.)71 819 y(Let)g(us)g(try)f(to)g(clarify)i
(the)f(nature)g(of)f(this)i(problem.)22 b(Consider)17
b(a)f Fo(d)p Fu(-degree)g(expander)g(with)g(v)o(ertex-set)0
882 y Fo(V)50 857 y Fk(def)55 882 y Fu(=)j Fn(f)p Fu(0)p
Fo(;)8 b Fu(1)p Fn(g)222 865 y Fm(n)243 882 y Fu(,)15
b(and)h(some)g Fo(d)p Fu(-coloring)g(of)f(its)h(edges.)22
b(F)l(or)15 b(ev)o(ery)g(t)o(w)o(o)g(sets)g(of)g(v)o(ertices,)h
Fo(A)g Fu(and)g Fo(B)r Fu(,)g(denote)g(b)o(y)0 944 y
Fo(E)34 951 y Fm(i)47 944 y Fu(\()p Fo(A;)8 b(B)r Fu(\))15
b(the)h(set)f(of)g(edges)h(of)f(color)h Fo(i)f Fu(that)f(connect)i(a)g
(v)o(ertex)f(in)h Fo(A)f Fu(to)g(a)h(v)o(ertex)f(in)h
Fo(B)r Fu(.)21 b(By)16 b(the)g(Expander)0 1006 y(Mixing)j(Lemma,)f(it)g
(follo)o(ws)g(that)f(the)h Fp(aver)n(age)g Fu(of)942
985 y Fi(j)p Fm(E)976 989 y Fh(i)989 985 y Fk(\()p Fm(A;B)q
Fk(\))p Fi(j)p 942 995 145 2 v 991 1021 a(j)p Fm(V)6
b Fi(j)1091 1006 y Fu(,)18 b(tak)o(en)g(o)o(v)o(er)f(all)i(1)9
b Fn(\024)h Fo(i)g Fn(\024)f Fo(d)p Fu(,)18 b(is)h(appro)o(ximately)6
1055 y Fi(j)p Fm(A)p Fi(j)p 5 1065 47 2 v 5 1091 a(j)p
Fm(V)7 b Fi(j)62 1076 y Fn(\001)86 1055 y Fi(j)p Fm(B)q
Fi(j)p 86 1065 V 86 1091 a(j)p Fm(V)f Fi(j)137 1076 y
Fu(.)19 b(The)14 b(question)f(is)h(whether)660 1055 y
Fi(j)p Fm(E)694 1059 y Fh(i)708 1055 y Fk(\()p Fm(A;B)q
Fk(\))p Fi(j)p 660 1065 145 2 v 709 1091 a(j)p Fm(V)7
b Fi(j)823 1076 y Fu(is)13 b(appro)o(ximately)1170 1055
y Fi(j)p Fm(A)p Fi(j)p 1169 1065 47 2 v 1169 1091 a(j)p
Fm(V)6 b Fi(j)1226 1076 y Fn(\001)1250 1055 y Fi(j)p
Fm(B)q Fi(j)p 1250 1065 V 1250 1091 a(j)p Fm(V)g Fi(j)1301
1076 y Fu(,)13 b Fp(for)i(almost)f(al)r(l)f Fu(1)5 b
Fn(\024)g Fo(i)g Fn(\024)g Fo(d)p Fu(.)19 b(One)14 b(can)0
1138 y(easily)g(v)o(erify)f(that,)g(in)h(general,)f(the)h(answ)o(er)e
(is)i(negativ)o(e.)19 b(Sp)q(eci\014cally)l(,)e(for)12
b(Ca)o(yley)h(Graph)g(expanders)h(\(e.g.,)0 1200 y([25)o(,)i(5)o(,)g
(16)o(,)g(24)o(]\),)g(there)g(are)f(sets)h Fo(A)g Fu(and)g
Fo(B)j Fu(for)c(whic)o(h)i Fp(ther)n(e)g(exist)g(no)g
Fo(i)e Fu(suc)o(h)i(that)1519 1179 y Fi(j)p Fm(E)1553
1183 y Fh(i)1566 1179 y Fk(\()p Fm(A;B)q Fk(\))p Fi(j)p
1519 1189 145 2 v 1568 1216 a(j)p Fm(V)7 b Fi(j)1684
1200 y Fu(appro)o(ximates)6 1249 y Fi(j)p Fm(A)p Fi(j)p
5 1259 47 2 v 5 1286 a(j)p Fm(V)g Fi(j)64 1270 y Fn(\001)90
1249 y Fi(j)p Fm(B)q Fi(j)p 90 1259 V 90 1286 a(j)p Fm(V)f
Fi(j)155 1270 y Fu(\(e.g.,)13 b(consider)i(the)f(cosets)g(obtained)h(b)
o(y)f(omitting)g(one)g(generator\).)k(The)d(problem)f(raised)h(b)o(y)f
(Nati)0 1332 y(Linial)k(w)o(as)c(to)h(construct)g(an)h(expander)g(for)f
(whic)o(h)h(the)f(mixing)i(prop)q(ert)o(y)e(holds)h(for)f(most)g
(colors)g(\(and)g(not)0 1394 y(only)h(on)f(the)g(a)o(v)o(erage\).)71
1456 y(W)l(e)h(resolv)o(e)g(the)h(ab)q(o)o(v)o(e)f(problem)g(b)o(y)h
(presen)o(ting)g(a)e(transformation)g(whic)o(h)j(tak)o(es)d(an)h
(arbitrary)g(\(edge-)0 1518 y(colored\))g(expander)h(and)f(pro)q(duces)
h(an)f(\(edge-colored\))g(expanders)h(for)e(whic)o(h)i(the)f(mixing)i
(prop)q(ert)o(y)d(holds)0 1580 y(for)e(most)f(colors)i(\(as)e(required)
j(ab)q(o)o(v)o(e\).)j(Our)c(transformation)e(preserv)o(es)h(the)h(v)o
(ertex)f(set)g(and)h(the)f(expansion)0 1643 y(prop)q(erties)18
b(of)g(the)g(original)g(expander,)h(but)f(increases)g(the)g(degree)g(b)
o(y)g(a)g(p)q(olynomial)h(factor)e(\(i.e.,)g(from)g Fo(d)0
1705 y Fu(to)e(p)q(oly)q(\()p Fo(d)p Fu(\)\).)k(Although)c(the)h
(transformation)d(is)j(not)f(explicitly)j(presen)o(ted)d(in)h(this)g
(pap)q(er,)f(it)h(can)f(b)q(e)h(easily)0 1767 y(deriv)o(ed)g(from)f
(the)g(description)i(ab)q(o)o(v)o(e.)0 1917 y Fq(4)67
b(Tin)n(y)24 b(F)-6 b(amilies)24 b(Extracting)g(High)f(Min-en)n(trop)n
(y)0 2025 y Fu(Recall)18 b(that)e(the)h Fp(extr)n(action)f
Fu(prop)q(ert)o(y)l(,)h(for)f(a)g(family)h(of)f(functions)i(eac)o(h)e
(mapping)h Fo(n)p Fu(-bit)h(strings)e(to)g Fo(m)p Fu(-bit)0
2087 y(strings,)e(means)g(that)g(eac)o(h)g(subset)h(of)e
Fo(K)f Fn(\001)7 b Fu(2)779 2070 y Fm(m)825 2087 y Fu(strings)14
b(in)h Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1136 2070 y
Fm(n)1171 2087 y Fu(is)15 b(mapp)q(ed)g(almost)f(uniformly)h(to)f
Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1908 2070 y Fm(m)1937
2087 y Fu(,)0 2149 y(b)o(y)17 b(all)i(but)e(a)g(small)h(fraction)f(of)g
(the)h(functions)g(in)g(the)f(family)l(.)28 b(W)l(e)17
b(consider)h(the)g(extraction)f(problem)h(in)0 2211 y(t)o(w)o(o)g(sp)q
(ecial)k(cases:)29 b(the)19 b(case)h(where)g Fo(m)g Fu(is)g(v)o(ery)f
(small)i(\(in)f(the)g(next)f(section\))h(and)g(the)g(case)g
Fo(m)f Fu(is)h(v)o(ery)0 2273 y(close)g(to)e Fo(n)i Fu(\(in)f(this)h
(section\).)32 b(Actually)l(,)21 b(w)o(e)e(consider)h(a)e
(generalization)j(of)e(the)g(extraction)g(problem)h(to)0
2335 y(random)g(v)m(ariables)h(with)f(an)g(upp)q(er)h(b)q(ound,)h(of)
930 2317 y Fk(1)p 896 2324 84 2 v 896 2351 a Fm(K)r Fi(\001)p
Fk(2)953 2342 y Fh(m)985 2335 y Fu(,)f(on)f(the)g(probabilit)o(y)h
(function.)35 b(Suc)o(h)21 b(a)f(b)q(ound)g(is)0 2397
y(called)d Fp(min-entr)n(opy)e Fu(\(cf.,)f(Chor)h(and)g(Goldreic)o(h)h
([10)o(]\).)0 2509 y Fv(De\014nition)j(4.1)k Fu(\(min-en)o(trop)o(y\):)
e Fp(L)n(et)c Fo(X)j Fp(b)n(e)d(a)g(r)n(andom)g(variable.)23
b(We)18 b(say)f(that)g Fo(X)k Fp(has)c Fe(min-entrop)o(y)e
Fo(k)j Fp(if)0 2571 y Fu(Prob\()p Fo(X)8 b Fu(=)d Fo(x)p
Fu(\))12 b Fn(\024)h Fu(2)328 2555 y Fi(\000)p Fm(k)374
2571 y Fp(,)k(for)f(e)n(ach)g Fo(x)p Fp(.)p 0 2613 780
2 v 52 2643 a Fj(4)69 2659 y Fs(A)d(m)o(ulti-graph)i(is)f(a)f(graph)g
(in)h(whic)o(h)g(parallel)i(edges)d(are)g(allo)o(w)o(ed.)952
2795 y Fu(10)p eop
%%Page: 11 12
11 11 bop 71 42 a Fu(Here)15 b(w)o(e)h(treat)e(the)i(case)f(of)g
(random)h(v)m(ariables)g(with)g(min-en)o(trop)o(y)g Fo(n)11
b Fn(\000)f Fo(k)17 b Fu(with)f Fo(k)e Fn(\034)g Fo(n)p
Fu(.)21 b(W)l(e)15 b(construct)0 104 y(a)h(family)i(of)e(p)q(oly)q(\(2)
360 87 y Fm(k)380 104 y Fo(=\017)p Fu(\))g(functions)i(mapping)f
Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)954 87 y Fm(n)991
104 y Fu(to)16 b Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1161
87 y Fm(m)1191 104 y Fu(,)16 b(where)h Fo(m)e Fu(=)g
Fo(n)c Fn(\000)h Fo(O)q Fu(\()p Fo(k)q Fu(\).)23 b(F)l(or)16
b(eac)o(h)h(suc)o(h)0 166 y(random)f(v)m(ariable,)h(all)g(but)f(a)f
Fo(\017)i Fu(fraction)f(of)f(the)h(functions,)h(when)f(applied)i(to)e
(it,)g(yield)h(a)f(random)g(v)m(ariable)0 228 y(whic)o(h)f(is)f
Fo(\017)p Fu(-close)i(to)d(uniform)h(\(in)h(norm-1\).)j(Lo)q(osely)d
(sp)q(eaking,)g(this)f(means)g(that)g(these)g(functions)h(are)e(able)0
290 y(to)h(\\smo)q(othen")h(almost)f(the)h(en)o(tire)h(min-en)o(trop)o
(y;)f(sp)q(eci\014cally)l(,)i(min-en)o(trop)o(y)f Fo(n)10
b Fn(\000)g Fo(k)16 b Fu(is)f(mapp)q(ed)h(to)e(almost)0
352 y(uniform)i(distribution)g(o)o(v)o(er)f(the)g(strings)g(of)g
(length)h Fo(n)10 b Fn(\000)h Fo(O)q Fu(\()p Fo(k)q Fu(\).)71
414 y(In)18 b(a)f(t)o(ypical)i(use)f(of)f(this)h(extraction,)g(most)f
(notably)h(the)g(applications)h(of)e(the)h(lefto)o(v)o(er)f(hash)h
(lemma,)0 476 y Fo(\017)d Fu(=)f(2)105 460 y Fi(\000)p
Fk(\012\()p Fm(k)q Fk(\))201 476 y Fu(.)23 b(In)17 b(these)f(cases)g
(the)g(size)h(of)f(our)g(family)h(is)g(p)q(oly)q(\(1)p
Fo(=\017)p Fu(\))f(whic)o(h)h(is)f(optimal)h(b)o(y)f(the)g(lo)o(w)o(er)
g(b)q(ound)0 538 y(b)q(elo)o(w.)0 666 y Fl(4.1)56 b(Main)18
b(Result)0 757 y Fv(Theorem)f(4.2)22 b Fu(\(Extractors)c(for)h(High)h
(Min-En)o(trop)o(y\):)28 b Fp(L)n(et)19 b Fo(k)i(<)f(n)g
Fp(and)g Fo(m)g(<)g(n)13 b Fn(\000)g Fo(k)22 b Fp(b)n(e)d(inte)n(gers,)
h(and)0 819 y Fo(\017)d(>)f Fu(max)p Fn(f)p Fu(2)217
803 y Fi(\000)p Fk(\()p Fm(m)p Fi(\000)p Fm(O)q Fk(\()p
Fm(k)q Fk(\)\))p Fm(=O)q Fk(\(1\))482 819 y Fo(;)8 b
Fu(2)526 803 y Fi(\000)p Fk(\()p Fm(n)p Fi(\000)p Fm(m)p
Fi(\000)p Fm(O)q Fk(\()p Fm(k)q Fk(\)\))p Fm(=O)q Fk(\(1\))837
819 y Fn(g)p Fp(.)27 b Fu(\(In)17 b(particular,)h Fo(m)f(<)f(n)c
Fn(\000)g Fo(O)q Fu(\()p Fo(k)q Fu(\).\))26 b Fp(Then,)18
b(ther)n(e)g(exists)g(a)0 882 y(family)e(of)h(functions,)e(e)n(ach)h
(mapping)h Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)802 865
y Fm(n)839 882 y Fp(to)16 b Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p
Fn(g)1006 865 y Fm(m)1036 882 y Fp(,)16 b(satisfying)f(the)i(fol)r
(lowing)f(pr)n(op)n(erties.)68 981 y Fn(\017)23 b Fu(succinctness:)f
Fp(the)17 b(family)g(c)n(ontains)e(a)i(p)n(olynomial)f(in)1113
963 y Fk(2)1130 951 y Fh(k)p 1113 970 35 2 v 1123 997
a Fm(\017)1169 981 y Fp(numb)n(er)g(of)h(functions,)e(and)i(e)n(ach)f
(function)114 1043 y(is)f(r)n(epr)n(esente)n(d)g(by)i(a)f(unique)g
(string)g(of)g(length)g Fo(l)q Fu(\()p Fo(k)q(;)8 b(\017)p
Fu(\))j(=)i Fo(O)q Fu(\()p Fo(k)e Fu(+)f(log)1341 1025
y Fk(1)p 1341 1032 17 2 v 1342 1059 a Fm(\017)1362 1043
y Fu(\))p Fp(.)68 1143 y Fn(\017)23 b Fu(e\016cien)o(t)11
b(ev)m(aluation:)19 b Fp(Ther)n(e)12 b(exists)f(a)h(lo)n(gsp)n(ac)n(e)f
(algorithm)h(that,)h(on)f(input)g(a)g(description)g(of)g(a)g(function)
114 1205 y Fo(f)21 b Fp(and)16 b(a)h(string)e Fo(x)p
Fp(,)i(r)n(eturns)e Fo(f)5 b Fu(\()p Fo(x)p Fu(\))p Fp(.)68
1304 y Fn(\017)23 b Fu(extraction)14 b(prop)q(ert)o(y:)20
b Fp(F)m(or)15 b(every)h(r)n(andom)g(variable)f Fo(X)h
Fn(2)d(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1281 1288 y Fm(n)1317
1304 y Fp(of)16 b(min-entr)n(opy)g Fo(n)9 b Fn(\000)g
Fo(k)q Fp(,)16 b(al)r(l)f(but)h(an)114 1366 y Fo(\017)g
Fp(fr)n(action)g(of)h(the)f(functions)g Fo(f)21 b Fp(in)16
b(the)g(family)h(satisfy)659 1454 y Fu(1)p 659 1474 23
2 v 659 1516 a(2)697 1485 y Fn(\001)764 1444 y Fd(X)719
1535 y Fm(\013)p Fi(2f)p Fk(0)p Fm(;)p Fk(1)p Fi(g)841
1527 y Fh(m)875 1485 y Fn(j)p Fu(Prob\()p Fo(f)5 b Fu(\()p
Fo(X)t Fu(\))g(=)g Fo(\013)p Fu(\))k Fn(\000)1275 1454
y Fu(1)p 1259 1474 55 2 v 1259 1516 a(2)1282 1503 y Fm(m)1318
1485 y Fn(j)j(\024)h Fo(\017)0 1649 y Fu(That)f(is,)h(w)o(e)f(ma)o(y)g
(extract)g Fo(m)g Fu(=)h Fo(n)5 b Fn(\000)g Fo(O)q Fu(\()p
Fo(k)q Fu(\))g Fn(\000)g Fo(O)q Fu(\(log\(1)p Fo(=\017)p
Fu(\)\))12 b(bits)h(from)e(min-en)o(trop)o(y)i Fo(n)5
b Fn(\000)g Fo(k)q Fu(,)13 b(and)g(the)f(extracted)0
1712 y(distribution)20 b(is)e Fo(\017)p Fu(-close)h(to)e(uniform.)29
b(Using)18 b(Raman)o(ujan)g(Graphs)f(as)h(expanders)g(and)g(T)l(o)q
(eplitz)i(matrices)0 1774 y(as)14 b(Univ)o(ersal)i(Hashing,)f(in)h(the)
e(construction)h(b)q(elo)o(w,)g(one)g(ma)o(y)f(obtain)h
Fo(l)q Fu(\()p Fo(k)q(;)8 b(\017)p Fu(\))j(=)i(4)p Fo(k)d
Fu(+)g(20)e(log)1684 1785 y Fk(2)1703 1774 y Fu(\(1)p
Fo(=\017)p Fu(\))g(+)i Fo(O)q Fu(\(1\))0 1836 y(and)15
b Fo(m)e Fu(=)g Fo(n)d Fn(\000)h Fu(2)p Fo(k)g Fn(\000)f
Fu(12)e(log)487 1847 y Fk(2)505 1836 y Fu(\(1)p Fo(=\017)p
Fu(\))i Fn(\000)g Fo(O)q Fu(\(1\).)0 1963 y Fl(4.2)56
b(The)18 b(construction)0 2055 y Fu(Again,)h(w)o(e)f(use)h(univ)o
(ersal)g(hashing)g(functions)g(and)g(expander)f(graphs.)29
b(This)19 b(time)g(w)o(e)f(use)g(an)h(e\016cien)o(tly)0
2117 y(constructible)f(expander)g(graph,)f Fo(G)p Fu(,)f(of)h(degree)g
Fo(d)g Fu(\(p)q(o)o(w)o(er)f(of)g(t)o(w)o(o\),)g(second)h(eigen)o(v)m
(alue)i Fo(\025)p Fu(,)e(and)g(v)o(ertex)f(set)0 2179
y Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)113 2162 y Fm(m)143
2179 y Fu(,)19 b(so)f(that)341 2161 y Fm(\025)p 341 2168
20 2 v 342 2195 a(d)384 2179 y Fn(\024)473 2161 y Fm(\017)487
2149 y Fc(2)p 443 2168 91 2 v 443 2195 a Fk(4)p Fi(\001)p
Fk(2)487 2186 y Fh(k=)p Fc(2)557 2179 y Fu(\(and)h Fo(d)f
Fu(=)h(p)q(oly)q(\(2)890 2162 y Fm(k)910 2179 y Fo(=\017)p
Fu(\)\).)30 b(As)19 b(b)q(efore,)g(for)f(ev)o(ery)h Fo(i)f
Fn(2)h Fu([)p Fo(d)p Fu(])1601 2154 y Fk(def)1606 2179
y Fu(=)24 b Fn(f)p Fu(1)p Fo(;)8 b Fu(2)p Fo(:::;)g(d)o
Fn(g)16 b Fu(and)0 2241 y Fo(v)f Fn(2)e(f)p Fu(0)p Fo(;)8
b Fu(1)p Fn(g)193 2225 y Fm(m)223 2241 y Fu(,)15 b(denote)h(b)o(y)f
Fo(g)483 2248 y Fm(i)497 2241 y Fu(\()p Fo(v)r Fu(\))f(the)i(v)o(ertex)
f(reac)o(hed)h(b)o(y)f(mo)o(ving)h(along)f(the)h Fo(i)1393
2225 y Fk(th)1442 2241 y Fu(edge)f(of)h(the)f(v)o(ertex)g
Fo(v)r Fu(.)21 b(The)0 2303 y(univ)o(ersal)16 b(family)l(,)f(denoted)h
Fo(H)t Fu(,)d(con)o(tains)i(hash)g(functions)h(eac)o(h)e(mapping)i(\()p
Fo(n)9 b Fn(\000)h Fo(m)p Fu(\)-bit)15 b(long)g(strings)f(to)g([)p
Fo(d)p Fu(].)0 2429 y Fv(Our)22 b(construction.)46 b
Fu(W)l(e)20 b(no)o(w)e(de\014ne)j(the)e(functions)h(in)h(our)e(family)l
(,)i(denoted)e Fo(F)6 b Fu(.)33 b(F)l(or)19 b(eac)o(h)g(hashing)0
2491 y(function)d Fo(h)d Fn(2)g Fo(H)t Fu(,)h(w)o(e)h(in)o(tro)q(duce)h
(a)f(function)h Fo(f)i Fn(2)13 b Fo(F)21 b Fu(de\014ned)c(b)o(y)729
2599 y Fo(f)5 b Fu(\()p Fo(x)p Fu(\))830 2574 y Fk(def)835
2599 y Fu(=)18 b Fo(g)910 2606 y Fm(h)p Fk(\(lsb)q(\()p
Fm(x)p Fk(\)\))1043 2599 y Fu(\(msb\()p Fo(x)p Fu(\)\))952
2795 y(11)p eop
%%Page: 12 13
12 12 bop 0 42 a Fu(where)14 b(lsb)q(\()p Fo(x)p Fu(\))f(returns)h(the)
g Fo(n)7 b Fn(\000)g Fo(m)15 b Fu(least)f(signi\014can)o(t)h(bits)f(of)
f Fo(x)g Fn(2)g(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1277
25 y Fm(n)1298 42 y Fu(,)13 b(and)h(msb\()p Fo(x)p Fu(\))f(returns)h
(the)g Fo(m)g Fu(most)0 104 y(signi\014can)o(t)j(bits)f(of)f
Fo(x)p Fu(.)22 b(Namely)l(,)16 b Fo(f)5 b Fu(\()p Fo(x)p
Fu(\))16 b(is)g(the)g(v)o(ertex)f(reac)o(hed)h(from)f(the)h(v)o(ertex)g
Fo(v)1492 79 y Fk(def)1497 104 y Fu(=)j(msb\()p Fo(x)p
Fu(\))c(b)o(y)h(follo)o(wing)0 166 y(the)f Fo(i)94 149
y Fk(th)141 166 y Fu(edge)g(of)g Fo(v)r Fu(,)f(where)h
Fo(i)f Fu(is)h(the)g(image)g(of)f(the)h Fo(n)9 b Fn(\000)h
Fo(m)k Fu(least)h(signi\014can)o(t)h(bits)f(of)f Fo(x)h
Fu(under)g(the)g(function)g Fo(h)p Fu(.)0 228 y(\(Again,)g(our)g(c)o
(hoice)h(of)f(using)h(the)f Fo(n)10 b Fn(\000)h Fo(m)k
Fu(least)g(signi\014can)o(t)h(bits)g(is)g(arbitrary)l(.\))71
290 y(W)l(e)e(remark)g(that)g(one)h(ma)o(y)f(use)h(an)o(y)f(family)i
(of)e(extractors)f(with)i(the)g(appropriate)g(parameters)e(instead)0
352 y(of)g(the)g(univ)o(ersal)h(family)g Fo(H)i Fu(used)e(ab)q(o)o(v)o
(e.)19 b(In)13 b(fact,)g(in)h(preliminary)g(v)o(ersions)g(of)e(this)i
(w)o(ork)e(w)o(e)h(ha)o(v)o(e)f(used)i(the)0 414 y(extractors)i(of)g
([29])g(in)i(order)f(to)f(deriv)o(e)i(alternativ)o(e)g(constructions)f
(with)g(size)h Fo(k)1454 398 y Fm(O)q Fk(\(log)q(\(1)p
Fm(=\017)p Fk(\))1612 414 y Fu(.)26 b(Ho)o(w)o(ev)o(er,)16
b(these)0 476 y(alternativ)o(e)f(constructions)h(are)f(subsumed)h(b)o
(y)f(Zuc)o(k)o(erman's)f(recen)o(t)h(w)o(ork)g([37)o(].)0
604 y Fl(4.3)56 b(Analysis)0 695 y Fu(Despite)13 b(the)g(apparen)o(t)g
(similarit)o(y)h(to)e(the)h(construction)h(for)e(mixing,)i(the)f
(analysis)h(of)e(the)h(curren)o(t)g(construc-)0 757 y(tion)j(is)h
(completely)g(di\013eren)o(t.)23 b(In)16 b(particular,)h(it)f(is)h
(based)f(on)g(\\stronger")f(tec)o(hnical)i(to)q(ols:)k(the)c(Expander)0
819 y(Smo)q(othing)f(Lemma)f(and)g(the)g(Lefto)o(v)o(er)g(Hash)g
(Lemma.)71 882 y(Clearly)l(,)20 b(the)f(family)h Fo(F)25
b Fu(satis\014es)19 b(the)g(succinctness)h(and)g(e\016ciency)g
(requiremen)o(ts.)31 b(W)l(e)19 b(no)o(w)g(turn)g(to)0
944 y(pro)o(v)o(e)14 b(that)g(it)g(satis\014es)h(the)g(extraction)f
(prop)q(ert)o(y)l(.)19 b(W)l(e)c(\014x)g(an)f(arbitrary)g(random)g(v)m
(ariable)i Fo(X)g Fn(2)c(f)p Fu(0)p Fo(;)c Fu(1)p Fn(g)1865
927 y Fm(n)1886 944 y Fu(,)14 b(of)0 1006 y(min-en)o(trop)o(y)g
Fo(n)7 b Fn(\000)g Fo(k)q Fu(,)15 b(and)f(consider)h(the)f
(distribution)h(\()p Fo(f)r(;)8 b(f)d Fu(\()p Fo(X)t
Fu(\)\),)12 b(when)i Fo(f)19 b Fu(is)14 b(randomly)g(c)o(hosen)g(in)h
Fo(F)6 b Fu(.)20 b(Once)0 1068 y(w)o(e)c(b)q(ound)h(the)g(statistical)f
(di\013erence)i(b)q(et)o(w)o(een)e(\()p Fo(f)r(;)8 b(f)d
Fu(\()p Fo(X)t Fu(\)\))14 b(and)j(\()p Fo(f)r(;)8 b(U)1262
1075 y Fm(m)1293 1068 y Fu(\))16 b(b)o(y)g Fo(\017)1409
1051 y Fk(2)1428 1068 y Fu(,)g(where)h Fo(U)1621 1075
y Fm(m)1668 1068 y Fu(is)g(the)g(uniform)0 1130 y(distribution)g(o)o(v)
o(er)d Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)458 1113 y
Fm(m)488 1130 y Fu(,)15 b(the)g(theorem)g(follo)o(ws)g(\(b)o(y)g(a)g
(coun)o(ting)g(argumen)o(t\).)0 1242 y Fv(Lemma)i(4.3)23
b Fp(L)n(et)16 b Fo(X)j Fp(and)e Fo(f)h Fn(2)c Fo(F)23
b Fp(b)n(e)16 b(as)h(ab)n(ove.)k(Then,)16 b(the)h(statistic)n(al)f
(di\013er)n(enc)n(e)f(b)n(etwe)n(en)g Fu(\()p Fo(f)r(;)8
b(f)d Fu(\()p Fo(X)t Fu(\)\))15 b Fp(and)0 1304 y Fu(\()p
Fo(f)r(;)8 b(U)94 1311 y Fm(m)125 1304 y Fu(\))16 b Fp(is)g(b)n(ounde)n
(d)g(ab)n(ove)g(by)g Fo(\017)581 1287 y Fk(2)600 1304
y Fp(.)0 1416 y Fv(Pro)q(of:)25 b Fu(Let)18 b Fo(Z)j
Fu(b)q(e)d(a)f(random)h(v)m(ariable)h(represen)o(ting)f(the)g
(distribution)h(on)f(the)f Fo(m)h Fu(most)f(signi\014can)o(t)i(bits)0
1478 y(of)i Fo(X)t Fu(;)i(i.e.,)f Fo(Z)k Fu(=)d(msb\()p
Fo(X)t Fu(\).)36 b(F)l(or)21 b(eac)o(h)g Fo(z)k Fn(2)e(f)p
Fu(0)p Fo(;)8 b Fu(1)p Fn(g)960 1461 y Fm(m)989 1478
y Fu(,)23 b(let)e Fo(Y)1122 1485 y Fm(z)1163 1478 y Fu(b)q(e)h(a)f
(random)f(v)m(ariable)j(represen)o(ting)f(the)0 1540
y(distribution)16 b(on)d(lsb)q(\()p Fo(X)t Fu(\))f(conditioned)k(on)e
Fo(Z)i Fu(=)d Fo(z)r Fu(;)h(i.e.,)g Fo(X)j Fu(is)d(the)g(concatenation)
g(of)f Fo(Z)k Fu(and)e Fo(Y)1667 1547 y Fm(Z)1694 1540
y Fu(.)k(W)l(e)14 b(call)h Fp(b)n(ad)0 1602 y Fu(those)h
Fo(z)r Fu('s)g(in)g Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)356
1586 y Fm(m)402 1602 y Fu(for)16 b(whic)o(h)g Fo(Y)629
1609 y Fm(z)665 1602 y Fu(has)g(`to)q(o)f(small')h(min-en)o(trop)o(y)l
(.)23 b(Namely)l(,)17 b(for)e Fo(\016)h(>)f Fu(0)g(to)h(b)q(e)h
(\014xed)f(later,)0 1664 y(let)g(the)f(set)g(of)g(bad)g(pre\014xes)h(b)
q(e)g(denoted)f(b)o(y)536 1772 y Fo(B)570 1779 y Fm(\016)601
1747 y Fk(def)607 1772 y Fu(=)j Fn(f)p Fo(z)7 b Fn(2)e(f)p
Fu(0)p Fo(;)j Fu(1)p Fn(g)859 1753 y Fm(m)901 1772 y
Fu(:)k Fn(9)p Fo(y)j Fu(s.t.)d(Prob)o(\()p Fo(Y)1202
1779 y Fm(z)1227 1772 y Fu(=)5 b Fo(y)r Fu(\))12 b Fo(>)h(\016)r
Fn(g)0 1880 y Fu(The)i(reader)h(can)f(easily)h(v)o(erify)l(,)f(using)h
(the)g(min-en)o(trop)o(y)f(b)q(ound)h(on)f Fo(X)t Fu(,)f(that)725
2007 y(Prob)o(\()p Fo(Z)8 b Fn(2)d Fo(B)947 2014 y Fm(\016)966
2007 y Fu(\))13 b Fo(<)1050 1977 y Fu(2)1073 1960 y Fm(m)p
Fi(\000)p Fk(\()p Fm(n)p Fi(\000)p Fm(k)q Fk(\))p 1050
1997 171 2 v 1124 2039 a Fo(\016)1892 2007 y Fu(\(7\))0
2123 y(\(As)e(otherwise,)h(there)g(exists)f Fo(z)k Fn(2)e
Fo(B)640 2130 y Fm(\016)670 2123 y Fu(\(the)e(most)g(probable)h
Fo(z)j Fn(2)e Fo(B)1166 2130 y Fm(\016)1185 2123 y Fu(\))e(and)g
Fo(y)k Fn(2)e(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1491 2106
y Fm(n)p Fi(\000)p Fm(m)1578 2123 y Fu(\(the)k(most)e(probable)0
2185 y Fo(y)18 b Fu(for)d Fo(Y)136 2192 y Fm(z)155 2185
y Fu(\))g(so)h(that)f(Prob)o(\()p Fo(X)9 b Fu(=)d Fo(z)r(y)r
Fu(\))13 b(=)h(Prob\()p Fo(Z)8 b Fu(=)e Fo(z)r Fu(\))11
b Fn(\001)f Fu(Prob\()p Fo(Y)1084 2192 y Fm(z)1109 2185
y Fu(=)c Fo(y)r Fu(\))13 b Fo(>)1258 2167 y Fk(2)1275
2155 y Fh(m)p Fb(\000)p Fc(\()p Fh(n)p Fb(\000)p Fh(k)p
Fc(\))p 1258 2174 146 2 v 1296 2201 a Fm(\016)q Fi(\001)p
Fk(2)1339 2192 y Fh(m)1420 2185 y Fn(\001)d Fo(\016)15
b Fu(=)f(2)1550 2169 y Fi(\000)p Fk(\()p Fm(n)p Fi(\000)p
Fm(k)q Fk(\))1668 2185 y Fu(.\))21 b(Also,)16 b(it)g(can)0
2247 y(b)q(e)g(v)o(eri\014ed)g(that)f(for)f(ev)o(ery)h
Fo(z)752 2309 y Fu(Prob\()p Fo(Z)8 b Fu(=)d Fo(z)r Fu(\))13
b Fn(\024)g Fu(2)1071 2290 y Fi(\000)p Fk(\()p Fm(m)p
Fi(\000)p Fm(k)q Fk(\))1892 2309 y Fu(\(8\))0 2398 y(\(As)i(otherwise,)
g(there)g(exists)h Fo(y)e Fn(2)f(f)p Fu(0)p Fo(;)8 b
Fu(1)p Fn(g)734 2382 y Fm(n)p Fi(\000)p Fm(m)825 2398
y Fu(so)15 b(that)g(Prob)o(\()p Fo(X)8 b Fu(=)d Fo(z)r(y)r
Fu(\))13 b Fo(>)g Fu(2)1329 2382 y Fi(\000)p Fk(\()p
Fm(m)p Fi(\000)p Fm(k)q Fk(\))1467 2398 y Fn(\001)c Fu(2)1512
2382 y Fi(\000)p Fk(\()p Fm(n)p Fi(\000)p Fm(m)p Fk(\))1655
2398 y Fu(=)k(2)1726 2382 y Fi(\000)p Fk(\()p Fm(n)p
Fi(\000)p Fm(k)q Fk(\))1844 2398 y Fu(.\))0 2460 y(W)l(e)k(no)o(w)g
(turn)h(to)e(b)q(ound)j(the)e(statistical)h(di\013erence)h(b)q(et)o(w)o
(een)e(the)h(distributions)h(\()p Fo(f)r(;)8 b(f)d Fu(\()p
Fo(X)t Fu(\)\))15 b(and)j(\()p Fo(f)r(;)8 b(U)1889 2467
y Fm(m)1920 2460 y Fu(\),)0 2523 y(where)17 b Fo(f)22
b Fu(is)c(uniformly)g(distributed)g(in)g Fo(F)6 b Fu(.)26
b(Denote)16 b(the)h(statistical)h(di\013erence)g(b)q(et)o(w)o(een)f
(distributions)i Fo(D)1932 2530 y Fk(1)0 2585 y Fu(and)j
Fo(D)133 2592 y Fk(2)174 2585 y Fu(b)o(y)g(\001[)p Fo(D)333
2592 y Fk(1)351 2585 y Fo(;)8 b(D)410 2592 y Fk(2)427
2585 y Fu(])22 b(\(i.e.,)h(\001[)p Fo(D)663 2592 y Fk(1)681
2585 y Fo(;)8 b(D)740 2592 y Fk(2)757 2585 y Fu(])794
2560 y Fk(def)799 2585 y Fu(=)869 2567 y Fk(1)p 869 2574
17 2 v 869 2600 a(2)898 2553 y Fd(P)942 2596 y Fm(\013)973
2585 y Fn(j)p Fu(Prob)o(\()p Fo(D)1138 2592 y Fk(1)1173
2585 y Fu(=)17 b Fo(\013)p Fu(\))d Fn(\000)i Fu(Prob)o(\()p
Fo(D)1489 2592 y Fk(2)1524 2585 y Fu(=)h Fo(\013)p Fu(\))p
Fn(j)p Fu(\).)40 b(Some)22 b(useful)952 2795 y(12)p eop
%%Page: 13 14
13 13 bop 0 42 a Fu(inequalities)21 b(used)f(b)q(elo)o(w)f(are)f(\001[)
p Fo(g)r Fu(\()p Fo(D)692 49 y Fk(1)709 42 y Fu(\))p
Fo(;)8 b(g)r Fu(\()p Fo(D)828 49 y Fk(2)844 42 y Fu(\)])18
b Fn(\024)h Fu(\001[)p Fo(D)1036 49 y Fk(1)1054 42 y
Fo(;)8 b(D)1113 49 y Fk(2)1130 42 y Fu(],)19 b(for)f(ev)o(ery)g
(function)i Fo(g)r Fu(,)e(and)h(\001[)p Fo(D)1789 49
y Fk(1)1807 42 y Fo(;)8 b(D)1866 49 y Fk(3)1884 42 y
Fu(])18 b Fn(\024)0 104 y Fu(\001[)p Fo(D)89 111 y Fk(1)107
104 y Fo(;)8 b(D)166 111 y Fk(2)183 104 y Fu(])i(+)g(\001[)p
Fo(D)340 111 y Fk(2)358 104 y Fo(;)e(D)417 111 y Fk(3)435
104 y Fu(].)19 b(W)l(e)c(ha)o(v)o(e)280 204 y(Exp)360
215 y Fm(f)s Fi(2)p Fm(F)430 204 y Fu(\(\001[\()p Fo(f)r(;)8
b(f)d Fu(\()p Fo(X)t Fu(\)\))p Fo(;)i Fu(\()p Fo(f)r(;)g(U)798
211 y Fm(m)827 204 y Fu(\)]\))40 b(=)i(Exp)1073 215 y
Fm(f)s Fi(2)p Fm(F)1143 204 y Fu(\(\001[)p Fo(f)5 b Fu(\()p
Fo(X)t Fu(\))p Fo(;)j(U)1369 211 y Fm(m)1398 204 y Fu(]\))463
b(\(9\))916 279 y Fn(\024)42 b Fu(Exp)1073 290 y Fm(f)s
Fi(2)p Fm(F)1143 279 y Fu(\(\001[)p Fo(f)5 b Fu(\()p
Fo(X)1299 260 y Fi(0)1309 279 y Fu(\))p Fo(;)j(U)1379
286 y Fm(m)1409 279 y Fu(]\))i(+)g(\001[)p Fo(X)q(;)e(X)1648
260 y Fi(0)1657 279 y Fu(])199 b(\(10\))0 380 y(where)18
b Fo(X)176 363 y Fi(0)204 380 y Fu(is)g(the)g(random)f(v)m(ariable)i
(induced)h(b)o(y)e Fo(X)i Fu(conditioned)g(on)d Fo(Z)j
Fn(62)d Fo(B)1416 387 y Fm(\016)1435 380 y Fu(.)27 b(By)18
b(Eq.)f(\(7\),)g(\001[)p Fo(X)q(;)8 b(X)1876 363 y Fi(0)1885
380 y Fu(])17 b Fo(<)5 424 y Fk(2)22 412 y Fb(\000)p
Fc(\()p Fh(n)p Fb(\000)p Fh(m)p Fb(\000)p Fh(k)p Fc(\))p
5 431 169 2 v 81 458 a Fm(\016)178 442 y Fu(,)e(and)h(it)f(is)h(left)f
(only)h(to)f(b)q(ound)h(the)f(other)g(term)f(in)j(Eq.)d(\(10\).)71
504 y(Let)k Fo(A)h Fu(b)q(e)g(the)f(matrix)g(represen)o(ting)i(the)e
(transition)h(probabilities)h(in)g(a)e(random)g(step)g(on)h(the)f
(graph)0 566 y Fo(G)p Fu(;)g(i.e.,)f Fo(Ap)g Fu(describ)q(es)i(the)f
(probabilit)o(y)g(distribution)h(after)e(one)g(random)g(step)h(on)f
(the)g(graph)g Fo(G)p Fu(,)h(starting)0 628 y(with)f(the)g
(distribution)h Fo(p)p Fu(.)24 b(Here)16 b(and)h(in)h(the)e(sequel,)i
(w)o(e)e(abuse)h(notation)f(and)h(refer)g(to)e(random)i(v)m(ariables)0
690 y(and)g(distributions)h(as)f(v)o(ectors)f(in)i(the)f(natural)f
(manner)h(\(i.e.,)g(the)g Fo(i)1234 674 y Fk(th)1284
690 y Fu(comp)q(onen)o(t)g(of)f(the)h(v)o(ector)f Fo(p)h
Fu(is)g Fo(p)p Fu(\()p Fo(i)p Fu(\))0 752 y(and)h(the)g
Fo(i)188 736 y Fk(th)239 752 y Fu(comp)q(onen)o(t)g(of)f(the)h(v)o
(ector)f Fo(X)k Fu(is)e(the)f(probabilit)o(y)h(that)e
Fo(X)j Fu(=)e Fo(i)p Fu(\).)27 b(Eac)o(h)17 b(column)i(in)g
Fo(A)f Fu(has)f Fo(d)0 815 y Fu(non-zero)d(en)o(tries)g(and)g(eac)o(h)f
(suc)o(h)h(en)o(try)g(holds)g(the)g(v)m(alue)1048 797
y Fk(1)p 1047 804 18 2 v 1047 830 a Fm(d)1070 815 y Fu(.)19
b(F)l(or)13 b(ev)o(ery)g Fo(h)g Fn(2)g Fo(H)t Fu(,)g(let)h
Fo(A)1548 822 y Fm(h)1583 815 y Fu(b)q(e)h(the)e(matrix)h(that)0
877 y(results)i(from)e Fo(A)h Fu(b)o(y)g(mo)q(difying)i(the)e(non-zero)
g(en)o(tries)h(as)f(follo)o(ws.)k(The)d Fo(i)1318 860
y Fk(th)1366 877 y Fu(non-zero)f(en)o(try)g(in)h(column)g
Fo(z)h Fu(is)0 939 y(c)o(hanged)e(from)287 921 y Fk(1)p
287 928 V 287 954 a Fm(d)324 939 y Fu(to)f(Prob\()p Fo(h)p
Fu(\()p Fo(Y)564 946 y Fm(z)583 939 y Fu(\))5 b(=)g Fo(i)p
Fu(\).)19 b(Note)14 b(that)g Fo(A)953 946 y Fm(h)975
939 y Fo(Z)k Fu(equals)d Fo(g)1183 946 y Fm(h)p Fk(\()p
Fm(Y)1235 950 y Fh(Z)1258 946 y Fk(\))1273 939 y Fu(\()p
Fo(Z)s Fu(\))f(whic)o(h)i(in)f(turn)g(equals)g Fo(f)5
b Fu(\()p Fo(X)t Fu(\))14 b(for)0 1001 y(the)h(function)h
Fo(f)21 b Fu(asso)q(ciated)15 b(with)h(the)f(hashing)h(function)g
Fo(h)p Fu(.)k(Th)o(us,)15 b(letting)h Fo(Z)1407 984 y
Fi(0)1431 1001 y Fu(=)d(msb\()p Fo(X)1620 984 y Fi(0)1631
1001 y Fu(\),)h(w)o(e)h(get)223 1102 y(Exp)303 1113 y
Fm(f)s Fi(2)p Fm(F)372 1102 y Fu(\(\001[)p Fo(f)5 b Fu(\()p
Fo(X)528 1083 y Fi(0)538 1102 y Fu(\))p Fo(;)j(U)608
1109 y Fm(m)639 1102 y Fu(]\))40 b(=)i(Exp)867 1113 y
Fm(h)p Fi(2)p Fm(H)941 1102 y Fu(\(\001[)p Fo(A)1044
1109 y Fm(h)1064 1102 y Fo(Z)1098 1083 y Fi(0)1110 1102
y Fo(;)8 b(U)1162 1109 y Fm(m)1193 1102 y Fu(]\))645
b(\(11\))710 1176 y Fn(\024)42 b Fu(\001[)p Fo(AZ)906
1158 y Fi(0)918 1176 y Fo(;)8 b(U)970 1183 y Fm(m)1001
1176 y Fu(])h(+)i(Exp)1149 1187 y Fm(h)p Fi(2)p Fm(H)1222
1176 y Fu(\(\001[)p Fo(A)1325 1183 y Fm(h)1346 1176 y
Fo(Z)1380 1158 y Fi(0)1392 1176 y Fo(;)d(AZ)1481 1158
y Fi(0)1492 1176 y Fu(]\))346 b(\(12\))710 1251 y Fn(\024)42
b Fu(\001[)p Fo(Z)872 1232 y Fi(0)884 1251 y Fo(;)8 b(Z)s
Fu(])h(+)h(\001[)p Fo(AZ)q(;)e(U)1175 1258 y Fm(m)1205
1251 y Fu(])i(+)h(Exp)1354 1262 y Fm(h)p Fi(2)p Fm(H)1427
1251 y Fu(\(\001[)p Fo(A)1530 1258 y Fm(h)1551 1251 y
Fo(Z)1585 1232 y Fi(0)1597 1251 y Fo(;)d(AZ)1686 1232
y Fi(0)1697 1251 y Fu(]\))141 b(\(13\))0 1356 y(The)13
b(\014rst)g(term)g(in)h(Eq.)e(\(13\))g(is)i(b)q(ounded)h(b)o(y)e(Eq.)f
(\(7\).)18 b(Fixing)c Fo(\016)1149 1331 y Fk(def)1154
1356 y Fu(=)1214 1338 y Fm(\017)1228 1326 y Fc(6)p 1212
1345 34 2 v 1212 1372 a Fk(8)p Fm(d)1264 1356 y Fu(\(as)f(the)g(min-en)
o(trop)o(y)g(b)q(ound)h(of)f(go)q(o)q(d)0 1418 y Fo(Y)26
1425 y Fm(z)46 1418 y Fu('s\))h(and)h(using)h(the)g(Lefto)o(v)o(er)e
(Hash)h(Lemma)g(\(Lemma)g(2.1\))f(w)o(e)h(get,)g(for)f(eac)o(h)h
Fo(z)g Fn(62)e Fo(B)1560 1425 y Fm(\016)1579 1418 y Fu(,)647
1539 y(Exp)727 1549 y Fm(h)p Fi(2)p Fm(H)800 1539 y Fu(\(\001[)p
Fo(h)p Fu(\()p Fo(Y)939 1546 y Fm(z)958 1539 y Fu(\))p
Fo(;)8 b(\021)r Fu(]\))i Fo(<)1123 1514 y Fc(3)1112 1498
y Fn(p)p 1150 1498 46 2 v 41 x Fo(\016)r(d)i Fu(=)1261
1508 y Fo(\017)1279 1491 y Fk(2)p 1261 1528 38 2 v 1268
1570 a Fu(2)0 1647 y(where)j Fo(\021)g Fu(is)h(uniformly)f(distributed)
h(o)o(v)o(er)e Fn(f)p Fu(1)p Fo(;)8 b(:::;)g(d)o Fn(g)p
Fu(.)17 b(Recalling)g(the)d(de\014nition)j(of)d Fo(A)1514
1654 y Fm(h)1535 1647 y Fu(,)h(this)g(means)f(that)g(the)0
1709 y Fp(exp)n(e)n(cte)n(d)19 b Fu(di\013erence)c(b)q(et)o(w)o(een)g
(corresp)q(onding)h(en)o(tries)f(in)g(the)g(matrices)g
Fo(A)f Fu(and)h Fo(A)1475 1716 y Fm(h)1511 1709 y Fu(is)g(at)f(most)g
Fo(\017)1740 1692 y Fk(2)1759 1709 y Fo(=)p Fu(2.)19
b(Th)o(us,)0 1771 y(for)c(ev)o(ery)g(probabilit)o(y)h(v)o(ector)f
Fo(p)g Fu(\(and)g(in)h(particular)g(for)e Fo(p)h Fu(corresp)q(onding)h
(to)f Fo(Z)1455 1754 y Fi(0)1467 1771 y Fu(\),)717 1891
y(Exp)798 1902 y Fm(h)p Fi(2)p Fm(H)871 1891 y Fu(\(\001[)p
Fo(A)974 1898 y Fm(h)995 1891 y Fo(p;)8 b(Ap)p Fu(]\))j
Fo(<)1191 1860 y(\017)1209 1844 y Fk(2)p 1191 1881 V
1198 1922 a Fu(2)0 1999 y(This)19 b(yields)i(a)d(b)q(ound)i(on)f(the)g
(third)g(term)f(in)i(Eq.)e(\(13\).)30 b(It)19 b(is)g(left)g(to)g(b)q
(ound)g(the)g(second)h(term;)f(that)f(is)0 2061 y(\001[)p
Fo(AZ)q(;)8 b(U)169 2068 y Fm(m)199 2061 y Fu(].)19 b(This)14
b(is)f(done)h(using)g(the)f(Expander)h(Smo)q(othing)f(Lemma)h(\(Lemma)f
(2.3\),)f(while)i(relying)h(on)e(the)0 2124 y(min-en)o(trop)o(y)i(b)q
(ound)i(of)d(Eq.)h(\(8\).)k(W)l(e)c(get)685 2244 y(\001[)p
Fo(AZ)q(;)8 b(U)854 2251 y Fm(m)884 2244 y Fu(])25 b
Fo(<)987 2213 y(\025)p 987 2233 27 2 v 988 2275 a(d)1029
2244 y Fn(\001)1051 2202 y(p)p 1089 2202 44 2 v 42 x
Fu(2)1112 2231 y Fm(k)1158 2244 y Fn(\024)1223 2213 y
Fo(\017)1241 2197 y Fk(2)p 1223 2233 38 2 v 1231 2275
a Fu(4)0 2349 y(Com)o(bining)16 b(all)g(the)f(ab)q(o)o(v)o(e)g(b)q
(ounds,)h(w)o(e)f(get)385 2470 y(Exp)465 2481 y Fm(f)s
Fi(2)p Fm(F)534 2470 y Fu(\(\001[\()p Fo(f)r(;)8 b(f)d
Fu(\()p Fo(X)t Fu(\)\))p Fo(;)i Fu(\()p Fo(f)r(;)h(U)903
2477 y Fm(m)931 2470 y Fu(\)]\))41 b Fo(<)h Fu(2)10 b
Fn(\001)1158 2440 y Fu(2)1181 2423 y Fi(\000)p Fk(\()p
Fm(n)p Fi(\000)p Fm(m)p Fi(\000)p Fm(k)q Fk(\))p 1158
2460 197 2 v 1246 2501 a Fo(\016)1370 2470 y Fu(+)1421
2440 y Fo(\017)1439 2423 y Fk(2)p 1421 2460 38 2 v 1428
2501 a Fu(2)1473 2470 y(+)1523 2440 y Fo(\017)1541 2423
y Fk(2)p 1523 2460 V 1531 2501 a Fu(4)1869 2470 y(\(14\))0
2589 y(Recalling)17 b(that)e Fo(d)d Fu(=)h(\()403 2571
y Fk(2)420 2559 y Fh(k)p 403 2578 35 2 v 413 2605 a Fm(\017)442
2589 y Fu(\))460 2573 y Fm(c)492 2589 y Fu(and)i Fo(\016)g
Fu(=)670 2571 y Fm(\017)684 2559 y Fc(6)p 668 2578 34
2 v 668 2605 a Fk(8)p Fm(d)719 2589 y Fu(=)782 2571 y
Fm(\017)796 2559 y Fc(6+)p Fh(c)p 772 2578 84 2 v 772
2605 a Fk(2)789 2597 y Fc(3+)p Fh(ck)861 2589 y Fu(,)f(and)i(using)g
Fo(n)10 b Fn(\000)g Fo(m)g Fn(\000)g Fo(k)k Fu(=)f(6)c(+)h
Fo(ck)h Fu(+)f(\(8)f(+)i Fo(c)p Fu(\))e Fn(\001)g Fu(log)q(\(1)p
Fo(=\017)p Fu(\),)14 b(the)0 2651 y(\014rst)h(term)g(in)h(Eq.)e(\(14\))
g(is)i(b)q(ounded)h(b)o(y)e Fo(\017)747 2635 y Fk(2)766
2651 y Fo(=)p Fu(4,)f(and)h(the)h(lemma)f(follo)o(ws.)p
1369 2657 34 40 v 952 2795 a(13)p eop
%%Page: 14 15
14 14 bop 0 42 a Fl(4.4)56 b(Lo)n(w)n(er)19 b(Bound)0
133 y Fu(W)l(e)14 b(conclude)h(b)o(y)e(observing)h(that)f(a)g(lo)o(w)o
(er)g(b)q(ound)h(of)f([29)o(])g(\(i.e.,)g([29)o(,)h(Thm)f(3]\))g
(implies)i(that,)e(for)g Fo(\017)g Fu(=)g(2)1842 116
y Fi(\000)p Fk(\012\()p Fm(k)q Fk(\))1937 133 y Fu(,)0
195 y(our)i(construction)h(is)g(optimal.)22 b(This)16
b(holds)h(ev)o(en)f(when)g(trying)g(to)f(extract)f(just)i(one)g(bit.)21
b(Sp)q(eci\014call)q(y)l(,)d Fp(any)0 257 y(family)e(of)g(functions)f
(fr)n(om)h Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)608 241
y Fm(n)645 257 y Fp(to)16 b Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p
Fn(g)p Fp(,)14 b(with)i(extr)n(action)g(pr)n(op)n(erty)h(of)f(ac)n(cur)
n(acy)g Fo(\017)d(<)g Fu(0)p Fo(:)p Fu(5)i Fp(with)h(r)n(esp)n(e)n(ct)0
319 y(to)h(r)n(andom)f(variables)g(of)g(min-entr)n(opy)g
Fo(n)11 b Fn(\000)f Fo(k)k Fn(\024)f Fo(n)d Fn(\000)h
Fu(1)p Fp(,)16 b(must)g(have)h(size)e(at)i(le)n(ast)e
Fu(max)p Fn(f)p Fo(k)c Fu(+)f(1)p Fo(;)e Fu(\(1)p Fo(=\017)p
Fu(\))g Fn(\000)j Fu(1)p Fn(g)p Fu(.)0 467 y Fq(5)67
b(Tin)n(y)24 b(F)-6 b(amilies)24 b(Extracting)g(Lo)n(w)e(Min-En)n(trop)
n(y)0 574 y Fu(Here)16 b(w)o(e)f(treat)f(the)i(case)f(of)g(random)g(v)m
(ariables)i(with)f(min-en)o(trop)o(y)f Fo(k)q Fu(,)g(for)g
Fo(k)f Fn(\034)g Fo(n)p Fu(.)21 b(W)l(e)15 b(construct)g(a)g(family)0
636 y(of)i(p)q(oly)q(\(2)181 619 y Fm(k)201 636 y Fo(n=\017)p
Fu(\))h(functions)g(mapping)g Fn(f)p Fu(0)p Fo(;)8 b
Fu(1)p Fn(g)805 619 y Fm(n)843 636 y Fu(to)16 b Fn(f)p
Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1013 619 y Fm(m)1043 636
y Fu(,)18 b(where)f Fo(m)f Fu(=)h(\012\()p Fo(k)q Fu(\).)25
b(\(Again,)18 b Fo(\017)g Fu(is)f(the)h(accuracy)0 698
y(parameter.\))j(Lo)q(osely)16 b(sp)q(eaking,)h(this)f(means)g(that)f
(these)h(functions)g(are)g(able)g(to)f(\\smo)q(othen")g(a)h(constan)o
(t)0 760 y(fraction)h(of)f(the)i(min-en)o(trop)o(y;)f(sp)q
(eci\014cally)l(,)j(min-en)o(trop)o(y)e Fo(k)g Fu(is)f(mapp)q(ed)h(to)e
(almost)h(uniform)g(distribution)0 822 y(o)o(v)o(er)d(the)i(strings)f
(of)f(length)i(\012\()p Fo(k)q Fu(\).)0 946 y Fl(5.1)56
b(Main)18 b(Result)0 1038 y Fv(Theorem)f(5.1)22 b Fu(\(Extractors)15
b(for)h(Lo)o(w)h(Min-En)o(trop)o(y\):)23 b Fp(L)n(et)17
b Fu(3)p Fo(m)d(<)i(k)g(<)g(n)i Fp(and)f Fo(\017)f(>)f
Fu(2)1576 1021 y Fi(\000)p Fk(\()p Fm(k)q Fi(\000)p Fk(2)p
Fm(m)p Fi(\000)p Fk(9\))p Fm(=)p Fk(5)1797 1038 y Fp(.)25
b(Ther)n(e)0 1100 y(exists)15 b(a)i(family)f(of)h(functions,)e(e)n(ach)
h(mapping)h Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)966 1083
y Fm(n)1003 1100 y Fp(to)16 b Fn(f)p Fu(0)p Fo(;)8 b
Fu(1)p Fn(g)1170 1083 y Fm(m)1200 1100 y Fp(,)16 b(satisfying)f(the)i
(fol)r(lowing)f(pr)n(op)n(erties.)68 1182 y Fn(\017)23
b Fu(succinctness:)e Fp(the)14 b(family)g(c)n(ontains)f(a)h(p)n
(olynomial)g(in)1097 1164 y Fk(2)1114 1152 y Fh(m)1141
1164 y Fm(n)p 1097 1171 65 2 v 1122 1198 a(\017)1180
1182 y Fp(numb)n(er)g(of)h(functions,)e(and)h(e)n(ach)g(function)114
1244 y(is)h(r)n(epr)n(esente)n(d)g(by)i(a)f(unique)g(string)g(of)g
(length)g Fo(l)q Fu(\()p Fo(m;)8 b(\017;)g(n)p Fu(\))j(=)h
Fo(O)q Fu(\()p Fo(m)e Fu(+)h(log)1418 1226 y Fm(n)p 1418
1233 21 2 v 1421 1260 a(\017)1443 1244 y Fu(\))p Fp(.)68
1337 y Fn(\017)23 b Fu(e\016cien)o(t)11 b(ev)m(aluation:)19
b Fp(Ther)n(e)12 b(exists)f(a)h(lo)n(gsp)n(ac)n(e)f(algorithm)h(that,)h
(on)f(input)g(a)g(description)g(of)g(a)g(function)114
1399 y Fo(f)21 b Fp(and)16 b(a)h(string)e Fo(x)p Fp(,)i(r)n(eturns)e
Fo(f)5 b Fu(\()p Fo(x)p Fu(\))p Fp(.)68 1491 y Fn(\017)23
b Fu(extraction)17 b(prop)q(ert)o(y:)25 b Fp(F)m(or)18
b(every)g(r)n(andom)h(variable)f Fo(X)i Fn(2)c(f)p Fu(0)p
Fo(;)8 b Fu(1)p Fn(g)1307 1475 y Fm(n)1347 1491 y Fp(of)18
b(min-entr)n(opy)g Fo(k)q Fp(,)h(al)r(l)g(but)f(an)h
Fo(\017)114 1553 y Fp(fr)n(action)d(of)g(the)h(functions)e
Fo(f)21 b Fp(in)16 b(the)h(family)f(satisfy)659 1620
y Fu(1)p 659 1640 23 2 v 659 1682 a(2)697 1651 y Fn(\001)764
1610 y Fd(X)719 1701 y Fm(\013)p Fi(2f)p Fk(0)p Fm(;)p
Fk(1)p Fi(g)841 1693 y Fh(m)875 1651 y Fn(j)p Fu(Prob\()p
Fo(f)5 b Fu(\()p Fo(X)t Fu(\))g(=)g Fo(\013)p Fu(\))k
Fn(\000)1275 1620 y Fu(1)p 1259 1640 55 2 v 1259 1682
a(2)1282 1669 y Fm(m)1318 1651 y Fn(j)j(\024)h Fo(\017)0
1794 y Fu(That)j(is,)h(w)o(e)f(ma)o(y)f(extract)h Fo(m)e
Fu(=)608 1776 y Fk(1)p 608 1783 17 2 v 608 1809 a(2)641
1794 y Fn(\001)c Fu(\()p Fo(k)i Fn(\000)f Fu(9)g Fn(\000)g
Fu(5)d(log)933 1804 y Fk(2)952 1794 y Fu(\(1)p Fo(=\017)p
Fu(\)\))15 b(bits)i(from)e(min-en)o(trop)o(y)i Fo(k)q
Fu(,)f(and)h(the)f(extracted)0 1856 y(distribution)h(is)f
Fo(\017)p Fu(-close)g(to)f(uniform.)20 b(In)c(fact,)e
Fo(l)q Fu(\()p Fo(m;)8 b(\017;)g(n)p Fu(\))i(=)j(4)p
Fo(m)d Fu(+)h(10)d(log)1305 1867 y Fk(2)1324 1856 y Fu(\(1)p
Fo(=\017)p Fu(\))h(+)i(2)d(log)1568 1867 y Fk(2)1594
1856 y Fo(n)i Fu(+)h Fo(O)q Fu(\(1\).)0 1980 y Fl(5.2)56
b(The)18 b(Construction)0 2071 y Fu(W)l(e)d(use)g(a)f(construction)h
(of)f(small)i(probabilit)o(y)g(spaces)f(with)g(small)g(bias.)20
b(In)o(tuitiv)o(ely)l(,)d(w)o(e)d(consider)i(a)e(prime)0
2133 y Fo(p)e Fn(\031)h Fu(2)106 2117 y Fm(m)150 2133
y Fu(and)g(a)g(construction)g(of)f Fo(t)608 2108 y Fk(def)613
2133 y Fu(=)676 2116 y Fm(n)p 671 2123 30 2 v 671 2149
a(m)719 2133 y Fu(random)g(v)m(ariables,)i(\()p Fo(\030)1120
2140 y Fk(1)1138 2133 y Fo(;)8 b(:::;)g(\030)1239 2140
y Fm(t)1251 2133 y Fu(\),)k(eac)o(h)h(distributed)i(o)o(v)o(er)c
Fo(GF)6 b Fu(\()p Fo(p)p Fu(\))13 b(with)0 2196 y(the)i(follo)o(wing)h
Fp(smal)r(l)g(bias)f Fu(prop)q(ert)o(y:)114 2278 y(for)10
b(ev)o(ery)g Fo(t)p Fu(-long)h(sequence)h(\()p Fo(a)645
2285 y Fk(1)663 2278 y Fo(;)c(:::;)g(a)768 2285 y Fm(t)780
2278 y Fu(\))i(of)h(elemen)o(ts)g(in)g Fo(GF)6 b Fu(\()p
Fo(p)p Fu(\),)11 b(so)f(that)g(not)g(all)h Fo(a)1542
2285 y Fm(i)1556 2278 y Fu('s)f(are)g(zero,)h(the)114
2340 y(random)g(v)m(ariable)443 2307 y Fd(P)487 2318
y Fm(t)487 2351 y(i)p Fk(=1)550 2340 y Fo(a)574 2347
y Fm(i)588 2340 y Fo(\030)608 2347 y Fm(i)633 2340 y
Fu(is)h(almost)f(uniformly)h(distributed)h(o)o(v)o(er)d
Fo(GF)c Fu(\()p Fo(p)p Fu(\))11 b(\(i.e.,)g(its)h(statistical)114
2402 y(distance)k(from)e(uniform)i(is)f(small\).)0 2484
y(The)k(actual)f(condition)i(is)f(giv)o(en)g(in)g(De\014nition)h(2.4.)
29 b(T)o(ypically)l(,)20 b(suc)o(h)f(random)f(v)m(ariables)i(are)e
(de\014ned)i(b)o(y)0 2546 y(the)e(uniform)g(distribution)h(o)o(v)o(er)e
(some)g(sample)h(space)g Fo(S)h Fn(\022)e Fo(GF)6 b Fu(\()p
Fo(p)p Fu(\))1227 2529 y Fm(t)1242 2546 y Fu(.)27 b(W)l(e)17
b(will)j(use)e(suc)o(h)g(a)f(sample)h(space,)0 2608 y
Fo(S)s Fu(,)c(for)h(bias)h Fo(\017)240 2591 y Fi(0)265
2608 y Fu(=)d(p)q(oly)q(\()p Fo(\017=p)p Fu(\))i(\(to)f(b)q(e)i(sp)q
(eci\014ed)h(later\).)j(Hence,)c(using)g(the)f(sample)h(space)f(of)g
([4,)f(14],)g(w)o(e)h(ha)o(v)o(e)0 2670 y Fn(j)p Fo(S)s
Fn(j)c Fu(=)i(p)q(oly)r(\()p Fo(n=\017)289 2654 y Fi(0)301
2670 y Fu(\))f(=)h(p)q(oly)q(\()p Fo(n)d Fn(\001)g Fo(p=\017)p
Fu(\))j(=)g(p)q(oly)q(\()p Fo(n)p Fu(2)840 2654 y Fm(m)871
2670 y Fo(=\017)p Fu(\).)952 2795 y(14)p eop
%%Page: 15 16
15 15 bop 0 42 a Fv(Our)21 b(construction.)46 b Fu(The)18
b(functions)i(in)f(our)f(family)l(,)h(denoted)g Fo(F)6
b Fu(,)19 b(corresp)q(ond)g(to)f(the)g(samples)h(in)g(the)0
104 y(small-bias)e(space.)j(Namely)l(,)15 b(for)g(eac)o(h)g(\()p
Fo(s)738 111 y Fk(1)757 104 y Fo(;)8 b(:::;)g(s)859 111
y Fm(t)871 104 y Fu(\))k Fn(2)h Fo(S)s Fu(,)h(w)o(e)h(in)o(tro)q(duce)h
(the)g(function)g Fo(f)i Fn(2)13 b Fo(F)21 b Fu(de\014ned)c(b)o(y)824
219 y Fo(f)5 b Fu(\()p Fo(x)p Fu(\))925 194 y Fk(def)930
219 y Fu(=)1007 166 y Fm(t)983 179 y Fd(X)987 267 y Fm(i)p
Fk(=1)1051 219 y Fo(s)1072 226 y Fm(i)1086 219 y Fo(x)1112
226 y Fm(i)0 337 y Fu(where)19 b Fo(x)161 344 y Fm(i)193
337 y Fu(is)g(the)f Fo(i)339 320 y Fk(th)390 337 y Fu(co)q(ordinate)h
(in)g Fo(x)f Fn(2)g Fo(GF)6 b Fu(\()p Fo(p)p Fu(\))895
320 y Fm(t)928 337 y Fu(and)18 b(the)h(arithmetic)f(is)h(in)h
Fo(GF)6 b Fu(\()p Fo(p)p Fu(\).)28 b(The)19 b(functions,)g(so)0
399 y(de\014ned,)14 b(map)e Fo(GF)6 b Fu(\()p Fo(p)p
Fu(\))395 382 y Fm(t)421 399 y Fu(to)12 b Fo(GF)6 b Fu(\()p
Fo(p)p Fu(\).)18 b(Standard)13 b(mo)q(di\014cations)g(can)f(b)q(e)h
(applied)i(to)c(deriv)o(e)i(functions)g(mapping)0 461
y Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)113 445 y Fm(n)149
461 y Fu(to)14 b Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)317
445 y Fm(m)362 461 y Fu(\(recall)16 b Fo(p)d Fn(\031)g
Fu(2)608 445 y Fm(m)654 461 y Fu(and)i Fo(p)765 445 y
Fm(t)793 461 y Fn(\031)e Fu(2)864 445 y Fm(n)886 461
y Fu(\).)0 585 y Fl(5.3)56 b(Analysis)0 677 y Fu(Our)12
b(analysis)f(uses)h(the)f(fact)g(that)f(the)i(construction)f(of)g
(small{bias)h(spaces)f(of)g([4)o(,)g(14])f(satis\014es)i(a)f(b)q(ound)h
(on)f(an)0 739 y(exp)q(onen)o(tial)k(sum)f(related)g(to)f(the)h(ab)q(o)
o(v)o(e)f(in)o(tuitiv)o(e)i(motiv)m(ation)f(to)f(small{bias)i(spaces)f
(\(see)g(De\014nition)h(2.4\).)0 801 y(W)l(e)j(then)g(pro)o(v)o(e)f(a)h
(Lindsey{lik)o(e)i(lemma)e(on)g(near-orthogonal)f(v)o(ectors)g(and)h
(com)o(bine)h(it)f(with)g(the)g(b)q(ound)0 863 y(ab)q(o)o(v)o(e)d(to)f
(giv)o(e)i(the)f(result.)71 925 y(Supp)q(ose,)25 b(on)d(the)h(con)o
(trary)e(to)h(the)h(extraction)f(prop)q(ert)o(y)l(,)i(that)e(for)g
(some)g(random)h(v)m(ariable)g Fo(X)29 b Fu(=)0 987 y(\()p
Fo(X)56 994 y Fk(1)74 987 y Fo(;)8 b(:::;)g(X)193 994
y Fm(t)205 987 y Fu(\))23 b Fn(2)h Fo(GF)6 b Fu(\()p
Fo(p)p Fu(\))430 971 y Fm(t)467 987 y Fu(with)22 b(min-en)o(trop)o(y)g
Fo(k)q Fu(,)h(and)g(for)e(an)h Fo(\017)g Fu(fraction)g(of)g(the)g
Fo(f)5 b Fu('s)22 b(in)g Fo(F)6 b Fu(,)24 b(the)e(random)0
1049 y(v)m(ariable)15 b Fo(f)5 b Fu(\()p Fo(X)t Fu(\))12
b(is)j Fo(\017)p Fu(-a)o(w)o(a)o(y)d(\(in)j(v)m(ariation)f(distance\))g
(from)f(the)h(uniform)g(distribution.)21 b(Then,)14 b(it)g(follo)o(ws)g
(that)0 1112 y(there)19 b(is)h(a)f(subset)h Fo(S)387
1095 y Fi(0)417 1112 y Fn(\022)g Fo(S)i Fu(of)d Fo(\017)13
b Fn(\001)g(j)p Fo(S)s Fn(j)18 b Fu(sequences)i(so)f(that,)g(for)g(eac)
o(h)i(\026)-25 b Fo(s)1317 1087 y Fk(def)1322 1112 y
Fu(=)25 b(\()p Fo(s)1421 1119 y Fk(1)1440 1112 y Fo(;)8
b(:::;)g(s)1542 1119 y Fm(t)1554 1112 y Fu(\))19 b Fn(2)h
Fo(S)1672 1095 y Fi(0)1683 1112 y Fu(,)g(the)f(random)0
1174 y(v)m(ariable)170 1141 y Fd(P)214 1152 y Fm(t)214
1185 y(i)p Fk(=1)278 1174 y Fo(X)316 1181 y Fm(i)329
1174 y Fo(s)350 1181 y Fm(i)379 1174 y Fu(is)d Fo(\017)p
Fu(-a)o(w)o(a)o(y)e(from)g(the)h(uniform)h(distribution.)21
b(Namely)l(,)16 b(for)e(ev)o(ery)j(\026)-25 b Fo(s)1592
1149 y Fk(def)1597 1174 y Fu(=)18 b(\()p Fo(s)1689 1181
y Fk(1)1708 1174 y Fo(;)8 b(:::;)g(s)1810 1181 y Fm(t)1822
1174 y Fu(\))k Fn(2)h Fo(S)1926 1157 y Fi(0)1937 1174
y Fu(,)575 1261 y(1)p 575 1281 23 2 v 575 1323 a(2)613
1292 y Fn(\001)636 1237 y Fm(p)p Fi(\000)p Fk(1)636 1251
y Fd(X)637 1340 y Fm(j)r Fk(=0)703 1218 y Fd(\014)703
1243 y(\014)703 1268 y(\014)703 1293 y(\014)703 1318
y(\014)717 1292 y Fu(Prob)821 1220 y Fd( )878 1239 y
Fm(t)854 1251 y Fd(X)857 1340 y Fm(i)p Fk(=1)922 1292
y Fo(X)960 1299 y Fm(i)973 1292 y Fo(s)994 1299 y Fm(i)1021
1292 y Fu(=)g Fo(j)1091 1220 y Fd(!)1146 1292 y Fn(\000)1209
1261 y Fu(1)p 1209 1281 V 1209 1323 a Fo(p)1237 1218
y Fd(\014)1237 1243 y(\014)1237 1268 y(\014)1237 1293
y(\014)1237 1318 y(\014)1288 1292 y Fn(\025)39 b Fo(\017)489
b Fu(\(15\))0 1413 y(Let)22 b Fo(v)i Fu(b)q(e)e(a)g Fp(zer)n(o-sum)j
Fo(p)p Fu(-dimensional)f(v)o(ector)e(with)g(norm-1)f(greater)g(than)h
(2)p Fo(\017)g Fu(\(here)g Fo(v)i Fu(represen)o(ts)e(the)0
1475 y(di\013erence)17 b(b)q(et)o(w)o(een)f(the)g(probabilit)o(y)i
(function)e(of)924 1443 y Fd(P)968 1453 y Fm(t)968 1486
y(i)p Fk(=1)1031 1475 y Fo(X)1069 1482 y Fm(i)1082 1475
y Fo(s)1103 1482 y Fm(i)1133 1475 y Fu(and)h(the)f(uniform)g
(distributional)i(function\).)0 1537 y(Then,)e(the)g(norm-2)f(of)g
Fo(v)457 1512 y Fk(def)463 1537 y Fu(=)k(\()p Fo(v)557
1544 y Fk(1)575 1537 y Fo(;)8 b(:::;)g(v)678 1544 y Fm(p)694
1537 y Fu(\))16 b(is)g(at)f(least)937 1501 y Fd(p)p 978
1501 197 2 v 978 1537 a Fo(p)10 b Fn(\001)g Fu(\(2)p
Fo(\017=p)p Fu(\))1157 1524 y Fk(2)1188 1537 y Fu(=)k(2)p
Fo(\017=)1301 1509 y Fn(p)p 1339 1509 23 2 v 28 x Fo(p)p
Fu(.)21 b(P)o(assing)16 b(to)f(the)h(F)l(ourier)g(basis)0
1599 y(\(i.e.,)d(in)g(whic)o(h)h(the)e Fo(j)377 1583
y Fk(th)423 1599 y Fu(v)o(ector)g(is)h Fo(p)623 1583
y Fi(\000)p Fk(1)p Fm(=)p Fk(2)706 1599 y Fn(\001)5 b
Fu(\()p Fo(!)772 1583 y Fm(j)789 1599 y Fo(;)j(!)840
1583 y Fk(2)p Fm(j)873 1599 y Fo(;)g(:::;)g(!)984 1583
y Fm(pj)1016 1599 y Fu(\))k(with)h Fo(!)h Fu(b)q(eing)g(a)f
Fo(p)1368 1583 y Fk(th)1414 1599 y Fu(ro)q(ot)e(of)i(unit)o(y\),)g(w)o
(e)f(represen)o(t)0 1661 y Fo(v)20 b Fu(b)o(y)h(^)-25
b Fo(v)21 b Fu(=)e(\()8 b(^)-31 b Fo(v)246 1668 y Fk(1)264
1661 y Fo(;)8 b(:::;)15 b Fu(^)-30 b Fo(v)367 1668 y
Fm(p)384 1661 y Fu(\),)19 b(where)27 b(^)-31 b Fo(v)591
1668 y Fm(j)627 1661 y Fu(=)700 1643 y Fk(1)p 686 1650
45 2 v 686 1659 a Fi(p)p 713 1659 18 2 v 18 x Fm(p)743
1629 y Fd(P)787 1673 y Fm(i)808 1661 y Fo(!)838 1645
y Fm(ij)880 1661 y Fn(\001)12 b Fo(v)927 1668 y Fm(i)941
1661 y Fu(.)31 b(The)19 b(norm-2)f(of)h Fo(v)h Fu(and)h(^)-25
b Fo(v)20 b Fu(are)f(equal,)h(and)f(th)o(us)g(the)0 1735
y(max-norm)13 b(of)i(^)-24 b Fo(v)15 b Fu(is)g(at)e(least)514
1712 y Fk(2)p Fm(\017=)562 1694 y Fi(p)p 589 1694 V 18
x Fm(p)p 514 1725 92 2 v 538 1733 a Fi(p)p 565 1733 18
2 v 18 x Fm(p)623 1735 y Fu(=)g(2)p Fo(\017=p)p Fu(.)20
b(Let)14 b Fn(k)p Fo(c)p Fn(k)f Fu(denote)h(the)g(magnitude)g(of)g(the)
g(complex)h(n)o(um)o(b)q(er)f Fo(c)p Fu(.)19 b(It)0 1798
y(follo)o(ws)c(that)g(there)g(exists)h(a)f Fo(j)j Fu(so)d(that)719
1769 y Fn(p)p 757 1769 23 2 v 29 x Fo(p)10 b Fn(\001)g(k)e
Fu(^)-31 b Fo(v)858 1805 y Fm(j)875 1798 y Fn(k)13 b
Fu(=)g Fn(k)990 1765 y Fd(P)1033 1809 y Fm(i)1054 1798
y Fo(v)1076 1805 y Fm(i)1090 1798 y Fo(!)1120 1781 y
Fm(j)r(i)1149 1798 y Fn(k)g(\025)g Fu(2)p Fo(\017=)1297
1769 y Fn(p)p 1335 1769 V 29 x Fo(p)i Fu(and)h(this)f
Fo(j)j Fu(cannot)d(b)q(e)h Fo(p)f Fu(\(since)0 1828 y
Fd(P)44 1871 y Fm(i)65 1860 y Fo(v)87 1867 y Fm(i)101
1860 y Fo(!)131 1843 y Fm(pi)178 1860 y Fu(=)229 1828
y Fd(P)272 1871 y Fm(i)294 1860 y Fo(v)316 1867 y Fm(i)345
1860 y Fu(=)h(0\).)25 b(Applying)19 b(this)f(argumen)o(t)e(to)g(the)h
(v)o(ector)g(represen)o(ting)g(the)g(di\013erence)i(b)q(et)o(w)o(een)0
1922 y(the)14 b(probabilit)o(y)h(function)g(of)535 1890
y Fd(P)579 1900 y Fm(t)579 1933 y(i)p Fk(=1)642 1922
y Fo(X)680 1929 y Fm(i)694 1922 y Fo(s)715 1929 y Fm(i)743
1922 y Fu(and)f(the)g(uniform)g(distributional)i(function,)e(w)o(e)g
(conclude)h(that)f(for)0 1984 y(ev)o(ery)j(\026)-25 b
Fo(s)154 1959 y Fk(def)159 1984 y Fu(=)18 b(\()p Fo(s)251
1991 y Fk(1)270 1984 y Fo(;)8 b(:::;)g(s)372 1991 y Fm(t)384
1984 y Fu(\))k Fn(2)h Fo(S)488 1967 y Fi(0)515 1984 y
Fu(there)i(exists)g(some)g Fo(j)g Fn(2)e(f)p Fu(1)p Fo(;)8
b(:::;)g(p)r Fn(\000)s Fu(1)p Fn(g)p Fu(,)j(so)k(that)492
2068 y(2)p Fo(\017)p 482 2089 61 2 v 482 2102 a Fn(p)p
520 2102 23 2 v 28 x Fo(p)589 2099 y Fn(\024)666 2026
y Fd(\015)666 2050 y(\015)666 2075 y(\015)666 2100 y(\015)666
2125 y(\015)689 2059 y(X)713 2147 y Fm(i)757 2027 y Fd( )790
2099 y Fu(Prob)894 2027 y Fd( )951 2046 y Fm(t)927 2059
y Fd(X)927 2148 y Fm(k)q Fk(=1)995 2099 y Fo(X)1033 2106
y Fm(k)1053 2099 y Fo(s)1074 2106 y Fm(k)1107 2099 y
Fu(=)e Fo(i)1171 2027 y Fd(!)1226 2099 y Fn(\000)1290
2068 y Fu(1)p 1290 2089 V 1290 2130 a Fo(p)1317 2027
y Fd(!)1360 2099 y Fn(\001)d Fo(!)1413 2080 y Fm(ij)1442
2026 y Fd(\015)1442 2050 y(\015)1442 2075 y(\015)1442
2100 y(\015)1442 2125 y(\015)589 2244 y Fu(=)666 2170
y Fd(\015)666 2195 y(\015)666 2220 y(\015)666 2245 y(\015)666
2270 y(\015)689 2203 y(X)713 2292 y Fm(i)757 2244 y Fu(Prob)861
2172 y Fd( )918 2191 y Fm(t)894 2203 y Fd(X)894 2293
y Fm(k)q Fk(=1)962 2244 y Fo(X)1000 2251 y Fm(k)1020
2244 y Fo(s)1041 2251 y Fm(k)1074 2244 y Fu(=)j Fo(i)1138
2172 y Fd(!)1181 2244 y Fn(\001)d Fo(!)1234 2225 y Fm(ij)1263
2170 y Fd(\015)1263 2195 y(\015)1263 2220 y(\015)1263
2245 y(\015)1263 2270 y(\015)589 2364 y Fu(=)666 2316
y Fd(\015)666 2341 y(\015)666 2366 y(\015)689 2364 y
Fu(Exp)777 2317 y Fd(\020)802 2364 y Fo(!)832 2344 y
Fm(j)853 2316 y Fd(P)897 2326 y Fh(t)897 2360 y(k)p Fc(=1)956
2344 y Fm(X)983 2348 y Fh(k)1001 2344 y Fm(s)1017 2348
y Fh(k)1037 2317 y Fd(\021)1062 2316 y(\015)1062 2341
y(\015)1062 2366 y(\015)0 2465 y Fu(It)15 b(follo)o(ws)h(that)e(for)h
(some)g Fo(j)g Fn(2)e(f)p Fu(1)p Fo(;)8 b(:::;)g(p)f
Fn(\000)k Fu(1)p Fn(g)k Fu(there)g(exists)h(a)f(subset)g
Fo(S)1272 2448 y Fi(00)1306 2465 y Fn(\022)e Fo(S)1385
2448 y Fi(0)1411 2465 y Fu(of)i(cardinalit)o(y)1699 2444
y Fi(j)p Fm(S)1731 2431 y Fb(0)1742 2444 y Fi(j)p 1696
2454 60 2 v 1696 2480 a Fm(p)p Fi(\000)p Fk(1)1773 2465
y Fo(>)1828 2447 y Fm(\017)p 1827 2454 18 2 v 1827 2480
a(p)1859 2465 y Fn(\001)10 b(j)p Fo(S)s Fn(j)p Fu(,)0
2540 y(so)15 b(that)f(for)h(ev)o(ery)i(\026)-25 b Fo(s)378
2515 y Fk(def)383 2540 y Fu(=)18 b(\()p Fo(s)475 2547
y Fk(1)494 2540 y Fo(;)8 b(:::;)g(s)596 2547 y Fm(t)608
2540 y Fu(\))k Fn(2)h Fo(S)712 2523 y Fi(00)711 2591
y Fd(\015)711 2616 y(\015)711 2641 y(\015)734 2640 y
Fu(Exp)822 2593 y Fd(\020)847 2640 y Fo(!)877 2620 y
Fm(j)898 2592 y Fd(P)942 2602 y Fh(t)942 2635 y(i)p Fc(=1)996
2620 y Fm(X)1023 2624 y Fh(i)1036 2620 y Fm(s)1052 2624
y Fh(i)1067 2593 y Fd(\021)1092 2591 y(\015)1092 2616
y(\015)1092 2641 y(\015)1128 2640 y Fn(\025)1190 2609
y Fu(2)p Fo(\017)p 1181 2629 61 2 v 1181 2643 a Fn(p)p
1219 2643 23 2 v 28 x Fo(p)1869 2640 y Fu(\(16\))952
2795 y(15)p eop
%%Page: 16 17
16 16 bop 0 42 a Fu(Assume,)18 b(without)g(loss)g(of)f(generalit)o(y)i
(that)e Fo(j)i Fu(=)e(1.)28 b(By)18 b(partitioning)h(these)f(sequences)
g(according)h(to)e(the)0 104 y(appro)o(ximate)g(direction)i(of)e(the)g
(exp)q(onen)o(tial)i(sum)e(and)h(applying)h(a)e(pigeon-hole)i(argumen)o
(t,)e(w)o(e)g(obtain)g(a)0 166 y(set)e Fo(B)g Fn(\022)e
Fo(S)199 149 y Fi(00)235 166 y Fu(of)i(cardinalit)o(y)h(\012\()p
Fo(\017)p Fn(j)p Fo(S)s Fn(j)p Fo(=p)p Fu(\))d(so)i(that)532
215 y Fd(\015)532 240 y(\015)532 265 y(\015)532 289 y(\015)532
314 y(\015)532 339 y(\015)580 270 y Fu(1)p 560 290 63
2 v 560 332 a Fn(j)p Fo(B)r Fn(j)697 260 y Fd(X)635 351
y Fk(\()p Fm(s)664 355 y Fc(1)679 351 y Fm(;:::;s)745
355 y Fh(t)758 351 y Fk(\))p Fi(2)p Fm(B)828 301 y Fu(Exp)915
254 y Fd(\020)940 301 y Fo(!)970 252 y Fd(P)1014 263
y Fh(t)1014 296 y(i)p Fc(=1)1068 280 y Fm(X)1095 284
y Fh(i)1109 280 y Fm(s)1125 284 y Fh(i)1139 254 y Fd(\021)1164
215 y(\015)1164 240 y(\015)1164 265 y(\015)1164 289 y(\015)1164
314 y(\015)1164 339 y(\015)1200 301 y Fu(=)e(\012\()p
Fo(\017=)1340 270 y Fn(p)p 1377 270 23 2 v 1377 301 a
Fo(p)p Fu(\))0 433 y(Sp)q(eci\014cally)l(,)k(w)o(e)12
b(ma)o(y)h(partition)g(the)g(v)o(ectors)g(according)g(quarters)g(of)f
(the)i(plain)g(and)f(consider)h(the)g(direction)0 495
y(whic)o(h)i(resides)g(in)g(the)f(middle)i(of)e(the)h(quarter)e(with)i
(the)f(largest)g(n)o(um)o(b)q(er)g(of)g(v)o(ectors.)k(This)d(yields,)g
Fo(B)g Fn(\022)d Fo(S)1939 479 y Fi(0)0 558 y Fu(so)i(that)969
653 y Fn(j)p Fo(B)r Fn(j)42 b(\025)1155 622 y Fu(1)p
1155 643 V 1155 684 a(4)1193 653 y Fn(\001)9 b(j)p Fo(S)1259
634 y Fi(00)1280 653 y Fn(j)25 b(\025)1397 622 y Fo(\017)p
1383 643 46 2 v 1383 684 a Fu(4)p Fo(p)1444 653 y Fn(\001)9
b(j)p Fo(S)s Fn(j)346 b Fu(\(17\))376 710 y Fd(\015)376
735 y(\015)376 760 y(\015)376 784 y(\015)376 809 y(\015)376
834 y(\015)424 765 y Fu(1)p 404 785 63 2 v 404 827 a
Fn(j)p Fo(B)r Fn(j)541 755 y Fd(X)479 846 y Fk(\()p Fm(s)508
850 y Fc(1)524 846 y Fm(;:::;s)590 850 y Fh(t)602 846
y Fk(\))p Fi(2)p Fm(B)672 796 y Fu(Exp)759 749 y Fd(\020)784
796 y Fo(!)814 747 y Fd(P)858 758 y Fh(t)858 791 y(i)p
Fc(=1)912 775 y Fm(X)939 779 y Fh(i)953 775 y Fm(s)969
779 y Fh(i)984 749 y Fd(\021)1008 710 y(\015)1008 735
y(\015)1008 760 y(\015)1008 784 y(\015)1008 809 y(\015)1008
834 y(\015)1073 796 y Fn(\025)1155 728 y(p)p 1193 728
23 2 v 37 x Fu(2)p 1155 785 61 2 v 1174 827 a(2)1230
796 y Fn(\001)1268 765 y Fu(2)p Fo(\017)p 1258 785 V
1258 799 a Fn(p)p 1296 799 23 2 v 28 x Fo(p)1349 796
y Fu(=)1410 756 y Fn(p)p 1448 756 V 40 x Fu(2)9 b Fn(\001)1529
765 y Fo(\017)p 1508 785 61 2 v 1508 799 a Fn(p)p 1546
799 23 2 v 28 x Fo(p)1869 796 y Fu(\(18\))0 931 y(Con)o(tradiction)j
(follo)o(ws)f(b)o(y)h(con)o(trasting)f(Eq.)g(\(18\))g(with)h(the)f
(follo)o(wing)i(lemma,)f(whic)o(h)h(generalizes)g(Lindsey's)0
993 y(Lemma)i(\(cf.,)f([13,)g(p.)20 b(88])15 b(and)g([2)o(]\).)0
1091 y Fv(Lemma)i(5.2)23 b Fu(\(A)13 b(Generalized)i(Lindsey's)g
(Lemma\):)k Fp(L)n(et)14 b Fo(A)h Fp(b)n(e)f(an)g Fo(N)5
b Fp(-by-)p Fo(M)20 b Fp(matrix)15 b(of)g(c)n(omplex)g(numb)n(ers,)0
1153 y(so)20 b(that)i(e)n(ach)e(r)n(ow)h(has)f(inner-pr)n(o)n(duct)708
1136 y Fk(5)748 1153 y Fp(with)h(itself)e(e)n(qual)h(to)h
Fo(M)26 b Fp(and)20 b(e)n(ach)h(p)n(air)g(of)f(di\013er)n(ent)g(r)n
(ows)h(have)0 1215 y(inner-pr)n(o)n(duct)d(b)n(ounde)n(d)f
Fu(\(in)g(magnitude\))h Fp(by)f Fo(\017)853 1198 y Fi(0)865
1215 y Fo(M)5 b Fp(.)25 b(L)n(et)17 b Fo(u)g Fp(b)n(e)g(an)h
Fo(N)5 b Fp(-dimensional)16 b(pr)n(ob)n(ability)h(ve)n(ctor)h(with)0
1277 y(e)n(ach)23 b(c)n(omp)n(onents)e(b)n(ounde)n(d)h(ab)n(ove)h(by)f
Fo(\016)r Fp(,)i(and)f Fo(v)h Fp(b)n(e)e(an)g Fo(M)5
b Fp(-dimensional)22 b(pr)n(ob)n(ability)h(ve)n(ctor)f(with)h(e)n(ach)0
1339 y(c)n(omp)n(onents)15 b(b)n(eing)g(either)502 1321
y Fk(1)p 496 1328 30 2 v 496 1355 a Fm(K)547 1339 y Fp(or)i(zer)n(o.)j
(Then,)728 1476 y Fn(k)p Fo(uAv)835 1457 y Fi(>)862 1476
y Fn(k)13 b(\024)946 1396 y Fd(s)p 987 1396 235 2 v 987
1476 a Fu(\()p Fo(\017)1023 1463 y Fi(0)1045 1476 y Fu(+)e
Fo(\016)r Fu(\))e Fn(\001)1168 1445 y Fo(M)p 1168 1465
50 2 v 1172 1507 a(K)0 1669 y Fu(Lindsey's)21 b(Lemma)e(is)h(obtained)g
(from)f(the)g(ab)q(o)o(v)o(e)g(b)o(y)g(requiring)i(the)e(ro)o(ws)g(of)g
Fo(A)g Fu(to)g(b)q(e)h(orthogonal)f(\(i.e.,)0 1731 y
Fo(\017)18 1715 y Fi(0)43 1731 y Fu(=)13 b(0\))h(and)i(considering)g
(only)g(\\\015at")e(distributions)j(\(i.e.,)e(eac)o(h)g
Fo(u)1196 1738 y Fm(i)1225 1731 y Fu(b)q(eing)h(either)g
Fo(\016)h Fu(or)e(0\).)1623 1715 y Fk(6)0 1835 y Fv(Pro)q(of:)20
b Fu(Denote,)14 b(\001)381 1810 y Fk(def)386 1835 y Fu(=)k
Fn(k)p Fo(uAv)546 1818 y Fi(>)574 1835 y Fn(k)p Fu(.)h(Then,)c(using)h
(Cauc)o(h)o(y)f(Sc)o(h)o(w)o(artz)g(Inequalit)o(y)l(,)h(w)o(e)f(get)606
1930 y(\001)644 1912 y Fk(2)705 1930 y Fn(\024)41 b Fu(\()p
Fo(v)12 b Fn(\001)e Fo(v)880 1912 y Fi(>)907 1930 y Fu(\))g
Fn(\001)g Fu(\(\()p Fo(uA)p Fu(\))f Fn(\001)h Fu(\()p
Fo(uA)p Fu(\))1200 1912 y Fi(>)1227 1930 y Fu(\))705
2018 y(=)796 1988 y(1)p 786 2008 42 2 v 786 2050 a Fo(K)843
2018 y Fn(\001)g Fu(\(\()902 1978 y Fd(X)925 2066 y Fm(i)969
2018 y Fo(u)995 2025 y Fm(i)1009 2018 y Fo(A)1043 2025
y Fm(i)1057 2018 y Fu(\))f Fn(\001)h Fu(\()1125 1978
y Fd(X)1149 2066 y Fm(i)1192 2018 y Fo(u)1218 2025 y
Fm(i)1232 2018 y Fo(A)1266 2025 y Fm(i)1280 2018 y Fu(\))1298
2000 y Fi(>)1326 2018 y Fu(\))0 2145 y(where)k Fo(A)164
2152 y Fm(i)191 2145 y Fu(is)g(the)f Fo(i)327 2128 y
Fk(th)374 2145 y Fu(ro)o(w)f(of)h(the)h(matrix)f Fo(A)g
Fu(and)h Fo(u)892 2152 y Fm(i)919 2145 y Fu(is)g(the)g
Fo(i)1056 2128 y Fk(th)1102 2145 y Fu(en)o(try)f(of)g(the)h(v)o(ector)e
Fo(u)p Fu(.)20 b(Using)14 b(the)f(h)o(yp)q(othesis)0
2207 y(concerning)j(the)g(inner-pro)q(duct)h(of)d(the)i(ro)o(ws)e(of)h
Fo(A)g Fu(w)o(e)g(obtain)g(the)g(b)q(ound)582 2340 y(\001)620
2321 y Fk(2)680 2340 y Fn(\024)771 2309 y Fu(1)p 762
2329 V 762 2371 a Fo(K)819 2340 y Fn(\001)841 2255 y
Fd(0)841 2330 y(@)878 2299 y(X)881 2388 y Fm(i)p Fi(6)p
Fk(=)p Fm(j)945 2340 y Fo(u)971 2347 y Fm(i)985 2340
y Fo(u)1011 2347 y Fm(j)1028 2340 y Fo(\017)1046 2321
y Fi(0)1059 2340 y Fo(M)g Fu(+)1163 2299 y Fd(X)1187
2388 y Fm(i)1231 2340 y Fo(u)1257 2321 y Fk(2)1257 2351
y Fm(i)1275 2340 y Fo(M)1324 2255 y Fd(1)1324 2330 y(A)p
0 2430 780 2 v 52 2456 a Fj(5)69 2472 y Fs(Note)g(that)g(inner-pro)q
(duct)i(of)e(complex)h(v)o(ectors)g(is)f(de\014ned)h(as)f(comp)q(onen)o
(t-wise)i(complex)f(m)o(ultiplica)q(tion)j(of)14 b(one)i(v)o(ector)0
2523 y(b)o(y)d(the)g(conjugate)h(of)f(the)g(other.)52
2554 y Fj(6)69 2570 y Fs(The)e(standard)i(form)o(ulation)g(refers)e(to)
g(matrices)h(with)g Fa(\006)p Fs(1)f(en)o(tries)h(and)g(asserts)g(that)
g(the)f(sum)g(of)g(elemen)o(ts)i(in)f(an)o(y)f Fg(L)p
Fs(-times-)0 2620 y Fg(K)k Fs(generalized)h(sub-matrix)f(is)f(b)q
(ounded)i(b)o(y)693 2588 y Fa(p)p 725 2588 103 2 v 32
x Fg(LK)s(M)t Fs(.)i(Instead,)c(our)g(form)o(ulation)h(b)q(ounds)g(the)
f(sum)f(normalized)j(b)o(y)e(the)g(area)0 2670 y(of)f(the)g(sub-matrix)
h(\(i.e.,)f(divided)i(b)o(y)f Fg(L)8 b Fa(\001)g Fg(K)s
Fs(,)k(with)i Fg(L)c Fs(=)h(1)p Fg(=\016)q Fs(\).)952
2795 y Fu(16)p eop
%%Page: 17 18
17 17 bop 680 86 a Fo(<)762 55 y(M)p 762 76 50 2 v 766
117 a(K)826 86 y Fn(\001)848 2 y Fd(0)848 76 y(@)885
86 y Fo(\017)903 67 y Fi(0)922 46 y Fd(X)934 134 y Fm(i;j)990
86 y Fo(u)1016 93 y Fm(i)1030 86 y Fo(u)1056 93 y Fm(j)1083
86 y Fu(+)1129 46 y Fd(X)1153 134 y Fm(i)1196 86 y Fo(u)1222
67 y Fk(2)1222 97 y Fm(i)1241 2 y Fd(1)1241 76 y(A)0
237 y Fu(Using)128 205 y Fd(P)172 248 y Fm(i;j)219 237
y Fo(u)245 244 y Fm(i)258 237 y Fo(u)284 244 y Fm(j)315
237 y Fu(=)13 b(\()381 205 y Fd(P)425 248 y Fm(i)446
237 y Fo(u)472 244 y Fm(i)486 237 y Fu(\))504 221 y Fk(2)535
237 y Fu(=)h(1)h(and)711 205 y Fd(P)754 248 y Fm(i)776
237 y Fo(u)802 221 y Fk(2)802 248 y Fm(i)834 237 y Fn(\024)882
205 y Fd(P)926 248 y Fm(i)947 237 y Fo(u)973 244 y Fm(i)997
237 y Fn(\001)10 b Fo(\016)15 b Fu(=)f Fo(\016)r Fu(,)h(w)o(e)g(get)g
(\001)1335 221 y Fk(2)1366 237 y Fn(\024)1420 219 y Fm(M)p
1420 226 35 2 v 1422 253 a(K)1470 237 y Fn(\001)10 b
Fu(\()p Fo(\017)1529 221 y Fi(0)1551 237 y Fu(+)g Fo(\016)r
Fu(\))15 b(and)h(the)g(lemma)0 299 y(follo)o(ws.)p 217
305 34 40 v 0 423 a(Con)o(tradiction)h(to)f(Eq.)h(\(18\))e(follo)o(ws)i
(b)o(y)g(considering)i(the)e Fo(p)1089 406 y Fm(t)1103
423 y Fu(-b)o(y-)p Fn(j)p Fo(S)s Fn(j)f Fu(matrix)h(with)g(ro)o(ws)f
(corresp)q(onding)i(to)0 485 y(elemen)o(ts)e(of)f Fo(GF)6
b Fu(\()p Fo(p)p Fu(\))368 468 y Fm(t)397 485 y Fu(and)16
b(columns)g(corresp)q(onding)g(to)e(elemen)o(ts)i(of)f
Fo(S)s Fu(.)20 b(The)15 b(\()p Fo(x;)8 b(s)p Fu(\))1509
468 y Fk(th)1557 485 y Fu(en)o(try)15 b(in)h(this)f(matrix)0
551 y(consists)j(of)f Fo(!)254 503 y Fd(P)297 513 y Fh(t)297
546 y(i)p Fc(=1)352 531 y Fm(x)371 535 y Fh(i)384 531
y Fm(s)400 535 y Fh(i)414 551 y Fu(,)h(where)g Fo(x)e
Fu(=)g(\()p Fo(x)716 558 y Fk(1)735 551 y Fo(;)8 b(:::;)g(x)842
558 y Fm(t)854 551 y Fu(\))16 b Fn(2)g Fo(GF)6 b Fu(\()p
Fo(p)p Fu(\))1064 535 y Fm(t)1096 551 y Fu(and)17 b Fo(s)g
Fu(=)f(\()p Fo(s)1314 558 y Fk(1)1333 551 y Fo(;)8 b(:::;)g(s)1435
558 y Fm(t)1447 551 y Fu(\))16 b Fn(2)h Fo(S)s Fu(.)26
b(Let)17 b Fo(u)h Fu(b)q(e)g(a)f(v)o(ector)0 613 y(describing)22
b(the)e(probabilit)o(y)i(distribution)g(of)e(the)g(random)g(v)m
(ariable)i Fo(X)h Fu(\(i.e.,)e Fo(u)1479 620 y Fm(x)1521
613 y Fu(=)g(Prob\()p Fo(X)16 b Fu(=)e Fo(x)p Fu(\)\))20
b(and)0 676 y Fo(\016)c Fu(=)f(2)109 659 y Fi(\000)p
Fm(k)171 676 y Fu(\(the)h(upp)q(er)h(b)q(ound)g(on)f(probabilit)o(y)i
(for)d Fo(X)t Fu(\).)22 b(Let)16 b Fo(v)i Fu(b)q(e)f(the)f
(\(normalized\))h(v)o(ector)e(c)o(haracterizing)0 738
y(the)d(set)h Fo(B)i Fu(\(i.e.,)d Fo(v)316 745 y Fm(s)346
738 y Fu(equals)501 720 y Fk(1)p 486 727 47 2 v 486 753
a Fi(j)p Fm(B)q Fi(j)550 738 y Fu(if)h Fo(s)g Fn(2)g
Fo(B)i Fu(and)d(0)g(otherwise\).)19 b(Note)12 b(that)g(the)g(inner-pro)
q(duct)i(of)e(di\013eren)o(t)h(ro)o(ws)0 816 y(corresp)q(onding)g(to)e
(sequences)h Fo(x)h Fu(=)g(\()p Fo(x)674 823 y Fk(1)692
816 y Fo(;)8 b(:::;)g(x)799 823 y Fm(t)811 816 y Fu(\))j(and)h
Fo(y)j Fu(=)e(\()p Fo(y)1050 823 y Fk(1)1068 816 y Fo(;)8
b(:::;)g(y)1171 823 y Fm(t)1183 816 y Fu(\))k(equals)1347
784 y Fd(P)1391 827 y Fm(s)p Fi(2)p Fm(S)1460 816 y Fo(!)1490
767 y Fd(P)1534 777 y Fh(t)1534 811 y(i)p Fc(=1)1583
795 y Fk(\()p Fm(x)1615 799 y Fh(i)1628 795 y Fi(\000)p
Fm(y)1671 799 y Fh(i)1684 795 y Fk(\))p Fi(\001)p Fm(s)1723
799 y Fh(i)1737 816 y Fu(,)g(whic)o(h,)h(b)o(y)0 885
y(construction)j(of)g(the)g(sample)h(space)f Fo(S)s Fu(,)f(has)h
(magnitude)h(b)q(ounded)g(b)o(y)f Fo(\017)1303 868 y
Fi(0)1315 885 y Fn(j)p Fo(S)s Fn(j)p Fu(.)22 b(Letting)16
b Fo(A)e Fu(=)h(\()p Fo(!)1713 836 y Fd(P)1756 846 y
Fh(t)1756 880 y(i)p Fc(=1)1810 864 y Fm(x)1829 868 y
Fh(i)1842 864 y Fm(s)1858 868 y Fh(i)1873 885 y Fu(\))1891
892 y Fm(x;s)1937 885 y Fu(,)0 947 y(w)o(e)g(ha)o(v)o(e)409
1082 y Fn(k)p Fo(uAv)516 1064 y Fi(>)544 1082 y Fn(k)41
b Fu(=)685 996 y Fd(\015)685 1021 y(\015)685 1046 y(\015)685
1071 y(\015)685 1096 y(\015)685 1121 y(\015)753 1042
y(X)708 1133 y Fm(x)p Fi(2)p Fm(GF)t Fk(\()p Fm(p)p Fk(\))843
1125 y Fh(t)865 1042 y Fd(X)865 1131 y Fm(s)p Fi(2)p
Fm(S)932 1082 y Fo(u)958 1089 y Fm(x)989 1082 y Fn(\001)10
b Fo(!)1042 1034 y Fd(P)1086 1044 y Fh(t)1086 1078 y(i)p
Fc(=1)1140 1062 y Fm(x)1159 1066 y Fh(i)1172 1062 y Fm(s)1188
1066 y Fh(i)1213 1082 y Fn(\001)g Fo(v)1258 1089 y Fm(s)1276
996 y Fd(\015)1276 1021 y(\015)1276 1046 y(\015)1276
1071 y(\015)1276 1096 y(\015)1276 1121 y(\015)608 1249
y Fu(=)685 1162 y Fd(\015)685 1187 y(\015)685 1212 y(\015)685
1237 y(\015)685 1262 y(\015)685 1287 y(\015)753 1208
y(X)708 1299 y Fm(x)p Fi(2)p Fm(GF)t Fk(\()p Fm(p)p Fk(\))843
1291 y Fh(t)867 1208 y Fd(X)865 1297 y Fm(s)p Fi(2)p
Fm(B)962 1218 y Fu(1)p 942 1238 63 2 v 942 1280 a Fn(j)p
Fo(B)r Fn(j)1019 1249 y(\001)g Fu(Prob)o(\()p Fo(X)e
Fu(=)d Fo(x)p Fu(\))11 b Fn(\001)e Fo(!)1349 1200 y Fd(P)1393
1210 y Fh(t)1393 1244 y(i)p Fc(=1)1447 1228 y Fm(x)1466
1232 y Fh(i)1479 1228 y Fm(s)1495 1232 y Fh(i)1510 1162
y Fd(\015)1510 1187 y(\015)1510 1212 y(\015)1510 1237
y(\015)1510 1262 y(\015)1510 1287 y(\015)608 1402 y Fu(=)685
1329 y Fd(\015)685 1353 y(\015)685 1378 y(\015)685 1403
y(\015)685 1428 y(\015)710 1362 y(X)708 1451 y Fm(s)p
Fi(2)p Fm(B)805 1371 y Fu(1)p 785 1392 V 785 1433 a Fn(j)p
Fo(B)r Fn(j)862 1402 y(\001)h Fu(Exp\()p Fo(!)1013 1354
y Fd(P)1056 1364 y Fh(t)1056 1397 y(i)p Fc(=1)1111 1382
y Fm(X)1138 1386 y Fh(i)1151 1382 y Fm(s)1167 1386 y
Fh(i)1182 1402 y Fu(\))1200 1329 y Fd(\015)1200 1353
y(\015)1200 1378 y(\015)1200 1403 y(\015)1200 1428 y(\015)0
1541 y Fu(W)l(e)22 b(are)g(no)o(w)g(ready)g(to)g(apply)h(Lemma)f(5.2:)
33 b(Here,)24 b Fo(M)29 b Fu(=)c Fn(j)p Fo(S)s Fn(j)p
Fu(,)d Fo(K)28 b Fu(=)c Fn(j)p Fo(B)r Fn(j)h(\025)1512
1523 y Fm(\017)p 1502 1530 34 2 v 1502 1557 a Fk(4)p
Fm(p)1556 1541 y Fn(\001)14 b(j)p Fo(S)s Fn(j)21 b Fu(\(b)o(y)h(Eq.)g
(\(17\)\),)0 1616 y Fo(\016)j Fu(=)e(2)126 1599 y Fi(\000)p
Fm(k)195 1616 y Fn(\024)268 1598 y Fm(\017)282 1585 y
Fc(3)p 258 1605 50 2 v 258 1631 a Fk(4)p Fm(p)292 1623
y Fc(2)334 1616 y Fu(\(b)o(y)e(Theorem's)g(h)o(yp)q(othesis)h(and)f
(assuming)h Fo(p)h Fn(\031)g Fu(2)1305 1599 y Fm(m)1336
1616 y Fu(\),)f(and)g Fo(\017)1502 1599 y Fi(0)1537 1591
y Fk(def)1542 1616 y Fu(=)1620 1598 y Fm(\017)1634 1585
y Fc(3)p 1610 1605 V 1610 1631 a Fk(5)p Fm(p)1644 1623
y Fc(2)1686 1616 y Fu(\(de\014ned)h(here)0 1678 y(preserving)16
b Fo(\017)237 1661 y Fi(0)262 1678 y Fu(=)d(p)q(oly)q(\()p
Fo(\017=p)p Fu(\)\).)19 b(Applying)e(the)e(lemma)h(w)o(e)f(get)453
1739 y Fd(\015)453 1764 y(\015)453 1789 y(\015)453 1814
y(\015)453 1839 y(\015)453 1864 y(\015)500 1794 y Fu(1)p
481 1815 63 2 v 481 1856 a Fn(j)p Fo(B)r Fn(j)618 1785
y Fd(X)555 1876 y Fk(\()p Fm(s)584 1880 y Fc(1)600 1876
y Fm(;:::;s)666 1880 y Fh(t)679 1876 y Fk(\))p Fi(2)p
Fm(B)748 1825 y Fu(Exp\()p Fo(!)876 1777 y Fd(P)920 1787
y Fh(t)920 1820 y(i)p Fc(=1)974 1805 y Fm(X)1001 1809
y Fh(i)1014 1805 y Fm(s)1030 1809 y Fh(i)1045 1825 y
Fu(\))1063 1739 y Fd(\015)1063 1764 y(\015)1063 1789
y(\015)1063 1814 y(\015)1063 1839 y(\015)1063 1864 y(\015)1128
1825 y Fn(\024)1204 1745 y Fd(s)p 1246 1745 235 2 v 80
x Fu(\()p Fo(\017)1282 1812 y Fi(0)1304 1825 y Fu(+)10
b Fo(\016)r Fu(\))g Fn(\001)1427 1794 y Fo(M)p 1427 1815
50 2 v 1431 1856 a(K)1128 1992 y(<)1204 1908 y Fd(v)1204
1932 y(u)1204 1957 y(u)1204 1981 y(t)p 1248 1908 250
2 v 1267 1961 a Fo(\017)1285 1948 y Fk(3)p 1253 1981
65 2 v 1253 2023 a Fu(2)p Fo(p)1299 2010 y Fk(2)1333
1992 y Fn(\001)1398 1961 y(j)p Fo(S)s Fn(j)p 1360 1981
133 2 v 1375 2005 a Fm(\017)p 1365 2012 34 2 v 1365 2039
a Fk(4)p Fm(p)1414 2023 y Fn(\001)g(j)p Fo(S)s Fn(j)1128
2123 y Fu(=)1204 2083 y Fn(p)p 1242 2083 23 2 v 40 x
Fu(2)g Fn(\001)1324 2092 y Fo(\017)p 1303 2113 61 2 v
1303 2126 a Fn(p)p 1341 2126 23 2 v 28 x Fo(p)0 2254
y Fu(whic)o(h)16 b(con)o(tradicts)f(Eq.)g(\(18\).)71
2316 y(There)j(is)h(still)h(a)e(minor)g(tec)o(hnicalit)o(y)i(to)d(b)q
(e)i(addressed:)27 b(ho)o(w)18 b(do)g(w)o(e)g(ac)o(hiev)o(e)h(a)f
(mapping)h(to)e Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)1920
2299 y Fm(m)0 2378 y Fu(\(rather)13 b(than)h(to)g Fo(GF)6
b Fu(\()p Fo(p)p Fu(\)\).)19 b(This)14 b(is)h(resolv)o(ed)g(b)o(y)f
(letting)h Fo(p)d Fn(\031)h Fu(2)1125 2361 y Fm(m)1157
2378 y Fo(=\017)h Fu(\(still)h Fo(p)1340 2361 y Fm(t)1367
2378 y Fn(\031)e Fu(2)1438 2361 y Fm(n)1461 2378 y Fu(\))h(and)g
(mapping)h Fo(GF)6 b Fu(\()p Fo(p)p Fu(\))14 b(to)0 2440
y Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)113 2423 y Fm(m)156
2440 y Fu(in)15 b(the)f(natural)g(manner)f(\(i.e.,)h(eac)o(h)g(range)f
(elemen)o(t)i(ha)o(ving)f Fo(\017)1261 2423 y Fi(\000)p
Fk(1)1313 2440 y Fn(\006)7 b Fu(1)14 b(elemen)o(ts\).)20
b(The)14 b(increase)h(in)f Fo(p)0 2502 y Fu(only)f(e\013ects)f(the)h
(condition)g(2)527 2486 y Fi(\000)p Fm(k)586 2502 y Fn(\024)649
2484 y Fm(\017)663 2472 y Fc(3)p 639 2491 50 2 v 639
2518 a Fk(4)p Fm(p)673 2509 y Fc(2)706 2502 y Fu(whic)o(h)h(still)g
(holds)f(assuming)f Fo(p)h(<)g Fu(2)1337 2486 y Fm(m)p
Fk(+1)1410 2502 y Fo(=\017)g Fu(and)f(using)i(the)e(h)o(yp)q(othesis)0
2564 y Fo(\017)i(>)g Fu(2)104 2548 y Fi(\000)p Fk(\()p
Fm(k)q Fi(\000)p Fk(2)p Fm(m)p Fi(\000)p Fk(4\))p Fm(=)p
Fk(5)324 2564 y Fu(.)22 b(The)16 b(appro)o(ximate)f(mapping)i(of)e
Fo(GF)6 b Fu(\()p Fo(p)p Fu(\))16 b(to)f Fn(f)p Fu(0)p
Fo(;)8 b Fu(1)p Fn(g)1271 2548 y Fm(m)1316 2564 y Fu(yields)18
b(and)e(additional)h(extraction)0 2626 y(deviation)f(of)f
Fo(\017)h Fu(\(totaling)e(to)h(2)p Fo(\017)p Fu(\).)20
b(Substituting)c Fo(\017=)p Fu(2)f(for)g Fo(\017)p Fu(,)g(the)g
(theorem)g(follo)o(ws.)952 2795 y(17)p eop
%%Page: 18 19
18 18 bop 0 42 a Fl(5.4)56 b(Lo)n(w)n(er)19 b(Bound)0
133 y Fu(T)l(o)h(illustrate)h(that)e(our)h(construction)g(is)h(near)e
(optimal)i(when)f Fo(k)i Fu(=)f Fo(O)q Fu(\(log)8 b Fo(n)p
Fu(\))20 b(w)o(e)g(recall)h(a)e(lo)o(w)o(er)h(b)q(ound)0
195 y(of)e([29)o(])g(\(already)g(men)o(tioned)h(in)g(Section)h(4.4\).)
27 b(W)l(e)19 b(stress)f(that)f(the)i(b)q(ound)g(holds)g(ev)o(en)f
(when)h(trying)g(to)0 257 y(extract)14 b(just)h(one)h(bit.)0
369 y Fv(Theorem)h(5.3)22 b Fu([29,)14 b(Thm.)g(3]:)20
b Fp(A)c(family)f(of)i(functions)e(fr)n(om)h Fn(f)p Fu(0)p
Fo(;)8 b Fu(1)p Fn(g)1247 353 y Fm(n)1283 369 y Fp(to)16
b Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)p Fp(,)14 b(with)j(extr)n(action)f
(pr)n(op)n(erty)0 431 y(of)i(ac)n(cur)n(acy)f Fo(\017)e(<)h
Fu(0)p Fo(:)p Fu(5)g Fp(with)i(r)n(esp)n(e)n(ct)e(to)i(r)n(andom)g
(variables)f(of)g(min-entr)n(opy)h Fo(k)e Fn(\024)f Fo(n)c
Fn(\000)g Fu(1)p Fp(,)18 b(must)f(have)h(size)f(at)0
493 y(le)n(ast)e Fu(max)p Fn(f)p Fo(n)10 b Fn(\000)h
Fo(k)g Fu(+)f(1)p Fo(;)e Fu(\(1)p Fo(=\017)p Fu(\))g
Fn(\000)j Fu(1)p Fn(g)p Fp(.)0 605 y Fu(W)l(e)k(note)g(that)g(the)g
Fo(B)r(P)6 b(P)23 b Fu(sim)o(ulation)16 b(of)f([33)o(])g(men)o(tioned)h
(in)g(the)f(in)o(tro)q(duction)i(uses)e(an)g(extracting)g(family)0
667 y(for)g(this)g(v)m(alue)i(of)d(the)i(parameter)e
Fo(k)q Fu(.)0 818 y Fq(6)67 b(A)22 b(New)f(Sampling)k(Pro)r(cedure)0
925 y Fu(In)17 b(man)o(y)g(settings)f(rep)q(eated)i(sampling)f(is)h
(used)f(to)f(estimate)h(the)f(a)o(v)o(erage)g(v)m(alue)i(of)e(a)h(h)o
(uge)f(set)h(of)f(v)m(alues.)0 987 y(Namely)l(,)e(there)g(is)f(a)h(v)m
(alue)g(function)g Fo(\027)j Fu(de\014ned)e(o)o(v)o(er)d(a)i(h)o(uge)f
(space,)h(sa)o(y)f Fo(\027)8 b Fu(:)d Fn(f)p Fu(0)p Fo(;)j
Fu(1)p Fn(g)1462 971 y Fm(n)1487 987 y Fn(7!)d Fu([0)p
Fo(;)j Fu(1],)k(and)i(one)f(wishes)0 1049 y(to)h(appro)o(ximate)j(\026)
-25 b Fo(\027)356 1024 y Fk(def)361 1049 y Fu(=)429 1031
y Fk(1)p 419 1038 37 2 v 419 1065 a(2)436 1057 y Fh(n)468
1017 y Fd(P)512 1061 y Fm(x)p Fi(2f)p Fk(0)p Fm(;)p Fk(1)p
Fi(g)631 1052 y Fh(n)660 1049 y Fo(\027)s Fu(\()p Fo(x)p
Fu(\).)20 b(T)l(o)14 b(this)i(end,)f(one)g(ma)o(y)f(randomly)h(select)h
(a)f(small)g(sample)h(set)f Fo(S)0 1111 y Fu(and)c(compute)282
1094 y Fk(1)p 269 1101 42 2 v 269 1127 a Fi(j)p Fm(S)r
Fi(j)323 1079 y Fd(P)367 1123 y Fm(x)p Fi(2)p Fm(S)440
1111 y Fo(\027)s Fu(\()p Fo(x)p Fu(\).)18 b(Using)11
b(a)g(sample)g(of)g Fo(O)q Fu(\(1)p Fo(=\017)1028 1095
y Fk(2)1046 1111 y Fu(\))f(uniformly)i(and)f(indep)q(enden)o(tly)j
(selected)e(p)q(oin)o(ts,)0 1174 y(one)18 b(gets,)f(with)h(constan)o(t)
e(probabilit)o(y)l(,)j(an)f(appro)o(ximation)f(that)g(is)h(within)g(an)
g(additiv)o(e)g(factor)f(of)g Fo(\017)h Fu(from)0 1236
y(the)c(correct)f(a)o(v)o(erage.)18 b(In)c(fact,)f(a)g(set)g(of)g
Fo(O)q Fu(\(1)p Fo(=\017)836 1219 y Fk(2)854 1236 y Fu(\))g(p)q(oin)o
(ts)h(selected)h(in)f(a)f(pairwise-indep)q(end)q(en)o(t)j(and)e
(uniform)0 1298 y(manner)d(yields)h(the)e(same)g(qualit)o(y)h(of)f
(appro)o(ximation.)18 b(Whereas)11 b(generating)f Fo(t)h
Fu(totally)g(indep)q(enden)o(t)h(random)0 1360 y(p)q(oin)o(ts)j(in)g
Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)301 1343 y Fm(n)337
1360 y Fu(requires)15 b Fo(t)9 b Fn(\001)g Fo(n)14 b
Fu(un)o(biased)i(coin)f(\015ips,)h(one)e(can)h(generate)f
Fo(t)h Fu(pairwise-indep)q(en)q(den)o(t)i(random)0 1422
y(p)q(oin)o(ts)f(using)g(only)g(2)10 b Fn(\001)g Fo(n)16
b Fu(un)o(biased)h(coin)f(\015ips)h([11)o(].)j(Using)d(the)e(new)h
(family)g(of)f(mixing)i(functions,)f(w)o(e)f(w)o(ere)0
1484 y(able)g(to)f(reduce)h(the)g(randomness)f(complexit)o(y)h(of)f
(the)h(appro)o(ximation)f(problem)h(to)f Fo(n)9 b Fu(+)g
Fo(O)q Fu(\(log\(1)p Fo(=\017)p Fu(\)\),)k(while)0 1546
y(main)o(taining)j(the)g(n)o(um)o(b)q(er)f(of)g(sample)h(p)q(oin)o(ts)f
(\(up-to)g(a)g(m)o(ultiplicativ)o(e)j(constan)o(t\).)0
1658 y Fv(De\014nition)h(6.1)k Fu(\(sampler\):)18 b Fp(A)11
b Fe(sampler)e Fp(is)i(a)h(r)n(andomize)n(d)f(algorithm)i(that)f(on)f
(input)h(p)n(ar)n(ameters)g Fo(n)g Fu(\(length\))p Fp(,)0
1720 y Fo(\017)k Fu(\(accuracy\))g Fp(and)f Fo(\016)j
Fu(\(error\))p Fp(,)d(and)g(or)n(acle)h(ac)n(c)n(ess)e(to)i
Fu(an)o(y)g Fp(function)f Fo(\027)9 b Fu(:)c Fn(f)p Fu(0)p
Fo(;)j Fu(1)p Fn(g)1381 1704 y Fm(n)1406 1720 y Fn(7!)d
Fu([0)p Fo(;)j Fu(1])p Fp(,)14 b(outputs,)j(with)g(pr)n(ob-)0
1782 y(ability)f(at)h(le)n(ast)e Fu(1)10 b Fn(\000)g
Fo(\016)r Fp(,)16 b(a)h(value)f(that)h(is)f(at)h(most)f
Fo(\017)h Fp(away)g(fr)n(om)i Fu(\026)-26 b Fo(\027)1184
1757 y Fk(def)1189 1782 y Fu(=)1258 1764 y Fk(1)p 1247
1771 37 2 v 1247 1798 a(2)1264 1790 y Fh(n)1297 1750
y Fd(P)1341 1794 y Fm(x)p Fi(2f)p Fk(0)p Fm(;)p Fk(1)p
Fi(g)1460 1785 y Fh(n)1489 1782 y Fo(\027)s Fu(\()p Fo(x)p
Fu(\))p Fp(.)20 b(Namely,)603 1890 y Fu(Prob\()p Fn(j)p
Fu(sampler)885 1869 y Fm(\027)905 1890 y Fu(\()p Fo(n;)8
b(\017;)g(\016)r Fu(\))h Fn(\000)k Fu(\026)-26 b Fo(\027)t
Fn(j)12 b Fo(>)h(\017)p Fu(\))25 b Fo(<)h(\016)0 1998
y Fp(wher)n(e)16 b(the)h(pr)n(ob)n(ability)f(is)g(taken)g(over)g(the)h
(internal)e(c)n(oin)g(tosses)h(of)g(the)h(sampler.)0
2110 y Fv(Theorem)g(6.2)22 b Fp(Ther)n(e)16 b(exists)g(a)g
Fu(p)q(oly)r(\()p Fo(n;)8 b(\017)777 2093 y Fi(\000)p
Fk(1)821 2110 y Fo(;)g(\016)864 2093 y Fi(\000)p Fk(1)907
2110 y Fu(\))p Fp(-time)17 b(sampler)f(which)68 2209
y Fn(\017)23 b Fp(makes)16 b Fo(O)q Fu(\()324 2191 y
Fk(1)p 309 2198 47 2 v 309 2225 a Fm(\016)q(\017)339
2216 y Fc(2)360 2209 y Fu(\))g Fp(or)n(acle)g(queries;)g(and)68
2309 y Fn(\017)23 b Fp(tosses)15 b Fo(n)10 b Fu(+)h Fo(O)q
Fu(\(log\(1)p Fo(=\017)p Fu(\)\))e(+)i Fo(O)q Fu(\(log\(1)p
Fo(=\016)r Fu(\)\))k Fp(c)n(oins.)0 2421 y Fu(Our)22
b(original)h(pro)q(of)f(of)f(Theorem)h(6.2)f(used)h(the)g(mixing)h
(functions)g(guaran)o(teed)e(b)o(y)h(Theorem)f(3.1.)39
b(In)0 2483 y(retrosp)q(ect,)12 b(w)o(e)f(realized)i(that)e(the)g
(construction)h(amoun)o(ts)f(to)g(uniformly)h(selecting)h(a)e(v)o
(ertex)g(in)i(an)e(expander)0 2545 y(ha)o(ving)g(v)o(ertex)g(set)g
Fn(f)p Fu(0)p Fo(;)d Fu(1)p Fn(g)456 2528 y Fm(n)487
2545 y Fu(and)k(degree)f(p)q(oly)r(\(1)p Fo(=\017\016)r
Fu(\),)f(and)i(a)o(v)o(eraging)e(o)o(v)o(er)g(the)h(v)m(alues)h(observ)
o(ed)f(in)h(a)f Fp(p)n(airwise)0 2607 y(indep)n(endent)20
b Fo(O)q Fu(\(1)p Fo(=\017)375 2590 y Fk(2)393 2607 y
Fo(\016)r Fu(\)-long)e(sequence)g(of)f(the)g(neigh)o(b)q(ors)h(of)f
(this)h(v)o(ertex.)25 b(F)l(urthermore,)17 b(for)g(the)g(sp)q(ecial)0
2669 y(case)e(of)g(Bo)q(olean)g(functions,)h(an)f(optimal)g(sampler)h
(utilizes)h(a)d(Raman)o(ujan)h(graph)g(of)g(degree)g
Fo(O)q Fu(\(1)p Fo(=\017)1804 2652 y Fk(2)1822 2669 y
Fo(\016)r Fu(\))g(and)952 2795 y(18)p eop
%%Page: 19 20
19 19 bop 0 2 2066 2 v -1 64 2 63 v 230 64 V 256 45 a
Fu(lo)o(w)o(er)14 b(b)q(ound)i([8])p 1037 64 V 500 w(upp)q(er)g(b)q
(ound)g([8])p 1437 64 V 81 w(algorithms)f(\(this)g(pap)q(er\))p
2065 64 V 0 65 2066 2 v 0 75 V -1 137 2 63 v 25 119 a(Bo)q(olean)p
230 137 V 72 w Fo(n)10 b Fu(+)g(log)397 130 y Fk(2)416
119 y Fu(\(1)p Fo(=\016)r Fu(\))p 1037 137 V 543 w Fo(n)h
Fu(+)f(2)e(log)1235 130 y Fk(2)1254 119 y Fu(\(2)p Fo(=\016)r
Fu(\))p 1437 137 V 105 w Fo(n)i Fu(+)h Fo(O)q Fu(\(log\(1)p
Fo(=\016)r Fu(\)\))44 b(\(Thm.)14 b(6.5\))p 2065 137
V -1 200 V 25 181 a(functions)p 230 200 V 96 w Fn(\000)p
Fu(2)8 b(log)425 192 y Fk(2)444 181 y Fu(\(1)p Fo(=\017)p
Fu(\))h Fn(\000)i Fu(log)657 192 y Fk(2)684 181 y Fu(log)742
192 y Fk(2)761 181 y Fu(\(1)p Fo(=\016)r Fu(\))e Fn(\000)h
Fo(O)q Fu(\(1\))p 1037 200 V 1437 200 V 2065 200 V 0
201 2066 2 v -1 263 2 63 v 25 245 a(general)p 230 263
V 90 w Fo(n)g Fu(+)g(log)397 256 y Fk(2)416 245 y Fu(\(1)p
Fo(=\016)r Fu(\))p 1037 263 V 543 w Fo(n)h Fu(+)f(2)e(log)1235
256 y Fk(2)1254 245 y Fu(\(2)p Fo(=\016)r Fu(\))p 1437
263 V 105 w Fo(n)i Fu(+)h Fo(O)q Fu(\(log\(1)p Fo(=\016)r
Fu(\)\))p 2065 263 V -1 325 V 25 307 a(functions)p 230
325 V 96 w Fn(\000)p Fu(2)d(log)425 318 y Fk(2)444 307
y Fu(\(1)p Fo(=\017)p Fu(\))h Fn(\000)i Fu(log)657 318
y Fk(2)684 307 y Fu(log)742 318 y Fk(2)761 307 y Fu(\(1)p
Fo(=\016)r Fu(\))e Fn(\000)h Fo(O)q Fu(\(1\))p 1037 325
V 95 w(+)e(log)1210 318 y Fk(2)1236 307 y Fu(log)1295
318 y Fk(2)1314 307 y Fu(\(1)p Fo(=\017)p Fu(\))p 1437
325 V 94 w(+)p Fo(O)q Fu(\(log)q(\(1)p Fo(=\017)p Fu(\)\))44
b(\(Cor.)14 b(6.3\))p 2065 325 V 0 327 2066 2 v 0 416
a(Figure)k(1:)26 b(The)19 b(randomness)f(complexit)o(y)h(of)f(samplers)
g(whic)o(h)h(mak)o(e)f(\002\()1348 395 y Fk(log\(1)p
Fm(=\016)q Fk(\))p 1348 405 119 2 v 1393 431 a Fm(\017)1407
423 y Fc(2)1472 416 y Fu(\))g(queries.)30 b(The)18 b(previous)0
478 y(b)q(est)d(algorithm)h(\(b)o(y)e([7]\))g(had)i(randomness)f
(complexit)o(y)h(2)p Fo(n)10 b Fu(+)g Fo(O)q Fu(\(log)q(\(1)p
Fo(=\016)r Fu(\)\),)j(pro)o(vided)j Fo(\017)d(>)g Fu(2)1689
461 y Fi(\000)p Fm(n)1737 478 y Fu(.)0 608 y(consists)i(of)f(uniformly)
h(selecting)h(a)e(single)i(v)o(ertex)e(and)g(outputting)h(the)f(a)o(v)o
(erage)g(of)g(the)g(function)h(v)m(alues)h(on)0 670 y
Fp(al)r(l)k Fu(its)15 b(neigh)o(b)q(ors.)21 b(Both)15
b(these)g(constructions)g(are)g(analyzed)h(b)q(elo)o(w.)71
733 y(The)d(sampler)h(guaran)o(teed)f(in)i(Theorem)e(6.2)g(is)h(not)f
(optimal)h(in)h(its)e(query)h(complexit)o(y)h(\(sp)q(eci\014cally)l(,)h
(the)0 795 y(dep)q(endency)i(on)e Fo(\016)h Fu(is)f(bad\).)22
b(Ho)o(w)o(ev)o(er,)14 b(this)i(can)g(b)q(e)h(easily)f(redeemed)h
(using)f(a)g(generic)g(metho)q(d)g(of)g(Bellare)0 857
y(et)f(al)h([7)o(]:)k(Giv)o(en)c(a)f(sampler)g(of)g(query)h(complexit)o
(y)g Fo(q)r Fu(\()p Fo(n;)8 b(\017;)g(\016)r Fu(\))13
b(and)j(randomness)f(complexit)o(y)h Fo(r)q Fu(\()p Fo(n;)8
b(\017;)g(\016)r Fu(\))14 b(they)0 919 y(obtain)21 b(a)g(sampler)h(ha)o
(ving)g(query)f(complexit)o(y)h Fo(O)q Fu(\()p Fo(q)r
Fu(\()p Fo(n;)8 b(\017;)g Fu(0)p Fo(:)p Fu(1\))j Fn(\001)j
Fu(log)q(\(1)p Fo(=\016)r Fu(\)\))19 b(and)j(randomness)f(complexit)o
(y)0 981 y Fo(r)q Fu(\()p Fo(n;)8 b(\017;)g Fu(0)p Fo(:)p
Fu(1\))f(+)k Fo(O)q Fu(\(log\(1)p Fo(=\016)r Fu(\)\).)18
b(Applying)f(their)f(metho)q(d)g(to)e(the)h(sampler)h(of)f(Theorem)g
(6.2,)f(w)o(e)h(obtain)0 1083 y Fv(Corollary)j(6.3)k
Fp(Ther)n(e)16 b(exists)f(a)i Fu(p)q(oly)q(\()p Fo(n;)8
b(\017)787 1067 y Fi(\000)p Fk(1)831 1083 y Fo(;)g Fu(log\(1)p
Fo(=\016)r Fu(\)\))p Fp(-time)15 b(sampler)i(which)68
1176 y Fn(\017)23 b Fp(makes)16 b Fo(O)q Fu(\()310 1155
y Fk(log)o(\(1)p Fm(=\016)q Fk(\))p 309 1166 V 354 1192
a Fm(\017)368 1184 y Fc(2)433 1176 y Fu(\))g Fp(or)n(acle)g(queries;)g
(and)68 1273 y Fn(\017)23 b Fp(tosses)15 b Fo(n)10 b
Fu(+)h Fo(O)q Fu(\(log\(1)p Fo(=\017)p Fu(\)\))e(+)i
Fo(O)q Fu(\(log\(1)p Fo(=\016)r Fu(\)\))k Fp(c)n(oins.)0
1376 y Fu(This)j(sampler)f(is)h(optimal)g(\(up)f(to)f(a)h(m)o
(ultiplicativ)o(e)j(factor\))c(in)i(its)f(sample-complexit)o(y)i([8)o
(,)e(Thm.)g(1],)g(and)0 1438 y(among)f(the)h(samplers)g(with)g(nearly)h
(optimal)f(sample)g(complexit)o(y)h(it)f(is)h(optimal)f(\(up)g(to)f
(the)h(additiv)o(e)h(log-)0 1500 y(arithmic)j(factors\))e(in)i(its)f
(randomness-complexit)o(y)h([8)o(,)g(Thm.)f(2].)34 b(Previously)l(,)22
b Fp(e\016cient)i Fu(samplers)d(with)0 1562 y(optimal)14
b(sample-complexit)o(y)i(w)o(ere)e(kno)o(wn)g(only)g(for)f(t)o(wice)i
(the)f(optimal)g(randomness-complexit)o(y)h([7])e(\(y)o(et,)0
1624 y([8)o(])g(ha)o(v)o(e)g(pro)o(v)o(ed)g(non-constructiv)o(ely)i
(that)d(\\samplers")i(with)f(sample)h(and)g(randomness)f(complexities)i
(as)e(in)0 1686 y(the)i(corollary)h(do)f(exist)422 1670
y Fk(7)441 1686 y Fu(\).)k(The)d(kno)o(wn)f(results)g(are)g(summarized)
h(in)g(Figure)g(1.)0 1811 y Fv(Remark:)44 b Fu(The)16
b(randomness-complexit)o(y)h(of)e(the)h(sampler)g(asserted)g(in)g
(Corollary)g(6.3)f(can)g(b)q(e)i(impro)o(v)o(ed)0 1873
y(from)12 b Fo(n)5 b Fu(+)g Fo(O)q Fu(\(log)q(\(1)p Fo(=\017)p
Fu(\)\))g(+)g Fo(O)q Fu(\(log\(1)p Fo(=\016)r Fu(\)\))11
b(to)h Fo(n)5 b Fu(+)g Fo(O)q Fu(\(log)q(\(1)p Fo(=\017)p
Fu(\)\))g(+)g(\(2)g(+)g Fo(o)p Fu(\(1\)\))g Fn(\001)g
Fu(log)1364 1884 y Fk(2)1383 1873 y Fu(\(1)p Fo(=\016)r
Fu(\)\).)17 b(This)c(is)g(done)h(b)o(y)e(using)0 1935
y(a)k(Raman)o(ujan)f(Expander)h(\(rather)f(than)h(an)g(arbitrary)f
(expander\))h(in)h(the)f(metho)q(d)g(of)f(Bellare)i(et)f(al)g([7].)21
b(A)0 1997 y(similar)14 b(commen)o(t)f(holds)g(with)h(resp)q(ect)f(to)g
(Theorem)g(6.5)f(\(b)q(elo)o(w\),)h(where)g(the)g(randomness)g
(complexit)o(y)h(can)0 2059 y(b)q(e)i(impro)o(v)o(ed)f(to)g
Fo(n)10 b Fu(+)h(\(2)e(+)i Fo(o)p Fu(\(1\)\))d Fn(\001)i
Fu(log)684 2070 y Fk(2)703 2059 y Fu(\(1)p Fo(=\016)r
Fu(\))j(\(rather)i(than)g Fo(n)10 b Fu(+)h Fo(O)q Fu(\(log\(1)p
Fo(=\016)r Fu(\)\)\).)0 2185 y Fl(6.1)56 b(A)19 b(Sampler)e(for)h(the)g
(Bo)r(olean)g(Case)0 2277 y Fu(W)l(e)f(start)e(b)o(y)i(presen)o(ting)g
(a)f(sampler)h(for)f(the)g(sp)q(ecial)j(case)d(of)h(Bo)q(olean)g
(functions.)25 b(This)17 b(simpler)h(sampler)0 2339 y(has)g(ev)o(en)g
(lo)o(w)o(er)f(randomness)g(complexit)o(y)i(\(sp)q(eci\014cally)h
Fo(n)e Fu(instead)g(of)g Fo(n)12 b Fu(+)g Fo(O)q Fu(\(log\(1)p
Fo(=\017)p Fu(\)\)\).)26 b(Our)18 b(sampling)0 2401 y(pro)q(cedure)d
(is)f(exactly)g(the)g(one)g(suggested)f(b)o(y)h(Karp,)g(Pippinger)h
(and)f(Sipser)h(for)e(hitting)i(a)e(witness)h(set)g([23)o(],)0
2463 y(y)o(et)i(the)g(analysis)h(is)g(somewhat)e(more)h(in)o(v)o(olv)o
(ed.)24 b(F)l(urthermore,)15 b(to)h(get)g(an)g(algorithm)g(whic)o(h)h
(samples)g(the)0 2525 y(univ)o(erse)e(only)g(on)f Fo(O)q
Fu(\(1)p Fo(=\016)r(\017)477 2509 y Fk(2)495 2525 y Fu(\))g(p)q(oin)o
(ts,)h(it)f(is)h(crucial)g(to)f(use)h(a)e(Raman)o(ujan)i(graph)f(in)h
(role)f(of)g(the)g(expander)h(in)0 2587 y(the)g(Karp-Pippinger-Sip)q
(ser)j(metho)q(d.)p 0 2624 780 2 v 52 2654 a Fj(7)69
2670 y Fs(Actually)m(,)c(the)f(non-constructiv)o(e)j(upp)q(er)e(b)q
(ound)g(of)f([8,)f(Cor.)h(2])f(is)i(sligh)o(tly)i(b)q(etter)d(than)g
(the)h(result)g(of)e(Corollary)j(6.3.)952 2795 y Fu(19)p
eop
%%Page: 20 21
20 20 bop 0 42 a Fv(De\014nition)19 b(6.4)k Fu(\(Bo)q(olean)g
(sampler\):)34 b Fp(A)23 b Fe(Bo)q(olean)g(sampler)d
Fp(is)j(a)g(r)n(andomize)n(d)g(algorithm,)i(denote)n(d)e
Fo(A)p Fp(,)0 104 y(which)17 b(satis\014es)663 166 y
Fu(Prob\()p Fn(j)p Fo(A)825 147 y Fm(\027)845 166 y Fu(\()p
Fo(n;)8 b(\017;)g(\016)r Fu(\))h Fn(\000)k Fu(\026)-26
b Fo(\027)s Fn(j)13 b Fo(>)g(\017)p Fu(\))25 b Fo(<)h(\016)0
251 y Fp(for)17 b(every)f(Bo)n(ole)n(an)f(function)h
Fo(\027)8 b Fu(:)d Fn(f)p Fu(0)p Fo(;)j Fu(1)p Fn(g)704
234 y Fm(n)730 251 y Fn(7!)d(f)p Fu(0)p Fo(;)j Fu(1)p
Fn(g)p Fp(.)0 353 y Fv(Theorem)17 b(6.5)22 b Fp(Ther)n(e)16
b(exists)g(a)g Fu(p)q(oly)r(\()p Fo(n;)8 b(\017)777 337
y Fi(\000)p Fk(1)821 353 y Fo(;)g Fu(log\(1)p Fo(=\016)r
Fu(\)\))p Fp(-time)15 b(Bo)n(ole)n(an)g(sampler)h(which)68
446 y Fn(\017)23 b Fp(makes)16 b Fo(O)q Fu(\()310 425
y Fk(log)o(\(1)p Fm(=\016)q Fk(\))p 309 435 119 2 v 354
462 a Fm(\017)368 453 y Fc(2)433 446 y Fu(\))g Fp(or)n(acle)g(queries;)
g(and)68 543 y Fn(\017)23 b Fp(tosses)15 b Fo(n)10 b
Fu(+)h Fo(O)q Fu(\(log\(1)p Fo(=\016)r Fu(\)\))k Fp(c)n(oins.)0
645 y Fu(As)21 b(in)h(the)e(general)i(case,)g(Theorem)e(6.5)g(will)j
(follo)o(w)e(b)o(y)g(emplo)o(ying)g(the)g(metho)q(d)g(of)g(Bellare)h
(et.)e(al.)h([7)o(])0 707 y(to)c(an)g(ev)o(en)g(simpler)i(sampler)e
(asserted)g(in)h(Lemma)g(6.6)e(\(b)q(elo)o(w\).)26 b(This)18
b(latter)f(sampling)h(algorithm)f(uses,)0 769 y(in)23
b(an)g(essen)o(tial)g(w)o(a)o(y)l(,)g(an)f(e\016cien)o(tly)i
(constructible)g(Raman)o(ujan)e(\(expander\))g(Graph)g([24)o(];)k
(namely)l(,)e Fo(d)p Fu(-)0 831 y(regular)15 b(expanders)g(with)g
(second)h(eigen)o(v)m(alue,)g Fo(\025)p Fu(,)e(satisfying)h
Fo(\025)d Fn(\024)h Fu(2)1212 793 y Fn(p)p 1250 793 24
2 v 38 x Fo(d)h Fu(\(rather)g(than)h(merely)g Fo(\025)d(<)h(d)1806
810 y Fk(1)p Fi(\000)1876 799 y Fc(1)p 1854 804 60 2
v 1854 821 a Fh(O)q Fc(\(1\))1920 831 y Fu(\).)0 893
y(Sp)q(eci\014cally)l(,)20 b(w)o(e)c(use)h(an)f(expander)i(of)e(degree)
h Fo(d)d Fu(=)h(4)p Fo(=\016)r(\017)1028 877 y Fk(2)1063
893 y Fu(and)i(asso)q(ciate)g(the)f(v)o(ertex)g(set)h(of)f(the)g
(expander)0 956 y(with)i Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p
Fn(g)219 939 y Fm(n)240 956 y Fu(.)28 b(\(This)18 b(is)h(sligh)o(tly)g
(inaccurate)f(as)g(w)o(e)f(do)h(not)g(ha)o(v)o(e)f(explicit)j
(constructions)f(of)e(Raman)o(ujan)0 1018 y(Graphs)h(of)h(size)g(2)331
1001 y Fm(n)354 1018 y Fu(.)30 b(Still)20 b(w)o(e)f(can)g(e\016cien)o
(tly)h(construct)e(a)h(Raman)o(ujan)f(Graph)g(with)i
Fo(N)j Fu(v)o(ertices)c(where)0 1080 y(\(1)7 b Fn(\000)g
Fo(\017)108 1063 y Fi(0)120 1080 y Fu(\))g Fn(\001)g
Fu(2)188 1063 y Fm(n)222 1080 y Fo(<)13 b(N)k Fn(\024)c
Fu(2)395 1063 y Fm(n)418 1080 y Fu(.)19 b(Using)14 b(this)g(graph)g(in)
g(our)f(construction)h(causes)g(us)g(to)f(estimate)h(the)f(a)o(v)o
(erage)g(v)m(alue)0 1142 y(of)i Fo(\027)j Fu(tak)o(en)d(o)o(v)o(er)f
Fo(N)20 b Fu(of)15 b(the)g(2)523 1125 y Fm(n)561 1142
y Fu(p)q(ossible)h(strings,)f(but)h(this)f(a)o(v)o(erage)f(v)m(alue)j
(is)e(within)i Fo(\017)1549 1125 y Fi(0)1576 1142 y Fu(of)g(\026)-25
b Fo(\027)s Fu(.\))0 1266 y Fv(The)26 b(algorithm.)46
b Fu(The)22 b(sampling)h(algorithm)f(consists)g(of)g(uniformly)h
(selecting)g(a)f(v)o(ertex,)h Fo(v)r Fu(,)g(\(of)e(the)0
1328 y(expander\))16 b(and)f(a)o(v)o(eraging)f(o)o(v)o(er)h(the)g(v)m
(alues)h(assigned)g(\(b)o(y)f Fo(\027)s Fu(\))g(to)g(all)h(the)f(neigh)
o(b)q(ors)h(of)f Fo(v)r Fu(;)f(namely)l(,)803 1441 y(~)-25
b Fo(\027)839 1417 y Fk(def)844 1441 y Fu(=)902 1411
y(1)p 902 1431 24 2 v 902 1473 a Fo(d)967 1401 y Fd(X)938
1492 y Fm(u)p Fi(2N)t Fk(\()p Fm(v)q Fk(\))1063 1441
y Fo(\027)s Fu(\()p Fo(u)p Fu(\))0 1579 y(where)15 b
Fn(N)7 b Fu(\()p Fo(v)r Fu(\))15 b(denotes)g(the)g(set)g(of)g(neigh)o
(b)q(ors)h(of)f(v)o(ertex)f Fo(v)r Fu(.)0 1682 y Fv(Lemma)j(6.6)23
b Fp(The)e(ab)n(ove)g(sampling)f(algorithm)i(c)n(onstitutes)e(a)i(Bo)n
(ole)n(an)e(sampler.)36 b(It)21 b(makes)1776 1664 y Fk(4)p
1761 1671 47 2 v 1761 1697 a Fm(\017)1775 1689 y Fc(2)1791
1697 y Fm(\016)1833 1682 y Fp(or)n(acle)0 1744 y(queries,)16
b(tosses)f Fo(n)i Fp(c)n(oins,)e(and)h(runs)g(in)g(time)830
1723 y Fk(p)q(oly)q(\()p Fm(n)p Fk(\))p 830 1733 110
2 v 862 1759 a Fm(\017)876 1751 y Fc(2)892 1759 y Fm(\016)945
1744 y Fp(.)0 1846 y Fv(Pro)q(of:)j Fu(The)c(complexit)o(y)h(b)q(ounds)
g(are)e(ob)o(vious)h(from)f(the)h(description)h(of)f(the)f(algorithm.)
20 b(W)l(e)15 b(turn)g(to)f(the)0 1908 y(analysis)i(of)f(its)g
(estimates.)71 1970 y(W)l(e)h(denote)g(b)o(y)g Fo(B)j
Fu(the)e(set)f(of)f Fp(b)n(ad)i Fu(c)o(hoices)g(for)e(the)h(algorithm;)
h(namely)l(,)g(the)f(set)g(of)g(v)o(ertices)g(that)g(once)0
2033 y(selected)g(b)o(y)g(the)f(algorithm)g(yield)i(a)e(wrong)f
(estimate.)20 b(That)15 b(is,)g Fo(v)f Fn(2)f Fo(B)18
b Fu(if)755 2083 y Fd(\014)755 2108 y(\014)755 2133 y(\014)755
2158 y(\014)755 2183 y(\014)755 2208 y(\014)774 2138
y Fu(1)p 774 2159 24 2 v 774 2200 a Fo(d)838 2129 y Fd(X)810
2220 y Fm(u)p Fi(2N)t Fk(\()p Fm(v)q Fk(\))935 2169 y
Fo(\027)s Fu(\()p Fo(u)p Fu(\))9 b Fn(\000)k Fu(\026)-25
b Fo(\027)1102 2083 y Fd(\014)1102 2108 y(\014)1102 2133
y(\014)1102 2158 y(\014)1102 2183 y(\014)1102 2208 y(\014)1129
2169 y Fo(>)13 b(\017)0 2307 y Fu(Denote)i(b)o(y)g Fo(B)255
2290 y Fi(0)283 2307 y Fu(the)g(subset)g(of)g Fo(v)f
Fn(2)f Fo(B)18 b Fu(for)d(whic)o(h)733 2387 y(1)p 732
2407 V 732 2449 a Fo(d)797 2377 y Fd(X)768 2468 y Fm(u)p
Fi(2N)t Fk(\()p Fm(v)q Fk(\))893 2417 y Fo(\027)s Fu(\()p
Fo(u)p Fu(\))d Fo(>)k Fu(\026)-26 b Fo(\027)14 b Fu(+)c
Fo(\017)730 b Fu(\(19\))0 2570 y(It)15 b(follo)o(ws)g(that)g(eac)o(h)g
Fo(v)g Fn(2)d Fo(B)514 2554 y Fi(0)542 2570 y Fu(has)j
Fo(\017d)g Fu(to)q(o)g(man)o(y)g(neigh)o(b)q(ors)h(in)g(the)f(set)g
Fo(A)1338 2545 y Fk(def)1343 2570 y Fu(=)j Fn(f)p Fo(u)12
b Fu(:)g Fo(\027)s Fu(\()p Fo(u)p Fu(\))5 b(=)g(1)p Fn(g)p
Fu(;)15 b(namely)l(,)585 2670 y Fn(jf)p Fo(u)5 b Fn(2)g(N)i
Fu(\()p Fo(v)r Fu(\))k(:)h Fo(u)5 b Fn(2)g Fo(A)p Fn(gj)13
b Fo(>)g Fu(\()p Fo(\032)p Fu(\()p Fo(A)p Fu(\))8 b(+)j
Fo(\017)p Fu(\))f Fn(\001)g Fo(d)586 b Fu(\(20\))952
2795 y(20)p eop
%%Page: 21 22
21 21 bop 0 49 a Fu(where)15 b Fo(\032)p Fu(\()p Fo(A)p
Fu(\))237 24 y Fk(def)242 49 y Fu(=)300 28 y Fi(j)p Fm(A)p
Fi(j)p 300 39 45 2 v 308 65 a Fm(N)365 49 y Fu(and)h
Fo(N)508 24 y Fk(def)513 49 y Fu(=)i(2)589 33 y Fm(n)611
49 y Fu(.)i(Using)c(the)f(Expander)h(Mixing)g(Lemma)f(\(Lemma)g(2.2\))f
(ones)h(gets)g(that)503 178 y Fo(\017)c Fn(\001)f Fo(\032)p
Fu(\()p Fo(B)633 159 y Fi(0)644 178 y Fu(\))41 b(=)780
117 y Fd(\014)780 142 y(\014)780 166 y(\014)780 191 y(\014)799
147 y Fn(j)p Fo(B)848 131 y Fi(0)860 147 y Fn(j)10 b(\001)g
Fu(\()p Fo(\032)p Fu(\()p Fo(A)p Fu(\))e(+)j Fo(\017)p
Fu(\))p Fo(d)p 799 167 333 2 v 933 209 a(dN)1147 178
y Fn(\000)f Fo(\032)p Fu(\()p Fo(B)1270 159 y Fi(0)1282
178 y Fu(\))f Fn(\001)h Fo(\032)p Fu(\()p Fo(A)p Fu(\))1426
117 y Fd(\014)1426 142 y(\014)1426 166 y(\014)1426 191
y(\014)703 298 y Fn(\024)780 237 y Fd(\014)780 261 y(\014)780
286 y(\014)780 311 y(\014)799 267 y Fn(j)p Fu(\()p Fo(B)866
250 y Fi(0)888 267 y Fn(\002)g Fo(A)p Fu(\))g Fn(\\)h
Fo(E)s Fn(j)p 799 287 286 2 v 911 329 a(j)p Fo(E)s Fn(j)1100
298 y(\000)1151 267 y(j)p Fo(A)p Fn(j)p 1150 287 62 2
v 1150 329 a(j)p Fo(V)e Fn(j)1227 298 y(\001)1255 267
y(j)p Fo(B)1304 250 y Fi(0)1316 267 y Fn(j)p 1255 287
74 2 v 1261 329 a(j)p Fo(V)g Fn(j)1333 237 y Fd(\014)1333
261 y(\014)1333 286 y(\014)1333 311 y(\014)703 427 y
Fn(\024)785 397 y Fo(\025)p 785 417 27 2 v 786 459 a(d)827
427 y Fn(\001)855 360 y Fd(p)p 896 360 166 2 v 896 397
a Fn(j)p Fo(A)p Fn(j)h(\001)f(j)p Fo(B)1037 384 y Fi(0)1049
397 y Fn(j)p 855 417 208 2 v 938 459 a Fo(N)703 535 y
Fn(\024)805 504 y Fu(2)p 785 525 62 2 v 785 533 a Fn(p)p
823 533 24 2 v 39 x Fo(d)862 535 y Fn(\001)885 484 y
Fd(q)p 926 484 234 2 v 926 535 a Fo(\032)p Fu(\()p Fo(A)p
Fu(\))g Fn(\001)h Fo(\032)p Fu(\()p Fo(B)1130 522 y Fi(0)1142
535 y Fu(\))0 659 y(and)j Fo(\032)p Fu(\()p Fo(B)164
642 y Fi(0)176 659 y Fu(\))f Fn(\024)h Fo(\016)c Fn(\001)d
Fo(\032)p Fu(\()p Fo(A)p Fu(\))12 b(follo)o(ws.)19 b(Using)14
b(a)f(similar)h(argumen)o(t,)f(w)o(e)g(can)g(sho)o(w)g(that)f
Fo(\032)p Fu(\()p Fo(B)d Fn(n)d Fo(B)1601 642 y Fi(0)1613
659 y Fu(\))12 b Fn(\024)h Fo(\016)c Fn(\001)d Fu(\(1)g
Fn(\000)g Fo(\032)p Fu(\()p Fo(A)p Fu(\)\).)0 721 y(Th)o(us,)15
b Fo(\032)p Fu(\()p Fo(B)r Fu(\))d Fn(\024)h Fo(\016)k
Fu(and)e(the)h(claim)g(follo)o(ws.)p 825 727 34 40 v
0 858 a Fl(6.2)56 b(Pro)r(of)18 b(of)h(Theorem)e(6.2)0
950 y Fu(Here)i(w)o(e)g(ma)o(y)f(use)h(w)o(eak)o(er)f(expanders)i(than)
f(in)g(the)h(simpler)g(sample)f(\(ab)q(o)o(v)o(e\).)30
b(Sp)q(eci\014cally)m(,)22 b(w)o(e)d(use)g(an)0 1018
y(e\016cien)o(tly)j(constructible)f(expander)g(graph)f(of)g(degree)g
Fo(d)g Fu(and)g(second)h(eigen)o(v)m(alue)h Fo(\025)e
Fu(so)g(that)1750 1000 y Fm(\025)p 1750 1007 20 2 v 1751
1034 a(d)1796 1018 y Fn(\024)1852 969 y Fd(q)p 1894 969
57 2 v 1899 1000 a Fm(\017)1913 992 y Fc(5)1929 1000
y Fm(\016)p 1899 1007 47 2 v 1905 1034 a Fk(64)0 1080
y Fu(and)d Fo(d)d Fu(=)h(p)q(oly)r(\(1)p Fo(=\017\016)r
Fu(\).)22 b(As)17 b(ab)q(o)o(v)o(e,)f(w)o(e)g(asso)q(ciate)g(the)h(v)o
(ertex)f(set)g(of)g(the)g(expander)i(with)e Fn(f)p Fu(0)p
Fo(;)8 b Fu(1)p Fn(g)1732 1064 y Fm(n)1753 1080 y Fu(.)24
b(Here)16 b(w)o(e)0 1142 y(also)i(use)h(a)g(sample)g(space)f(for)g
(sequences)i(of)e(pairwise)i(indep)q(enden)o(t)g(elemen)o(ts)g
(uniformly)f(distributed)h(in)0 1205 y([)p Fo(d)p Fu(].)30
b(In)20 b(particular,)g(w)o(e)e(use)i(sequences)g(of)e(length)i
Fo(m)1001 1180 y Fk(def)1006 1205 y Fu(=)1085 1187 y
Fk(8)p 1070 1194 V 1070 1220 a Fm(\017)1084 1212 y Fc(2)1100
1220 y Fm(\016)1140 1205 y Fu(whic)o(h)g(can)f(b)q(e)h(e\016cien)o(tly)
g(generated)f(using)0 1267 y(2)8 b(log)89 1278 y Fk(2)115
1267 y Fo(d)k Fu(=)h Fo(O)q Fu(\(log)q(\(1)p Fo(=\017\016)r
Fu(\)\))h(random)h(bits)g(\(cf.,)f([11)o(]\).)0 1392
y Fv(The)24 b(sampler.)45 b Fu(W)l(e)20 b(select)i(uniformly)g(a)e(v)o
(ertex)g Fo(v)i Fu(\(in)g(the)e(expander)i(graph\))e(and)h(a)f
(sequence)i(of)e Fo(m)0 1454 y Fu(pairwise-indep)q(end)q(en)o(t)e
(elemen)o(ts)e(in)g([)p Fo(d)p Fu(],)e(denoted,)i Fo(i)952
1461 y Fk(1)970 1454 y Fo(;)8 b(:::;)g(i)1067 1461 y
Fm(m)1096 1454 y Fu(.)20 b(T)l(ogether)15 b(these)h(de\014ne)g
Fo(m)g Fu(sampling)g(p)q(oin)o(ts,)0 1516 y(denoted)k
Fo(u)202 1523 y Fk(1)220 1516 y Fo(;)8 b(:::;)g(u)327
1523 y Fm(j)342 1516 y Fu(,)20 b(where)g Fo(u)537 1523
y Fm(j)573 1516 y Fu(is)g(the)g Fo(i)722 1500 y Fk(th)722
1527 y Fm(j)774 1516 y Fu(neigh)o(b)q(or)g(of)f Fo(v)r
Fu(.)32 b(The)19 b(estimate)g(of)g(the)g(sampler)h(is)g(merely)g(the)0
1578 y(a)o(v)o(erage)14 b(\(of)g Fo(\027)s Fu(\))i(o)o(v)o(er)e(the)h
(v)m(alues)i(at)d(these)i Fo(u)800 1585 y Fm(j)817 1578
y Fu('s;)e(namely)l(,)815 1708 y(~)-26 b Fo(\027)850
1683 y Fk(def)855 1708 y Fu(=)922 1678 y(1)p 914 1698
40 2 v 914 1740 a Fo(m)981 1655 y Fm(m)966 1668 y Fd(X)967
1756 y Fm(j)r Fk(=1)1034 1708 y Fo(\027)s Fu(\()p Fo(u)1103
1715 y Fm(j)1120 1708 y Fu(\))0 1848 y(Clearly)l(,)13
b(the)f(complexities)i(are)e(as)g(claimed)h(in)g(Theorem)f(6.2.)17
b(It)c(is)f(left)g(to)g(sho)o(w)f(that)g(for)h(ev)o(ery)g
Fo(\027)c Fu(:)d Fn(f)p Fu(0)p Fo(;)j Fu(1)p Fn(g)1879
1832 y Fm(n)1905 1848 y Fn(7!)0 1910 y Fu([0)p Fo(;)g
Fu(1],)750 1972 y(Prob\()p Fn(j)s Fu(~)-26 b Fo(\027)12
b Fn(\000)h Fu(\026)-25 b Fo(\027)s Fn(j)12 b Fo(>)h(\017)p
Fu(\))26 b Fo(<)f(\016)0 2061 y Fu(W)l(e)18 b(\014rst)f(observ)o(e)g
(that)g(it)g(su\016ces)h(to)f(ev)m(aluate)h(the)g(b)q(eha)o(vior)g(of)f
(the)h(sampler)f(on)h(functions)g(of)f(the)h(form)0 2123
y Fo(\027)e Fu(:)d Fn(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)177
2106 y Fm(n)211 2123 y Fn(7!)13 b(f)p Fu(\()p Fo(i)g
Fn(\000)393 2105 y Fk(1)p 393 2112 17 2 v 393 2138 a(2)414
2123 y Fu(\))g Fn(\001)477 2105 y Fm(\017)p 476 2112
V 476 2138 a Fk(2)518 2123 y Fu(:)20 b Fo(i)13 b Fu(=)h(1)p
Fo(;)8 b(::;)g Fu(2)p Fo(\017)761 2106 y Fi(\000)p Fk(1)803
2123 y Fn(g)p Fu(.)34 b(Sp)q(eci\014cally)l(,)24 b(for)19
b(ev)o(ery)h Fo(\027)d Fu(:)12 b Fn(f)p Fu(0)p Fo(;)c
Fu(1)p Fn(g)1501 2106 y Fm(n)1535 2123 y Fn(7!)14 b Fu([0)p
Fo(;)8 b Fu(1],)19 b(there)h(exists)0 2185 y(a)f Fo(\026)12
b Fu(:)f Fn(f)p Fu(0)p Fo(;)d Fu(1)p Fn(g)218 2168 y
Fm(n)250 2185 y Fn(7!)k(f)p Fu(\()p Fo(i)f Fn(\000)428
2167 y Fk(1)p 428 2174 V 428 2200 a(2)450 2185 y Fu(\))h
Fn(\001)512 2167 y Fm(\017)p 510 2174 V 510 2200 a Fk(2)551
2185 y Fu(:)18 b Fo(i)12 b Fu(=)f(1)p Fo(;)d(::;)g Fu(2)p
Fo(\017)788 2168 y Fi(\000)p Fk(1)830 2185 y Fn(g)19
b Fu(so)g(that)f Fn(j)p Fo(\027)s Fu(\()p Fo(x)p Fu(\))12
b Fn(\000)h Fo(\026)p Fu(\()p Fo(x)p Fu(\))p Fn(j)18
b(\024)1375 2167 y Fm(\017)p 1374 2174 V 1374 2200 a
Fk(4)1395 2185 y Fu(,)i Fn(8)p Fo(x)f Fn(2)g(f)p Fu(0)p
Fo(;)8 b Fu(1)p Fn(g)1660 2168 y Fm(n)1681 2185 y Fu(.)31
b(Th)o(us,)20 b(b)q(oth)0 2247 y Fn(j)s Fu(\026)-26 b
Fo(\027)s Fu(\()p Fo(x)p Fu(\))9 b Fn(\000)14 b Fu(\026)-26
b Fo(\026)p Fu(\()p Fo(x)p Fu(\))p Fn(j)12 b(\024)323
2229 y Fm(\017)p 322 2236 V 322 2263 a Fk(4)358 2247
y Fu(and)k Fn(j)s Fu(~)-26 b Fo(\027)s Fu(\()p Fo(x)p
Fu(\))9 b Fn(\000)14 b Fu(~)-26 b Fo(\026)p Fu(\()p Fo(x)p
Fu(\))p Fn(j)12 b(\024)770 2229 y Fm(\017)p 769 2236
V 769 2263 a Fk(4)790 2247 y Fu(,)j(and)g(so)g(it)h(su\016ces)f(to)g
(pro)o(v)o(e)f(the)i(follo)o(wing)0 2357 y Fv(Lemma)h(6.7)23
b Fp(F)m(or)16 b(every)g Fo(\026)5 b Fu(:)g Fn(f)p Fu(0)p
Fo(;)j Fu(1)p Fn(g)643 2340 y Fm(n)669 2357 y Fn(7!)d(f)p
Fu(\()p Fo(i)k Fn(\000)836 2339 y Fk(1)p 836 2346 V 836
2372 a(2)857 2357 y Fu(\))h Fn(\001)914 2339 y Fm(\017)p
913 2346 V 913 2372 a Fk(2)947 2357 y Fu(:)i Fo(i)5 b
Fu(=)g(1)p Fo(;)j(::;)g Fu(2)p Fo(\017)1165 2340 y Fi(\000)p
Fk(1)1208 2357 y Fn(g)724 2481 y Fu(Prob)828 2422 y Fd(\022)859
2481 y Fn(j)t Fu(~)-27 b Fo(\026)10 b Fn(\000)k Fu(\026)-27
b Fo(\026)q Fn(j)12 b Fo(>)1062 2451 y(\017)p 1060 2471
23 2 v 1060 2513 a Fu(2)1088 2422 y Fd(\023)1144 2481
y Fo(<)25 b(\016)0 2606 y Fv(Pro)q(of:)32 b Fu(W)l(e)22
b(\014rst)f(mimic)h(the)g(pro)q(of)f(of)g(Lemma)g(6.6)g(and)g(sho)o(w)g
(that)g(for)g(all)h(but)f Fo(\016)r(=)p Fu(2)g(of)g(the)h(v)o(ertices)0
2668 y Fo(v)15 b Fn(2)e(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)193
2651 y Fm(n)214 2668 y Fu(,)248 2650 y Fk(1)p 247 2657
18 2 v 247 2684 a Fm(d)280 2668 y Fn(\001)303 2636 y
Fd(P)347 2679 y Fm(u)p Fi(2N)t Fk(\()p Fm(v)q Fk(\))474
2668 y Fo(\026)p Fu(\()p Fo(u)p Fu(\))15 b(is)h(within)h
Fo(\017=)p Fu(4)e(of)k(\026)-27 b Fo(\026)q Fu(,)15 b(where)h
Fn(N)7 b Fu(\()p Fo(v)r Fu(\))14 b(denotes)i(the)g(set)f(of)g(neigh)o
(b)q(ors)h(of)f(v)o(ertex)952 2795 y(21)p eop
%%Page: 22 23
22 22 bop 0 42 a Fo(v)r Fu(.)20 b(W)l(e)c(conclude)h(b)o(y)f(recalling)
h(that)e(with)h(probabilit)o(y)g(at)f(least)h(1)10 b
Fn(\000)1251 24 y Fm(\016)p 1251 31 17 2 v 1251 57 a
Fk(2)1288 42 y Fu(a)15 b(pairwise)h(indep)q(enden)o(t)i(sample)e(of)0
104 y Fo(m)f Fu(v)o(ertices)h(in)g Fn(N)7 b Fu(\()p Fo(v)r
Fu(\))14 b(appro)o(ximates)g(the)i(ab)q(o)o(v)o(e)f(up-to)g
Fo(\017=)p Fu(4.)71 166 y(Supp)q(ose,)g(for)g(simplicit)o(y)l(,)i(that)
e Fo(\017)g Fu(is)h(suc)o(h)g(that)e Fo(t)955 141 y Fk(def)960
166 y Fu(=)k(2)p Fo(\017)1054 149 y Fi(\000)p Fk(1)1114
166 y Fu(is)e(an)f(in)o(teger.)20 b(F)l(or)15 b(ev)o(ery)g
Fo(i)5 b Fu(=)g(1)p Fo(;)j(::;)g(t)p Fu(,)13 b(let)554
257 y Fo(A)588 264 y Fm(i)643 232 y Fk(def)649 257 y
Fu(=)47 b Fn(f)p Fo(v)14 b Fu(:)e Fo(\026)p Fu(\()p Fo(v)r
Fu(\))g(=)h(\()p Fo(i)c Fn(\000)i Fu(0)p Fo(:)p Fu(5\))e
Fn(\001)g Fo(\017=)p Fu(2)p Fn(g)623 b Fu(\(21\))554
351 y Fo(B)588 358 y Fm(i)643 326 y Fk(def)649 351 y
Fu(=)731 291 y Fd(\032)762 351 y Fo(v)14 b Fu(:)823 289
y Fd(\014)823 314 y(\014)823 339 y(\014)823 364 y(\014)842
320 y Fn(jN)7 b Fu(\()p Fo(v)r Fu(\))i Fn(\\)h Fo(A)1042
327 y Fm(i)1056 320 y Fn(j)p 842 340 227 2 v 944 382
a Fo(d)1084 351 y Fn(\000)g Fo(\032)p Fu(\()p Fo(A)1205
358 y Fm(i)1218 351 y Fu(\))1236 289 y Fd(\014)1236 314
y(\014)1236 339 y(\014)1236 364 y(\014)1263 351 y Fo(>)1316
320 y(\017)1334 303 y Fk(2)p 1316 340 38 2 v 1323 382
a Fu(4)1358 291 y Fd(\033)1869 351 y Fu(\(22\))0 456
y(Observ)o(e)16 b(that)e(if)i Fo(v)e Fn(62)f([)424 440
y Fm(t)424 468 y(i)p Fk(=1)480 456 y Fo(B)514 463 y Fm(i)544
456 y Fu(then)473 545 y(1)p 473 565 24 2 v 473 607 a
Fo(d)512 575 y Fn(\001)563 535 y Fd(X)534 626 y Fm(u)p
Fi(2N)t Fk(\()p Fm(v)q Fk(\))659 575 y Fo(\026)p Fu(\()p
Fo(u)p Fu(\))41 b(=)872 545 y(1)p 871 565 V 871 607 a
Fo(d)910 575 y Fn(\001)956 523 y Fm(t)933 535 y Fd(X)936
623 y Fm(i)p Fk(=1)993 575 y Fu(\()p Fo(i)9 b Fn(\000)h
Fu(0)p Fo(:)p Fu(5\))f Fn(\001)1197 545 y Fo(\017)p 1195
565 23 2 v 1195 607 a Fu(2)1233 575 y Fn(\001)h(jN)d
Fu(\()p Fo(v)r Fu(\))h Fn(\\)j Fo(A)1456 582 y Fm(i)1470
575 y Fn(j)789 729 y Fu(=)890 676 y Fm(t)866 689 y Fd(X)869
777 y Fm(i)p Fk(=1)926 729 y Fu(\()p Fo(i)f Fn(\000)g
Fu(0)p Fo(:)p Fu(5\))f Fn(\001)1131 698 y Fo(\017)p 1129
718 V 1129 760 a Fu(2)1166 729 y Fn(\001)1189 669 y Fd(\022)1220
729 y Fo(\032)p Fu(\()p Fo(A)1296 736 y Fm(i)1309 729
y Fu(\))g Fn(\006)1387 698 y Fo(\017)1405 682 y Fk(2)p
1387 718 38 2 v 1394 760 a Fu(4)1429 669 y Fd(\023)789
880 y Fu(=)46 b(\026)-27 b Fo(\026)11 b Fn(\006)955 824
y Fk(2)p Fm(=\017)949 840 y Fd(X)952 928 y Fm(i)p Fk(=1)1009
880 y Fu(\()p Fo(i)e Fn(\000)i Fu(0)p Fo(:)p Fu(5\))e
Fn(\001)1211 849 y Fo(\017)1229 833 y Fk(3)p 1211 870
V 1219 911 a Fu(8)789 999 y(=)46 b(\026)-27 b Fo(\026)11
b Fn(\006)956 968 y Fo(\017)p 954 989 23 2 v 954 1030
a Fu(4)0 1098 y(Th)o(us,)k(the)g(a)o(v)o(erage)f(v)m(alue)i(of)f(the)g
(neigh)o(b)q(ors)h(of)f(eac)o(h)g Fo(v)f Fn(62)f([)1084
1081 y Fm(t)1084 1109 y(i)p Fk(=1)1141 1098 y Fo(B)1175
1105 y Fm(i)1204 1098 y Fu(is)j(within)1397 1080 y Fm(\017)p
1396 1087 17 2 v 1396 1113 a Fk(4)1433 1098 y Fu(of)e(the)i(the)f
(correct)g(v)m(alue)20 b(\026)-27 b Fo(\026)p Fu(.)71
1160 y(Next,)13 b(w)o(e)g(use)h(the)f(Expander)h(Mixing)h(Lemma)e
(\(Lemma)g(2.2\))f(as)h(in)i(the)e(pro)q(of)g(of)g(Lemma)h(6.6)e(to)h
(deriv)o(e)0 1222 y(a)18 b(b)q(ound)g(on)g(the)g(densit)o(y)g(of)g(eac)
o(h)g Fo(B)683 1229 y Fm(i)697 1222 y Fu(.)28 b(Analogously)l(,)19
b(w)o(e)e(partition)h Fo(B)1303 1229 y Fm(i)1336 1222
y Fu(in)o(to)f Fo(B)1466 1203 y Fk(+)1464 1232 y Fm(i)1512
1197 y Fk(def)1517 1222 y Fu(=)22 b Fn(f)p Fo(v)d Fu(:)d
Fn(jN)7 b Fu(\()p Fo(v)r Fu(\))k Fn(\\)h Fo(A)1871 1229
y Fm(i)1885 1222 y Fn(j)17 b Fo(>)0 1284 y Fu(\()p Fo(\032)p
Fu(\()p Fo(A)94 1291 y Fm(i)107 1284 y Fu(\))10 b(+)g
Fo(\017)198 1267 y Fk(2)217 1284 y Fo(=)p Fu(4\))f Fn(\001)h
Fo(d)p Fn(g)15 b Fu(and)g Fo(B)499 1265 y Fi(\000)497
1294 y Fm(i)540 1259 y Fk(def)546 1284 y Fu(=)j Fo(B)633
1291 y Fm(i)657 1284 y Fn(n)10 b Fo(B)726 1265 y Fk(+)724
1294 y Fm(i)754 1284 y Fu(,)15 b(and)g(get)429 1374 y
Fo(\017)447 1358 y Fk(2)p 429 1394 38 2 v 436 1436 a
Fu(4)481 1405 y Fn(\001)10 b Fo(\032)p Fu(\()p Fo(B)582
1386 y Fk(+)580 1416 y Fm(i)610 1405 y Fu(\))41 b(=)746
1331 y Fd(\014)746 1356 y(\014)746 1381 y(\014)746 1406
y(\014)746 1431 y(\014)764 1372 y Fn(j)p Fo(B)813 1353
y Fk(+)811 1382 y Fm(i)841 1372 y Fn(j)10 b(\001)g Fu(\()p
Fo(\032)p Fu(\()p Fo(A)981 1379 y Fm(i)994 1372 y Fu(\))f(+)1072
1354 y Fm(\017)1086 1342 y Fc(2)p 1072 1361 31 2 v 1079
1388 a Fk(4)1107 1372 y Fu(\))h Fn(\001)f Fo(d)p 764
1394 417 2 v 940 1436 a(dN)1196 1405 y Fn(\000)i Fo(\032)p
Fu(\()p Fo(B)1320 1386 y Fk(+)1318 1416 y Fm(i)1347 1405
y Fu(\))f Fn(\001)g Fo(\032)p Fu(\()p Fo(A)1474 1412
y Fm(i)1487 1405 y Fu(\))1505 1331 y Fd(\014)1505 1356
y(\014)1505 1381 y(\014)1505 1406 y(\014)1505 1431 y(\014)669
1546 y Fn(\024)746 1472 y Fd(\014)746 1497 y(\014)746
1522 y(\014)746 1547 y(\014)746 1572 y(\014)764 1515
y Fn(j)p Fu(\()p Fo(B)831 1496 y Fk(+)829 1526 y Fm(i)869
1515 y Fn(\002)h Fo(A)949 1522 y Fm(i)963 1515 y Fu(\))e
Fn(\\)i Fo(E)s Fn(j)p 764 1536 316 2 v 891 1577 a(j)p
Fo(E)s Fn(j)1095 1546 y(\000)1145 1515 y(j)p Fo(A)1192
1522 y Fm(i)1206 1515 y Fn(j)p 1145 1536 74 2 v 1151
1577 a(j)p Fo(V)e Fn(j)1233 1546 y(\001)1261 1515 y(j)p
Fo(B)1310 1496 y Fk(+)1308 1526 y Fm(i)1338 1515 y Fn(j)p
1261 1536 90 2 v 1275 1577 a(j)p Fo(V)g Fn(j)1356 1472
y Fd(\014)1356 1497 y(\014)1356 1522 y(\014)1356 1547
y(\014)1356 1572 y(\014)669 1707 y Fn(\024)751 1676 y
Fo(\025)p 751 1697 27 2 v 752 1738 a(d)792 1707 y Fn(\001)820
1617 y Fd(q)p 861 1617 196 2 v 861 1667 a Fn(j)p Fo(A)908
1674 y Fm(i)922 1667 y Fn(j)h(\001)f(j)p Fo(B)1016 1648
y Fk(+)1014 1678 y Fm(i)1044 1667 y Fn(j)p 820 1697 238
2 v 918 1738 a Fo(N)669 1839 y Fn(\024)746 1758 y Fd(s)p
787 1758 69 2 v 792 1808 a Fo(\017)810 1795 y Fk(5)829
1808 y Fo(\016)p 792 1828 59 2 v 799 1870 a Fu(64)866
1839 y Fn(\001)889 1786 y Fd(q)p 930 1786 249 2 v 930
1839 a Fo(\032)p Fu(\()p Fo(A)p Fu(\))h Fn(\001)f Fo(\032)p
Fu(\()p Fo(B)1134 1820 y Fk(+)1132 1849 y Fm(i)1162 1839
y Fu(\))0 1952 y(It)j(follo)o(ws)g(that)f Fo(\032)p Fu(\()p
Fo(B)365 1933 y Fk(+)363 1962 y Fm(i)392 1952 y Fu(\))h
Fn(\024)475 1934 y Fm(\017\016)p 475 1941 30 2 v 482
1968 a Fk(4)513 1952 y Fn(\001)s Fo(\032)p Fu(\()p Fo(A)605
1959 y Fm(i)618 1952 y Fu(\))g(and)f(similarly)j Fo(\032)p
Fu(\()p Fo(B)993 1933 y Fi(\000)991 1962 y Fm(i)1021
1952 y Fu(\))e Fn(\024)1104 1934 y Fm(\017\016)p 1104
1941 V 1111 1968 a Fk(4)1142 1952 y Fn(\001)s Fu(\(1)s
Fn(\000)s Fo(\032)p Fu(\()p Fo(A)1316 1959 y Fm(i)1329
1952 y Fu(\)\).)18 b(Th)o(us,)12 b Fo(\032)p Fu(\()p
Fn([)1593 1936 y Fm(t)1593 1963 y(i)p Fk(=1)1648 1952
y Fo(B)1682 1959 y Fm(i)1697 1952 y Fu(\))g Fn(\024)h
Fo(t)s Fn(\001)1815 1934 y Fm(\017\016)p 1815 1941 V
1822 1968 a Fk(4)1863 1952 y Fu(=)1916 1934 y Fm(\016)p
1916 1941 17 2 v 1916 1968 a Fk(2)1937 1952 y Fu(.)0
2014 y(Com)o(bined)j(with)f(the)h(ab)q(o)o(v)o(e)f(w)o(e)f(ha)o(v)o(e)
163 2133 y(Prob)260 2140 y Fm(v)q Fi(2f)p Fk(0)p Fm(;)p
Fk(1)p Fi(g)378 2132 y Fh(n)407 2049 y Fd(0)407 2124
y(@)443 2047 y(\014)443 2072 y(\014)443 2097 y(\014)443
2122 y(\014)443 2147 y(\014)443 2172 y(\014)463 2103
y Fu(1)p 462 2123 24 2 v 462 2165 a Fo(d)501 2133 y Fn(\001)552
2093 y Fd(X)524 2184 y Fm(u)p Fi(2N)t Fk(\()p Fm(v)q
Fk(\))648 2133 y Fo(\026)p Fu(\()p Fo(u)p Fu(\))c Fn(\000)k
Fu(\026)-26 b Fo(\026)820 2047 y Fd(\014)820 2072 y(\014)820
2097 y(\014)820 2122 y(\014)820 2147 y(\014)820 2172
y(\014)847 2133 y Fo(>)902 2103 y(\017)p 900 2123 23
2 v 900 2165 a Fu(4)927 2049 y Fd(1)927 2124 y(A)1001
2133 y Fn(\024)39 b Fu(Prob)1171 2140 y Fm(v)q Fi(2f)p
Fk(0)p Fm(;)p Fk(1)p Fi(g)1289 2132 y Fh(n)1311 2133
y Fu(\()p Fo(v)14 b Fn(2)f([)1438 2115 y Fm(t)1438 2145
y(i)p Fk(=1)1494 2133 y Fo(B)1528 2140 y Fm(i)1542 2133
y Fu(\))38 b Fn(\024)1676 2103 y Fo(\016)p 1676 2123
V 1676 2165 a Fu(2)1869 2133 y(\(23\))0 2264 y(where)15
b Fo(v)i Fu(is)f(uniformly)g(selected)h(in)f Fn(f)p Fu(0)p
Fo(;)8 b Fu(1)p Fn(g)759 2247 y Fm(n)779 2264 y Fu(.)71
2326 y(Finally)l(,)15 b(w)o(e)g(note)f(that,)f(for)h(ev)o(ery)h
Fo(v)f Fn(2)f(f)p Fu(0)p Fo(;)8 b Fu(1)p Fn(g)891 2309
y Fm(n)912 2326 y Fu(,)14 b(if)h Fo(u)1006 2333 y Fm(i)1020
2326 y Fo(;)8 b(:::;)g(u)1127 2333 y Fm(m)1170 2326 y
Fu(are)14 b(uniformly)h(distributed)h(in)g(a)e(pairwise-)0
2388 y(indep)q(enden)o(t)k(manner)d(in)h Fn(N)7 b Fu(\()p
Fo(v)r Fu(\))14 b(then)214 2518 y(Prob)311 2525 y Fm(u)331
2529 y Fc(1)347 2525 y Fm(;:::;u)417 2529 y Fh(m)453
2434 y Fd(0)453 2509 y(@)489 2432 y(\014)489 2457 y(\014)489
2482 y(\014)489 2507 y(\014)489 2532 y(\014)489 2557
y(\014)517 2488 y Fu(1)p 508 2508 40 2 v 508 2550 a Fo(m)563
2518 y Fn(\001)601 2465 y Fm(m)586 2478 y Fd(X)589 2566
y Fm(i)p Fk(=1)653 2518 y Fo(\026)p Fu(\()p Fo(u)724
2525 y Fm(i)738 2518 y Fu(\))c Fn(\000)817 2488 y Fu(1)p
816 2508 24 2 v 816 2550 a Fo(d)855 2518 y Fn(\001)906
2478 y Fd(X)878 2569 y Fm(u)p Fi(2N)t Fk(\()p Fm(v)q
Fk(\))1003 2518 y Fo(\026)p Fu(\()p Fo(u)p Fu(\))1092
2432 y Fd(\014)1092 2457 y(\014)1092 2482 y(\014)1092
2507 y(\014)1092 2532 y(\014)1092 2557 y(\014)1118 2518
y Fo(>)1173 2488 y(\017)p 1171 2508 23 2 v 1171 2550
a Fu(4)1198 2434 y Fd(1)1198 2509 y(A)1273 2518 y Fn(\024)1378
2486 y Fo(m)g Fn(\001)1456 2468 y Fk(1)p 1456 2475 17
2 v 1456 2501 a(4)p 1351 2508 154 2 v 1351 2550 a Fu(\()p
Fo(m)g Fn(\001)1448 2532 y Fm(\017)p 1446 2539 17 2 v
1446 2565 a Fk(4)1468 2550 y Fu(\))1486 2536 y Fk(2)1547
2518 y Fu(=)1626 2488 y Fo(\016)p 1625 2508 23 2 v 1625
2550 a Fu(2)1869 2518 y(\(24\))0 2651 y(where)15 b(the)h(equalit)o(y)g
(is)f(due)h(to)f Fo(m)d Fu(=)690 2633 y Fk(8)p 675 2640
47 2 v 675 2667 a Fm(\017)689 2659 y Fc(2)705 2667 y
Fm(\016)726 2651 y Fu(.)20 b(Com)o(bining)c(Eq.)f(\(23\))f(and)h(Eq.)g
(\(24\),)f(the)h(lemma)g(follo)o(ws.)p 1890 2657 34 40
v 952 2795 a(22)p eop
%%Page: 23 24
23 23 bop 0 42 a Fq(Ac)n(kno)n(wledgmen)n(ts)0 149 y
Fu(W)l(e)14 b(are)g(grateful)g(to)f(the)h(anon)o(ymous)g(referees)g
(for)g(their)g(useful)h(commen)o(ts,)f(and)g(to)f(Noga)h(Alon)g(for)g
(helpful)0 211 y(discussions.)0 361 y Fq(References)72
469 y Fu([1])22 b(M.)e(Ajtai,)h(J.)g(Komlos,)h(E.)e(Szemer)o(\023)-21
b(edi,)22 b(\\Deterministic)g(Sim)o(ulation)g(in)g(LogSpace",)f
Fp(Pr)n(o)n(c.)g(19th)143 531 y(STOC)p Fu(,)13 b(1987,)h(pp.)h
(132{140.)72 630 y([2])22 b(N.)f(Alon,)j(\\Eigen)o(v)m(alues,)g
(Geometric)d(Expanders,)j(Sorting)d(in)i(Rounds)f(and)g(Ramsey)g
(Theory",)143 692 y Fp(Combinatoric)n(a)p Fu(,)14 b(6)h(\(1986\),)e
(pp.)i(231{243.)72 792 y([3])22 b(N.)15 b(Alon,)h(J.)f(Bruc)o(k,)h(J.)f
(Naor,)g(M.)f(Naor)h(and)h(R.)f(Roth,)h(\\Construction)f(of)g
(Asymptotically)h(Go)q(o)q(d,)143 854 y(Lo)o(w-Rate)j(Error-Correcting)
g(Co)q(des)g(through)g(Pseudo-Random)h(Graphs",)g Fp(IEEE)f(T)m(r)n
(ansactions)143 916 y(on)d(Information)g(The)n(ory)f
Fv(38)g Fu(\(1992\),)e(pp.)i(509{516.)72 1015 y([4])22
b(N.)g(Alon,)i(O.)e(Goldreic)o(h,)j(J.)d(Hastad,)i(R.)e(P)o(eralta,)h
(\\Simple)h(Constructions)e(of)g(Almost)g Fo(k)q Fu(-wise)143
1077 y(Indep)q(enden)o(t)15 b(Random)e(V)l(ariables",)g
Fp(Journal)h(of)g(R)n(andom)h(structur)n(es)f(and)g(A)o(lgorithms)p
Fu(,)e(V)l(ol.)h(3,)g(No.)143 1140 y(3,)h(\(1992\),)f(pp.)j(289{304.)72
1239 y([5])22 b(N.)10 b(Alon)i(and)f(V.D.)f(Milman,)i
Fo(\025)699 1246 y Fk(1)717 1239 y Fu(,)f(Isop)q(erimetric)h
(Inequalities)i(for)c(Graphs)g(and)h(Sup)q(erconcen)o(trators,)143
1301 y Fp(J.)16 b(Combinatorial)g(The)n(ory,)g(Ser.)g(B)f
Fu(38)f(\(1985\),)f(pp.)j(73{88.)72 1401 y([6])22 b(N.)15
b(Alon)g(and)h(J.H.)f(Sp)q(encer,)h Fp(The)g(Pr)n(ob)n(abilistic)f
(Metho)n(d)p Fu(,)g(John)g(Wiley)i(&)e(Sons,)g(Inc.,)g(1992.)72
1500 y([7])22 b(M.)15 b(Bellare,)i(O.)e(Goldreic)o(h,)i(and)f(S.)g
(Goldw)o(asser)f(\\Randomness)h(in)h(In)o(teractiv)o(e)f(Pro)q(ofs",)e
Fp(Compu-)143 1562 y(tational)i(Complexity)p Fu(,)f(V)l(ol.)g(4,)g(No.)
f(4)h(\(1993\),)e(pp.)i(319{354.)72 1662 y([8])22 b(R.)17
b(Canetti,)h(G.)f(Ev)o(en)g(and)h(O.)f(Goldreic)o(h,)i(\\Lo)o(w)o(er)d
(Bounds)i(for)f(Sampling)i(Algorithms)f(for)f(Esti-)143
1724 y(mating)e(the)g(Av)o(erage",)f Fp(IPL)p Fu(,)g(V)l(ol.)h(53,)f
(pp.)i(17{25,)d(1995.)72 1823 y([9])22 b(L.)12 b(Carter)f(and)i(M.)e(W)
l(egman,)i(\\Univ)o(ersal)g(Classes)f(of)g(Hash)g(F)l(unctions",)h
Fp(J.)h(Computer)g(and)g(System)143 1885 y(Scienc)n(es)t
Fu(,)e(V)l(ol.)k(18,)e(pp.)h(143{154)e(\(1979\).)49 1985
y([10])22 b(B.)17 b(Chor)h(and)g(O.)f(Goldreic)o(h,)j(\\Un)o(biased)e
(Bits)g(from)g(Sources)g(of)f(W)l(eak)h(Randomness)g(and)g(Prob-)143
2047 y(abilistic)i(Comm)o(unication)f(Complexit)o(y",)g
Fp(SIAM)f(J.)h(Comput.)p Fu(,)h(V)l(ol.)e(17,)h(No.)f(2,)g(April)i
(1988,)e(pp.)143 2109 y(230{261.)49 2208 y([11])k(B.)16
b(Chor)g(and)g(O.)g(Goldreic)o(h,)h(\\On)g(the)f(P)o(o)o(w)o(er)f(of)h
(Tw)o(o{P)o(oin)o(t)f(Based)h(Sampling,")i Fp(Jour.)f(of)h(Com-)143
2270 y(plexity)p Fu(,)c(V)l(ol)i(5,)f(1989,)e(pp.)j(96{106.)49
2370 y([12])22 b(A.)14 b(Cohen)h(and)f(A.)g(Wigderson,)h(\\Disp)q
(ensers,)g(Deterministic)h(Ampli\014cation,)g(and)e(W)l(eak)h(Random)
143 2432 y(Sources",)g Fp(30th)i(F)o(OCS)p Fu(,)c(1989,)h(pp.)h(14{19.)
49 2531 y([13])22 b(P)l(.)15 b(Erd\177)-23 b(os)14 b(and)i(J.)f(Sp)q
(encer,)h Fp(Pr)n(ob)n(abilistic)f(Metho)n(ds)h(in)g(Combinatorics)p
Fu(,)e(Academic)i(Press,)f(1974.)952 2795 y(23)p eop
%%Page: 24 25
24 24 bop 49 42 a Fu([14])22 b(G.)17 b(Ev)o(en,)h(\\Construction)g(of)g
(Small)h(Probabilit)o(y)g(Spaces)f(for)g(Deterministic)h(Sim)o
(ulation",)h(M.Sc.)143 104 y(thesis,)d(Computer)f(Science)j(Departmen)o
(t,)d(T)l(ec)o(hnion,)i(Haifa,)f(Israel,)g(1991.)f(\(In)h(Hebrew,)g
(abstract)143 166 y(in)f(English\))49 265 y([15])22 b(G.)14
b(Ev)o(en,)i(O.)f(Goldreic)o(h,)h(M.)f(Lub)o(y)l(,)h(N.)f(Nisan,)g(and)
h(B.)f(V)l(eli)o(\024)-21 b(ck)o(o)o(vi)o(\023)g(c,)15
b(\\Appro)o(ximations)h(of)f(General)143 327 y(Indep)q(enden)o(t)i
(Distributions",)f Fp(24th)h(STOC)p Fu(,)c(pp.)i(10{16,)f(1992.)49
427 y([16])22 b(O.)16 b(Gab)q(er)g(and)h(Z.)e(Galil,)j(\\Explicit)g
(Constructions)e(of)g(Linear)h(Size)h(Sup)q(erconcen)o(trators",)e
Fp(JCSS)p Fu(,)143 489 y Fv(22)f Fu(\(1981\),)e(pp.)i(407-420.)49
588 y([17])22 b(O.)13 b(Goldreic)o(h,)h(H.)f(Kra)o(w)o(cyzk)f(and)h(M.)
g(Lub)o(y)l(,)h(\\On)f(the)g(Existence)h(of)f(Pseudorandom)g
(Generators",)143 650 y Fp(SIAM)i(J.)h(on)g(Computing)p
Fu(,)f(V)l(ol.)g(22-6)f(\(Dec.)h(1993\),)e(pp.)j(1163{1175.)49
750 y([18])22 b(S.)15 b(Goldw)o(asser)h(and)g(M.)f(Sipser,)i(\\Priv)m
(ate)f(Coins)g(v)o(ersus)g(Public)h(Coins)g(in)f(In)o(teractiv)o(e)g
(Pro)q(of)g(Sys-)143 812 y(tems",)d Fp(A)n(dvanc)n(es)g(in)h(Computing)
h(R)n(ese)n(ar)n(ch:)k(a)c(r)n(ese)n(ar)n(ch)f(annual)p
Fu(,)f(V)l(ol.)h(5)f(\(Randomness)h(and)f(Com-)143 874
y(putation,)i(S.)g(Micali,)h(ed.\),)e(pp.)i(73{90,)d(1989.)49
974 y([19])22 b(R.)g(Impagliazzo)i(and)e(M.)g(Lub)o(y)l(,)j(\\One-W)l
(a)o(y)d(F)l(unctions)i(are)e(Essen)o(tial)h(for)f(Complexit)o(y)h
(Based)143 1036 y(Cryptograph)o(y",)13 b Fp(30th)k(F)o(OCS)p
Fu(,)c(pp.)j(230{235,)c(1989.)49 1135 y([20])22 b(R.)e(Impagliazzo)i
(and)f(L.A.)f(Levin,)k(\\No)c(Better)g(W)l(a)o(ys)g(to)g(Generate)h
(Hard)f(NP)h(Instances)g(than)143 1197 y(Pic)o(king)16
b(Uniformly)g(at)f(Random)g(",)f Fp(31st)j(F)o(OCS)p
Fu(,)c(pp.)i(812-821,)f(1990.)49 1297 y([21])22 b(R.)12
b(Impagliazzo,)h(L.A.)f(Levin,)h(and)g(M.G.)d(Lub)o(y)l(,)j
(\\Pseudorandom)f(Generators)f(from)g(an)o(y)h(One-W)l(a)o(y)143
1359 y(F)l(unctions",)j Fp(21st)h(STOC)p Fu(,)e(pp.)h(12{24,)e(1989.)49
1458 y([22])22 b(R.)15 b(Impagliazzo)i(and)e(D.)g(Zuc)o(k)o(erman,)g
(\\Ho)o(w)g(to)f(Recycle)k(Random)e(Bits",)f Fp(30th)i(F)o(OCS)p
Fu(,)d(1989,)g(pp.)143 1520 y(248-253.)49 1620 y([23])22
b(R.M.)12 b(Karp,)i(N.)f(Pippinger)i(and)e(M.)g(Sipser,)h(\\A)f
(Time-Randomness)i(T)l(radeo\013)t(",)c Fp(AMS)j(Confer)n(enc)n(e)143
1682 y(on)i(Pr)n(ob)n(abilistic)f(Computational)h(Complexity)t
Fu(,)f(Durham,)g(New)g(Hampshire)h(\(1982\).)49 1781
y([24])22 b(A.)12 b(Lub)q(otzky)l(,)i(R.)f(Phillips,)j(P)l(.)c(Sarnak,)
h(\\Explicit)i(Expanders)e(and)g(the)g(Raman)o(ujan)g(Conjectures",)143
1843 y Fp(Pr)n(o)n(c.)i(18th)i(STOC)p Fu(,)d(1986,)f(pp.)j(240-246.)49
1943 y([25])22 b(G.A.)c(Margulis,)i(\\Explicit)h(Construction)e(of)g
(Concen)o(trators",)f Fp(Pr)n(ob.)i(Per.)f(Infor.)g Fu(9)g(\(4\))f
(\(1973\),)143 2005 y(71{80.)13 b(\(In)j(Russian,)g(English)g
(translation)f(in)h Fp(Pr)n(oblems)g(of)g(Infor.)g(T)m(r)n(ans.)d
Fu(\(1975\),)g(325{332.\))49 2104 y([26])22 b(J.)14 b(Naor)f(and)i(M.)e
(Naor,)g(\\Small-bias)j(Probabilit)o(y)f(Spaces:)20 b(E\016cien)o(t)15
b(Constructions)f(and)g(Applica-)143 2167 y(tions",)g
Fp(SIAM)h(J.)h(on)g(Computing)p Fu(,)f(V)l(ol)h(22,)e(1993,)g(pp.)h
(838{856.)49 2266 y([27])22 b(N.)13 b(Nisan,)h(\\Pseudorandom)f
(Generators)g(for)g(Space)h(Bounded)h(Computation",)e
Fp(Combinatoric)n(a)h Fu(12)143 2328 y(\(4\),)g(1992,)f(pp.)j(449{461.)
49 2428 y([28])22 b(N.)15 b(Nisan,)g(\\)p Fn(RL)d(\022)h(S)s(C)s
Fu(",)h Fp(Journal)j(of)f(Computational)h(Complexity)e
Fu(4,)f(1994,)g(pp.)h(1{11.)49 2527 y([29])22 b(N.)14
b(Nisan)g(and)h(D.)f(Zuc)o(k)o(erman,)f(\\Randomness)i(is)f(Linear)i
(in)f(Space",)f(to)g(app)q(ear)g(in)h Fp(JCSS)p Fu(.)e(Prelim-)143
2589 y(inary)i(v)o(ersion)h(in)g Fp(25th)h(STOC)p Fu(,)c(pp.)i
(235{244,)e(1993.)952 2795 y(24)p eop
%%Page: 25 26
25 25 bop 49 42 a Fu([30])22 b(M.)e(Sipser,)k(\\A)d(Complexit)o(y)h
(Theoretic)g(Approac)o(h)f(to)g(Randomness",)i Fp(15th)g(STOC)p
Fu(,)c(1983,)j(pp.)143 104 y(330{335.)49 203 y([31])g(M.)c(Sipser,)i
(\\Expanders,)f(Randomness)h(or)e(Time)h(vs)g(Space",)h
Fp(Structur)n(e)g(in)f(Complexity)g(The)n(ory)143 265
y Fu(\(pro)q(ceedings\),)c(1986.)49 365 y([32])22 b(L.)10
b(Sto)q(c)o(kmey)o(er,)h(\\The)g(Complexit)o(y)g(of)f(Appro)o(ximate)g
(Coun)o(ting",)h Fp(15th)i(STOC)p Fu(,)c(1983,)h(pp.)g(118{126.)49
464 y([33])22 b(A.)e(Sriniv)m(asan)j(and)e(D.)f(Zuc)o(k)o(erman,)i
(\\Computing)f(with)g(V)l(ery)g(W)l(eak)g(Random)g(Sources",)h
Fp(35th)143 526 y(F)o(OCS)p Fu(,)13 b(pp.)i(264{275,)e(1994.)49
626 y([34])22 b(L.)16 b(V)l(alian)o(t)i(and)f(V.V.)f(V)l(azirani,)h
(\\NP)g(is)g(as)f(Easy)h(as)f(Detecting)h(Unique)h(Solutions",)f
Fp(The)n(or)n(etic)n(al)143 688 y(Computer)g(Scienc)n(e)p
Fu(,)c(V)l(ol.)i(47,)f(1986,)g(pp.)h(85{93.)49 787 y([35])22
b(A.)14 b(Wigderson,)h(D.)f(Zuc)o(k)o(erman,)g(\\)h(Expanders)g(that)g
(Beat)f(the)h(Eigen)o(v)m(alue)i(Bound,)e(Explicit)i(Con-)143
849 y(struction)k(and)h(Applications",)j Fp(Pr)n(o)n(c.)d(of)g(the)g
(25th)h(STOC)p Fu(,)d(pp.)i(245{251,)f(1993.)f(T)l(o)h(app)q(ear)h(in)
143 911 y Fp(Combinatoric)n(a)p Fu(.)49 1011 y([36])g(D.)d(Zuc)o(k)o
(erman,)h(\\Sim)o(ulating)h(BPP)e(Using)i(a)e(General)h(W)l(eak)g
(Random)g(Source,")h Fp(A)o(lgorithmic)n(a)p Fu(,)143
1073 y(V)l(ol.)15 b(16,)f(pp.)i(367{391,)c(1996.)49 1172
y([37])22 b(D.)11 b(Zuc)o(k)o(erman,)h(\\Randomness-Optimal)i
(Sampling,)g(Extractors,)d(and)h(Constructiv)o(e)g(Leader)h(Elec-)143
1235 y(tion",)h Fp(28th)k(STOC)p Fu(,)13 b(1996,)g(pp.)j(286{295.)952
2795 y(25)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF