path: root/other/burneye2/doc-external/hash/dietzfelbinger90dynamic.ps

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Fj(X)605 920 y Fl(x)625 930 y Fm(1)645 920 y Fl(;:::)o(;x)714
932 y(k)737 913 y Fo(=)e(1\))g Fi(\024)f Fj(c=s)969 895 y Fl(k)991
913 y Fo(,)i(since)g Fi(H)h Fo(is)f(\()p Fj(c;)8 b(k)r Fo(\)-univ)o(ersal.)21
b(On)c(the)f(other)46 973 y(hand,)g(it)g(is)g(clear)g(that)656
1033 y(\()p Fj(b)696 1040 y Fl(j)714 1033 y Fo(\))733 1040
y Fl(k)768 1033 y Fo(=)916 992 y Fg(X)820 1086 y Fm(\()p Fl(x)854
1096 y Fm(1)873 1086 y Fl(;:::)o(;x)942 1097 y(k)963 1086 y
Fm(\))p Ff(2)p Fm(\()p Fl(S)r Fm(\))1052 1097 y Fl(k)1081 1033
y Fj(X)1121 1040 y Fl(x)1141 1050 y Fm(1)1161 1040 y Fl(;:::)o(;x)1230
1052 y(k)1253 1033 y Fj(:)46 1163 y Fo(Consequen)o(tly)l(,)318
1273 y Fj(E)s Fo(\(\()p Fj(b)416 1280 y Fl(j)434 1273 y Fo(\))453
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y(k)703 1326 y Fm(\))p Ff(2)p Fm(\()p Fl(S)r Fm(\))792 1338
y Fl(k)820 1273 y Fj(E)s Fo(\()p Fj(X)918 1280 y Fl(x)938 1291
y Fm(1)958 1280 y Fl(;:::;x)1028 1293 y(k)1051 1273 y Fo(\))h
Fi(\024)f(j)p Fo(\()p Fj(S)s Fo(\))1221 1280 y Fl(k)1242 1273
y Fi(j)e(\001)1309 1240 y Fj(c)p 1297 1262 45 2 v 1297 1308
a(s)1320 1293 y Fl(k)1360 1273 y Fo(=)j(\()p Fj(n)p Fo(\))1479
1280 y Fl(k)1511 1273 y Fi(\001)1553 1240 y Fj(c)p 1541 1262
V 1541 1308 a(s)1564 1293 y Fl(k)1590 1273 y Fj(:)46 1427 y
Fo(\(b\))19 b(The)g(pro)q(of)i(is)e(similar)e(to)i(the)g(one)g(giv)o(en)g(in)
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1409 y Fl(j)1686 1439 y(x)1706 1449 y Fm(1)1726 1439 y Fl(;:::)o(;x)1795
1451 y(k)q Ff(\000)p Fm(1)1863 1427 y Fo(,)46 1496 y(\()p Fj(x)93
1503 y Fm(1)112 1496 y Fj(;)8 b(:)g(:)g(:)g(;)g(x)250 1503
y Fl(k)q Ff(\000)p Fm(1)316 1496 y Fo(\))13 b Fi(2)i Fo(\()p
Fj(S)s Fo(\))467 1503 y Fl(k)q Ff(\000)p Fm(1)533 1496 y Fo(,)h(0)e
Fi(\024)f Fj(j)k(<)d(s)p Fo(,)i(b)o(y)359 1620 y Fj(Y)399 1602
y Fl(j)387 1632 y(x)407 1642 y Fm(1)427 1632 y Fl(;:::;x)497
1644 y(k)q Ff(\000)p Fm(1)579 1620 y Fo(=)630 1559 y Fg(\032)670
1591 y Fo(1)p Fj(;)48 b Fo(if)16 b Fj(h)p Fo(\()p Fj(x)876
1598 y Fm(1)896 1591 y Fo(\))d(=)h Fj(:)8 b(:)g(:)13 b Fo(=)h
Fj(h)p Fo(\()p Fj(x)1178 1598 y Fl(k)q Ff(\000)p Fm(1)1244
1591 y Fo(\))g(=)g Fj(h)p Fo(\()p Fj(x)1404 1598 y Fm(0)1423
1591 y Fo(\))g(=)g Fj(j;)670 1651 y Fo(0)p Fj(;)48 b Fo(otherwise.)46
1747 y(Then)16 b Fj(E)s Fo(\()p Fj(Y)270 1729 y Fl(j)259 1760
y(x)279 1770 y Fm(1)299 1760 y Fl(;:::)o(;x)368 1771 y(k)q
Ff(\000)p Fm(1)436 1747 y Fo(\))e Fi(\024)g Fj(c=s)590 1729
y Fl(k)612 1747 y Fo(,)h(since)h Fi(H)h Fo(is)f(\()p Fj(c;)8
b(k)r Fo(\)-univ)o(ersal.)20 b(F)l(urther,)489 1866 y(\()p
Fj(b)529 1873 y Fl(x)549 1883 y Fm(0)571 1866 y Fo(\))590 1873
y Fl(k)q Ff(\000)p Fm(1)670 1866 y Fo(=)863 1824 y Fg(X)722
1918 y Fm(\()p Fl(x)756 1928 y Fm(1)775 1918 y Fl(;:::)o(;x)844
1930 y(k)q Ff(\000)p Fm(1)910 1918 y(\))p Ff(2)p Fm(\()p Fl(S)r
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1915 y Fm(0)p Ff(\024)p Fl(j)r(<s)1214 1866 y Fj(Y)1253 1847
y Fl(j)1242 1878 y(x)1262 1888 y Fm(1)1282 1878 y Fl(;:::)o(;x)1351
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(ected)g(v)m(alues,)f(w)o(e)h(get)475 2127 y Fj(E)s Fo(\(\()p
Fj(b)573 2134 y Fl(x)593 2145 y Fm(0)614 2127 y Fo(\))633 2134
y Fl(k)q Ff(\000)p Fm(1)700 2127 y Fo(\))e Fi(\024)f(j)p Fo(\()p
Fj(S)s Fo(\))870 2134 y Fl(k)q Ff(\000)p Fm(1)936 2127 y Fi(j)e(\001)g
Fj(s)g Fi(\001)1062 2094 y Fj(c)p 1050 2116 V 1050 2161 a(s)1073
2147 y Fl(k)1113 2127 y Fo(=)j(\()p Fj(n)p Fo(\))1232 2134
y Fl(k)q Ff(\000)p Fm(1)1310 2127 y Fi(\001)1374 2094 y Fj(c)p
1339 2116 90 2 v 1339 2161 a(s)1362 2147 y Fl(k)q Ff(\000)p
Fm(1)1434 2127 y Fj(;)46 2241 y Fo(as)j(claimed.)1554 b Fd(2)46
2361 y Fo(A)15 b(hash)h(function)f Fj(h)h Fo(is)f(called)g
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2342 y Fl(h)1010 2373 y(j)1046 2361 y Fi(\024)e Fj(l)j Fo(for)g(all)f
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b Fo(is)g(\()p Fj(k)g Fi(\000)d Fo(1\)-p)q(erfect\))20 b Fi(\025)g
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Fm(1)1253 2787 y Fw(.)33 b(In)21 b(c)n(ase)g Fj(s)f Fo(=)g
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%%Page: 13 13
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313 y(X)259 404 y Fm(0)p Ff(\024)p Fl(j)r(<s)364 354 y Fo(\()p
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686 y(\(a\))e(F)l(or)g Fo(0)d Fi(\024)g Fj(j)j(<)c(s)18 b Fw(arbitr)n(ary:)
385 827 y Fo(Pr\()p Fj(b)477 834 y Fl(j)509 827 y Fi(\025)c
Fj(u)p Fo(\))f Fi(\024)675 754 y Fg(\()716 798 y Fj(c)e Fi(\001)g
Fo(\()p Fj(e)815 780 y Fl(u)p Ff(\000)p Fm(1)883 798 y Fj(=u)935
780 y Fl(u)957 798 y Fo(\))g Fi(\001)g Fo(\()p Fj(n=s)p Fo(\))1126
780 y Fl(u)1149 798 y Fj(;)49 b Fw(for)17 b Fo(1)d Fi(\024)g
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840 y Fl(k)q Ff(\000)p Fm(1)881 858 y Fj(=u)933 840 y Fl(k)955
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979 y Fl(j)1118 972 y Fi(\025)13 b Fj(u)p Fo(\))h(=)g Fj(O)q
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1091 y Fw(\(b\))287 1182 y Fo(Pr\()p Fj(b)379 1189 y Fl(x)399
1200 y Fm(0)435 1182 y Fi(\025)f Fj(u)p Fo(\))h Fi(\024)600
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1778 y(Clearly)168 1888 y(Pr\()p Fi(9)p Fj(j)17 b Fo(:)c Fj(b)352
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1895 y Fl(k)823 1888 y Fi(\025)h Fj(k)r Fo(!\))f Fi(\024)h
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y Fl(j)1306 1888 y Fo(\))1325 1895 y Fl(k)1360 1888 y Fi(\025)i
Fj(k)r Fo(!\))h Fi(j)f Fo(0)i Fi(\024)e Fj(j)k(<)d(s)p Fi(g)p
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2238 y(considerations)14 b(that)465 2205 y Fg(P)509 2248 y
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2218 y Fl(n)p 1228 2226 V 1228 2255 a(k)q Ff(\000)p Fm(1)1298
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2220 y Fl(k)1545 2238 y Fi(\024)13 b Fj(n)7 b Fi(\001)g Fo(\()p
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%%Page: 14 14
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1518 y Fl(i)1527 1511 y Fo(is)g(inserted)8 b Fi(g)p Fj(:)46
1621 y Fo(Clearly)l(,)20 b(for)g(eac)o(h)g Fj(x)450 1628 y
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Fo(2)8 b(log)i Fj(n)20 b Fo(in)f(the)h(computation)f(for)h
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b(log)i Fj(n)p Fo(,)46 1982 y(and)20 b Fj(x)172 1989 y Fl(i)184
1994 y Fe(1)203 1982 y Fj(;)8 b(:)g(:)g(:)f(;)h(x)340 1989
y Fl(i)352 2000 y Fa(n)371 1993 y Fc(0)406 1982 y Fo(are)20
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(considered)f(in)46 2043 y(Lemma)f(5.4.)33 b(Th)o(us)20 b(w)o(e)g(ma)o(y)e
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2050 y Fl(i)1583 2055 y Fe(1)1602 2043 y Fj(;)8 b(:)g(:)g(:)g(;)g(x)1740
2050 y Fl(i)1752 2060 y Fa(n)1771 2053 y Fc(0)1806 2043 y Fo(has)46
2103 y(cost)16 b(at)h(least)f(\()p Fj(n)368 2085 y Ff(0)391
2103 y Fo(+)11 b(1\))d(log)r(\()p Fj(n)603 2085 y Ff(0)625
2103 y Fo(+)j(1\).)22 b(Altogether)15 b(w)o(e)h(get)330 2213
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2315 y Fm(0)p Ff(\024)p Fl(y)q Ff(\024)p Fl(n)p Ff(\000)p Fm(1)712
2286 y Fo(\()p Fj(y)12 b Fi(\001)f Fo(2)d(log)i Fj(n)h Fo(+)g(\()p
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%%Page: 21 21
20 bop 46 188 a Fx(5)49 b(PR)o(OOFS)16 b(OF)g(THE)g(LO)o(WER)h(BOUNDS)869
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1310 y Fj(T)6 b Fo(\()p Fj(n)p Fo(\))17 b Fi(\024)f Fo(3\()p
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1370 y(desired.)46 1489 y(This)e(\014nishes)g(the)g(pro)q(of)i(of)e
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1927 V 35 x Fj(n)p Fo(\))818 1920 y Ff(p)p 847 1920 22 2 v
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%%Page: 22 22
21 bop 46 188 a Fx(5)49 b(PR)o(OOFS)16 b(OF)g(THE)g(LO)o(WER)h(BOUNDS)869
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%%Page: 23 23
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b Fo(lim)168 1118 y Fl(k)q Ff(!1)285 1055 y Fj(d)310 1062 y
Fl(k)p 271 1077 75 2 v 271 1122 a Fj(k)r(=e)364 1088 y Fo(=)14
b(1)p Fj(:)46 1268 y Fw(Pr)n(o)n(of:)47 b Fo(\(a\))17 b(\(Induction)e(on)i
Fj(k)r Fo(.\))46 1387 y(Initial)e(step)h(\()p Fj(k)g Fo(=)e(1\):)21
b(Ob)o(viously)l(,)733 1374 y(~)723 1387 y Fj(T)752 1394 y
Fm(1)771 1387 y Fo(\()p Fj(n)p Fo(\))14 b(=)g(\()p Fj(n)d Fo(+)g(1\)\()p
Fj(n)h Fo(+)f(2\))p Fj(=)p Fo(2)h Fi(\000)f Fo(1)j Fi(\024)f
Fj(n)1436 1369 y Fm(2)1473 1387 y Fo(for)j Fj(n)e Fi(\025)g
Fo(3.)46 1520 y(Induction)19 b(step)h(\()p Fj(k)j(>)d Fo(1\):)29
b(Assume)782 1507 y(~)772 1520 y Fj(T)801 1527 y Fl(k)q Ff(\000)p
Fm(1)887 1520 y Fi(\024)20 b Fj(d)971 1527 y Fl(k)q Ff(\000)p
Fm(1)1051 1520 y Fi(\001)14 b Fj(n)1108 1502 y Fl(k)q(=)p Fm(\()p
Fl(k)q Ff(\000)p Fm(1\))1259 1520 y Fo(for)20 b(all)f Fj(n)i
Fi(\025)f Fj(n)1546 1527 y Fl(k)q Ff(\000)p Fm(1)1612 1520
y Fo(.)32 b(No)o(w)20 b(let)f Fj(n)46 1580 y Fo(b)q(e)f(\014xed,)f
Fj(n)h Fo(large)g(enough.)26 b(De\014ne)18 b Fj(t)777 1587
y Fm(0)812 1580 y Fo(=)f(min)n Fi(f)p Fj(t)f Fi(\025)g Fo(1)h
Fi(j)f Fj(i)1150 1587 y Fl(t)1181 1580 y Fj(>)g(n)p Fi(g)p
Fo(,)i(for)g(the)g(sequence)e Fj(i)1703 1587 y Fl(t)1718 1580
y Fo(,)h Fj(t)f Fi(\025)h Fo(1,)46 1640 y(de\014ned)f(in)h(the)f(strategy)l
(.)24 b(W)l(e)16 b(\014rst)h(estimate)e Fj(t)961 1647 y Fm(0)980
1640 y Fo(.)23 b(Clearly)l(,)16 b(b)o(y)g(the)h(de\014nition)f(of)h
Fj(i)1640 1647 y Fl(t)1671 1640 y Fo(and)h Fj(t)1785 1647 y
Fm(0)1821 1640 y Fo(w)o(e)46 1700 y(ha)o(v)o(e)535 1730 y Fl(t)548
1735 y Fe(0)565 1730 y Ff(\000)p Fm(1)543 1743 y Fg(X)542 1834
y Fl(s)p Fm(=1)618 1785 y Fj(s)641 1764 y Fl(k)q Ff(\000)p
Fm(1)735 1785 y Fi(\024)28 b Fo(\()p Fj(k)13 b Fi(\001)e Fj(b)905
1792 y Fl(k)q Ff(\000)p Fm(1)971 1785 y Fo(\))990 1764 y Fl(k)q
Ff(\000)p Fm(1)1067 1785 y Fi(\001)g Fj(n)28 b(<)1230 1730
y Fl(t)1243 1735 y Fe(0)1216 1743 y Fg(X)1215 1834 y Fl(s)p
Fm(=1)1284 1785 y Fj(s)1307 1764 y Fl(k)q Ff(\000)p Fm(1)1374
1785 y Fj(;)46 1902 y Fo(hence)15 b(\(b)o(y)h(estimating)f(the)h(sums)f(b)o
(y)h(in)o(tegrals)g(and)h(taking)f Fj(k)r Fo(-th)h(ro)q(ots\),)570
2008 y Fj(t)588 2015 y Fm(0)619 2008 y Fi(\000)11 b Fo(1)28
b Fi(\024)f Fj(k)13 b Fi(\001)e Fj(b)871 1982 y Fm(\()p Fl(k)q
Ff(\000)p Fm(1\))p Fl(=k)871 2020 y(k)q Ff(\000)p Fm(1)1013
2008 y Fi(\001)f Fj(n)1066 1987 y Fm(1)p Fl(=k)1151 2008 y
Fj(<)27 b(t)1234 2015 y Fm(0)1265 2008 y Fo(+)11 b(1)476 b(\(3\))46
2112 y(In)15 b(the)h(follo)o(wing,)f(w)o(e)h(estimate)685 2100
y(~)676 2112 y Fj(T)705 2119 y Fl(k)726 2112 y Fo(\()p Fj(i)762
2119 y Fl(t)775 2124 y Fe(0)804 2112 y Fi(\000)10 b Fo(1\),)16
b(whic)o(h)f(certainly)g(is)g(an)i(upp)q(er)f(b)q(ound)h(for)1756
2100 y(~)1746 2112 y Fj(T)1775 2119 y Fl(k)1796 2112 y Fo(\()p
Fj(n)p Fo(\).)46 2172 y(W)l(e)e(let)g Fj(i)216 2179 y Fm(0)249
2172 y Fo(=)f(0.)21 b(Then)16 b(inserting)f(the)h(elemen)o(t)c
Fj(x)978 2179 y Fl(i)990 2183 y Fa(t)1022 2172 y Fo(\(b)o(y)j(global)h
(rehashing\))g(has)h(cost)e Fj(i)1692 2179 y Fl(t)1717 2172
y Fo(+)10 b(1,)15 b(for)46 2233 y Fj(t)i Fo(=)h(1)p Fj(;)8
b Fo(2)p Fj(;)g(:)g(:)g(:)g(;)g(t)335 2240 y Fm(0)367 2233
y Fi(\000)k Fo(1;)20 b(inserting)e(the)g(elemen)o(ts)e Fj(x)995
2240 y Fl(i)1007 2245 y Fa(t)p Fc(\000)p Fe(1)1060 2240 y Fm(+1)1107
2233 y Fj(;)8 b(:)g(:)g(:)f(;)h(x)1244 2240 y Fl(i)1256 2244
y Fa(t)1270 2240 y Ff(\000)p Fm(1)1336 2233 y Fo(has)19 b(cost)g(\()p
Fj(i)1564 2240 y Fl(t)1591 2233 y Fi(\000)12 b Fj(i)1659 2240
y Fl(t)p Ff(\000)p Fm(1)1731 2233 y Fi(\000)h Fo(1\))g(+)55
2285 y(~)46 2298 y Fj(T)75 2305 y Fl(k)q Ff(\000)p Fm(1)141
2298 y Fo(\()p Fj(i)177 2305 y Fl(t)202 2298 y Fi(\000)e Fj(i)269
2305 y Fl(t)p Ff(\000)p Fm(1)340 2298 y Fi(\000)g Fo(1\),)16
b(for)g Fj(t)e Fo(=)f(1)p Fj(;)8 b Fo(2)p Fj(;)g(:)g(:)g(:)h(;)f(t)819
2305 y Fm(0)838 2298 y Fo(.)21 b(Th)o(us,)294 2421 y(~)284
2434 y Fj(T)313 2441 y Fl(k)334 2434 y Fo(\()p Fj(n)p Fo(\))k
Fi(\024)505 2379 y Fl(t)518 2384 y Fe(0)490 2392 y Fg(X)491
2483 y Fl(t)p Fm(=1)558 2434 y Fo(\(\()p Fj(i)613 2441 y Fl(t)639
2434 y Fi(\000)10 b Fj(i)705 2441 y Fl(t)p Ff(\000)p Fm(1)776
2434 y Fi(\000)h Fo(1\))g(+)939 2421 y(~)929 2434 y Fj(T)958
2441 y Fl(k)q Ff(\000)p Fm(1)1024 2434 y Fo(\()p Fj(i)1060
2441 y Fl(t)1086 2434 y Fi(\000)g Fj(i)1153 2441 y Fl(t)p Ff(\000)p
Fm(1)1223 2434 y Fi(\000)g Fo(1\)\))g(+)1395 2379 y Fl(t)1408
2384 y Fe(0)1426 2379 y Ff(\000)p Fm(1)1403 2392 y Fg(X)1404
2483 y Fl(t)p Fm(=1)1471 2434 y Fo(\()p Fj(i)1507 2441 y Fl(t)1532
2434 y Fo(+)g(1\))p Fj(;)46 2565 y Fo(or,)16 b(after)g(a)h(trivial)e
(transformation,)362 2684 y(~)352 2697 y Fj(T)381 2704 y Fl(k)402
2697 y Fo(\()p Fj(n)p Fo(\))25 b Fi(\024)573 2642 y Fl(t)586
2647 y Fe(0)558 2655 y Fg(X)559 2746 y Fl(t)p Fm(=1)626 2697
y Fo(\(\()p Fj(t)682 2704 y Fm(0)713 2697 y Fo(+)11 b(1)g Fi(\000)g
Fj(t)p Fo(\)\()p Fj(i)920 2704 y Fl(t)945 2697 y Fi(\000)g
Fj(i)1012 2704 y Fl(t)p Ff(\000)p Fm(1)1071 2697 y Fo(\))g(+)1160
2684 y(~)1150 2697 y Fj(T)1179 2704 y Fl(k)q Ff(\000)p Fm(1)1245
2697 y Fo(\()p Fj(i)1281 2704 y Fl(t)1307 2697 y Fi(\000)g
Fj(i)1374 2704 y Fl(t)p Ff(\000)p Fm(1)1444 2697 y Fi(\000)g
Fo(1\)\))p Fj(:)46 2839 y Fo(Substituting)17 b(the)g(induction)g(h)o(yp)q
(othesis)879 2826 y(~)870 2839 y Fj(T)899 2846 y Fl(k)q Ff(\000)p
Fm(1)965 2839 y Fo(\()p Fj(n)1013 2821 y Ff(0)1025 2839 y Fo(\))e
Fi(\024)g Fj(d)1138 2846 y Fl(k)q Ff(\000)p Fm(1)1216 2839
y Fi(\001)d Fo(\()p Fj(n)1290 2821 y Ff(0)1301 2839 y Fo(\))1320
2821 y Fl(k)q(=)p Fm(\()p Fl(k)q Ff(\000)p Fm(1\))1451 2839
y Fo(,)17 b(for)g Fj(n)1586 2821 y Ff(0)1613 2839 y Fi(\025)f
Fj(n)1697 2846 y Fl(k)q Ff(\000)p Fm(1)1763 2839 y Fo(,)h(in)o(to)46
2899 y(this)f(inequalit)o(y)e(yields)66 3018 y(~)56 3031 y
Fj(T)85 3038 y Fl(k)106 3031 y Fo(\()p Fj(n)p Fo(\))g Fi(\024)254
2976 y Fl(t)267 2981 y Fe(0)239 2989 y Fg(X)240 3080 y Fl(t)p
Fm(=1)308 3031 y Fo(\(\()p Fj(t)364 3038 y Fm(0)394 3031 y
Fo(+)d(1)g Fi(\000)g Fj(t)p Fo(\)\()p Fj(i)601 3038 y Fl(t)626
3031 y Fi(\000)g Fj(i)693 3038 y Fl(t)p Ff(\000)p Fm(1)764
3031 y Fi(\000)g Fo(1\))g(+)g Fj(d)942 3038 y Fl(k)q Ff(\000)p
Fm(1)1009 3031 y Fo(\()p Fj(i)1045 3038 y Fl(t)1070 3031 y
Fi(\000)g Fj(i)1137 3038 y Fl(t)p Ff(\000)p Fm(1)1208 3031
y Fi(\000)g Fo(1\))1301 3010 y Fl(k)q(=)p Fm(\()p Fl(k)q Ff(\000)p
Fm(1\))1432 3031 y Fo(\))g(+)1516 3011 y Fm(1)p 1516 3019 18
2 v 1516 3048 a(2)1538 3031 y Fj(t)1556 3038 y Fm(0)1576 3031
y Fo(\()p Fj(t)1613 3038 y Fm(0)1643 3031 y Fo(+)g(1\))g(+)g
Fj(E)1831 3038 y Fl(k)1853 3031 y Fj(;)p eop
%%Page: 25 25
24 bop 46 188 a Fx(5)49 b(PR)o(OOFS)16 b(OF)g(THE)g(LO)o(WER)h(BOUNDS)869
b Fo(25)46 342 y(for)18 b(some)e(constan)o(t)i Fj(E)479 349
y Fl(k)518 342 y Fo(\(needed)f(to)g(mak)o(e)f(up)i(for)f(the)h(error)f
(caused)h(b)o(y)f(replacing)1690 329 y(~)1680 342 y Fj(T)1709
349 y Fl(k)q Ff(\000)p Fm(1)1776 342 y Fo(\()p Fj(i)1812 349
y Fl(t)1838 342 y Fi(\000)46 402 y Fj(i)63 409 y Fl(t)p Ff(\000)p
Fm(1)134 402 y Fi(\000)11 b Fo(1\))18 b(b)o(y)f Fj(d)339 409
y Fl(k)q Ff(\000)p Fm(1)405 402 y Fo(\()p Fj(i)441 409 y Fl(t)467
402 y Fi(\000)12 b Fj(i)535 409 y Fl(t)p Ff(\000)p Fm(1)606
402 y Fi(\000)g Fo(1\))700 384 y Fl(k)q(=)p Fm(\()p Fl(k)q
Ff(\000)p Fm(1\))848 402 y Fo(for)17 b Fj(t)g Fo(so)h(small)d(that)j
Fj(i)1271 409 y Fl(t)1297 402 y Fi(\000)11 b Fj(i)1364 409
y Fl(t)p Ff(\000)p Fm(1)1436 402 y Fi(\000)g Fo(1)16 b Fj(<)f(n)1608
409 y Fl(k)q Ff(\000)p Fm(1)1675 402 y Fo(\).)23 b(By)17 b(the)46
462 y(de\014nition)i(of)h Fj(i)342 469 y Fl(t)376 462 y Fo(w)o(e)g(clearly)e
(ha)o(v)o(e)h Fj(i)744 469 y Fl(t)772 462 y Fi(\000)13 b Fj(i)841
469 y Fl(t)p Ff(\000)p Fm(1)914 462 y Fi(\000)g Fo(1)21 b Fi(\024)e
Fo(\()p Fj(t=)p Fo(\()p Fj(k)r(b)1197 469 y Fl(k)q Ff(\000)p
Fm(1)1263 462 y Fo(\)\))1301 438 y Fl(k)q Ff(\000)p Fm(1)1368
462 y Fo(;)i(furthermore,)d(from)h(the)46 522 y(b)q(ounds)e(on)g
Fj(t)301 529 y Fm(0)337 522 y Fo(in)f(\()p Fi(z)p Fo(\))g(it)g(follo)o(ws)g
(that)h Fj(t)809 504 y Fm(2)803 535 y(0)842 522 y Fo(=)c Fj(O)q
Fo(\()q(\()p Fj(n)999 504 y Fm(1)p Fl(=k)1056 522 y Fo(\))1075
504 y Fm(2)1094 522 y Fo(\))h(=)g Fj(O)q Fo(\()p Fj(n)p Fo(\).)22
b(Th)o(us,)251 651 y(~)241 663 y Fj(T)270 670 y Fl(k)291 663
y Fo(\()p Fj(n)p Fo(\))42 b Fi(\024)495 608 y Fl(t)508 613
y Fe(0)480 622 y Fg(X)481 712 y Fl(t)p Fm(=1)540 603 y Fg(\024\022)646
630 y Fj(t)p 598 652 115 2 v 598 697 a(k)r(b)646 704 y Fl(k)q
Ff(\000)p Fm(1)717 603 y Fg(\023)747 614 y Fl(k)q Ff(\000)p
Fm(1)825 663 y Fi(\001)11 b Fo(\()p Fj(t)887 670 y Fm(0)917
663 y Fo(+)g(1)g Fi(\000)g Fj(t)p Fo(\))g(+)g Fj(d)1173 670
y Fl(k)q Ff(\000)p Fm(1)1251 663 y Fi(\001)1276 603 y Fg(\022)1360
630 y Fj(t)p 1311 652 V 1311 697 a(k)r(b)1359 704 y Fl(k)q
Ff(\000)p Fm(1)1430 603 y Fg(\023)1461 614 y Fl(k)1493 603
y Fg(\025)1526 663 y Fo(+)g Fj(O)q Fo(\()p Fj(n)p Fo(\))400
813 y(=)42 b(\()p Fj(k)r(b)547 820 y Fl(k)q Ff(\000)p Fm(1)613
813 y Fo(\))632 792 y Fm(1)p Ff(\000)p Fl(k)710 813 y Fi(\001)735
752 y Fg(\022)780 758 y Fl(t)793 763 y Fe(0)765 771 y Fg(X)766
862 y Fl(t)p Fm(=1)825 813 y Fo(\()p Fj(t)862 820 y Fm(0)893
813 y Fo(+)11 b(1\))p Fj(t)1003 792 y Fl(k)q Ff(\000)p Fm(1)1080
813 y Fi(\000)1145 758 y Fl(t)1158 763 y Fe(0)1130 771 y Fg(X)1131
862 y Fl(t)p Fm(=1)1198 813 y Fj(t)1216 792 y Fl(k)1237 813
y Fj(=k)1288 752 y Fg(\023)1330 813 y Fo(+)g Fj(O)q Fo(\()p
Fj(n)p Fo(\))p Fj(:)46 958 y Fo(W)l(e)j(substitute)h(the)g(t)o(w)o(o)f
(inequalities)783 925 y Fg(P)827 937 y Fl(t)840 942 y Fe(0)827
969 y Fl(t)p Fm(=1)895 958 y Fj(t)913 940 y Fl(k)q Ff(\000)p
Fm(1)993 958 y Fi(\024)g Fo(\()p Fj(t)1083 965 y Fm(0)1110
958 y Fo(+)8 b(1\))1199 940 y Fl(k)1221 958 y Fj(=k)17 b Fo(and)1381
925 y Fg(P)1424 937 y Fl(t)1437 942 y Fe(0)1424 969 y Fl(t)p
Fm(=1)1493 958 y Fj(t)1511 940 y Fl(k)1560 958 y Fi(\025)d
Fj(t)1637 938 y Fl(k)q Fm(+1)1631 969 y(0)1703 958 y Fj(=)p
Fo(\()p Fj(k)f Fo(+)e(1\))46 1019 y(\(obtained)17 b(b)o(y)f(replacing)g(the)h
(sums)f(b)o(y)g(in)o(tegrals\),)g(and)i(simplify)l(,)13 b(noting)18
b(that)f(\()p Fj(t)1635 1026 y Fm(0)1665 1019 y Fo(+)11 b(1\))1757
1001 y Fl(k)q Fm(+1)1839 1019 y Fo(=)46 1079 y Fj(t)70 1058
y Fl(k)q Fm(+1)64 1090 y(0)147 1079 y Fo(+)g Fj(O)q Fo(\()p
Fj(t)277 1061 y Fl(k)271 1091 y Fm(0)298 1079 y Fo(\).)21 b(In)16
b(this)h(w)o(a)o(y)e(w)o(e)h(get)437 1193 y(~)427 1206 y Fj(T)456
1213 y Fl(k)477 1206 y Fo(\()p Fj(n)p Fo(\))25 b Fi(\024)681
1172 y Fo(1)p 638 1194 112 2 v 638 1240 a Fj(k)13 b Fo(+)e(1)765
1206 y Fi(\001)g Fo(\()p Fj(k)r(b)857 1213 y Fl(k)q Ff(\000)p
Fm(1)923 1206 y Fo(\))942 1185 y Fm(1)p Ff(\000)p Fl(k)1020
1206 y Fi(\001)g Fj(t)1069 1185 y Fl(k)q Fm(+1)1063 1218 y(0)1146
1206 y Fo(+)g Fj(O)q Fo(\()p Fj(t)1276 1185 y Fl(k)1270 1218
y Fm(0)1297 1206 y Fo(\))g(+)g Fj(O)q Fo(\()p Fj(n)p Fo(\))p
Fj(:)46 1343 y Fo(By)18 b(\()p Fi(z)p Fo(\),)h(w)o(e)g(ha)o(v)o(e)f
Fj(t)432 1325 y Fl(k)426 1356 y Fm(0)472 1343 y Fo(=)g Fj(O)q
Fo(\()q(\()p Fj(n)634 1325 y Fm(1)p Fl(=k)690 1343 y Fo(\))709
1325 y Fl(k)731 1343 y Fo(\))g(=)h Fj(O)q Fo(\()p Fj(n)p Fo(\))g(and)h
(furthermore)d(that)j Fj(t)1452 1323 y Fl(k)q Fm(+1)1446 1354
y(0)1536 1343 y Fo(=)f Fj(k)1620 1325 y Fl(k)q Fm(+1)1699 1343
y Fi(\001)13 b Fj(b)1747 1318 y Fl(k)q Ff(\000)p Fm(1)p Fl(=k)1747
1356 y(k)q Ff(\000)p Fm(1)1863 1343 y Fi(\001)46 1404 y Fj(n)75
1386 y Fm(1+1)p Fl(=k)188 1404 y Fo(+)e Fj(O)q Fo(\()p Fj(t)318
1386 y Fl(k)312 1416 y Fm(0)339 1404 y Fo(\).)21 b(Hence)352
1518 y(~)343 1531 y Fj(T)372 1538 y Fl(k)393 1531 y Fo(\()p
Fj(n)p Fo(\))41 b Fi(\024)630 1497 y Fo(1)p 587 1519 V 587
1565 a Fj(k)13 b Fo(+)e(1)714 1531 y Fi(\001)g Fo(\()p Fj(k)r(b)806
1538 y Fl(k)q Ff(\000)p Fm(1)872 1531 y Fo(\))891 1510 y Fm(1)p
Ff(\000)p Fl(k)969 1531 y Fi(\001)g Fj(k)1021 1510 y Fl(k)q
Fm(+1)1098 1531 y Fi(\001)g Fj(b)1144 1505 y Fl(k)q Ff(\000)p
Fm(1)p Fl(=k)1144 1543 y(k)q Ff(\000)p Fm(1)1258 1531 y Fi(\001)g
Fj(n)1312 1510 y Fm(1+1)p Fl(=k)1425 1531 y Fo(+)g Fj(O)q Fo(\()p
Fj(n)p Fo(\))502 1659 y(=)619 1625 y Fj(k)646 1607 y Fm(2)p
587 1647 V 587 1693 a Fj(k)i Fo(+)e(1)714 1659 y Fi(\001)g
Fj(b)760 1633 y Fm(\()p Fl(k)q Ff(\000)p Fm(1\))p Fl(=k)760
1671 y(k)q Ff(\000)p Fm(1)902 1659 y Fi(\001)f Fj(n)955 1638
y Fm(1+1)p Fl(=k)1068 1659 y Fo(+)h Fj(O)q Fo(\()p Fj(n)p Fo(\))p
Fj(:)46 1780 y Fo(F)l(or)16 b Fj(n)h Fo(large)f(enough,)g(this)h(implies)441
1895 y(~)432 1907 y Fj(T)461 1914 y Fl(k)482 1907 y Fo(\()p
Fj(n)p Fo(\))25 b Fi(\024)f Fj(k)13 b Fi(\001)705 1873 y Fj(k)g
Fo(+)e(1)p 705 1896 V 705 1941 a Fj(k)i Fo(+)e(2)833 1907 y
Fi(\001)g Fj(b)879 1882 y Fm(\()p Fl(k)q Ff(\000)p Fm(1\))p
Fl(=k)879 1920 y(k)q Ff(\000)p Fm(1)1020 1907 y Fi(\001)g Fj(n)1074
1887 y Fm(1+1)p Fl(=k)1201 1907 y Fo(=)25 b Fj(d)1289 1914
y Fl(k)1321 1907 y Fi(\001)11 b Fj(n)1375 1887 y Fm(1+1)p Fl(=k)1477
1907 y Fj(;)46 2029 y Fo(and)17 b(this)f(is)g(what)h(w)o(e)f(w)o(an)o(ted)g
(to)g(sho)o(w.)46 2173 y(\(b\))g(By)g(de\014nition,)439 2139
y Fj(d)464 2146 y Fl(k)p 439 2161 47 2 v 449 2207 a Fj(k)504
2173 y Fo(=)556 2113 y Fg(\022)608 2119 y Fl(k)591 2132 y Fg(Y)587
2222 y Fl(q)q Fm(=2)649 2113 y Fg(\022)684 2139 y Fj(q)d Fo(+)e(1)p
684 2161 108 2 v 684 2207 a Fj(q)i Fo(+)e(2)797 2113 y Fg(\023)828
2124 y Fl(q)847 2113 y Fg(\023)877 2124 y Fm(1)p Fl(=k)934
2173 y Fo(.)21 b(Recall)15 b(that)608 2292 y Fg( )646 2331
y Fj(q)d Fo(+)f(1)p 646 2353 V 646 2399 a Fj(q)h Fo(+)f(2)759
2292 y Fg(!)791 2303 y Fl(q)q Fm(+2)880 2365 y Fi(\024)949
2331 y Fo(1)p 949 2353 25 2 v 950 2399 a Fj(e)1003 2365 y Fi(\024)1067
2292 y Fg( )1105 2331 y Fj(q)h Fo(+)f(1)p 1105 2353 108 2 v
1105 2399 a Fj(q)h Fo(+)f(2)1218 2292 y Fg(!)1251 2303 y Fl(q)q
Fm(+1)46 2498 y Fo(for)16 b(all)g Fj(q)r Fo(,)f(and)i(hence)292
2604 y Fj(d)317 2611 y Fl(k)p 292 2626 47 2 v 302 2672 a Fj(k)355
2638 y Fi(\001)380 2577 y Fg(\022)432 2584 y Fl(k)415 2597
y Fg(Y)411 2687 y Fl(q)q Fm(=2)473 2577 y Fg(\022)508 2604
y Fj(q)c Fo(+)e(1)p 508 2626 108 2 v 508 2672 a Fj(q)i Fo(+)e(2)621
2577 y Fg(\023)652 2589 y Fm(2)680 2577 y Fg(\023)710 2589
y Fm(1)p Fl(=k)792 2638 y Fi(\024)855 2577 y Fg(\022)891 2604
y Fo(1)p 891 2626 25 2 v 892 2672 a Fj(e)920 2577 y Fg(\023)951
2589 y Fm(\()p Fl(k)q Ff(\000)p Fm(1\))p Fl(=k)1106 2638 y
Fi(\024)1175 2604 y Fj(d)1200 2611 y Fl(k)p 1175 2626 47 2
v 1185 2672 a Fj(k)1238 2638 y Fi(\001)1263 2577 y Fg(\022)1315
2584 y Fl(k)1298 2597 y Fg(Y)1293 2687 y Fl(q)q Fm(=2)1355
2577 y Fg(\022)1391 2604 y Fj(q)h Fo(+)f(1)p 1391 2626 108
2 v 1391 2672 a Fj(q)h Fo(+)f(2)1504 2577 y Fg(\023)o(\023)1565
2589 y Fm(1)p Fl(=k)1621 2638 y Fj(:)46 2781 y Fo(Clearly)l(,)525
2876 y(lim)514 2906 y Fl(k)q Ff(!1)604 2816 y Fg(\022)656 2822
y Fl(k)639 2835 y Fg(Y)635 2925 y Fl(q)q Fm(=2)710 2843 y Fj(q)h
Fo(+)f(1)p 710 2865 V 710 2911 a Fj(q)h Fo(+)f(2)823 2816 y
Fg(\023)853 2827 y Fm(1)p Fl(=k)924 2876 y Fo(=)25 b(lim)976
2906 y Fl(k)q Ff(!1)1065 2816 y Fg(\022)1145 2843 y Fo(3)p
1101 2865 112 2 v 1101 2911 a Fj(k)13 b Fo(+)e(1)1217 2816
y Fg(\023)1248 2827 y Fm(1)p Fl(=k)1318 2876 y Fo(=)j(1)p Fj(;)46
3033 y Fo(and)j(th)o(us)27 b(lim)248 3063 y Fl(k)q Ff(!1)351
3000 y Fj(d)376 3007 y Fl(k)p 351 3022 47 2 v 361 3067 a Fj(k)417
3033 y Fo(=)473 3000 y(1)p 473 3022 25 2 v 474 3067 a Fj(e)503
3033 y Fo(,)16 b(as)g(claimed.)1068 b Fd(2)p eop
%%Page: 26 26
25 bop 46 188 a Fx(REFERENCES)1447 b Fo(26)46 342 y Fk(References)46
472 y Fo([AL86])101 b(Aho,)15 b(H.)f(V.,)h(and)h(Lee,)f(D.,)g(Storing)h(a)g
(dynamic)d(sparse)j(table,)f Fx(Pro)q(c.)h(of)g(the)f(27th)290
532 y(IEEE)h(F)o(OCS)p Fo(,)g(1986,)h(pp.)f(55{60.)46 634 y([BK88])95
b(Brassard,)17 b(G.,)f(and)h(Kannan,)g(S.,)e(The)i(generation)g(of)f(random)h
(p)q(erm)o(utations)e(on)290 694 y(the)h(\015y)l(,)f Fx(Information)h(Pro)q
(cessing)h(Letters)h Fh(28)e Fo(\(1988\))i(207{212.)46 796
y([CW79])83 b(Carter,)24 b(J.)e(L.,)i(and)g(W)l(egman,)f(M.)f(N.,)h(Univ)o
(ersal)e(classes)i(of)g(hash)h(functions,)290 856 y Fx(J.)16
b(Comput.)f(Syst.)h(Sci.)f Fh(18)h Fo(\(1979\))i(143{154.)46
958 y([DKM88])48 b(Dietzfelbinger,)23 b(M.,)h(Karlin,)h(A.,)f(Mehlhorn,)h
(K.,)g(Mey)o(er)d(auf)i(der)g(Heide,)g(F.,)290 1018 y(Rohnert,)13
b(H.,)g(and)h(T)l(arjan,)h(R.)e(E.,)g(Dynamic)f(p)q(erfect)g(hashing:)21
b(Upp)q(er)13 b(and)h(lo)o(w)o(er)290 1078 y(b)q(ounds,)25
b Fx(Pro)q(c.)e(of)h(the)e(29th)i(IEEE)f(F)o(OCS)p Fo(,)g(1988,)i(pp.)e
(524{531;)29 b(also:)35 b(T)l(ec)o(h.)290 1138 y(Rep)q(ort)16
b(No.)g(282,)h(F)l(ac)o(h)o(b)q(ereic)o(h)d(Informatik,)g(Univ)o(ersit\177)
-24 b(at)15 b(Dortm)o(und,)g(1988.)46 1240 y([DM89])86 b(Dietzfelbinger,)11
b(M.,)i(and)h(Mey)o(er)e(auf)h(der)h(Heide,)e(F.,)h(An)g(optimal)f(parallel)g
(dictio-)290 1300 y(nary)l(,)i Fx(Pro)q(c.)g(of)g(A)o(CM)f(Symp.)f(on)j(P)o
(arallel)e(Algorithms)f(and)j(Arc)o(hitectures)p Fo(,)c(1989,)290
1360 y(pp.)16 b(360{368.)46 1462 y([DM90a])62 b(Dietzfelbinger,)12
b(M.,)i(Mey)o(er)f(auf)i(der)g(Heide,)e(F.,)h(Ho)o(w)g(to)h(distribute)f(a)h
(dictionary)290 1522 y(in)h(a)g(complete)e(net)o(w)o(ork,)h
Fx(Pro)q(c.)i(of)f(the)g(22nd)i(A)o(CM)d(STOC)p Fo(,)h(1990,)i(pp.)e
(117{127.)46 1624 y([DM90b])59 b(Dietzfelbinger,)16 b(M.,)h(and)i(Mey)o(er)e
(auf)h(der)g(Heide,)f(F.,)h(A)g(new)g(univ)o(ersal)f(class)h(of)290
1684 y(hash)i(functions,)g(and)h(dynamic)d(hashing)i(in)f(real)h(time,)e
Fx(Pro)q(c.)h(of)h(17th)h(ICALP)p Fo(,)290 1744 y(Springer)16
b(LNCS)g(443,)h(1990,)h(pp.)e(6{19.)46 1846 y([FKS84])71 b(F)l(redman,)12
b(M.)h(L.,)h(Koml\023)-24 b(os,)13 b(J.,)g(and)h(Szemer)o(\023)-23
b(edi,)11 b(E.,)j(Storing)g(a)g(sparse)g(table)g(with)290 1906
y Fj(O)q Fo(\(1\))j(w)o(orst)g(case)f(access)g(time,)e Fx(J.)i(A)o(CM)k
Fh(31)p Fo(\(3\),)c(1984,)h(538{544.)46 2008 y([G81])130 b(Gonnet,)15
b(Gaston)i(H.,)d(Exp)q(ected)g(length)h(of)h(the)f(longest)g(prob)q(e)h
(sequence)e(in)h(hash)290 2068 y(co)q(de)h(searc)o(hing,)g
Fx(J.)g(A)o(CM)k Fh(28)p Fo(\(3\))d(\(1981\))g(289{304.)46
2170 y([M84])123 b(Mehlhorn,)23 b(K.,)h Fw(Data)f(Structur)n(es)h(and)g(A)o
(lgorithms)p Fo(,)g(V)l(ol.)d(1,)k(Springer)e(V)l(erlag,)290
2230 y(Berlin,)14 b(1984.)46 2332 y([MNR90])50 b(Mehlhorn,)22
b(K.,)g(N\177)-24 b(aher,)22 b(S.,)h(and)f(Rauc)o(h,)h(M.,)f(On)f(the)h
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g(problem,)f Fx(SIAM)h(J.)g(Comput.)g Fh(19)p Fo(\(5\))h(\(1990\))h(902{906.)
46 2494 y([MV84])86 b(Mehlhorn,)15 b(K.,)f(and)j(Vishkin,)d(U.,)g(Randomized)
h(and)h(deterministic)c(sim)o(ulations)290 2554 y(of)h(PRAMs)f(b)o(y)h
(parallel)f(mac)o(hines)f(with)i(restricted)f(gran)o(ularit)o(y)g(of)h
(parallel)f(mem-)290 2614 y(ory)l(,)k Fx(Acta)f(Informatica)g
Fh(21)i Fo(\(1984\))g(339{374.)46 2716 y([S89])141 b(Siegel,)24
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Fx(Pro)q(c.)g(of)g(the)g(30th)h(IEEE)f(F)o(OCS)p Fo(,)f(1989,)j(pp.)d(20{)290
2836 y(25.)46 2938 y([T83])133 b(T)l(routman,)18 b(J.)g(L.,)h
Fw(V)l(ariational)h(c)n(alculus)h(with)f(elementary)h(c)n(onvexity)p
Fo(,)f(Springer)290 2998 y(V)l(erlag,)15 b(New)h(Y)l(ork,)f(1983.)p
eop
%%Page: 27 27
26 bop 46 188 a Fx(REFERENCES)1447 b Fo(27)46 342 y([W)o(C79])84
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(20th)i(IEEE)e(F)o(OCS)s Fo(,)f(1979,)j(175{182.)46 504 y([W90])118
b(W)l(enzel,)13 b(M.,)h(Eine)g(Impleme)o(n)o(ti)o(erung)e(v)o(on)j(Dynamic)e
(P)o(erfect)h(Hashing,)h(Diplom-)290 564 y(arb)q(eit,)h(Univ)o(ersit\177)-24
b(at)14 b(des)i(Saarlandes,)h(1990.)p eop
%%Page: 28 28
27 bop 46 188 a Fx(A)48 b(APPENDIX)1432 b Fo(28)46 342 y Fk(A)83
b(App)r(endix)46 525 y Fb(A.1)69 b(Pro)r(of)24 b(of)f(Lemm)n(a)d(5.6)46
722 y Fo(W)l(e)g(sho)o(w)i(the)e(follo)o(wing:)30 b(If)20 b(the)h(functions)f
Fj(T)967 729 y Fl(k)988 722 y Fj(;)8 b(k)24 b Fi(\025)d Fo(1,)h(satisfy)f
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Fj(T)509 789 y Fl(k)530 782 y Fo(\(0\))f(=)g(0)j(for)f(all)g
Fj(k)g Fi(\025)e Fo(1,)i(and)81 936 y(\(a\))25 b Fj(T)197 943
y Fm(1)216 936 y Fo(\()p Fj(n)p Fo(\))14 b Fi(\025)g Fo(\()p
Fj(n)d Fo(+)g(1\)\()p Fj(n)g Fo(+)g(2\))p Fj(=)p Fo(2)h Fi(\000)f
Fo(1)p Fj(;)30 b Fo(for)17 b(all)f Fj(n)d Fi(\025)h Fo(1)p
Fj(;)78 1073 y Fo(\(b\))25 b Fj(T)197 1080 y Fl(k)218 1073
y Fo(\()p Fj(n)p Fo(\))14 b Fi(\025)i Fo(min)437 1025 y Fg(n)471
1073 y Fo(1)11 b(+)581 1019 y Fl(l)556 1032 y Fg(X)555 1123
y Fl(j)r Fm(=1)617 1073 y Fo(\()p Fj(j)s(a)685 1080 y Fl(j)713
1073 y Fo(+)g Fj(T)791 1080 y Fl(k)q Ff(\000)p Fm(1)857 1073
y Fo(\()p Fj(a)902 1080 y Fl(j)931 1073 y Fi(\000)g Fo(1\)\))1057
1023 y Fg(\014)1057 1048 y(\014)1057 1073 y(\014)j Fj(l)h Fi(\025)e
Fo(1)p Fj(;)8 b(a)1239 1080 y Fm(1)1259 1073 y Fj(;)g(:)g(:)g(:)f(;)h(a)1394
1080 y Fl(l)1420 1073 y Fi(2)14 b Fj(I)-10 b(N)5 b(;)1570 1019
y Fl(l)1546 1032 y Fg(X)1545 1123 y Fl(j)r Fm(=1)1606 1073
y Fj(a)1632 1080 y Fl(j)1664 1073 y Fo(=)14 b Fj(n)1753 1025
y Fg(o)1781 1073 y Fj(;)168 1172 y Fo(for)i(all)g Fj(n)e Fi(\025)g
Fo(1)p Fj(;)8 b(k)16 b Fi(\025)d Fo(2)p Fj(;)46 1326 y Fo(then)j(the)g
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(of)h(Lemma)d(5.6;)i(that)h(is,)771 1428 y Fj(T)800 1435 y
Fl(k)821 1428 y Fo(\()p Fj(n)p Fo(\))d Fi(\025)g Fj(g)978 1435
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1686 y Fo(\()p Fj(y)r Fo(\))f(=)704 1587 y Fg(8)704 1624 y(>)704
1637 y(<)704 1711 y(>)704 1724 y(:)749 1625 y Fo(0)p Fj(;)300
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b Fo(if)15 b(0)g Fj(<)e(y)j Fi(\024)d Fj(e)1336 1667 y Fl(k)1357
1685 y Fo(;)749 1745 y(\()p Fj(k)r(=e)p Fo(\))e Fi(\001)g Fj(y)923
1727 y Fm(1+1)p Fl(=k)1024 1745 y Fj(;)49 b Fo(if)15 b Fj(e)1154
1727 y Fl(k)1189 1745 y Fj(<)f(y)r(:)46 1845 y Fo(W)l(e)23
b(pro)q(ceed)g(b)o(y)g(induction)g(on)h Fj(k)r Fo(.)42 b(F)l(or)24
b Fj(k)k Fo(=)d(1,)h(it)c(is)i(easily)e(c)o(hec)o(k)o(ed)f(that)j
Fj(g)1631 1852 y Fm(1)1651 1845 y Fo(\()p Fj(n)16 b Fo(+)g(1\))26
b Fi(\024)46 1905 y Fo(\()p Fj(n)12 b Fo(+)g(1\)\()p Fj(n)g
Fo(+)g(2\))p Fj(=)p Fo(2)h Fi(\000)f Fo(1)18 b(for)g(all)f
Fj(n)f Fi(\025)g Fo(0.)26 b(Th)o(us,)17 b(let)g Fj(k)h(>)e
Fo(1,)i(and)h(assume)d(the)i(claim)d(to)j(b)q(e)g(true)46 1965
y(for)f Fj(k)d Fi(\000)d Fo(1;)18 b(that)f(is,)g Fj(T)465 1972
y Fl(k)q Ff(\000)p Fm(1)531 1965 y Fo(\()p Fj(n)p Fo(\))e Fi(\025)g
Fj(g)690 1972 y Fl(k)q Ff(\000)p Fm(1)757 1965 y Fo(\()p Fj(n)d
Fo(+)f(1\))16 b(=)f Fj(g)r Fo(\()p Fj(n)d Fo(+)g(1\),)17 b(for)g(all)g
Fj(n)e Fi(\025)g Fo(0)p Fj(:)i Fo(\(F)l(rom)f(here)h(on,)g(w)o(e)46
2025 y(will)h(write)g Fj(g)j Fo(for)e Fj(g)412 2032 y Fl(k)q
Ff(\000)p Fm(1)479 2025 y Fj(:)p Fo(\))f(F)l(or)h Fj(n)g Fo(=)f(0,)i(the)f
(claim)d(is)j(trivially)e(satis\014ed.)29 b(Let)19 b Fj(n)g
Fi(\025)f Fo(1)h(b)q(e)g(\014xed.)46 2085 y(By)d(assumption)h(\(b\))g(ab)q(o)
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Fj(a)e Fo(=)h(\()p Fj(a)384 2153 y Fm(1)404 2146 y Fj(;)8 b(:)g(:)g(:)f(;)h
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2146 y Fj(a)1262 2153 y Fl(j)1294 2146 y Fo(=)f Fj(n)i Fo(and)376
2292 y Fj(T)405 2299 y Fl(k)426 2292 y Fo(\()p Fj(n)p Fo(\))11
b Fi(\000)g Fo(1)j Fi(\025)670 2238 y Fl(l)646 2250 y Fg(X)645
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Fi(\025)1225 2238 y Fl(l)1200 2250 y Fg(X)1200 2341 y Fl(j)r
Fm(=1)1261 2292 y Fo(\()p Fj(j)s(a)1329 2299 y Fl(j)1358 2292
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y Fj(a)723 2716 y Fl(j)741 2709 y Fj(;)48 b Fo(if)16 b Fj(j)e
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Fj(;)8 b(:)g(:)g(:)g(;)g(l)q(;)697 2770 y Fo(0)p Fj(;)68 b
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y(The)k(condition)365 2835 y Fg(P)409 2849 y Fl(l)409 2881
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Fl(a)1004 3024 y Fo(\()p Fj(x)p Fo(\))c(+)h Fj(g)r Fo(\()p
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2976 y Fg(\021)1336 3024 y Fj(dx)g Fo(+)1454 2990 y Fj(n)p
1454 3012 30 2 v 1456 3058 a Fo(2)1488 3024 y Fj(:)312 b Fo(\(2\))p
eop
%%Page: 29 29
28 bop 46 188 a Fx(A)48 b(APPENDIX)1432 b Fo(29)46 342 y(Our)15
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402 y(the)g(minimiz)o(ation)e(problem)i(a)h(little)e(further:)20
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462 y(as)i Fj(f)130 469 y Fl(a)167 462 y Fo(w)o(e)f(will)f(consider)h(con)o
(tin)o(uous)g(functions.)46 623 y Fh(De\014nition)h(A.1)79
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705 y Fm(+)1453 726 y Fw(\(strictly)17 b(p)n(ositive\))h(so)168
786 y(that)267 750 y Fg(R)295 764 y Ff(1)287 799 y Fm(0)332
786 y Fj(f)5 b Fo(\()p Fj(x)p Fo(\))j Fj(dx)15 b Fo(=)e Fj(n)18
b Fw(and)g(so)f(that)22 b Fo(lim)930 793 y Fl(x)p Ff(!1)1031
786 y Fj(e)1054 768 y Fl(x)1075 786 y Fj(f)5 b Fo(\()p Fj(x)p
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b(L)n(et)17 b Fj(G)p Fo(:)8 b Fj(I)-10 b(R)361 867 y Fm(+)361
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%%Page: 30 30
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%%Page: 31 31
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1212 810 30 2 v 1214 856 a Fo(2)1258 822 y(+)g(1)p Fj(:)445
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1072 y(inequalit)o(y)e Fj(T)301 1079 y Fl(k)322 1072 y Fo(\()p
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Fj(n)d Fo(+)g(1\))17 b(in)f(Case)h(1.)46 1191 y Fx(Case)g(2:)k
Fj(n)14 b Fi(\024)g Fj(e)341 1173 y Fl(k)q Ff(\000)p Fm(1)407
1191 y Fo(.)21 b(Then)16 b Fj(A)11 b Fi(\000)g Fj(k)16 b Fi(\024)d
Fo(0,)j(and)h(w)o(e)f(get)g(from)g(\(12\))h(and)f(\(8\))h(that)285
1317 y Fj(I)t Fo(\()p Fj(f)354 1324 y Fm(0)373 1317 y Fo(\))d(=)457
1258 y Fg(Z)499 1272 y Ff(1)481 1353 y Fm(0)536 1317 y Fj(x)d
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Ff(\000)p Fm(1)958 1317 y Fi(\001)g Fo(ln\()p Fj(e)1066 1296
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1444 y Fo(W)l(e)17 b(no)o(w)h(substitute)f(the)g(v)m(alue)g
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b(with)h(\(5\))g(and)h(\(4\))g(this)f(en)o(tails)f(that)713
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Fi(\025)g Fj(n)8 b Fo(ln)g Fj(n)j Fo(+)1077 1592 y Fj(n)p 1077
1614 V 1079 1660 a Fo(2)1122 1626 y(+)g(1)p Fj(:)46 1749 y
Fo(Elemen)o(tary)h(estimates)i(sho)o(w)h(that)h(the)e(righ)o(t)h(hand)h(side)
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Fm(0)1715 2404 y Fj(f)5 b Fo(\()p Fj(\030)r Fo(\))j Fj(d\030)r
Fo(,)46 2465 y(for)15 b Fj(f)k Fi(2)14 b(D)r Fo(,)h Fj(x)e
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Fj(e)991 2567 y Ff(\000)p Fl(x)1040 2585 y Fo(,)16 b(for)g(0)f
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%%Page: 32 32
31 bop 46 188 a Fx(A)48 b(APPENDIX)1432 b Fo(32)46 342 y Fh(De\014nition)17
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695 y Fm(1)p 562 703 58 2 v 562 732 a(1)p Ff(\000)p Fl(t)636
715 y Fi(\001)c Fj(G)p Fo(\()p Fi(\000)e Fo(ln)q(\(1)j Fi(\000)g
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b Fj(;)168 897 y Fw(wher)n(e)17 b Fj(G)p Fo(\()p Fj(x;)8 b(y)r
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Fj(t)d Fi(2)f Fo([0)p Fj(;)8 b Fo(1])18 b Fw(the)g(function)i
Fj(z)c Fi(7!)745 1273 y Fl(@)766 1261 y Fe(2)p 736 1281 56
2 v 736 1310 a Fl(@)r(z)774 1300 y Fe(2)797 1292 y Fj(H)t Fo(\()p
Fj(t;)8 b(z)r Fo(\))p Fw(,)17 b Fj(z)g Fi(2)e Fj(I)-10 b(R)p
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1353 y(exc)n(epting)h(for)e Fj(t)d Fi(6)p Fo(=)f(1)18 b Fw(and)g
Fj(y)d Fo(=)f Fj(e)668 1334 y Fl(k)q Ff(\000)p Fm(1)734 1353
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b Fo(Straigh)o(tforw)o(ard)17 b(v)o(eri\014cation.)1013 b Fd(2)46
1627 y Fh(Lemm)o(a)16 b(A.5)25 b Fw(Ther)n(e)14 b(is)h(a)g(bije)n(ction)h(b)n
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1688 y Fj(')g Fi(7!)f Fj(f)179 1695 y Fl(')204 1688 y Fw(,)18
b(wher)n(e)492 1805 y Fj(')524 1812 y Fl(f)546 1805 y Fo(\()p
Fj(t)p Fo(\))c(=)667 1706 y Fg(8)667 1743 y(>)667 1756 y(<)667
1830 y(>)667 1843 y(:)725 1721 y(Z)767 1735 y Ff(\000)6 b Fm(ln)o(\(1)p
Ff(\000)p Fl(t)p Fm(\))748 1816 y(0)908 1780 y Fj(f)f Fo(\()p
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Fj(t)13 b Fo(=)h(1)8 b Fj(;)46 1946 y Fw(and)356 2006 y Fj(f)380
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437 2584 56 2 v 437 2612 a Fl(@)r(z)475 2603 y Fe(2)497 2595
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%%Page: 33 33
32 bop 46 188 a Fx(A)48 b(APPENDIX)1432 b Fo(33)46 342 y(By)21
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443 y(\014)1106 468 y(\014)1119 495 y Fl(z)q Fm(=)p Fl(')1186
484 y Fc(0)1186 512 y Fm(0)1207 495 y(\()p Fl(t)p Fm(\))1276
468 y Fo(=)26 b Fj(A)d Fo(for)h(all)g Fj(t)i Fi(2)g Fo([0)p
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571 y Fm(0)837 553 y Fo(minim)o(iz)o(es)c Fj(I)t Fo(\()p Fj(f)5
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739 y Fj(H)t Fo(\()p Fj(t;)8 b(z)r Fo(\))13 b(=)964 719 y Fl(@)p
955 727 40 2 v 955 756 a(@)r(y)999 739 y Fj(G)p Fo(\()p Fi(\000)8
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39 2 v 268 908 a(@)r(z)312 891 y Fj(H)t Fo(\()p Fj(t;)8 b(z)r
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(\(14\))g(en)o(tails)e(that)60 1150 y Fl(@)p 51 1158 40 2 v
51 1187 a(@)r(y)95 1170 y Fj(G)p Fo(\()p Fj(x;)8 b(y)r Fo(\))247
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Fm(=)p Fl(f)322 1207 y Fm(0)343 1197 y(\()p Fl(x)p Fm(\))410
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b(R)811 1149 y Fm(+)811 1182 y(0)840 1170 y Fo(,)19 b(as)g(claimed.)24
b(This)19 b(\014nishes)f(the)g(pro)q(of)i(of)e(Prop)q(osi-)46
1245 y(tion)e(A.2.)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF