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Fm(\001)241 2474 y Fu(=)p Fv(12,)h(the)h(collision)j(probabilit)o(y)e (do)q(es)g(not)e(dep)q(end)j(on)e Fu(n)p Fv(,)g(either.)71 2531 y(Observ)o(e)21 b(that)f Fu(\025)p Fv(\()p Fu(N)5 b Fv(\))20 b(=)i Fs(N)7 b Fu(=)p Fv(\()p Fs(N)21 b(\000)14 b Fv(36\),)21 b(where)g Fs(N)28 b Fv(=)23 b Fu(N)5 b Fv(\()p Fu(N)t Fs(\000)h Fv(1\)\()p Fu(N)t Fs(\000)g Fv(2\).)33 b(By)21 b(our)g(assumption)g(that)0 2587 y Fu(N)c Fs(\025)c Fv(32)i(w)o(e)g(ha)o(v)o(e)g(that)g(1)d Fs(\024)h Fu(\025)p Fv(\()p Fu(N)5 b Fv(\))11 b Fs(\024)i Fv(1)p Fu(:)p Fv(002.)19 b(So)c(the)h(m)o(ultiplication)h(b)o(y)f Fu(\025)p Fv(\()p Fu(N)5 b Fv(\))13 b(can)j(e\013ectiv)o(ely)g(b)q(e)g (ignored;)0 2644 y Fu(B)r Fv(\()p Fu(N)5 b Fv(\))12 b Fs(\031)h Fu(\014)r Fv(\()p Fu(N)5 b Fv(\).)964 2824 y(6)p eop %%Page: 7 7 7 6 bop 217 1084 a currentpoint currentpoint translate 1.0 1.0 scale neg exch neg exch translate 217 1084 a @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 3600 @rwi @setspecial %%BeginDocument: plot.eps /gnudict 40 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /vshift -46 def /dl {10 mul} def /hpt 31.5 def /vpt 31.5 def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /P { stroke [] 0 setdash currentlinewidth 2 div sub M 0 currentlinewidth V stroke } def /D { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke P } def /A { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /B { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke P } def /C { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /T { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke P } def /S { 2 copy A C} def end gnudict begin gsave 50 50 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gsave translate 90 rotate 0 0 M (Upper bound on collision probablity, B\(N\)) Cshow grestore 3904 71 M (Number of buckets, N) Cshow LT0 840 4875 M 3 -6 V 3 -5 V 3 -5 V 3 -6 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -6 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -4 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -5 V 3 -4 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -4 V 3 -5 V 3 -5 V 3 -5 V 3 -4 V 3 -5 V 3 -5 V 3 -5 V 3 -4 V 3 -5 V 3 -5 V 3 -4 V 3 -5 V 3 -5 V 4 -4 V 3 -5 V 3 -5 V 3 -4 V 3 -5 V 3 -4 V 3 -5 V 3 -5 V 3 -4 V 3 -5 V 3 -4 V 3 -5 V 3 -4 V 3 -5 V 3 -5 V 4 -4 V 3 -5 V 3 -4 V 3 -5 V 3 -4 V 3 -5 V 3 -4 V 3 -5 V 3 -4 V 3 -5 V 3 -4 V 3 -4 V 3 -5 V 3 -4 V 3 -5 V 4 -4 V 3 -5 V 3 -4 V 3 -4 V 3 -5 V 3 -4 V 3 -5 V 3 -4 V 3 -4 V 3 -5 V 3 -4 V 3 -4 V 3 -5 V 3 -4 V 3 -4 V 4 -5 V 3 -4 V 3 -4 V 3 -5 V 3 -4 V 3 -4 V 3 -4 V 3 -5 V 3 -4 V 3 -4 V 3 -4 V 3 -5 V 3 -4 V 3 -4 V 3 -4 V 4 -5 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -5 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -5 V 3 -4 V 3 -4 V 4 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -5 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 4 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 4 -4 V 3 -3 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -3 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 4 -4 V 3 -4 V 3 -3 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -3 V 3 -4 V 3 -4 V 3 -4 V 3 -4 V 3 -3 V 3 -4 V 3 -4 V 4 -4 V 3 -3 V 3 -4 V 3 -4 V 3 -4 V 3 -3 V 3 -4 V 3 -4 V 3 -4 V 3 -3 V 3 -4 V 3 -4 V 3 -3 V 3 -4 V 3 -4 V 4 -4 V 3 -3 V 3 -4 V 3 -4 V 3 -3 V 3 -4 V 3 -4 V 3 -3 V 3 -4 V 3 -3 V 3 -4 V 3 -4 V 3 -3 V 3 -4 V 3 -4 V 4 -3 V 3 -4 V 3 -3 V 3 -4 V 3 -4 V 3 -3 V 3 -4 V 3 -3 V 3 -4 V 3 -3 V 3 -4 V 3 -4 V 3 -3 V 3 -4 V 3 -3 V 4 -4 V 3 -3 V 3 -4 V 3 -3 V 3 -4 V 3 -3 V 3 -4 V 3 -3 V 3 -4 V 3 -3 V 3 -4 V 3 -3 V 3 -4 V 3 -3 V 3 -4 V 3 -3 V 4 -4 V 3 -3 V 3 -4 V 3 -3 V 3 -4 V 3 -3 V 3 -3 V 3 -4 V 3 -3 V 3 -4 V 3 -3 V 3 -3 V 3 -4 V 3 -3 V 3 -4 V 4 -3 V 3 -4 V 3 -3 V 3 -3 V 3 -4 V 3 -3 V 3 -3 V 3 -4 V 3 -3 V 3 -3 V 3 -4 V 3 -3 V 3 -4 V 3 -3 V 3 -3 V 4 -4 V 3 -3 V 3 -3 V 3 -4 V 3 -3 V 3 -3 V 3 -3 V 3 -4 V 3 -3 V 3 -3 V 3 -4 V 3 -3 V 3 -3 V 3 -4 V 3 -3 V 4 -3 V 3 -3 V 3 -4 V 3 -3 V 3 -3 V 3 -3 V 3 -4 V 3 -3 V 3 -3 V 3 -3 V 3 -4 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 4 -4 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -4 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -4 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 4 -3 V 3 -4 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -4 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 4 -3 V 3 -3 V 3 -4 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 4 -3 V 3 -3 V 3 -4 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 4 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V currentpoint stroke M 3 -3 V 3 -3 V 4 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 4 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 4 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 4 -2 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -3 V 3 -2 V 4 -3 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 4 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 4 -2 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 4 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 4 -2 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 4 -3 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -2 V 3 -3 V 4 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 4 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 4 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 4 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 4 -3 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -3 V 4 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 4 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -3 V 4 -2 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 4 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 4 -2 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 4 -3 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -3 V 4 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 4 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -2 V 3 -2 V currentpoint stroke M 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -3 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 4 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 4 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 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-2 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 4 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 4 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 4 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 4 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 4 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V currentpoint stroke M 3 -1 V 3 -2 V 3 -1 V 3 -2 V 4 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 4 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 4 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 4 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 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3 -1 V 4 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 4 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 4 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 4 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 4 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 4 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 4 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 4 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 4 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 4 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 4 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 4 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 4 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 4 -2 V 3 -1 V currentpoint stroke M 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 4 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 4 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 4 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 4 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 V 4 -1 V 3 -2 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 3 -1 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V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 4 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -2 V 3 -1 V 4 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V 3 -1 V currentpoint stroke M stroke grestore end showpage %%EndDocument @endspecial 217 1084 a currentpoint currentpoint translate 1 1.0 div 1 1.0 div scale neg exch neg exch translate 217 1084 a 0 1182 a Fv(Figure)18 b(1:)26 b Ff(A)18 b(graphical)g(rep)o(resentation)i(of)e(Theo)o(rem)f (7.)28 b(W)o(e)18 b(plot)h(of)f Fu(N)k Ff(verses,)e Fu(B)r Fv(\()p Fu(N)5 b Fv(\))p Ff(,)18 b(our)f(b)q(ound)k(on)d(the)0 1238 y(collision)d(p)o(robabilit)o(y)g(of)g Fs(B)q Fv([)p Fu(w)q(;)8 b(n;)g(N)d Fv(])m Ff(.)71 1377 y Fv(W)l(e)10 b(b)q(eliev)o(e)i(that)e(it)h(is)f(p)q(ossible)i(to)e(relax)h(the)f (restriction)h Fu(n)i Fs(\024)1163 1343 y Fm(\000)1182 1357 y Fr(N)1189 1393 y Fp(3)1213 1343 y Fm(\001)1233 1377 y Fu(=)p Fv(12)c(all)i(the)g(w)o(a)o(y)e(to)h Fu(n)j(<)1669 1343 y Fm(\000)1688 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Fu(j)g Fv(=)e Fu(i)s Fv(+)s(1)470 2091 y(\()p Fu(N)p Fs(\000)s Fu(i)p Fv(\)\()p Fu(N)p Fs(\000)s Fu(i)-5 b Fs(\000)s Fv(1\)\()o Fu(N)p Fs(\000)s Fu(i)f Fs(\000)s Fv(2\))o(\))p Fu(=)p Fs(N)45 b Fv(if)16 b(3)c Fs(\024)h Fu(i)g Fs(\024)g Fu(N)p Fs(\000)s Fv(3)h(and)h Fu(j)g Fv(=)e Fu(i)s Fv(+)s(3)470 2148 y(0)590 b(otherwise)1892 1865 y(\(1\))0 2247 y(Let)14 b(us)g(giv)o(e)f(an)h(example)g(of)g(ho)o (w)f(the)g(ab)q(o)o(v)o(e)h(v)m(alues)g(are)g(computed.)19 b(Consider)14 b Fu(P)1465 2254 y Fr(ij)1509 2247 y Fv(for)f(the)h(case) g(asso)q(ciated)0 2303 y(to)g(3)e Fs(\024)h Fu(i)f Fs(\024)h Fu(N)n Fs(\000)q Fv(3)h(and)h Fu(j)g Fv(=)e Fu(i)8 b Fv(+)g(1.)20 b(In)15 b(order)f(to)f(go)h(from)g(state)f Fu(i)h Fv(to)g(state)g Fu(i)8 b Fv(+)g(1)14 b(in)i(a)e(single)h(step,)f (one)h(ball)g(of)0 2360 y(the)d(three)h(will)h(ha)o(v)o(e)e(to)g(land)h (in)g(one)f(of)g(the)h Fu(i)f Fv(buc)o(k)o(ets)g(that)g(has)g(an)g(o)q (dd)h(n)o(um)o(b)q(er)g(of)f(balls)h(already)l(,)g(while)h(the)0 2416 y(remaining)h(t)o(w)o(o)e(balls)i(m)o(ust)e(land)i(among)e(the)h Fu(N)e Fs(\000)c Fu(i)14 b Fv(remaining)h(buc)o(k)o(ets.)k(There)14 b(are)g(3)p Fu(i)p Fv(\()p Fu(N)d Fs(\000)d Fu(i)p Fv(\)\()p Fu(N)j Fs(\000)d Fu(i)g Fs(\000)g Fv(1\))0 2473 y(ordered)17 b(triples)h(of)e(buc)o(k)o(et)h(indices)i(that)d(will)j(accomplish)f (this)g(among)e(the)h Fs(N)23 b Fv(ordered)18 b(triples)g(of)e(buc)o(k) o(et)0 2529 y(indices.)29 b(\(The)17 b(\\3")g(tak)o(es)g(care)g(of)g (the)h(fact)f(that)g(there)g(are)g(3)p Fu(i)g Fv(w)o(a)o(ys)g(to)g(c)o (ho)q(ose)g(the)h(ball)g(whic)o(h)h(lands)f(in)0 2585 y(a)e(buc)o(k)o(et)h(with)g(an)f(o)q(dd)h(n)o(um)o(b)q(er)g(of)f (balls;)i(after)d(that)h(ball)i(is)f(selected,)h(the)e(remaining)i(t)o (w)o(o)d(balls)i(ha)o(v)o(e)g(to)0 2642 y(land)f(in)g(the)f(other)g Fu(N)g Fs(\000)10 b Fu(i)15 b Fv(buc)o(k)o(ets.\))k(The)d(reasoning)f (for)g(all)h(of)f(the)g(other)g Fu(P)1384 2649 y Fr(ij)1429 2642 y Fv(v)m(alues)i(is)f(similar.)952 2824 y(11)p eop %%Page: 12 12 12 11 bop 325 1092 a currentpoint currentpoint translate .5 .5 scale neg exch neg exch translate 325 1092 a @beginspecial 72 @llx 180 @lly 689 @urx 688 @ury 6170 @rwi @setspecial %%BeginDocument: 10.eps 1 setlinejoin /M { moveto } bind def /S { show } bind def /R { rmoveto } bind def /L { lineto } bind def /B { newpath 0 0 M 0 1 L 1 1 L 1 0 L closepath } bind def /CS { closepath stroke } bind def /S { /fixwidth exch def dup length /nchars exch def dup stringwidth pop fixwidth exch sub nchars div exch 0 exch ashow } def /bwproc { rgbproc dup length 3 idiv string 0 3 0 5 -1 roll { add 2 1 roll 1 sub dup 0 eq { pop 3 idiv 3 -1 roll dup 4 -1 roll dup 3 1 roll 5 -1 roll put 1 add 3 0 } { 2 1 roll } ifelse } forall pop pop pop } def systemdict /colorimage known not { /colorimage { pop pop /rgbproc exch def { bwproc } image } def } if 1 1 scale 0 setlinewidth /drawtri { /y3 exch def /x3 exch def /y2 exch def /x2 exch def /y1 exch def /x1 exch def 0 setgray newpath x1 y1 moveto x2 y2 lineto x3 y3 lineto closepath stroke } bind def /filltri { /y3 exch def /x3 exch def /y2 exch def /x2 exch def /y1 exch def /x1 exch def newpath x1 y1 moveto x2 y2 lineto x3 y3 lineto closepath fill } bind def /cliptri { /y3 exch def /x3 exch def /y2 exch def /x2 exch def /y1 exch def /x1 exch def 0 setgray newpath x1 y1 moveto x2 y2 lineto x3 y3 lineto closepath clip } bind def /imgscanrgb { gsave translate /scandy exch def /scandx exch def /istr scandx 3 mul string def scandx scandy scale scandx scandy 8 [scandx 0 0 scandy neg 0 scandy] {currentfile istr readhexstring pop} false 3 colorimage grestore } bind def /imgscanbw { gsave translate /scandy exch def /scandx exch def /istr scandx string def scandx scandy scale scandx scandy 8 [scandx 0 0 scandy neg 0 scandy] {currentfile istr readhexstring pop} image grestore } bind def /showcaseisoencoding [ /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef 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/caron /emdash /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def /showcasedingbatencoding [ /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /a1 /a2 /a202 /a3 /a4 /a5 /a119 /a118 /a117 /a11 /a12 /a13 /a14 /a15 /a16 /a105 /a17 /a18 /a19 /a20 /a21 /a22 /a23 /a24 /a25 /a26 /a27 /a28 /a6 /a7 /a8 /a9 /a10 /a29 /a30 /a31 /a32 /a33 /a34 /a35 /a36 /a37 /a38 /a39 /a40 /a41 /a42 /a43 /a44 /a45 /a46 /a47 /a48 /a49 /a50 /a51 /a52 /a53 /a54 /a55 /a56 /a57 /a58 /a59 /a60 /a61 /a62 /a63 /a64 /a65 /a66 /a67 /a68 /a69 /a70 /a71 /a72 /a73 /a74 /a203 /a75 /a204 /a76 /a77 /a78 /a79 /a81 /a82 /a83 /a84 /a97 /a98 /a99 /a100 /.notdef /a205 /a85 /a206 /a86 /a87 /a88 /a89 /a90 /a91 /a92 /a93 /a94 /a95 /a96 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /a101 /a102 /a103 /a104 /a106 /a107 /a108 /a112 /a111 /a110 /a109 /a120 /a121 /a122 /a123 /a124 /a125 /a126 /a127 /a128 /a129 /a130 /a131 /a132 /a133 /a134 /a135 /a136 /a137 /a138 /a139 /a140 /a141 /a142 /a143 /a144 /a145 /a146 /a147 /a148 /a149 /a150 /a151 /a152 /a153 /a154 /a155 /a156 /a157 /a158 /a159 /a160 /a161 /a163 /a164 /a196 /a165 /a192 /a166 /a167 /a168 /a169 /a170 /a171 /a172 /a173 /a162 /a174 /a175 /a176 /a177 /a178 /a179 /a193 /a180 /a199 /a181 /a200 /a182 /.notdef /a201 /a183 /a184 /a197 /a185 /a194 /a198 /a186 /a195 /a187 /a188 /a189 /a190 /a191 /.notdef ] def /Times-Roman findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding showcaseisoencoding def currentdict end /Times-Roman-SHOWISO exch definefont pop /Courier findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding showcaseisoencoding def currentdict end /Courier-SHOWISO exch definefont pop /newfont 10 dict def newfont begin /FontType 3 def /FontMatrix [1 0 0 1 0 0] def /FontBBox [0 0 1 1] def /Encoding 256 array def 0 1 255 {Encoding exch /.notdef put} for /CharProcs 1 dict def CharProcs begin /.notdef {} def end /BuildChar { 1 0 0 0 1 1 setcachedevice exch begin Encoding exch get CharProcs exch get end exec } def end /PatternFont newfont definefont pop gsave /saveit save def gsave gsave 1.000000 setlinewidth matrix currentmatrix [38.1838 0 0 38.1838 432 649] concat newpath 0 0 1 0 360 arc 0 0 0 setrgbcolor closepath setmatrix stroke grestore gsave 1.000000 setlinewidth matrix currentmatrix [38.1838 0 0 38.1838 541.184 540.184] concat newpath 0 0 1 0 360 arc 0 0 0 setrgbcolor closepath setmatrix stroke grestore gsave 1.000000 setlinewidth matrix currentmatrix [38.1838 0 0 38.1838 650.184 434.184] concat newpath 0 0 1 0 360 arc 0 0 0 setrgbcolor closepath setmatrix stroke grestore gsave 1.000000 setlinewidth matrix currentmatrix [38.1838 0 0 38.1838 326.368 540.368] concat newpath 0 0 1 0 360 arc 0 0 0 setrgbcolor closepath setmatrix stroke grestore gsave 1.000000 setlinewidth matrix currentmatrix [38.1838 0 0 38.1838 327.368 325.368] concat newpath 0 0 1 0 360 arc 0 0 0 setrgbcolor closepath setmatrix stroke grestore gsave 0 0 0 setrgbcolor 1.000000 setlinewidth newpath 251.252 453.276 M 249.984 455.32 L 248.568 457.309 L 247.008 459.228 L 245.307 461.06 L 243.473 462.792 L 241.516 464.41 L 239.445 465.9 L 237.276 467.252 L 235.02 468.455 L 232.695 469.503 L 230.315 470.389 L 227.898 471.111 L 225.46 471.668 L 223.018 472.06 L 220.588 472.292 L 218.184 472.368 L 218.184 472.368 L 215.779 472.292 L 213.349 472.06 L 210.907 471.668 L 208.469 471.111 L 206.053 470.389 L 203.673 469.503 L 201.347 468.455 L 199.092 467.252 L 196.922 465.9 L 194.852 464.41 L 192.895 462.792 L 191.061 461.06 L 189.36 459.228 L 187.799 457.309 L 186.383 455.32 L 185.116 453.276 L 185.116 453.276 L 183.979 451.156 L 182.964 448.935 L 182.083 446.624 L 181.347 444.235 L 180.763 441.781 L 180.341 439.277 L 180.086 436.739 L 180 434.184 L 180.086 431.629 L 180.341 429.091 L 180.763 426.587 L 181.347 424.133 L 182.083 421.744 L 182.964 419.433 L 183.979 417.212 L 185.116 415.092 L 185.116 415.092 L 186.383 413.048 L 187.799 411.059 L 189.36 409.14 L 191.061 407.307 L 192.895 405.575 L 194.852 403.957 L 196.922 402.467 L 199.092 401.116 L 201.347 399.912 L 203.673 398.865 L 206.053 397.978 L 208.469 397.256 L 210.907 396.7 L 213.349 396.307 L 215.779 396.076 L 218.184 396 L 218.184 396 L 220.588 396.076 L 223.018 396.307 L 225.46 396.7 L 227.898 397.256 L 230.315 397.978 L 232.695 398.865 L 235.02 399.912 L 237.276 401.116 L 239.445 402.467 L 241.516 403.957 L 243.473 405.575 L 245.307 407.307 L 247.008 409.14 L 248.568 411.059 L 249.984 413.048 L 251.252 415.092 L 251.252 415.092 L 252.388 417.212 L 253.403 419.433 L 254.284 421.744 L 255.021 424.133 L 255.604 426.587 L 256.026 429.091 L 256.282 431.629 L 256.368 434.184 L 256.282 436.739 L 256.026 439.277 L 255.604 441.781 L 255.021 444.235 L 254.284 446.624 L 253.403 448.935 L 252.388 451.156 L 251.252 453.276 L closepath stroke grestore gsave 1.000000 setlinewidth matrix currentmatrix [38.1838 0 0 38.1838 434.368 434.368] concat newpath 0 0 1 0 360 arc 0 0 0 setrgbcolor closepath setmatrix stroke grestore gsave 1.000000 setlinewidth matrix currentmatrix [38.1838 0 0 38.1838 543.368 327.368] concat newpath 0 0 1 0 360 arc 0 0 0 setrgbcolor closepath setmatrix stroke grestore gsave 0 0 0 setrgbcolor 1.000000 setlinewidth newpath 143.252 345.276 M 141.984 347.32 L 140.568 349.309 L 139.008 351.228 L 137.307 353.06 L 135.473 354.792 L 133.516 356.41 L 131.446 357.9 L 129.276 359.252 L 127.02 360.455 L 124.695 361.503 L 122.315 362.389 L 119.898 363.111 L 117.46 363.668 L 115.018 364.06 L 112.588 364.292 L 110.184 364.368 L 110.184 364.368 L 107.78 364.292 L 105.349 364.06 L 102.907 363.668 L 100.47 363.111 L 98.0526 362.389 L 95.673 361.503 L 93.3473 360.455 L 91.0919 359.252 L 88.922 357.9 L 86.852 356.41 L 84.8945 354.792 L 83.0608 353.06 L 81.36 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setrgbcolor 0 6.4 M (0) 16 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 318 524.602] concat newpath 0 0 M 0 32 L 18 32 L 18 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 316.25 522.852 M 316.25 559.002 L 337.75 559.002 L 337.75 522.852 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 318 524.602] concat /Times-Roman-SHOWISO findfont 32 scalefont setfont 0 0 0 setrgbcolor 0 6.4 M (3) 16 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 208 417.602] concat newpath 0 0 M 0 32 L 18 32 L 18 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 206.25 415.852 M 206.25 452.002 L 227.75 452.002 L 227.75 415.852 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 208 417.602] concat /Times-Roman-SHOWISO findfont 32 scalefont setfont 0 0 0 setrgbcolor 0 6.4 M (6) 16 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 103 308.602] concat newpath 0 0 M 0 32 L 18 32 L 18 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 101.25 306.852 M 101.25 343.002 L 122.75 343.002 L 122.75 306.852 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 103 308.602] concat /Times-Roman-SHOWISO findfont 32 scalefont setfont 0 0 0 setrgbcolor 0 6.4 M (9) 16 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 533 524.602] concat newpath 0 0 M 0 32 L 18 32 L 18 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 531.25 522.852 M 531.25 559.002 L 552.75 559.002 L 552.75 522.852 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 533 524.602] concat /Times-Roman-SHOWISO findfont 32 scalefont setfont 0 0 0 setrgbcolor 0 6.4 M (1) 16 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 641 416.602] concat newpath 0 0 M 0 32 L 18 32 L 18 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 639.25 414.852 M 639.25 451.002 L 660.75 451.002 L 660.75 414.852 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 641 416.602] concat /Times-Roman-SHOWISO findfont 32 scalefont setfont 0 0 0 setrgbcolor 0 6.4 M (2) 16 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 425 416.602] concat newpath 0 0 M 0 32 L 18 32 L 18 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 423.25 414.852 M 423.25 451.002 L 444.75 451.002 L 444.75 414.852 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 425 416.602] concat /Times-Roman-SHOWISO findfont 32 scalefont setfont 0 0 0 setrgbcolor 0 6.4 M (4) 16 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 533 308.602] concat newpath 0 0 M 0 32 L 18 32 L 18 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 531.25 306.852 M 531.25 343.002 L 552.75 343.002 L 552.75 306.852 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 533 308.602] concat /Times-Roman-SHOWISO findfont 32 scalefont setfont 0 0 0 setrgbcolor 0 6.4 M (5) 16 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 318 308.602] concat newpath 0 0 M 0 32 L 18 32 L 18 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 316.25 306.852 M 316.25 343.002 L 337.75 343.002 L 337.75 306.852 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 318 308.602] concat /Times-Roman-SHOWISO findfont 32 scalefont setfont 0 0 0 setrgbcolor 0 6.4 M (7) 16 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 425 200.602] concat newpath 0 0 M 0 32 L 18 32 L 18 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 423.25 198.852 M 423.25 235.002 L 444.75 235.002 L 444.75 198.852 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 425 200.602] concat /Times-Roman-SHOWISO findfont 32 scalefont setfont 0 0 0 setrgbcolor 0 6.4 M (8) 16 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 202 200.602] concat newpath 0 0 M 0 32 L 34 32 L 34 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 200.25 198.852 M 200.25 235.002 L 237.75 235.002 L 237.75 198.852 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 202 200.602] concat /Times-Roman-SHOWISO findfont 32 scalefont setfont 0 0 0 setrgbcolor 0 6.4 M (10) 32 S setmatrix grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 396.707 449.293 translate -45 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 342 504 M 390.343 455.657 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 288.707 341.293 translate -45 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 234 396 M 282.343 347.657 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 509.207 341.293 translate -45 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 454.5 396 M 502.843 347.657 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 401.207 233.293 translate -45 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 346.5 288 M 394.843 239.657 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 359.293 522.707 translate 135 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 365.657 516.343 M 414 468 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 251.293 414.707 translate 135 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 257.657 408.343 M 306 360 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 471.793 414.707 translate 135 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 478.157 408.343 M 526.5 360 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 359.793 306.707 translate 135 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 366.157 300.343 M 414.5 252 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 345.824 575.263 translate -132.51 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 396 630 M 351.906 581.897 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 237.824 467.263 translate -132.51 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 288 522 M 243.906 473.897 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 129.824 359.263 translate -132.51 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 180 414 M 135.906 365.897 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 414.676 612.737 translate 47.4896 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 408.594 606.103 M 364.5 558 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 306.676 504.737 translate 47.4896 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 300.594 498.103 M 256.5 450 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 198.676 396.737 translate 47.4896 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 192.594 390.103 M 148.5 342 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 453.824 467.263 translate -132.51 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 504 522 M 459.906 473.897 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 345.824 359.263 translate -132.51 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 396 414 M 351.906 365.897 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 237.824 251.263 translate -132.51 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 288 306 M 243.906 257.897 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 522.676 504.737 translate 47.4896 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 516.594 498.103 M 472.5 450 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 414.676 396.737 translate 47.4896 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 408.594 390.103 M 364.5 342 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 306.676 288.737 translate 47.4896 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 300.594 282.103 M 256.5 234 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 561.824 359.263 translate -132.51 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 612 414 M 567.906 365.897 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 453.824 251.263 translate -132.51 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 504 306 M 459.906 257.897 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 630.676 396.737 translate 47.4896 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 624.594 390.103 M 580.5 342 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 522.676 288.737 translate 47.4896 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 516.594 282.103 M 472.5 234 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 367 546 translate 180 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 376 546 M 431 546 L 539 438 L 611 438 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 607 422 translate 0 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 363 530 M 426 530 L 534 422 L 598 422 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 255 447 translate 180 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 264 447 M 319 447 L 427 339 L 499 339 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 502 321 translate 0 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 258 429 M 321 429 L 429 321 L 493 321 L stroke grestore gsave gsave matrix currentmatrix [1 0 0 1 251 507.57] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 249.25 505.82 M 249.25 527.999 L 281.75 527.999 L 281.75 505.82 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 251 507.57] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (210) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 151 397.57] concat newpath 0 0 M 0 18 L 20 18 L 20 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 149.25 395.82 M 149.25 417.999 L 172.75 417.999 L 172.75 395.82 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 151 397.57] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (24) 18 S setmatrix grestore gsave gsave matrix currentmatrix [1.03448 0 0 1 279 463] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 277.19 461.25 M 277.19 483.429 L 310.81 483.429 L 310.81 461.25 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1.03448 0 0 1 279 463] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (120) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 220 367.57] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 218.25 365.82 M 218.25 387.999 L 250.75 387.999 L 250.75 365.82 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 220 367.57] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (216) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 359 616] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 357.25 614.25 M 357.25 636.429 L 389.75 636.429 L 389.75 614.25 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 359 616] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (720) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 274 410.57] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 272.25 408.82 M 272.25 430.999 L 304.75 430.999 L 304.75 408.82 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 274 410.57] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (360) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 473 642.211] concat newpath 0 0 M 0 36 L 77.6 36 L 77.6 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 470.5 639.711 M 470.5 680.711 L 553.1 680.711 L 553.1 639.711 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 473 642.211] concat /Courier-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 21.7895 M ( start) 75.6 S 0 3.78947 M ( state) 64.8 S setmatrix grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 150 339 translate 180 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 159 339 M 214 339 L 322 231 L 394 231 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 395 214 translate 0 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 151 322 M 214 322 L 322 214 L 386 214 L stroke grestore gsave gsave matrix currentmatrix [1 0 0 1 383 512.727] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 381.25 510.977 M 381.25 533.155 L 413.75 533.155 L 413.75 510.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 383 512.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (126) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 335 468.727] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 333.25 466.977 M 333.25 489.155 L 365.75 489.155 L 365.75 466.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 335 468.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (378) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 470 513.727] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 468.25 511.977 M 468.25 534.155 L 500.75 534.155 L 500.75 511.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 470 513.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (504) 27 S setmatrix grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 613.707 451.293 translate -45 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 563 502 M 607.343 457.657 L stroke grestore gsave 0 0 0 setrgbcolor gsave 0 setlinejoin 576.293 520.707 translate 135 rotate 1 1 scale newpath 0 0 M -10 -3 L -10 3 L closepath fill grestore 0.500000 setlinewidth newpath 582.657 514.343 M 627 470 L stroke grestore gsave gsave matrix currentmatrix [1 0 0 1 615 480.727] concat newpath 0 0 M 0 18 L 20 18 L 20 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 613.25 478.977 M 613.25 501.155 L 636.75 501.155 L 636.75 478.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 615 480.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (48) 18 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 542 475.727] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 540.25 473.977 M 540.25 496.155 L 572.75 496.155 L 572.75 473.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 542 475.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (216) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 562 440.727] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 560.25 438.977 M 560.25 461.155 L 592.75 461.155 L 592.75 438.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 562 440.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (336) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 592 341.727] concat newpath 0 0 M 0 18 L 20 18 L 20 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 590.25 339.977 M 590.25 362.155 L 613.75 362.155 L 613.75 339.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 592 341.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (60) 18 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 568 392.727] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 566.25 390.977 M 566.25 413.155 L 598.75 413.155 L 598.75 390.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 568 392.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (336) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 483 232.727] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 481.25 230.977 M 481.25 253.155 L 513.75 253.155 L 513.75 230.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 483 232.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (336) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 468 284.727] concat newpath 0 0 M 0 18 L 20 18 L 20 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 466.25 282.977 M 466.25 305.155 L 489.75 305.155 L 489.75 282.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 468 284.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (60) 18 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 452 337.727] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 450.25 335.977 M 450.25 358.155 L 482.75 358.155 L 482.75 335.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 452 337.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (300) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 437 373.727] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 435.25 371.977 M 435.25 394.155 L 467.75 394.155 L 467.75 371.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 437 373.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (360) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 508 376.727] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 506.25 374.977 M 506.25 397.155 L 538.75 397.155 L 538.75 374.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 508 376.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (300) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 397 267.727] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 395.25 265.977 M 395.25 288.155 L 427.75 288.155 L 427.75 265.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 397 267.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (336) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 326 265.727] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 324.25 263.977 M 324.25 286.155 L 356.75 286.155 L 356.75 263.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 326 265.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (126) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 345 228.727] concat newpath 0 0 M 0 18 L 20 18 L 20 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 343.25 226.977 M 343.25 249.155 L 366.75 249.155 L 366.75 226.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 345 228.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (48) 18 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 165 303.727] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 163.25 301.977 M 163.25 324.155 L 195.75 324.155 L 195.75 301.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 165 303.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (216) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 260 222.727] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 258.25 220.977 M 258.25 243.155 L 290.75 243.155 L 290.75 220.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 260 222.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (720) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 270 291.727] concat newpath 0 0 M 0 18 L 11 18 L 11 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 268.25 289.977 M 268.25 312.155 L 282.75 312.155 L 282.75 289.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 270 291.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (6) 9 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 369 336.727] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 367.25 334.977 M 367.25 357.155 L 399.75 357.155 L 399.75 334.977 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 369 336.727] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (210) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 366 406.727] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 364.25 404.977 M 364.25 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currentmatrix [1 0 0 1 401 480] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (216) 27 S setmatrix grestore gsave gsave matrix currentmatrix [1 0 0 1 291 375] concat newpath 0 0 M 0 18 L 29 18 L 29 0 L closepath setmatrix 0 0 0 setrgbcolor grestore newpath 289.25 373.25 M 289.25 395.429 L 321.75 395.429 L 321.75 373.25 L closepath clip newpath 0 0 0 setrgbcolor matrix currentmatrix [1 0 0 1 291 375] concat /Times-Roman-SHOWISO findfont 18 scalefont setfont 0 0 0 setrgbcolor 0 3.42857 M (378) 27 S setmatrix grestore grestore showpage saveit restore grestore %%EndDocument @endspecial 325 1092 a currentpoint currentpoint translate 1 .5 div 1 .5 div scale neg exch neg exch translate 325 1092 a 0 1190 a Fv(Figure)19 b(2:)27 b Ff(The)20 b(Ma)o(rk)o(ov)e(chain)h Fu(M)24 b Ff(fo)o(r)18 b Fu(N)24 b Fv(=)19 b(10)f Ff(states.)33 b(The)20 b(sta)o(rt)f(state)i(is)e(state)h(0.)31 b(Divide)19 b(the)h(numb)q(er)0 1246 y(lab)q(eling)c(each)g(a)o(rc)e Fu(i)e Fs(!)h Fu(j)18 b Ff(b)o(y)d Fs(N)20 b Fv(=)13 b Fu(N)5 b Fv(\()p Fu(N)14 b Fs(\000)c Fv(1\)\()p Fu(N)k Fs(\000)c Fv(2\))i(=)h(720)h Ff(to)i(get)f(the)i(transition)f(p)o (robabilit)o(y)f Fu(P)1755 1253 y Fr(ij)1785 1246 y Ff(.)71 1388 y Fv(In)k(Figure)g(2)g(w)o(e)f(depict)i(the)f(Mark)o(o)o(v)e(c)o (hain)j Fu(M)k Fv(for)18 b(some)g(the)h(case)g(where)g(the)g(n)o(um)o (b)q(er)g(of)g(states)e(is)0 1445 y Fu(N)g Fv(=)c(10.)19 b(The)d(transition)f(probabilities)j(are)d(computed)g(from)g(Equation)g (1.)0 1526 y Fw(Using)23 b Fu(M)29 b Fw(to)23 b(bound)h Fu(\016)487 1533 y Fp(4)507 1526 y Fw(.)38 b Fv(W)l(e)22 b(are)f(no)o(w)g(ready)g(to)g(sho)o(w)g(that)f Fu(\016)1264 1533 y Fp(4)1307 1526 y Fs(\024)j Fu(B)r Fv(\()p Fu(N)5 b Fv(\).)38 b(Recall)23 b(that)e Fu(B)r Fv(\()p Fu(N)5 b Fv(\))23 b(=)0 1582 y Fu(\025)p Fv(\()p Fu(N)5 b Fv(\))j Fs(\001)i Fu(\014)r Fv(\()p Fu(N)5 b Fv(\),)14 b(where)h Fu(\025)p Fv(\()p Fu(N)5 b Fv(\))14 b(and)h Fu(\014)r Fv(\()p Fu(N)5 b Fv(\))14 b(are)h(giv)o(en)h(b)o(y)f(the)g(form)o(ulas) g(in)h(the)f(statemen)o(t)f(of)h(Theorem)g(7.)0 1688 y Fl(Lemma)i(10)23 b Fu(\016)280 1695 y Fp(4)326 1688 y Fs(\024)i Fu(B)r Fv(\()p Fu(N)5 b Fv(\))12 b Ft(.)0 1819 y(Pr)n(o)n(of)d Fv(:)26 b(First)13 b(some)f(notation.)19 b(Let)13 b Fu(t)g Fs(\024)g Fu(n)h Fv(b)q(e)f(a)g(n)o(um)o(b)q(er)h (and)f Fu(h)1141 1826 y Fp(1)1168 1819 y Fs(\001)8 b(\001)g(\001)e Fu(h)1255 1826 y Fr(t)1283 1819 y Fv(b)q(e)14 b(a)e(sequence)i(of)f (triples)h(of)f(distinct)0 1875 y(elemen)o(ts)21 b(dra)o(wn)f(from)g Fs(f)p Fv(1)p Fu(;)8 b(:)g(:)g(:)t(;)g(N)d Fs(g)p Fv(.)35 b(Then)21 b Ff(P)o(a)o(rit)o(y)o Fv(\()p Fu(h)986 1882 y Fp(1)1013 1875 y Fs(\001)8 b(\001)g(\001)e Fu(h)1100 1882 y Fr(t)1115 1875 y Fv(\))20 b(is)h(the)g Fu(N)5 b Fv(-v)o(ector)19 b(whose)i Fu(i)p Fv(-th)f(comp)q(onen)o(t,)0 1932 y Fu(i)e Fs(2)g(f)p Fv(1)p Fu(;)8 b(:)g(:)g(:)t(;)g(N)d Fs(g)p Fv(,)17 b(is)i(0)f(if)h Fu(i)f Fv(o)q(ccurs)h(an)f(ev)o(en)h(n)o 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y(the)j(state)f(of)h Fu(M)22 b Fv(one)17 b(is)h(in)g(after)e(hashing)i Fl(1)805 2108 y Fr(t)836 2101 y Fv(with)g Fu(h)e Fv(=)g Fu(h)1061 2108 y Fp(1)1088 2101 y Fs(\001)8 b(\001)g(\001)e Fu(h)1175 2108 y Fr(t)1197 2101 y Fs(\001)i(\001)g(\001)n Fv(.)26 b(F)l(or)16 b Fu(\033)j Fv(an)e Fu(N)5 b Fv(-v)o(ector)16 b(of)g(bits,)i(de\014ne)0 2158 y Ff(State)100 2165 y Fr(\033)124 2158 y Fv(\()p Fu(h)168 2165 y Fp(1)195 2158 y Fs(\001)8 b(\001)g(\001)d Fu(h)281 2165 y Fr(t)296 2158 y Fv(\))18 b(=)g Ff(NumOnes)r Fv(\()p Fu(\033)r Fs(\010)o Ff(P)o(a)o(rit)o(y)p Fv(\()p Fu(h)808 2165 y Fp(1)835 2158 y Fs(\001)8 b(\001)g(\001)d Fu(h)921 2165 y Fr(t)936 2158 y Fv(\)\).)29 b(Th)o(us)18 b Ff(State)1233 2165 y Fr(\033)1256 2158 y Fv(\()p Fu(h)1300 2165 y Fp(1)1327 2158 y Fu(:)8 b(:)g(:)e(h)1414 2165 y Fr(t)1429 2158 y Fv(\))18 b(captures)g(the)g(state)g(of)g Fu(M)0 2214 y Fv(one)f(is)h(in)g(after)e(hashing)i Fl(1)492 2221 y Fr(t)524 2214 y Fv(with)g Fu(h)e Fv(=)g Fu(h)749 2221 y Fp(1)776 2214 y Fs(\001)8 b(\001)g(\001)e Fu(h)863 2221 y Fr(t)885 2214 y Fs(\001)i(\001)g(\001)n Fv(,)18 b(giv)o(en)f(that)g(w)o(e)f(start)g(in)i(the)g(con\014guration)f(sp)q (eci\014ed)0 2271 y(b)o(y)e Fu(\033)r Fv(.)20 b(Let)218 2372 y Ff(Hist)q Fv(\()p Fu(h)339 2379 y Fp(1)359 2372 y Fu(h)385 2379 y Fp(2)405 2372 y Fu(h)431 2379 y Fp(3)450 2372 y Fu(h)476 2379 y Fp(4)496 2372 y Fv(\))42 b(=)f(0)13 b Ff(State)q Fv(\()p Fu(h)812 2379 y Fp(1)831 2372 y Fv(\))g Ff(State)q Fv(\()p Fu(h)1006 2379 y Fp(1)1025 2372 y Fu(h)1051 2379 y Fp(2)1071 2372 y Fv(\))g Ff(State)q Fv(\()p Fu(h)1246 2379 y Fp(1)1265 2372 y Fu(h)1291 2379 y Fp(2)1311 2372 y Fu(h)1337 2379 y Fp(3)1357 2372 y Fv(\))g Ff(State)q Fv(\()p Fu(h)1532 2379 y Fp(1)1551 2372 y Fu(h)1577 2379 y Fp(2)1597 2372 y Fu(h)1623 2379 y Fp(3)1643 2372 y Fu(h)1669 2379 y Fp(4)1689 2372 y Fv(\))f Fu(:)0 2474 y Fv(This)22 b(is)g(a)e(list)i(of)f(5)g(n)o(um)o(b) q(ers,)i(eac)o(h)e(in)h Fs(f)p Fv(0)p Fu(;)8 b Fs(\001)g(\001)g(\001)t Fu(;)g(N)d Fs(g)p Fv(,)21 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Fs(\001)g(\001)g(\001)d Fu(;)j(h)489 2651 y Fr(t)520 2644 y Fv(\(that)16 b(is,)i(eac)o(h)f Fu(h)829 2651 y Fr(i)861 2644 y Fv(is)h(a)f(random)f(triple)j(of)e(distinct)h(p) q(oin)o(ts)g(from)e Fs(f)p Fv(1)p Fu(;)8 b(:)g(:)g(:)t(;)g(N)d Fs(g)p Fv(\).)952 2824 y(12)p eop %%Page: 13 13 13 12 bop 0 79 a Fv(Let)15 b Fu(D)119 86 y Fr(t)149 79 y Fv(\(\\distinct"\))g(b)q(e)h(the)f(uniform)h(distribution)g(on)g (distinct)g Fu(h)1206 86 y Fp(1)1226 79 y Fu(;)8 b(:)g(:)g(:)d(;)j(h) 1354 86 y Fr(t)1383 79 y Fv(\(that)14 b(is,)h(eac)o(h)h Fu(h)1686 86 y Fr(i)1715 79 y Fv(is)g(a)e(random)0 136 y(triple)20 b(of)e(distinct)i(p)q(oin)o(ts)f(from)f Fs(f)p Fv(1)p Fu(;)8 b(:)g(:)g(:)d(;)j(N)d Fs(g)p Fv(,)17 b(and)i(no)g(t)o(w)o (o)f(of)g(these)h(triples)h(are)e(iden)o(tical\).)32 b(Let)19 b Fu(C)s Fv(\()p Fu(m;)8 b(t)p Fv(\))0 192 y(denote)14 b(the)g(probabilit)o(y)h(of)f(at)f(least)h(one)h(collision)h(in)f(the)f (exp)q(erimen)o(t)h(of)e(thro)o(wing)h Fu(t)g Fv(balls,)h(indep)q (enden)o(tly)0 249 y(and)g(at)g(random,)f(in)o(to)i Fu(m)f Fv(bins.)71 305 y(W)l(e)g(are)g(no)o(w)f(ready)i(to)e(pro)o(v)o(e)h (the)g(lemma.)74 464 y Fu(\016)94 471 y Fp(4)156 464 y Fv(=)41 b(Pr)234 493 y Fr(D)263 498 y Fj(4)289 464 y Fv([)o Ff(State)q Fv(\()p Fu(h)445 471 y Fp(1)465 464 y Fu(h)491 471 y Fp(2)511 464 y Fu(h)537 471 y Fp(3)557 464 y Fu(h)583 471 y Fp(4)603 464 y Fv(\))12 b(=)h(0])156 601 y(=)41 b(Pr)235 631 y Fr(R)262 636 y Fj(4)289 601 y Fv([)o Ff(State)q Fv(\()p Fu(h)445 608 y Fp(1)465 601 y Fu(h)491 608 y Fp(2)511 601 y Fu(h)537 608 y Fp(3)557 601 y Fu(h)583 608 y Fp(4)603 601 y Fv(\))12 b(=)h(0)f Fs(j)h Ff(Distinct)q Fv(\()p Fu(h)934 608 y Fp(1)954 601 y Fu(h)980 608 y Fp(2)1000 601 y Fu(h)1026 608 y Fp(3)1046 601 y Fu(h)1072 608 y Fp(4)1091 601 y Fv(\)])156 777 y(=)237 721 y(Pr)240 751 y Fr(R)267 756 y Fj(4)294 721 y Fv([)o Ff(State)q Fv(\()p Fu(h)450 728 y Fp(1)470 721 y Fu(h)496 728 y Fp(2)516 721 y Fu(h)542 728 y Fp(3)562 721 y Fu(h)588 728 y Fp(4)608 721 y Fv(\))f(=)h(0)f(and)h 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y Fp(43)940 1486 y Fu(P)969 1493 y Fp(30)1029 1486 y Fv(+)g Fu(P)1116 1493 y Fp(03)1154 1486 y Fu(P)1183 1493 y Fp(32)1220 1486 y Fu(P)1249 1493 y Fp(23)1287 1486 y Fu(P)1316 1493 y Fp(30)1354 1439 y Fm(\021)156 1637 y Fv(=)41 b Fu(\025)p Fv(\()p Fu(N)5 b Fv(\))k Fs(\001)369 1589 y Fm(\020)393 1637 y Fv(1)h Fs(\001)454 1606 y Fv(\()p Fu(N)p Fs(\000)s Fv(3\)\()p Fu(N)p Fs(\000)s Fv(4\)\()o Fu(N)p Fs(\000)s Fv(5)o(\))p 454 1626 398 2 v 631 1668 a Fs(N)866 1637 y(\001)894 1606 y Fv(120)p 894 1626 69 2 v 906 1668 a Fs(N)977 1637 y(\001)1015 1606 y Fv(6)p 1005 1626 44 2 v 1005 1668 a Fs(N)1077 1637 y Fv(+)23 b(1)9 b Fs(\001)1195 1606 y Fv(9\()p Fu(N)p Fs(\000)s Fv(3\)\()p Fu(N)p Fs(\000)s Fv(4\))p 1195 1626 288 2 v 1317 1668 a Fs(N)1498 1637 y(\001)1525 1606 y Fv(36\()p Fu(N)p Fs(\000)s Fv(4\))p 1525 1626 178 2 v 1592 1668 a Fs(N)1718 1637 y(\001)1757 1606 y Fv(6)p 1746 1626 44 2 v 1746 1668 a Fs(N)1818 1637 y Fv(+)414 1749 y(1)h Fs(\001)475 1718 y Fv(18\()p Fu(N)p Fs(\000)s Fv(3\)\()p Fu(N)p Fs(\000)s Fv(4)o(\))p 475 1739 311 2 v 608 1780 a Fs(N)800 1749 y(\001)828 1718 y Fv(6\()p Fu(N)p Fs(\000)s Fv(2\)\()p Fu(N)p Fs(\000)s Fv(3\))p 828 1739 288 2 v 950 1780 a Fs(N)1130 1749 y(\001)1169 1718 y Fv(6)p 1158 1739 44 2 v 1158 1780 a Fs(N)1207 1702 y Fm(\021)1892 1749 y Fv(\(3\))156 1903 y(=)41 b Fu(\025)p Fv(\()p Fu(N)5 b Fv(\))k Fs(\001)374 1873 y Fv(720\()p Fu(N)p Fs(\000)s Fv(3\)\()p Fu(N)o Fs(\000)s Fv(4\)\()o Fu(N)p Fs(\000)s Fv(5)o(\))r(+)s(1944)o(\()p Fu(N)o Fs(\000)s Fv(3\)\()o Fu(N)p Fs(\000)s Fv(4)o(\))1238 1856 y Fp(2)1257 1873 y Fv(+)s(648\()p Fu(N)p Fs(\000)s Fv(2\)\()p Fu(N)p Fs(\000)s Fv(3)o(\))1631 1856 y Fp(2)p 374 1893 1275 2 v 979 1934 a Fs(N)1023 1921 y Fp(3)156 2041 y Fv(=)41 b Fu(B)r Fv(\()p Fu(N)5 b Fv(\))0 2143 y(Equation)13 b(2)f(is)h(justi\014ed)h(b)o(y)e (referring)h(to)f(Figure)h(2:)18 b(the)13 b(only)g(length-4)g(routes)f (from)g(state)g(0)g(bac)o(k)h(to)f(state)f(0)0 2200 y(are)h(03630,)f (03430,)g(03230,)g(and)i(03030.)k(The)c(last)f(of)g(these)h(can)f(only) h(arise)g(from)e(non-distinct)j Fu(h)1719 2207 y Fp(1)1739 2200 y Fu(;)8 b(h)1786 2207 y Fp(2)1805 2200 y Fu(;)g(h)1852 2207 y Fp(3)1871 2200 y Fu(;)g(h)1918 2207 y Fp(4)1937 2200 y Fv(.)0 2256 y(F)l(or)18 b(the)g(other)g(three)g(w)o(e)g(simply)h (disregard)g(the)f(conjunction)h(with)g Ff(Distinct)q Fv(\()p Fu(h)1455 2263 y Fp(1)1475 2256 y Fu(h)1501 2263 y Fp(2)1521 2256 y Fu(h)1547 2263 y Fp(3)1566 2256 y Fu(h)1592 2263 y Fp(4)1612 2256 y Fv(\))f(b)q(ecause)h(w)o(e)f(are)0 2313 y(giving)e(an)f(upp)q(er)h(b)q(ound.)21 b(Equation)16 b(3)e(is)i(obtained)g(directly)h(from)d(Equation)h(1.)457 b Fs(})0 2394 y Fw(Using)19 b Fu(M)25 b Fw(to)19 b(bound)h Fu(\016)471 2401 y Fp(6)491 2394 y Fu(;)8 b(\016)532 2401 y Fp(8)551 2394 y Fu(;)g Fs(\001)g(\001)g(\001)m Fw(.)27 b Fv(Assume)18 b(that)f Fu(N)k Fv(is)d(ev)o(en)g(and)g Fu(N)j Fs(\025)c Fv(6.)26 b(W)l(e)18 b(will)h(sho)o(w,)e(in)h(this)g (case,)0 2450 y(that)13 b Fu(\016)117 2457 y Fr(t)145 2450 y Fs(\024)g Fu(B)r Fv(\()p Fu(N)5 b Fv(\).)18 b(Here)c(is)g(the)g (idea.)20 b(T)l(ak)o(e)13 b(a)h(random)f(function)h Fu(h)f Fs(2)g(B)i Fv(and)f(lo)q(ok)g(at)f(it's)g(last)h(6)f(maps|)h(sa)o(y)0 2507 y Fu(h)g Fv(=)g Fu(h)115 2514 y Fp(7)142 2507 y Fs(\001)8 b(\001)g(\001)e Fu(h)229 2514 y Fr(t)271 2507 y Fu(h)297 2514 y Fp(1)317 2507 y Fu(h)343 2514 y Fp(2)362 2507 y Fu(h)388 2514 y Fp(3)408 2507 y Fu(h)434 2514 y Fp(4)454 2507 y Fu(h)480 2514 y Fp(5)500 2507 y Fu(h)526 2514 y Fp(6)546 2507 y Fv(.)21 b(No)o(w)16 b Fu(h)711 2514 y Fp(1)730 2507 y Fu(;)8 b(:)g(:)g(:)d(;)j(h)858 2514 y Fp(6)893 2507 y Fv(are)15 b(statistically)i(correlated)f(to)f Fu(h)1509 2514 y Fp(7)1529 2507 y Fu(;)8 b(:)g(:)g(:)d(;)j(h)1657 2514 y Fr(t)1687 2507 y Fv(\(for)14 b(example,)0 2563 y Fu(h)26 2570 y Fp(1)64 2563 y Fs(6)p Fv(=)19 b Fu(h)144 2570 y Fp(7)164 2563 y Fv(\),)g(y)o(et)f Fu(h)319 2570 y Fp(1)339 2563 y Fu(;)8 b(:)g(:)g(:)d(;)j(h)467 2570 y Fp(6)505 2563 y Fv(are)18 b(not)g Ft(to)n(o)k Fv(far)c(from)g(b)q (eing)i(random)e(and)h(indep)q(enden)o(t,)j(in)d(the)g(sense)g(that,)0 2620 y(for)g(an)o(y)h Fu(h)191 2627 y Fp(7)219 2620 y Fs(\001)8 b(\001)g(\001)d Fu(h)305 2627 y Fr(t)320 2620 y Fv(,)21 b(a)f(uniformly)h(selected)h(sequence)f(of)f(maps)f Fu(h)1183 2603 y Fq(0)1183 2631 y Fp(1)1203 2620 y Fu(h)1229 2603 y Fq(0)1229 2631 y Fp(2)1249 2620 y Fu(h)1275 2603 y Fq(0)1275 2631 y Fp(3)1295 2620 y Fu(h)1321 2603 y Fq(0)1321 2631 y Fp(4)1341 2620 y Fu(h)1367 2603 y Fq(0)1367 2631 y Fp(5)1387 2620 y Fu(h)1413 2603 y Fq(0)1413 2631 y Fp(6)1453 2620 y Fv(w)o(ould)i(ha)o(v)o(e)e(b)q(een)i(a)f(v)m(alid) 952 2824 y(13)p eop %%Page: 14 14 14 13 bop 0 79 a Fv(con)o(tin)o(uation)18 b(with)f(probabilit)o(y)i(at) d(least)h(1)p Fu(=)p Fv(2.)25 b(\(This)18 b(follo)o(ws)f(from)g(our)g (assumption)g(that)f Fu(n)h Fs(\024)1782 45 y Fm(\000)1801 59 y Fr(N)1808 95 y Fp(3)1832 45 y Fm(\001)1852 79 y Fu(=)p Fv(12.\))0 136 y(Th)o(us,)g(up)h(to)f(a)g(factor)f(of)h(2,)h(w)o (e)f(can)g(b)q(ound)i(the)e(c)o(hance)h(of)f(landing)i(in)f(state)e(0)h (on)h(applying)h Fu(h)e Fv(to)g Fl(1)1870 143 y Fr(t)1902 136 y Fv(b)o(y)0 192 y(lo)q(oking)j(at)e(the)h(c)o(hance)h(of)e (landing)j(in)e(state)g(0)f(after)g(applying)j(a)d(uniformly)i (selected)h Fu(h)1643 199 y Fp(1)1670 192 y Fu(:)8 b(:)g(:)e(h)1757 199 y Fp(6)1796 192 y Fv(starting)0 249 y(from)14 b(some)h(in)h(some)f (arbitrary)g(\(unkno)o(wn\))g(state)f(of)h(the)g(Mark)o(o)o(v)f(c)o (hain.)71 305 y(T)l(o)g(formalize)h(the)g(ab)q(o)o(v)o(e)f(argumen)o (t,)f(let)i Fu(f)840 312 y Fr(i)855 305 y Fv(\()p Fu(t)p Fv(\))f(denote)h(the)f(maxim)o(um,)h(o)o(v)o(er)e(all)j(initial)h (states)c Fu(s)p Fv(,)i(of)f(the)0 362 y(probabilit)o(y)j(that)d(w)o(e) h(arriv)o(e)g(in)i(state)d Fu(i)h Fv(in)h(exactly)g Fu(t)g Fv(transitions,)f(giv)o(en)h(that)e(w)o(e)h(start)f(in)i(state)f Fu(s)p Fv(.)21 b(This)15 b(is)0 418 y(the)i(same)f(as)g(the)g(suprem)o (um,)h(o)o(v)o(er)f(all)h(distributions)h Fu(\031)g Fv(on)f(the)f (starting)g(state)g(of)g Fu(M)5 b Fv(,)16 b(of)g(the)h(probabilit)o(y)0 474 y(that)c(w)o(e)h(arriv)o(e)g(in)h(state)f Fu(i)g Fv(in)h(exactly)f Fu(t)h Fv(transitions,)f(giv)o(en)h(that)e(w)o(e)h (start)f(in)i(an)f(initial)i(state)e(as)g(c)o(hosen)g(b)o(y)0 531 y(sampling)i(from)f Fu(\031)r Fv(.)k(W)l(e)c(will)i(need)f(the)f (follo)o(wing)h(lemma)g(ab)q(out)f(the)g(b)q(eha)o(vior)h(of)f(Mark)o (o)o(v)e(c)o(hain)j Fu(M)5 b Fv(.)0 635 y Fl(Lemma)17 b(11)23 b Fu(f)282 642 y Fp(0)302 635 y Fv(\(6\))12 b Fs(\024)h Fv(\(25920)p Fu(N)596 619 y Fp(8)624 635 y Fv(+)d(154080)p Fu(N)849 619 y Fp(7)867 635 y Fv(\))p Fu(=)p Fs(N)952 619 y Fp(5)971 635 y Ft(.)0 739 y Fv(The)k(pro)q(of)g (is)h(a)f(tedious)h(but)f(straigh)o(tforw)o(ard)e(calculation)k(using)f (the)f(transition)g(probabilities)j(of)d Fu(M)5 b Fv(.)19 b(It)14 b(is)0 796 y(relegated)h(to)f(App)q(endix)j(B.)j(The)14 b(p)q(oin)o(t)h(isn't)g(the)g(sp)q(eci\014c)h(form)o(ula,)e(but)h(only) g(that)f Fu(f)1549 803 y Fp(0)1569 796 y Fv(\(6\))g(is)h(less)g(than)g (half)0 852 y Fu(B)r Fv(\()p Fu(N)5 b Fv(\))15 b(for)f(all)i (su\016cien)o(tly)h(large)e Fu(N)5 b Fv(.)0 956 y Fl(Lemma)17 b(12)23 b Ft(Assume)16 b Fv(6)c Fs(\024)h Fu(t)g Fs(\024)g Fu(n)p Ft(.)22 b(Then)15 b Fu(\016)791 963 y Fr(t)819 956 y Fs(\024)e Fu(B)r Fv(\()p Fu(N)5 b Fv(\))p Ft(.)0 1085 y(Pr)n(o)n(of)k Fv(:)30 b(W)l(e)16 b(use)f(the)g(same)g(notation)g (as)g(in)h(the)f(pro)q(of)g(of)g(Lemma)g(10.)24 1241 y Fu(\016)44 1248 y Fr(t)100 1241 y Fv(=)52 b(Pr)177 1271 y Fr(h)p Fq(2B)253 1241 y Fv([)o Fu(h)p Fv(\()p Fl(1)335 1248 y Fr(t)350 1241 y Fv(\))12 b(=)h Fl(0)p Fv(\)])100 1328 y(=)220 b(Pr)177 1358 y Fr(h)197 1363 y Fj(7)215 1358 y Fq(\001\001\001)o Fr(h)264 1362 y Fi(t)288 1358 y Fr(h)308 1363 y Fj(1)326 1358 y Fr(h)346 1363 y Fj(2)364 1358 y Fr(h)384 1363 y Fj(3)401 1358 y Fr(h)421 1363 y Fj(4)439 1358 y Fr(h)459 1363 y Fj(5)477 1358 y Fr(h)497 1363 y Fj(6)514 1358 y Fq(2)p Fr(D)567 1362 y Fi(t)588 1328 y Fv([)p Ff(State)q Fv(\()p Fu(h)745 1335 y Fp(7)772 1328 y Fs(\001)8 b(\001)g(\001)e Fu(h)859 1335 y Fr(t)899 1328 y Fu(h)925 1335 y Fp(1)945 1328 y Fu(h)971 1335 y Fp(2)991 1328 y Fu(h)1017 1335 y Fp(3)1037 1328 y Fu(h)1063 1335 y Fp(4)1083 1328 y Fu(h)1109 1335 y Fp(5)1129 1328 y Fu(h)1155 1335 y Fp(6)1174 1328 y Fv(\))13 b(=)g(0)o(])100 1466 y(=)121 b(Pr)177 1496 y Fr(h)197 1501 y Fj(7)215 1496 y Fq(\001\001\001)o Fr(h)264 1500 y Fi(t)279 1496 y Fq(2)p Fr(D)332 1501 y Fi(t)p Fk(\000)p Fj(6)560 1466 y Fv(Pr)503 1494 y Fi(h)521 1501 y Fj(1)538 1494 y Fk(\001\001\001)r Fi(h)585 1501 y Fj(6)602 1494 y Fk(2)p Fi(D)648 1501 y Fj(6)392 1531 y Fq(f)p Fr(h)430 1536 y Fj(1)447 1531 y Fr(;)p Fq(\001\001\001)o Fr(;h)516 1536 y Fj(6)534 1531 y Fq(g\\f)p Fr(h)614 1536 y Fj(7)630 1531 y Fr(;)p Fq(\001\001\001)o Fr(;h)699 1535 y Fi(t)713 1531 y Fq(g)p Fp(=)p Fq(;)784 1466 y Fv([)o Ff(State)r Fv(\()p Fu(h)941 1473 y Fp(7)968 1466 y Fs(\001)8 b(\001)g(\001)d Fu(h)1054 1473 y Fr(t)1095 1466 y Fu(h)1121 1473 y Fp(1)1140 1466 y Fu(h)1166 1473 y Fp(2)1186 1466 y Fu(h)1212 1473 y Fp(3)1232 1466 y Fu(h)1258 1473 y Fp(4)1278 1466 y Fu(h)1304 1473 y Fp(5)1324 1466 y Fu(h)1350 1473 y Fp(6)1370 1466 y Fv(\))12 b(=)h(0])100 1629 y Fs(\024)103 b Fv(max)177 1659 y Fr(h)197 1664 y Fj(7)215 1659 y Fq(\001\001\001)o Fr(h)264 1663 y Fi(t)279 1659 y Fq(2)p Fr(D)332 1664 y Fi(t)p Fk(\000)p Fj(6)560 1629 y Fv(Pr)503 1658 y Fi(h)521 1665 y Fj(1)538 1658 y Fk(\001\001\001)r Fi(h)585 1665 y Fj(6)602 1658 y Fk(2)p Fi(D)648 1665 y Fj(6)392 1695 y Fq(f)p Fr(h)430 1700 y Fj(1)447 1695 y Fr(;)p Fq(\001\001\001)o Fr(;h)516 1700 y Fj(6)534 1695 y Fq(g\\f)p Fr(h)614 1700 y Fj(7)630 1695 y Fr(;)p Fq(\001\001\001)o Fr(;h)699 1699 y Fi(t)713 1695 y Fq(g)p Fp(=)p Fq(;)784 1629 y Fv([)o Ff(NumOnes)r Fv(\()p Ff(P)o(a)o(rit)o(y)o Fv(\()p Fu(h)1156 1636 y Fp(7)1183 1629 y Fs(\001)8 b(\001)g(\001)e Fu(h)1270 1636 y Fr(t)1285 1629 y Fv(\))12 b Fs(\010)h Ff(P)o(a)o(rit)o(y)o Fv(\()p Fu(h)1518 1636 y Fp(1)1538 1629 y Fu(h)1564 1636 y Fp(2)1584 1629 y Fu(h)1610 1636 y Fp(3)1630 1629 y Fu(h)1656 1636 y Fp(4)1676 1629 y Fu(h)1702 1636 y Fp(5)1722 1629 y Fu(h)1748 1636 y Fp(6)1768 1629 y Fv(\)\))f(=)h(0)o(])100 1793 y(=)96 b(Pr)177 1823 y Fr(h)197 1828 y Fj(1)215 1823 y Fq(\001\001\001)o Fr(h)264 1828 y Fj(6)282 1823 y Fq(2)p Fr(E)341 1793 y Fv([)o Ff(NumOnes)r Fv(\()p Fu(\033)14 b Fs(\010)f Ff(P)o(a)o(rit)o(y)o Fv(\()p Fu(h)801 1800 y Fp(1)821 1793 y Fu(h)847 1800 y Fp(2)867 1793 y Fu(h)893 1800 y Fp(3)913 1793 y Fu(h)939 1800 y Fp(4)958 1793 y Fu(h)984 1800 y Fp(5)1004 1793 y Fu(h)1030 1800 y Fp(6)1050 1793 y Fv(\)\))f(=)h(0])45 b Fx(/)-8 b(/Fix)15 b Fy(h)1359 1799 y Fe(7)1384 1793 y Fd(\001)7 b(\001)g(\001)e Fy(h)1463 1799 y Fc(t)1478 1793 y Fx(;)15 b(de\014ne)h Fy(E)r Fx(,)e Fy(\033)i Fx(ac)n(c)n(or)n(dingly)100 1968 y Fv(=)244 1912 y(Pr)182 1942 y Fr(h)202 1947 y Fj(1)220 1942 y Fq(\001\001\001)o Fr(h)269 1947 y Fj(6)287 1942 y Fq(2)p Fr(R)338 1947 y Fj(6)362 1912 y Fv([)p Ff(State)475 1919 y Fr(\033)498 1912 y Fv(\()p Fu(h)542 1919 y Fp(1)562 1912 y Fu(h)588 1919 y Fp(2)608 1912 y Fu(h)634 1919 y Fp(3)654 1912 y Fu(h)680 1919 y Fp(4)699 1912 y Fu(h)725 1919 y Fp(5)745 1912 y Fu(h)771 1919 y Fp(6)791 1912 y Fv(\))d(=)g(0)h(and)i Ff(Distinct)q Fv(\()p Fu(h)1188 1919 y Fp(1)1215 1912 y Fs(\001)8 b(\001)g(\001)e Fu(h)1302 1919 y Fp(6)1347 1912 y Fu(h)1373 1919 y Fp(7)1401 1912 y Fs(\001)i(\001)g(\001)d Fu(h)1487 1919 y Fr(t)1502 1912 y Fv(\)])p 182 1958 1351 2 v 497 1999 a(Pr)434 2029 y Fr(h)454 2034 y Fj(1)472 2029 y Fq(\001\001\001)o Fr(h)521 2034 y Fj(6)539 2029 y Fq(2)p Fr(D)592 2034 y Fj(6)616 1999 y Fv([)p Ff(Distinct)q Fv(\()p Fu(h)821 2006 y Fp(1)841 1999 y Fu(h)867 2006 y Fp(2)887 1999 y Fu(h)913 2006 y Fp(3)933 1999 y Fu(h)959 2006 y Fp(4)978 1999 y Fu(h)1004 2006 y Fp(5)1024 1999 y Fu(h)1050 2006 y Fp(6)1095 1999 y Fu(h)1121 2006 y Fp(7)1149 1999 y Fs(\001)j(\001)g(\001)e Fu(h)1236 2006 y Fr(t)1251 1999 y Fv(\))o(])100 2161 y Fs(\024)244 2105 y Fv(Pr)182 2135 y Fr(h)202 2140 y Fj(1)220 2135 y Fq(\001\001\001)o Fr(h)269 2140 y Fj(6)287 2135 y Fq(2)p Fr(R)338 2140 y Fj(6)362 2105 y Fv([)p Ff(State)475 2112 y Fr(\033)498 2105 y Fv(\()p Fu(h)542 2112 y Fp(1)562 2105 y Fu(h)588 2112 y Fp(2)608 2105 y Fu(h)634 2112 y Fp(3)654 2105 y Fu(h)680 2112 y Fp(4)699 2105 y Fu(h)725 2112 y Fp(5)745 2105 y Fu(h)771 2112 y Fp(6)791 2105 y Fv(\))13 b(=)g(0)o(])p 182 2151 723 2 v 426 2199 a(1)d Fs(\000)g Fv(6)25 b Fu(t=)591 2164 y Fm(\000)611 2178 y Fr(N)617 2214 y Fp(3)642 2164 y Fm(\001)100 2299 y Fs(\024)42 b Fv(2)10 b Fs(\001)72 b Fv(Pr)233 2329 y Fr(h)253 2334 y Fj(1)270 2329 y Fq(\001\001\001)p Fr(h)320 2334 y Fj(6)338 2329 y Fq(2)p Fr(R)389 2334 y Fj(6)412 2299 y Fv([)p Ff(State)525 2306 y Fr(\033)549 2299 y Fv(\()p Fu(h)593 2306 y Fp(1)612 2299 y Fu(h)638 2306 y Fp(2)658 2299 y Fu(h)684 2306 y Fp(3)704 2299 y Fu(h)730 2306 y Fp(4)750 2299 y Fu(h)776 2306 y Fp(5)796 2299 y Fu(h)822 2306 y Fp(6)842 2299 y Fv(\))12 b(=)h(0])149 b Fx(/)-8 b(/F)m(r)n(om)15 b(assumption)g(that)g Fy(n)d Fd(\024)1633 2265 y Fm(\000)1652 2281 y Fc(N)1658 2313 y Fe(3)1681 2265 y Fm(\001)1700 2299 y Fy(=)p FC(12)100 2437 y Fs(\024)42 b Fv(2)10 b Fs(\001)g Fu(f)255 2444 y Fp(0)275 2437 y Fv(\(6\))749 b Fx(/)-8 b(/De\014nition)16 b(of)f Fy(f)100 2575 y Fs(\024)42 b Fv(2)10 b Fs(\001)g Fv(\(25920)p Fu(N)408 2556 y Fp(8)435 2575 y Fv(+)h(154080)p Fu(N)661 2556 y Fp(7)678 2575 y Fv(\))p Fu(=)p Fs(N)763 2556 y Fp(5)1083 2575 y Fx(/)-8 b(/By)15 b(L)n(emma)g(12)100 2644 y Fs(\024)42 b Fu(B)r Fv(\()p Fu(N)5 b Fv(\))15 b(for)g(all)h Fu(N)h Fs(\025)c Fv(32)952 2824 y(14)p eop %%Page: 15 15 15 14 bop 0 79 a Fv(F)l(or)17 b(the)g(last)g(inequalit)o(y:)26 b(it)17 b(is)h(easy)f(to)g(v)o(erify)g(that)g(this)h(holds)g(for)e (su\016cien)o(tly)j(large)e Fu(N)5 b Fv(.)26 b(The)17 b(crosso)o(v)o(er)0 136 y(p)q(oin)o(t)f(w)o(as)e(determined)j(n)o (umerically)l(.)1234 b Fs(})0 236 y Fv(W)l(e)19 b(ha)o(v)o(e)g(no)o(w)g (sho)o(wn)g(that,)h(under)g(the)f(conditions)i(of)e(the)g(theorem,)h Fu(B)r Fv(\()p Fu(N)5 b Fv(\))19 b Fs(\025)h 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Fu(\031)r Fv(\()p Fu(i)p Fv(\).)23 b(As)16 b(b)q(efore,)h(w)o(e)g(xor)f(the)h(con)o(ten)o(ts)f (of)g(eac)o(h)h(buc)o(k)o(et)f(and)h(output)g(their)g(concatenation)g (in)g(some)0 136 y(canonical)f(order.)k(W)l(e)15 b(call)i(the)e(ab)q(o) o(v)o(e)g(sc)o(heme)g(the)h(buc)o(k)o(et)f(hash)g(of)g(the)g(graph)g Fu(G)p Fv(.)71 192 y(F)l(or)e(a)h(graph)g Fu(G)g Fv(to)f(b)q(e)i(\\go)q (o)q(d")f(w)o(e)g(w)o(an)o(t)f(a)h(small)g(n)o(um)o(b)q(er)h(of)f(v)o (ertices)g Fu(N)5 b Fv(,)14 b(a)g(large)g(n)o(um)o(b)q(er)g(of)g(edges) g Fu(m)p Fv(,)0 249 y(and)j(suc)o(h)g(that)f(for)g(all)h Fu(k)h Fv(where)f(1)e Fs(\024)g Fu(k)i Fs(\024)e Fu(n)g Fs(\024)h Fu(m)p Fv(,)g(if)i Fu(k)f Fv(distinct)h(edges)f(are)f (selected)i(at)e(random)h(from)f Fu(G)p Fv(,)0 305 y(then)i(the)g (probabilit)o(y)h(that)e(their)i(union)g(\(with)f(m)o(ultiplicities\))j (comprises)d(a)g(union)h(of)e(cycles)i(is)g(at)e(most)0 362 y(some)e(tin)o(y)g(n)o(um)o(b)q(er)h Fu(\017)p Fv(.)71 418 y(One)22 b(p)q(ossible)i(c)o(hoice)f(of)f(graphs)f(in)i(this)g (regard)e(are)h(the)g(\()p Fu(d;)8 b(g)r Fv(\)-cages)20 b(\(see)i([8)o(]\).)40 b(A)22 b(\()p Fu(d;)8 b(g)r Fv(\)-cage)19 b(is)0 474 y(a)g(smallest)h Fu(d)p Fv(-regular)f(graph)g(whose)g (shortest)g(cycle)h(has)f Fu(g)i Fv(edges.)32 b(These)20 b(graphs)f(ha)o(v)o(e)g(b)q(een)h(explicitly)0 531 y(constructed)c(for) f(v)m(arious)h(v)m(alues)g(of)g(\()p Fu(d;)8 b(g)r Fv(\).)18 b(Though)e(\()p Fu(d;)8 b(g)r Fv(\)-cages)13 b(are)i(rather)g(large)g (\(for)g(ev)o(en)h Fu(g)h Fv(they)e(ha)o(v)o(e)0 587 y(at)g(least)g(\(2\()p Fu(d)9 b Fs(\000)s Fv(1\))333 571 y Fr(g)q(=)p Fp(2)390 587 y Fs(\000)s Fv(2\))p Fu(=)p Fv(\()p Fu(d)g Fs(\000)s Fv(2\))14 b(no)q(des\))i(some)e(\()p Fu(d;)8 b(g)r Fv(\)-cages)13 b(ma)o(y)i(giv)o(e)g(rise)h(to)e(useful)j (hash)e(families.)22 b(F)l(or)0 644 y(example,)17 b(assume)g Fu(d)10 b Fs(\000)h Fv(1)16 b(is)h(a)g(prime)g(p)q(o)o(w)o(er.)23 b(Let)16 b Fu(C)s Fv([)p Fu(d;)8 b Fv(6])14 b(b)q(e)k(the)e(\()p Fu(d;)8 b Fv(6\)-cage.)22 b(This)17 b(is)g(the)f(the)h(p)q(oin)o (t-line)0 700 y(incidence)g(graph)c(of)h(the)g(pro)s(jectiv)o(e)f (plane)i(of)f(order)g Fu(d)8 b Fs(\000)g Fv(1.)18 b(Buc)o(k)o(et)c (hashing)g(with)g Fu(C)s Fv([10)p Fu(;)8 b Fv(6])k(ma)o(y)h(b)q(e)h(a)g (go)q(o)q(d)0 757 y(w)o(a)o(y)g(to)h(hash)g(909)f(w)o(ords)h(do)o(wn)g (to)f(182)h(w)o(ords.)0 838 y Fw(Open)22 b(questions.)29 b Fv(The)19 b(generalized)h(notion)f(of)g(buc)o(k)o(et)f(hashing)i (amoun)o(ts)d(to)h(sa)o(ying)h(that)f(hashing)h(is)0 895 y(ac)o(hiev)o(ed)h(for)e(eac)o(h)h(bit)h(p)q(osition)g(1)8 b Fu(:)g(:)g(:)d(w)20 b Fv(b)o(y)f(matrix)f(m)o(ultiplication)k(with)d (a)g(sparse)g(Bo)q(olean)g(matrix)g Fu(H)t Fv(.)0 951 y(Expressing)e(the)f(metho)q(d)h(in)g(this)g(generalit)o(y)g(raises)f (questions)h(lik)o(e)h(the)e(follo)o(wing:)23 b(for)16 b(a)g(giv)o(en)h Fu(N)5 b Fv(,)15 b Fu(n)i Fv(and)0 1007 y Fu(k)q Fv(,)22 b(for)e(all)i(distributions)g Fs(D)g Fv(of)e(binary)i Fu(N)c Fs(\002)c Fu(n)21 b Fv(matrices)g Fu(H)j Fv(ha)o(ving)d Fu(k)h Fv(ones)f(p)q(er)g(column,)i(for)d(whic)o (h)h(is)0 1064 y(max)85 1074 y Fr(x)p Fq(2f)p Fp(0)p Fr(;)p Fp(1)p Fq(g)208 1059 y Fi(n)230 1074 y Fq(\000f)p Fp(0)293 1064 y Fi(n)314 1074 y Fq(g)358 1064 y Fv(Pr)341 1094 y Fr(H)s Fq(2D)432 1064 y Fv([)o Fu(H)t(x)12 b Fv(=)h Fl(0)q Fv(])20 b(minimized?)40 b(What)20 b(if)h(w)o(e)f(also)h(demand)g (that)f(eac)o(h)h(ro)o(w)e(has)i(a)f(\014xed)0 1135 y(n)o(um)o(b)q(er)d (of)e(ones?)24 b(What)16 b(if,)h(instead)f(of)g(sa)o(ying)g(that)g (there)g(are)g Fu(k)i Fv(ones)e(p)q(er)h(column,)g(w)o(e)f(cap)g(the)g (densit)o(y)0 1192 y(of)f(the)g(matrix)g(at)f(some)h(v)m(alue)i Fu(\032)p Fv(?)j(Answ)o(ers)15 b(ma)o(y)f(lead)i(to)f(faster)f(buc)o(k) o(et)h(hash)h(sc)o(hemes.)0 1314 y Fh(Ac)n(kno)n(wledgmen)n(ts)0 1399 y Fv(Man)o(y)c(thanks)g(to)g(the)h(t)o(w)o(o)e(anon)o(ymous)h (referees)h(for)f(their)h(careful)h(reviews.)19 b(Thanks)13 b(also)f(to)g(Mihir)i(Bellare,)0 1456 y(Don)h(Copp)q(ersmith,)g(Hugo)g (Kra)o(w)o(czyk,)g(and)g(Da)o(vid)g(Zuc)o(k)o(erman)g(for)f(their)i (commen)o(ts)f(and)g(suggestions.)0 1599 y FE(References)34 1701 y Fv([1])22 b(N.)g(Alon,)j(O.)d(Goldreic)o(h,)j(J.)e(H)-6 b(\027)-28 b(astad)22 b(and)h(R.)f(P)o(eralta,)i(Simple)g (constructions)f(of)f(almost)g Fu(k)q Fv(-wise)105 1757 y(indep)q(enden)o(t)15 b(random)d(v)m(ariables,)i(31st)e(Ann)o(ual)h (Symp)q(osium)h(on)e(F)l(oundations)h(of)f(Computer)g(Science,)105 1814 y(IEEE)j(Computer)g(So)q(ciet)o(y)l(,)h(1990,)d(pp.)j(544{553.)34 1907 y([2])22 b(R.)15 b(Arnold)h(and)f(D.)g(Copp)q(ersmith,)g(An)h (alternativ)o(e)f(to)g(p)q(erfect)g(hashing,)h(IBM)f(R)o(C)h(10332)d (\(1984\).)34 2001 y([3])22 b(M.)f(Bellare,)k(R.)d(Canetti)g(and)g(H.)g (Kra)o(w)o(czyk,)h(Keying)g(hash)g(functions)f(for)g(message)g(authen)o (tica-)105 2058 y(tion,)14 b Fb(Adv)m(ances)i(in)f(Cryptology)f({)g(CR) l(YPTO)h('96)p Fv(,)e(Lecture)i(Notes)f(in)h(Computer)f(Science,)i(v)o (ol.)e(1109,)105 2114 y(Springer-V)l(erlag,)i(1996,)e(pp.)h(1{15.)34 2208 y([4])22 b(M.)16 b(Bellare,)j(O.)e(Goldreic)o(h)i(and)e(S.)g (Goldw)o(asser,)g(Incremen)o(tal)i(cryptograph)o(y:)d(the)h(case)g(of)g (hashing)105 2264 y(and)i(signing,)h Fb(Adv)m(ances)g(in)g(Cryptology)e ({)h(CR)l(YPTO)g('94)p Fv(,)f(Lecture)i(Notes)e(in)i(Computer)e (Science,)105 2321 y(v)o(ol.)d(839,)f(Springer-V)l(erlag,)i(1994,)e (pp.)h(216{233.)34 2415 y([5])22 b(M.)d(Bellare,)k(J.)d(Kilian)i(and)e (P)l(.)g(Roga)o(w)o(a)o(y)l(,)g(The)g(securit)o(y)h(of)e(cipher)j(blo)q (c)o(k)f(c)o(haining,)h Fb(Adv)m(ances)f(in)105 2471 y(Cryptology)d({)g(CR)l(YPTO)h('94)p Fv(,)f(Lecture)h(Notes)f(in)h (Computer)f(Science,)j(v)o(ol.)d(839,)g(Springer-V)l(erlag,)105 2527 y(1994,)13 b(pp.)j(341{358.)952 2824 y(18)p eop %%Page: 19 19 19 18 bop 34 79 a Fv([6])22 b(J.)12 b(Bierbrauer,)i(T.)d(Johansson,)i (G.)f(Kabatianskii)i(and)e(B.)g(Smeets,)h(On)g(families)h(of)e(hash)g (functions)i(via)105 136 y(geometric)j(co)q(des)g(and)g(concatenation,) g Fb(Adv)m(ances)h(in)g(Cryptology)e({)g(CR)l(YPTO)i('93)p Fv(,)e(Lecture)h(Notes)105 192 y(in)f(Computer)f(Science,)h(v)o(ol.)f (773,)f(Springer-V)l(erlag,)i(1994,)e(pp.)h(331{342.)34 286 y([7])22 b(A.)e(Bosselaers,)h(R.)f(Go)o(v)m(aerts)f(and)h(J.)h(V)l (andew)o(alle,)h(F)l(ast)d(hashing)i(on)f(the)h(P)o(en)o(tium,)g Fb(Adv)m(ances)g(in)105 342 y(Cryptology)c({)g(CR)l(YPTO)h(96)p Fv(,)f(Lecture)i(Notes)e(in)h(Computer)f(Science,)j(v)o(ol.)d(1109,)g (Springer-V)l(erlag,)105 399 y(1996,)c(pp.)j(298{312.)34 493 y([8])22 b(J.)15 b(Bondy)h(and)f(U.)g(Murt)o(y,)f Fb(Graph)h(theory)f(with)i(Applications,)h Fv(North)d(Holland,)i(1976.) 34 587 y([9])22 b(G.)e(Brassard,)i(On)g(computationally)h(secure)f (authen)o(tication)g(tags)e(requiring)j(short)d(secret)i(shared)105 643 y(k)o(eys,)14 b Fb(Adv)m(ances)j(in)f(Cryptology)e({)h(CR)l(YPTO)h ('82)p Fv(,)e(Springer-V)l(erlag,)i(1983,)e(pp.)h(79{86.)11 737 y([10])22 b(L.)13 b(Carter)f(and)h(M.)f(W)l(egman,)h(Univ)o(ersal)h (hash)f(functions,)h Fb(J.)f(of)g(Computer)f(and)h(System)g(Sciences)k Fv(18,)105 793 y(1979,)c(pp.)j(143{154.)11 887 y([11])22 b(Y.)g(Desmedt,)h(Unconditionally)i(secure)e(authen)o(tication)g(sc)o (hemes)f(and)h(practical)g(and)f(theoretical)105 944 y(consequences,)17 b Fb(Adv)m(ances)h(in)f(Cryptology)f({)g(CR)l(YPTO)h ('85)p Fv(,)e(Lecture)i(Notes)f(in)h(Computer)f(Science,)105 1000 y(v)o(ol.)f(218,)f(Springer-V)l(erlag,)i(1985,)e(pp.)h(42{45.)11 1094 y([12])22 b(P)l(.)d(Gemmell)i(and)f(M.)f(Naor,)h(Co)q(des)g(for)f (in)o(teractiv)o(e)h(authen)o(tication,)h Fb(Adv)m(ances)g(in)f (Cryptology)f({)105 1150 y(CR)l(YPTO)d('93)p Fv(,)f(Lecture)i(Notes)f (in)g(Computer)g(Science,)i(v)o(ol.)d(773,)g(Springer-V)l(erlag,)i (1994,)e(pp.)h(355{)105 1207 y(367.)11 1301 y([13])22 b(O.)c(Goldreic)o(h,)i(S.)e(Goldw)o(asser)g(and)g(S.)g(Micali,)j(Ho)o (w)c(to)h(construct)g(random)g(functions,)h Fb(Journal)g(of)105 1357 y(the)c(A)o(CM)p Fv(,)f(V)l(ol.)i(33,)e(No.)g(4,)h(1986,)f(pp.)h (210{217.)11 1451 y([14])22 b(S.)c(Goldw)o(asser,)g(S.)g(Micali)i(and)e (R.)g(Riv)o(est,)h(A)g(digital)g(signature)g(sc)o(heme)f(secure)h (against)f(adaptiv)o(e)105 1507 y(c)o(hosen-message)12 b(attac)o(ks,)f Fb(SIAM)h(Journal)h(of)f(Computing,)g Fv(v)o(ol.)g(17,)g(no.)f(2)h(\(April)h(1988\),)e(pp.)h(281{308.)11 1601 y([15])22 b(S.)c(Halevi)i(and)f(H.)g(Kra)o(w)o(czyk,)g(MMH:)f (Message)g(authen)o(tication)h(in)h(soft)o(w)o(are)d(in)j(the)e (Gbit/second)105 1658 y(rates,)c Fb(Pro)q(ceedings)i(of)f(the)g(4th)g (W)l(orkshop)g(on)g(F)l(ast)f(Soft)o(w)o(are)g(Encryption)p Fv(,)i(Springer-V)l(erlag,)g(1997.)11 1751 y([16])22 b(T.)27 b(Johansson,)j(Buc)o(k)o(et)d(hashing)h(with)g(small)g(k)o(ey)g (size,)j Fb(Adv)m(ances)d(in)g(Cryptology)f({)g(EUR)o(O-)105 1808 y(CR)l(YPT)15 b('97,)f Fv(Lecture)i(Notes)f(in)h(Computer)f (Science,)h(Springer-V)l(erlag,)h(1997.)11 1902 y([17])22 b(T.)c(Johansson,)h(G.)e(Kabatianskii)j(and)f(B.)f(Smeets,)h(On)g(the)g (relation)g(b)q(et)o(w)o(een)g(A-co)q(des)g(and)f(co)q(des)105 1958 y(correcting)13 b(indep)q(enden)o(t)j(errors.)c Fb(Adv)m(ances)i(in)g(Cryptology)e({)h(EUR)o(OCR)l(YPT)h('93,)f Fv(Lecture)g(Notes)g(in)105 2014 y(Computer)i(Science,)h(v)o(ol.)f (765,)f(Springer-V)l(erlag,)i(1994,)e(pp.)h(1{11.)11 2108 y([18])22 b(H.)38 b(Kra)o(w)o(czyk,)k(LFSR-based)e(hashing)f(and)f (authen)o(tication,)44 b Fb(Adv)m(ances)39 b(in)g(Cryptology)e({)105 2165 y(CR)l(YPTO)16 b('94,)f Fv(Lecture)i(Notes)f(in)g(Computer)g (Science,)i(v)o(ol.)d(839,)g(Springer-V)l(erlag,)i(1994,)e(pp.)h(129{) 105 2221 y(139.)11 2315 y([19])22 b(M.)14 b(Lub)o(y)i(and)f(C.)g(Rac)o (k)o(o\013,)e(Ho)o(w)i(to)f(construct)h(pseudorandom)g(p)q(erm)o (utations)g(from)g(pseudorandom)105 2371 y(functions,")g Fb(SIAM)h(J.)f(Comput,)f Fv(v)o(ol.)h(17,)f(no.)h(2)g(\(April)h (1988\).)11 2465 y([20])22 b(X.)10 b(Lai,)i(R.)f(Ruepp)q(el)j(and)d(J.) f(W)l(o)q(ollv)o(en,)j(A)e(fast)f(cryptographic)h(c)o(hec)o(ksum)g (algorithm)g(based)g(on)g(stream)105 2522 y(ciphers,)k Fb(Adv)m(ances)g(in)f(Cryptology)l(,)g(Pro)q(ceedings)h(of)e(A)o(USCR)l (YPT)h(92,)g Fv(Lecture)g(Notes)g(in)g(Computer)105 2578 y(Science,)j(v)o(ol.)d(718,)g(Springer-V)l(erlag,)j(1992,)c(pp.)j (339{348.)952 2824 y(19)p eop %%Page: 20 20 20 19 bop 11 79 a Fv([21])22 b(P)l(.)12 b(P)o(earson,)g(F)l(ast)g (hashing)h(of)g(v)m(ariable-length)h(text)f(strings,)f Fb(Comm)o(unications)h(of)f(the)h(A)o(CM,)f Fv(v)o(ol.)g(33,)105 136 y(no.)j(6)f(\(1990\),)f(pp.)j(677{680.)11 230 y([22])22 b(R.)15 b(Riv)o(est,)g(The)h(MD5)e(message)h(digest)g(algorithm,)g (IETF)g(RF)o(C-1321,)f(1992.)11 323 y([23])22 b(P)l(.)15 b(Roga)o(w)o(a)o(y,)f(Buc)o(k)o(et)i(hashing)g(and)g(its)g(application) i(to)d(fast)g(message)g(authen)o(tication,)h Fb(Adv)m(ances)h(in)105 380 y(Cryptology)h({)g(CR)l(YPTO)h('95)p Fv(,)f(Lecture)h(Notes)f(in)h (Computer)f(Science,)j(v)o(ol.)d(963,)g(Springer-V)l(erlag,)105 436 y(1995,)13 b(pp.)j(313{328.)11 530 y([24])22 b(V.)10 b(Shoup,)i(On)g(fast)e(and)h(pro)o(v)m(ably)g(secure)h(message)e (authen)o(tication)i(based)f(on)f(univ)o(ersal)i(hashing,)h Fb(Ad-)105 587 y(v)m(ances)f(in)h(Cryptology)e({)h(CR)l(YPTO)g('96)p Fv(,)f(Lecture)i(Notes)e(in)i(Computer)e(Science,)j(v)o(ol.)e(1109,)f (Springer-)105 643 y(V)l(erlag,)k(1996,)f(pp.)h(74{85.)11 737 y([25])22 b(A.)13 b(Siegel)h(On)g(univ)o(ersal)g(classes)g(of)f (fast)f(high)i(p)q(erformance)f(hash)h(functions,)g(their)f(time-space) h(trade-)105 793 y(o\013,)20 b(and)g(their)g(applications,)i(30th)d (Ann)o(ual)i(Symp)q(osium)g(on)f(F)l(oundations)g(of)f(Computer)g (Science,)105 850 y(IEEE)c(Computer)g(So)q(ciet)o(y)l(,)h(1989,)d(pp.)j (20{25.)11 944 y([26])22 b(G.)c(Simmons,)h(A)g(surv)o(ey)f(of)g (information)h(authen)o(tication,)g(in)g Fb(Con)o(temp)q(orary)f (Cryptograph)o(y)l(,)g(The)105 1000 y(Science)f(of)e(Information)g(In)o (tegrit)o(y)l(,)g Fv(G.)f(Simmons,)h(editor,)g(IEEE)h(Press,)e(New)i(Y) l(ork,)e(1992.)11 1094 y([27])22 b(D.)17 b(Stinson,)h(Univ)o(ersal)h (hashing)f(and)g(authen)o(tication)g(co)q(des,)h Fb(Designs,)f(Co)q (des)g(and)f(Cryptograph)o(y)l(,)105 1150 y Fv(v)o(ol.)e(4)g(\(1994\),) e(pp.)i(369{380.)11 1244 y([28])22 b(R.)17 b(T)l(a)o(ylor,)f(An)h(in)o (tegrit)o(y)g(c)o(hec)o(k)g(v)m(alue)h(algorithm)f(for)f(stream)g (ciphers,)j Fb(Adv)m(ances)e(in)h(Cryptology)e({)105 1301 y(CR)l(YPTO)g(`93)p Fv(,)e(Lecture)h(Notes)g(in)h(Computer)f (Science,)i(v)o(ol.)e(773,)f(Springer-V)l(erlag,)i(1994,)d(40{48.)11 1394 y([29])22 b(J.)15 b(T)l(ouc)o(h,)g(P)o(erformance)f(analysis)i(of) f(MD5,)f(Pro)q(c.)h(Sigcomm)g('95,)f(A)o(CM,)g(1995,)g(pp.)h(77{86.)11 1488 y([30])22 b(G.)15 b(Tsudik,)h(Message)f(authen)o(tication)h(with)g (one-w)o(a)o(y)f(hash)g(functions,)h Fb(Pro)q(ceedings)h(of)e(Info)q (com)h(92,)105 1545 y Fv(IEEE)f(Press,)g(1992.)11 1638 y([31])22 b(U.)13 b(V)l(azirani,)i(E\016ciency)h(considerations)e(in)h (using)g(semi-random)f(sources,)g Fb(Pro)q(ceedings)h(of)f(the)g(Nine-) 105 1695 y(teen)o(th)h(Ann)o(ual)h(A)o(CM)f(Symp)q(osium)h(on)f(Theory) g(of)g(Computing)p Fv(,)g(A)o(CM)f(Press,)h(1987,)e(pp.)j(160{168.)11 1789 y([32])22 b(M.)9 b(W)l(egman)h(and)h(L.)f(Carter,)g(New)g(hash)g (functions)h(and)g(their)f(use)h(in)g(authen)o(tication)g(and)f(set)g (equalit)o(y)l(,)105 1845 y Fb(J.)15 b(of)g(Computer)f(and)i(System)f (Sciences,)i Fv(v)o(ol.)d(22,)h(1981,)e(pp.)j(265{279.)11 1939 y([33])22 b(A.)10 b(Zobrist,)g(A)h(new)f(hashing)h(metho)q(d)g (with)f(applications)i(for)e(game)g(pla)o(ying,)h(Univ)o(ersit)o(y)g (of)f(Wisconsin,)105 1995 y(Dept.)k(of)h(Computer)g(Science,)i(TR)e (#88)g(\(April)h(1970\).)0 2139 y FE(A)67 b(De\014nitions)23 b(of)f(Securit)n(y)0 2240 y Fw(MA)o(C)15 b(security.)k Fv(W)l(e)13 b(follo)o(w)h([14)o(,)f(5])g(and)h(de\014ne)g (deterministic,)i(coun)o(ter-based)d(message)g(authen)o(tication)0 2297 y(co)q(des.)35 b(A)20 b(MA)o(C)f(sc)o(heme)h Fs(M)g Fv(sp)q(eci\014es:)31 b(constan)o(ts)19 b Fu(L)h Fv(and)g Fu(c)p Fv(,)g(determining)i(Messages)d(=)i Fs(f)p Fv(0)p Fu(;)8 b Fv(1)p Fs(g)1803 2276 y Fq(\024)p Fr(L)1877 2297 y Fv(and)0 2353 y(T)l(ags)17 b(=)i Fs(f)p Fv(0)p Fu(;)8 b Fv(1)p Fs(g)275 2332 y Fr(c)292 2353 y Fv(;)20 b(a)e(set)g(of)g(strings)g(Keys;)i(a)e(n)o(um)o(b)q(er)h(MAX)f (\(alternativ)o(ely)l(,)h(MAX)f(=)g Fs(1)p Fv(\);)i(and)f(a)f(pair)g (of)0 2409 y(functions)e(\(MA)o(C)o Fu(;)8 b Fv(MA)o(CV)o(\),)15 b(where)399 2512 y(MA)o(C)d(:)83 b(Keys)10 b Fs(\002)h Fv(Messages)e Fs(\002)i(f)p Fv(1)p Fu(;)d(:)g(:)g(:)t(;)g Fv(MAX)o Fs(g)k(!)h Fv(T)l(ags)p Fu(;)52 b Fv(and)364 2580 y(MA)o(CV)13 b(:)83 b(Keys)10 b Fs(\002)h Fv(Messages)e Fs(\002)i Fv(T)l(ags)h Fs(!)h(f)p Fv(0)p Fu(;)8 b Fv(1)p Fs(g)i Fu(:)952 2824 y Fv(20)p eop %%Page: 21 21 21 20 bop 436 917 a currentpoint currentpoint translate .5 .5 scale neg exch neg exch translate 436 917 a @beginspecial 179 @llx 264 @lly 689 @urx 688 @ury 5100 @rwi @setspecial %%BeginDocument: M.eps 1 setlinejoin /M { moveto } bind def /S { show } bind def /R { rmoveto } bind def /L { lineto } bind def /B { newpath 0 0 M 0 1 L 1 1 L 1 0 L closepath } bind def /CS { closepath stroke } bind def /S { /fixwidth exch def dup length /nchars exch def dup stringwidth pop fixwidth exch sub nchars div exch 0 exch ashow } def /bwproc { rgbproc dup length 3 idiv string 0 3 0 5 -1 roll { add 2 1 roll 1 sub dup 0 eq { pop 3 idiv 3 -1 roll dup 4 -1 roll dup 3 1 roll 5 -1 roll put 1 add 3 0 } { 2 1 roll } ifelse } forall pop pop pop } def systemdict /colorimage known not { /colorimage { pop pop /rgbproc exch def { bwproc } image } def } if 1 1 scale 0 setlinewidth /drawtri { /y3 exch def /x3 exch def /y2 exch def /x2 exch def /y1 exch def /x1 exch def 0 setgray newpath x1 y1 moveto x2 y2 lineto x3 y3 lineto closepath stroke } bind def /filltri { /y3 exch def /x3 exch def /y2 exch def /x2 exch def /y1 exch def /x1 exch def newpath x1 y1 moveto x2 y2 lineto x3 y3 lineto closepath fill } bind def /cliptri { /y3 exch def /x3 exch def /y2 exch def /x2 exch def /y1 exch def /x1 exch def 0 setgray newpath x1 y1 moveto x2 y2 lineto x3 y3 lineto closepath clip } bind def /imgscanrgb { gsave translate /scandy exch def /scandx exch def /istr scandx 3 mul string def scandx scandy scale scandx scandy 8 [scandx 0 0 scandy neg 0 scandy] {currentfile istr readhexstring pop} false 3 colorimage grestore } bind def /imgscanbw { gsave translate /scandy exch def /scandx exch def /istr scandx string def scandx scandy scale scandx scandy 8 [scandx 0 0 scandy neg 0 scandy] {currentfile istr readhexstring pop} image grestore } bind def /showcaseisoencoding [ /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /minus /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /guilsinglright /fraction /florin /quotesingle /quotedblleft /guilsinglleft /fi /fl /endash /dagger /daggerdbl /bullet /quotesinglbase /quotedblbase /quotedblright /ellipsis /trademark /dotlessi /grave /acute /circumflex /tilde /macron /breve /dotaccent /dieresis /perthousand /ring /cedilla /Ydieresis /hungarumlaut /ogonek /caron /emdash /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def /showcasedingbatencoding [ /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /a1 /a2 /a202 /a3 /a4 /a5 /a119 /a118 /a117 /a11 /a12 /a13 /a14 /a15 /a16 /a105 /a17 /a18 /a19 /a20 /a21 /a22 /a23 /a24 /a25 /a26 /a27 /a28 /a6 /a7 /a8 /a9 /a10 /a29 /a30 /a31 /a32 /a33 /a34 /a35 /a36 /a37 /a38 /a39 /a40 /a41 /a42 /a43 /a44 /a45 /a46 /a47 /a48 /a49 /a50 /a51 /a52 /a53 /a54 /a55 /a56 /a57 /a58 /a59 /a60 /a61 /a62 /a63 /a64 /a65 /a66 /a67 /a68 /a69 /a70 /a71 /a72 /a73 /a74 /a203 /a75 /a204 /a76 /a77 /a78 /a79 /a81 /a82 /a83 /a84 /a97 /a98 /a99 /a100 /.notdef /a205 /a85 /a206 /a86 /a87 /a88 /a89 /a90 /a91 /a92 /a93 /a94 /a95 /a96 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /a101 /a102 /a103 /a104 /a106 /a107 /a108 /a112 /a111 /a110 /a109 /a120 /a121 /a122 /a123 /a124 /a125 /a126 /a127 /a128 /a129 /a130 /a131 /a132 /a133 /a134 /a135 /a136 /a137 /a138 /a139 /a140 /a141 /a142 /a143 /a144 /a145 /a146 /a147 /a148 /a149 /a150 /a151 /a152 /a153 /a154 /a155 /a156 /a157 /a158 /a159 /a160 /a161 /a163 /a164 /a196 /a165 /a192 /a166 /a167 /a168 /a169 /a170 /a171 /a172 /a173 /a162 /a174 /a175 /a176 /a177 /a178 /a179 /a193 /a180 /a199 /a181 /a200 /a182 /.notdef /a201 /a183 /a184 /a197 /a185 /a194 /a198 /a186 /a195 /a187 /a188 /a189 /a190 /a191 /.notdef ] def /Times-Roman findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding showcaseisoencoding def currentdict end /Times-Roman-SHOWISO exch definefont pop /DJB where { pop save true } { false } ifelse % %!PS-AdobeFont-2.0: DJB 001.001 % To load to VM, remove the '' character from the 'exitserver' line. % serverdict begin 0 exitserver % for VM installation 11 dict begin /FontInfo 10 dict dup begin version (001.000) readonly def /Notice (Adobe Type 1 format font) readonly def /Copyright (This is a spot for a copywrite notice) readonly def /FullName (SGI DJB) readonly def /FamilyName (SGI DJB) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -110 def /UnderlineThickness 49 def end readonly def /FontName /DJB def /Encoding StandardEncoding def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /UniqueID 40454 def /FontBBox{-603 -760 1281 861}readonly def currentdict end currentfile eexec 982e801e86ef83d068e1fb5368d790d3cc2410c853b1f3c26aa0d5f8d031daae d43da52924f123f98fb74610b415ba2336fea65815a394e33340bfeef042ab1b 7221f89ded48535249b758dd59e188dca286b48d42d71917de4c340838b4bf15 388d5a09bdaebc3c427aba60a52309bc6d0e21cf6527538b121e4862a23556b5 ac0701f35f3f4b097e0fbccf3eff81f1adec616c54503d6cd4d9c9fa46d0c407 87f4d2d6a2c92ae244675e21110016f1c777764a9c4fe13387178beb531e9e24 284c6f3a924d5c4f28d5ba42e3158128228d0fba9e8c5dc3439ea1b6395020ba 70824394eaa938c2c55f309193dff6b703b5714609089f57aa1451bc4efa7f9f 7b71fa61bb41d0a87e371aebe554bd9118d2a0b4ed5fb276bf6c10d491025d37 23e18f86f7dca6c806c9324db755ecfeb763754c9863779cc960944c51c13739 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forall /Encoding showcaseisoencoding def currentdict end /DJB-SHOWISO exch definefont pop /Haeberli where { pop save true } { false } ifelse % %!PS-AdobeFont-2.0: Haeberli 001.001 % To load to VM, remove the '' character from the 'exitserver' line. % serverdict begin 0 exitserver % for VM installation 11 dict begin /FontInfo 10 dict dup begin version (001.000) readonly def /Notice (Adobe Type 1 format font) readonly def /Copyright (This is a spot for a copywrite notice) readonly def /FullName (SGI Haeberli) readonly def /FamilyName (SGI Haeberli) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -110 def /UnderlineThickness 49 def end readonly def /FontName /Haeberli def /Encoding StandardEncoding def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /UniqueID 53457 def /FontBBox{-129 -396 747 707}readonly def currentdict end currentfile eexec 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forall /Encoding showcaseisoencoding def currentdict end /Helvetica-Narrow-SHOWISO exch definefont pop /newfont 10 dict def newfont begin /FontType 3 def /FontMatrix [1 0 0 1 0 0] def /FontBBox [0 0 1 1] def /Encoding 256 array def 0 1 255 {Encoding exch /.notdef put} for /CharProcs 1 dict def CharProcs begin /.notdef {} def end /BuildChar { 1 0 0 0 1 1 setcachedevice exch begin Encoding exch get CharProcs exch get end exec } def end /PatternFont newfont definefont pop gsave /saveit save def gsave gsave 1.000000 setlinewidth matrix currentmatrix [38.1838 0 0 38.1838 432 649] concat newpath 0 0 1 0 360 arc 0 0 0 setrgbcolor closepath setmatrix stroke grestore gsave 1.000000 setlinewidth matrix currentmatrix [38.1838 0 0 38.1838 541.184 540.184] concat newpath 0 0 1 0 360 arc 0 0 0 setrgbcolor closepath setmatrix stroke grestore gsave 1.000000 setlinewidth matrix currentmatrix [38.1838 0 0 38.1838 650.184 434.184] concat newpath 0 0 1 0 360 arc 0 0 0 setrgbcolor closepath setmatrix stroke grestore gsave 1.000000 setlinewidth matrix currentmatrix [38.1838 0 0 38.1838 326.368 540.368] concat newpath 0 0 1 0 360 arc 0 0 0 setrgbcolor closepath setmatrix stroke grestore gsave 0 0 0 setrgbcolor 1.000000 setlinewidth newpath 251.252 453.276 M 249.984 455.32 L 248.568 457.309 L 247.008 459.228 L 245.307 461.06 L 243.473 462.792 L 241.516 464.41 L 239.445 465.9 L 237.276 467.252 L 235.02 468.455 L 232.695 469.503 L 230.315 470.389 L 227.898 471.111 L 225.46 471.668 L 223.018 472.06 L 220.588 472.292 L 218.184 472.368 L 218.184 472.368 L 215.779 472.292 L 213.349 472.06 L 210.907 471.668 L 208.469 471.111 L 206.053 470.389 L 203.673 469.503 L 201.347 468.455 L 199.092 467.252 L 196.922 465.9 L 194.852 464.41 L 192.895 462.792 L 191.061 461.06 L 189.36 459.228 L 187.799 457.309 L 186.383 455.32 L 185.116 453.276 L 185.116 453.276 L 183.979 451.156 L 182.964 448.935 L 182.083 446.624 L 181.347 444.235 L 180.763 441.781 L 180.341 439.277 L 180.086 436.739 L 180 434.184 L 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Fd(\001)e Fy(P)1362 440 y Fe(63)436 523 y Fd(\024)525 495 y FC(6)p 515 513 41 2 v 515 551 a Fd(N)569 523 y(\001)h FC(1)g(+)666 495 y(3)p Fy(N)725 480 y Fe(2)753 495 y FC(+)g(10)p Fy(N)p 666 513 208 2 v 750 551 a Fd(N)888 523 y(\001)914 495 y FC(6\()p Fy(N)p Fd(\000)r FC(2\)\()p Fy(N)p Fd(\000)r FC(3\))p 914 513 263 2 v 1025 551 a Fd(N)1191 523 y FC(+)g(1)g Fd(\001)1288 495 y FC(36\()p Fy(N)p Fd(\000)r FC(4\))p 1288 513 163 2 v 1349 551 a Fd(N)1464 523 y FC(+)h(1)f Fd(\001)1562 495 y FC(120)p 1562 513 63 2 v 1573 551 a Fd(N)436 630 y(\024)515 602 y FC(54)p Fy(N)595 587 y Fe(4)p 515 620 99 2 v 534 658 a Fd(N)574 646 y Fe(2)303 754 y Fy(f)323 760 y Fe(0)342 754 y FC(\(4\))41 b(=)h Fy(f)530 760 y Fe(3)549 754 y FC(\(3\))9 b Fd(\001)g Fy(P)659 760 y Fe(30)436 843 y Fd(\024)515 815 y FC(54)p Fy(N)595 800 y Fe(4)p 515 834 V 534 872 a Fd(N)574 860 y Fe(3)627 843 y Fd(\001)663 815 y FC(6)p 653 834 41 2 v 653 872 a Fd(N)436 950 y(\024)515 922 y FC(324)p Fy(N)616 907 y Fe(4)p 515 940 119 2 v 545 979 a Fd(N)585 967 y Fe(3)303 1075 y Fy(f)323 1081 y Fe(2)342 1075 y FC(\(4\))41 b(=)h Fy(f)530 1081 y Fe(1)549 1075 y FC(\(3\))9 b Fd(\001)g Fy(P)659 1081 y Fe(12)703 1075 y FC(+)g Fy(f)764 1081 y Fe(3)783 1075 y FC(\(3\))h Fd(\001)e Fy(P)893 1081 y Fe(32)937 1075 y FC(+)i Fy(f)999 1081 y Fc(R)1026 1075 y FC(\(3\))p Fy(P)1106 1081 y Fc(R)p Fe(2)436 1163 y Fd(\024)515 1135 y FC(42)p Fy(N)595 1120 y Fe(3)p 515 1154 99 2 v 534 1192 a Fd(N)574 1180 y Fe(2)627 1163 y Fd(\001)653 1135 y FC(3\()p Fy(N)p Fd(\000)r FC(1\)\()p Fy(N)p Fd(\000)r FC(2\))p 653 1154 263 2 v 764 1192 a Fd(N)930 1163 y FC(+)976 1135 y(54)p Fy(N)1056 1120 y Fe(4)p 976 1154 99 2 v 996 1192 a Fd(N)1036 1180 y Fe(2)1088 1163 y Fd(\001)1114 1135 y FC(18\()p Fy(N)p Fd(\000)r FC(3\))p 1114 1154 163 2 v 1175 1192 a Fd(N)1291 1163 y FC(+)f(1)g Fd(\001)1388 1135 y FC(60)p 1388 1154 42 2 v 1389 1192 a Fd(N)436 1270 y(\024)515 1242 y FC(60)p 515 1261 V 516 1299 a Fd(N)571 1270 y FC(+)617 1242 y(1098)p Fy(N)739 1227 y Fe(5)p 617 1261 140 2 v 657 1299 a Fd(N)697 1287 y Fe(3)775 1270 y FC(\()p Fy(N)17 b Fd(\025)12 b FC(3\))303 1395 y Fy(f)323 1401 y Fe(4)342 1395 y FC(\(4\))41 b(=)h Fy(f)530 1401 y Fe(1)549 1395 y FC(\(3\))9 b Fd(\001)g Fy(P)659 1401 y Fe(14)703 1395 y FC(+)g Fy(f)764 1401 y Fe(3)783 1395 y FC(\(3\))h Fd(\001)e Fy(P)893 1401 y Fe(34)937 1395 y FC(+)i Fy(f)999 1401 y Fc(R)1026 1395 y FC(\(3\))g Fd(\001)e Fy(P)1136 1401 y Fc(R)p Fe(4)436 1484 y Fd(\024)515 1456 y FC(42)p Fy(N)595 1441 y Fe(3)p 515 1474 99 2 v 534 1512 a Fd(N)574 1500 y Fe(2)627 1484 y Fd(\001)h FC(1)g(+)724 1456 y(54)p Fy(N)804 1441 y Fe(4)p 724 1474 V 744 1512 a Fd(N)784 1500 y Fe(2)837 1484 y Fd(\001)862 1456 y FC(9\()p Fy(N)p Fd(\000)r FC(3\)\()p Fy(N)p Fd(\000)r FC(4\))p 862 1474 263 2 v 973 1512 a Fd(N)1139 1484 y FC(+)h(1)f Fd(\001)1236 1456 y FC(60\()p Fy(N)p Fd(\000)r FC(5\))h(+)f(210)p 1236 1474 276 2 v 1354 1512 a Fd(N)436 1591 y(\024)515 1563 y FC(60)p 515 1581 42 2 v 516 1619 a Fd(N)571 1591 y FC(+)617 1563 y(438)p Fy(N)718 1548 y Fe(6)p 617 1581 119 2 v 647 1619 a Fd(N)687 1607 y Fe(3)303 1715 y Fy(f)323 1721 y Fe(6)342 1715 y FC(\(4\))41 b(=)h Fy(f)530 1721 y Fe(3)549 1715 y FC(\(3\))9 b Fd(\001)g Fy(P)659 1721 y Fe(36)703 1715 y FC(+)g Fy(f)764 1721 y Fc(R)792 1715 y FC(\(3\))g Fd(\001)g Fy(P)902 1721 y Fc(R)p Fe(6)436 1804 y Fd(\024)515 1776 y FC(54)p Fy(N)595 1761 y Fe(4)p 515 1795 99 2 v 534 1833 a Fd(N)574 1821 y Fe(2)627 1804 y Fd(\001)g FC(1)g(+)g(1)g Fd(\001)775 1776 y FC(15\()p Fy(N)p Fd(\000)r FC(5\)\()p Fy(N)p Fd(\000)r FC(6\))i(+)e(126\()p Fy(N)p Fd(\000)r FC(7\))g(+)h(504)p 775 1795 631 2 v 1070 1833 a Fd(N)436 1911 y FC(=)515 1883 y(15)p Fy(N)595 1868 y Fe(2)p 515 1902 99 2 v 544 1940 a Fd(N)627 1911 y FC(+)674 1883 y(15)p Fy(N)754 1868 y Fe(4)781 1883 y FC(+)f(189)p Fy(N)923 1868 y Fe(3)950 1883 y Fd(\000)h FC(294)p Fy(N)1093 1868 y Fe(2)1120 1883 y Fd(\000)f FC(144)p Fy(N)p 674 1902 588 2 v 938 1940 a Fd(N)978 1928 y Fe(2)436 2018 y Fd(\024)515 1990 y FC(15)p Fy(N)595 1975 y Fe(4)p 515 2008 99 2 v 534 2046 a Fd(N)574 2034 y Fe(2)627 2018 y FC(+)674 1990 y(25)p Fy(N)754 1975 y Fe(2)p 674 2008 V 702 2046 a Fd(N)790 2018 y FC(\()p Fy(N)17 b Fd(\025)12 b FC(18\))303 2143 y Fy(f)323 2149 y Fe(3)342 2143 y FC(\(5\))41 b(=)h Fy(f)530 2149 y Fe(0)549 2143 y FC(\(4\))9 b Fd(\001)g Fy(P)659 2149 y Fe(03)703 2143 y FC(+)g Fy(f)764 2149 y Fe(2)783 2143 y FC(\(4\))h Fd(\001)e Fy(P)893 2149 y Fe(23)937 2143 y FC(+)i Fy(f)999 2149 y Fe(4)1018 2143 y FC(\(4\))f Fd(\001)g Fy(P)1128 2149 y Fe(43)1172 2143 y FC(+)g Fy(f)1233 2149 y Fe(6)1252 2143 y FC(\(4\))h Fd(\001)e Fy(P)1362 2149 y Fe(63)436 2231 y Fd(\024)515 2203 y FC(324)p Fy(N)616 2188 y Fe(4)p 515 2222 119 2 v 545 2260 a Fd(N)585 2248 y Fe(3)648 2231 y Fd(\001)h FC(1)g(+)740 2173 y Fm(\022)776 2203 y FC(60)p 776 2222 42 2 v 777 2260 a Fd(N)831 2231 y FC(+)878 2203 y(1098)p Fy(N)1000 2188 y Fe(5)p 878 2222 140 2 v 918 2260 a Fd(N)958 2248 y Fe(3)1022 2173 y Fm(\023)1062 2231 y Fd(\001)1088 2203 y FC(6\()p Fy(N)p Fd(\000)r FC(2\)\()p Fy(N)p Fd(\000)r FC(3\))p 1088 2222 263 2 v 1199 2260 a Fd(N)1365 2231 y FC(+)634 2291 y Fm(\022)670 2321 y FC(60)p Fy(N)p 670 2340 80 2 v 690 2378 a Fd(N)764 2349 y FC(+)810 2321 y(438)p Fy(N)911 2306 y Fe(6)p 810 2340 119 2 v 840 2378 a Fd(N)880 2366 y Fe(3)934 2291 y Fm(\023)974 2349 y Fd(\001)999 2321 y FC(36\()p Fy(N)p Fd(\000)r FC(4\))p 999 2340 163 2 v 1060 2378 a Fd(N)1176 2349 y FC(+)1217 2291 y Fm(\022)1253 2321 y FC(15)p Fy(N)1333 2306 y Fe(2)p 1253 2340 99 2 v 1282 2378 a Fd(N)1365 2349 y FC(+)1412 2321 y(25)p Fy(N)1492 2306 y Fe(4)p 1412 2340 V 1431 2378 a Fd(N)1471 2366 y Fe(2)1515 2291 y Fm(\023)1554 2349 y Fd(\001)1580 2321 y FC(120)p 1580 2340 63 2 v 1591 2378 a Fd(N)436 2467 y(\024)515 2439 y FC(4320)p Fy(N)637 2424 y Fe(8)664 2439 y FC(+)g(25680)p Fy(N)848 2424 y Fe(7)p 515 2458 351 2 v 661 2496 a Fd(N)701 2484 y Fe(4)303 2601 y Fy(f)323 2607 y Fe(0)342 2601 y FC(\(6\))41 b(=)h Fy(\031)534 2607 y Fe(3)552 2601 y FC(\(5\))9 b Fd(\001)650 2573 y FC(6)p 640 2591 41 2 v 640 2629 a Fd(N)952 2824 y Fv(23)p eop %%Page: 24 24 24 23 bop 436 95 a Fd(\024)515 67 y FC(4320)p Fy(N)637 52 y Fe(8)664 67 y FC(+)9 b(25680)p Fy(N)848 52 y Fe(7)p 515 86 351 2 v 661 124 a Fd(N)701 112 y Fe(4)880 95 y Fd(\001)915 67 y FC(6)p 905 86 41 2 v 905 124 a Fd(N)436 202 y FC(=)515 174 y(25920)p Fy(N)658 159 y Fe(8)684 174 y FC(+)h(154080)p Fy(N)890 159 y Fe(7)p 515 193 393 2 v 681 231 a Fd(N)721 219 y Fe(5)0 357 y Fv(This)16 b(completes)g(the)f(pro)q(of)g(of)g(the)g(lemma.)1130 b Fs(})952 2824 y Fv(24)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF