path: root/other/burneye2/doc-external/hash/gupta94randomization.ps

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1621 y(Ec)n(clesiastes)k FH(\(King)17 b(James)g(V)l(ersion\))0
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b Fz(I.1.2)11 b([)p FA(Computing)h(Metho)q(dologies)p Fz(]:)17
b(Algorithms;)122 1694 y(F.1.2)c([)p FA(Computation)j(b)o(y)f(Abstract)i
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1309 y FF(deterministic)s FH(.)73 1411 y(Randomized)i(algorithms)h(en)o
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b(ev)o(en)g(if)i(a)0 1657 y(comp)q(osite)i FB(n)f FH(p)q(ossesses)i(a)e
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y(\(3\))h(if)f(there)h(is)f(an)h Fn(i)f Fo(such)h(that)1022
2389 y Fw(n)p Ft(\000)p FI(1)p 1022 2396 67 2 v 1040 2424 a(2)1058
2414 y Fv(i)1106 2406 y Fz(=)13 b Fn(m)24 b Fo(is)f(integral,)381
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2491 y Fp(\000)d Fz(1)p Fn(;)e(n)p Fz(\))k Fn(<)h(n)p Fo(,)24
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 -75 758 a
 endTexFig
0 2018 a FH(Figure)18 b(2:)25 b(P)o(ercen)o(tage)17 b(of)h(witnesses)h(to)g
(the)e(comp)q(ositeness)i(of)g FB(n)f FH(in)g(the)g(range)g(10,000)i(to)e
(12,000.)0 2090 y(The)e(p)q(oin)o(ts)i(at)e FB(y)g FH(=)e(0)i(represen)o(t)g
(prime)g(n)o(um)o(b)q(ers.)951 2828 y(23)p eop
bop 0 213 a Fp(g)73 432 y FH(In)15 b(the)h(ab)q(o)o(v)o(e)f(algorithm,)i(if)f
(either)g(condition)h(\(2\))f(or)g(\(3\))g(is)g(satis\014ed)h(then)e
FB(n)h FH(is)g(comp)q(osite.)22 b(On)0 505 y(the)14 b(other)f(hand,)i(if)f
(\(2\))g(and)h(\(3\))f(are)f(not)i(satis\014ed)g(b)o(y)e FB(x)g
FH(then)h FB(n)g FH(ma)o(y)f(or)h(ma)o(y)g(not)g(b)q(e)g(comp)q(osite)g(and)0
577 y(the)g(pro)q(cedure)h(m)o(ust)f(b)q(e)h(rep)q(eated.)20
b(If)14 b FB(r)i FH(trials)g(are)f(used,)f(the)g(probabilit)o(y)j(that)e
FB(n)g FH(is)g(comp)q(osite)g(and)0 649 y(not)j(detected)f(is)i(less)f(than)h
(1)p FB(=)p FH(2)615 631 y Fw(r)635 649 y FH(.)26 b(Therefore,)18
b(with)g(v)o(ery)f(few)h(trials,)h(one)f(can)g(either)g(pro)o(v)o(e)f(that)i
(a)0 721 y(n)o(um)o(b)q(er)14 b(is)i(comp)q(osite)g(or)g(gain)g(a)g(high)g
(degree)e(of)i(con\014dence)e(that)i(it)f(is)h(prime.)21 b(See)15
b(also)h([BBC)1859 703 y FI(+)1888 721 y FH(88])0 794 y(for)g(some)f(in)o
(triguing)j(observ)m(ations)g(ab)q(out)f(the)e(p)q(erformance)g(of)h(Rabin's)
h(primalit)o(y)f(test)g(and)g(ab)q(out)0 866 y(its)h(reliabilit)o(y)i(when)d
(used)g(to)h(generate)f(a)h(random)f(in)o(teger)h(that)f(is)h(probably)h
(prime.)73 968 y(In)d(the)f(mid-sev)o(en)o(ties,)h(another)g(probabilisti)q
(c)i(primalit)o(y)f(testing)g(algorithm)h(w)o(as)e(disco)o(v)o(ered)g(b)o(y)0
1040 y(Solo)o(v)m(a)o(y)20 b(and)h(Strassen)f([SS77].)31 b(Some)20
b(basic)g(results)h(in)f(n)o(um)o(b)q(er)f(theory)g(are)h(needed)f(to)h
(describ)q(e)0 1113 y(their)15 b(algorithm.)22 b(F)l(or)15
b(an)o(y)f(prime)h(n)o(um)o(b)q(er)e FB(n)p FH(,)i(one)g(can)f(de\014ne)g
FB(Z)1252 1094 y Ft(\003)1248 1125 y Fw(n)1286 1113 y FH(=)g
Fx(f)p FH(1)p FB(;)8 b FH(.)g(.)g(.)g FB(;)g(n)f Fx(\000)g
FH(1)p Fx(g)p FH(,)15 b(a)g(cyclic)f(group)0 1185 y(under)j(m)o
(ultiplication)i(mo)q(d)e FB(n)p FH(.)23 b(The)16 b FF(L)n(e)n(gendr)n(e)i
(Symb)n(ol)f FH(for)g(an)o(y)f(elemen)o(t)g FB(x)e Fx(2)h FB(Z)1572
1167 y Ft(\003)1568 1197 y Fw(n)1592 1185 y FH(,)h(denoted)h(b)o(y)f(\()1894
1167 y Fw(x)p 1894 1174 22 2 v 1894 1197 a(n)1917 1185 y FH(\),)0
1257 y(is)g(de\014ned)g(to)g(b)q(e)g(1)g(or)g Fx(\000)p FH(1)f(dep)q(ending)i
(on)f(whether)g(or)g(not)g FB(x)f FH(is)i(a)f(p)q(erfect)f(square)g(\(i.e.,)g
(a)h(quadratic)0 1329 y(residue)g(mo)q(dulo)i FB(n)p FH(\))e(of)g(some)g
(other)g(elemen)o(t)f(in)i FB(Z)982 1311 y Ft(\003)978 1342
y Fw(n)1002 1329 y FH(.)k(More)16 b(precisely)l(,)g(\()1391
1311 y Fw(x)p 1391 1318 V 1391 1342 a(n)1414 1329 y FH(\))e(=)g(1)i(if)h
FB(x)c Fx(\021)h FB(y)1704 1311 y FI(2)1773 1329 y FH(\(mo)q(d)j
FB(n)p FH(\))0 1401 y(for)g(some)f FB(y)f Fx(2)f FB(Z)320 1383
y Ft(\003)316 1414 y Fw(n)340 1401 y FH(,)i Fx(\000)p FH(1)g(otherwise.)73
1504 y(If)h FB(x)f FH(is)i(a)f(p)q(erfect)g(square,)f(sa)o(y)h
FB(x)e Fx(\021)g FB(y)797 1486 y FI(2)866 1504 y FH(\(mo)q(d)i
FB(n)p FH(\),)g(then)g(it)g(is)h(not)f(di\016cult)h(to)f(see)g(that)g
FB(x)1831 1471 y Fv(n)p Fl(\000)p Fq(1)p 1831 1478 59 2 v 1852
1498 a(2)1911 1504 y Fx(\021)0 1576 y FB(y)26 1558 y FI(\()p
Fw(n)p Ft(\000)p FI(1\))135 1576 y Fx(\021)d FH(1)50 b(\(mo)q(d)17
b FB(n)p FH(\).)k(This)d(leads)f(to)f(a)g(fast)h(w)o(a)o(y)e(of)i(computing)g
(the)e(Legendre)h(sym)o(b)q(ol.)22 b(One)16 b(can)0 1648 y(extend)g(these)h
(concepts)f(to)h(a)g(general)h FB(n)f FH(whic)o(h)g(ma)o(y)f(or)h(ma)o(y)g
(not)g(b)q(e)g(prime.)22 b(In)17 b(this)h(case,)e(for)h(an)o(y)0
1720 y(n)o(um)o(b)q(er)g FB(n)p FH(,)g(one)h(can)f(de\014ne)g
FB(Z)599 1702 y Ft(\003)595 1733 y Fw(n)635 1720 y FH(=)f Fx(f)p
FB(x)p Fx(j)p FB(x)f Fx(2)h(f)p FH(1)p FB(;)8 b FH(.)g(.)g(.)g
FB(;)g(n)j Fx(\000)h FH(1)p Fx(g)p FH(,)18 b(and)g(gcd\()p
FB(x;)8 b(n)p FH(\))15 b(=)h(1)p Fx(g)p FH(.)25 b(Once)17 b(again,)i
FB(Z)1930 1702 y Ft(\003)1926 1733 y Fw(n)0 1793 y FH(is)h(a)f(group)h(under)
f(m)o(ultiplication)j(mo)q(d)d FB(n)p FH(.)29 b(The)19 b(Legendre)g(sym)o(b)q
(ol)h(is)g(generalized)f(to)h(the)e FF(Jac)n(obi)0 1865 y(symb)n(ol)5
b FH(:)22 b(if)17 b FB(n)f FH(is)i(prime,)e(the)g(Jacobi)i(sym)o(b)q(ol)f
(equals)g(the)f(Legendre)h(sym)o(b)q(ol;)g(when)f FB(n)h FH(is)g(comp)q
(osite,)0 1937 y(the)d(Jacobi)i(sym)o(b)q(ol)g(is)f(de\014ned)g(to)g(b)q(e)g
(the)f(pro)q(duct)i(of)f(all)h(the)e(Legendre)h(sym)o(b)q(ols)h(corresp)q
(onding)h(to)0 2009 y(the)f(prime)g(factors)h(of)g FB(n)p FH(,)e(i.e.,)h(if)g
FB(n)e FH(=)g(\005)p FB(p)788 2016 y Fw(i)802 2009 y FH(,)i(then)g(\()962
1991 y Fw(x)p 962 1998 22 2 v 962 2022 a(n)986 2009 y FH(\))d(=)h(\005\()1126
1991 y Fw(x)p 1126 1998 V 1126 2022 a(p)1144 2027 y Fv(i)1159
2009 y FH(\).)73 2124 y(In)22 b(the)g(algorithm)i(b)o(y)e(Solo)o(v)m(a)o(y)h
(and)f(Strassen,)i(for)f FB(x)g Fx(2)h(f)p FH(1)p FB(;)8 b
FH(.)g(.)g(.)g FB(;)g(n)15 b Fx(\000)g FH(1)p Fx(g)22 b FH(to)h(b)q(e)f(a)h
(witness)g(to)0 2209 y(comp)q(ositeness)d(of)f FB(n)p FH(,)f(either)h(gcd\()p
FB(x;)8 b(n)p FH(\))17 b FB(>)h FH(1)h(or)g FB(x)980 2176 y
Fv(n)p Fl(\000)p Fq(1)p 979 2183 59 2 v 1001 2203 a(2)1095
2209 y FH(\(mo)q(d)e FB(n)p FH(\))h Fx(6)p FH(=)f(\()1364 2191
y Fw(x)p 1364 2198 22 2 v 1364 2221 a(n)1388 2209 y FH(\).)28
b(Their)19 b(algorithm)h(can)f(b)q(e)0 2293 y(stated)e(as)f(follo)o(ws.)0
2441 y Fy(PrimeT)m(est)f(\(Solovay-Str)n(assen\))23 b Fp(f)h
Fo(\(*)f(another)g(algorithm)g(for)g(primality)g(testing*\))143
2525 y(Input)g Fn(n)143 2610 y Fo(REPEAT)g Fn(r)h Fo(times)p
Fp(f)285 2695 y Fo(\(1\))g(randomly)f(pick)g(an)g Fn(x)h Fo(between)f(1)h
(and)f Fn(n)951 2828 y FH(24)p eop
bop 285 213 a Fo(\(2\))24 b(if)f Fz(gcd)q(\()p Fn(x;)8 b(n)p
Fz(\))j Fn(>)i Fz(1)p Fo(,)23 b(output)h(`composite')e(and)h(HALT)285
298 y(\(3\))h(if)f Fn(x)483 265 y Fv(n)p Fl(\000)p Fq(1)p 483
272 59 2 v 505 293 a(2)594 298 y Fz(\(mo)q(d)14 b Fn(n)p Fz(\))f
Fp(6)p Fz(=)g(\()838 274 y Fw(x)p 838 281 22 2 v 838 303 a(n)861
298 y Fz(\))p Fo(,)23 b(output)g(`composite')g(and)g(HALT)285
382 y Fp(g)143 467 y Fo(output)g(`prime')g(and)g(HALT)0 552
y Fp(g)73 766 y FH(Determining)15 b(if)g FB(x)f FH(and)g FB(n)g
FH(are)h(relativ)o(ely)f(prime)g(\(e.g.)20 b(b)o(y)14 b(Euclid's)h
(algorithm\),)h(computing)f FB(x)1885 733 y Fv(n)p Fl(\000)p
Fq(1)p 1885 740 59 2 v 1906 760 a(2)0 838 y FH(\(mo)q(d)i FB(n)p
FH(\),)j(and)h(the)e(Jacobi)i(sym)o(b)q(ol)f(\()742 820 y Fw(x)p
742 827 22 2 v 742 850 a(n)766 838 y FH(\),)g(can)g(all)h(b)q(e)f
(accomplished)h(in)f(logarithmic)i(time.)31 b(If)19 b FB(n)h
FH(is)0 910 y(prime,)13 b(then)g(it)g(follo)o(ws)i(from)d(the)h(fact)f(that)i
FB(Z)883 892 y Ft(\003)879 922 y Fw(n)915 910 y FH(is)g(cyclic,)f(that)g
FB(x)1241 875 y Fq(\()p Fv(n)p Fl(\000)p Fq(1\))p 1241 884
83 2 v 1274 905 a(2)1344 910 y Fx(\021)g FH(\()1415 892 y Fw(x)p
1415 899 22 2 v 1415 922 a(n)1439 910 y FH(\))50 b(\(mo)q(d)17
b FB(n)p FH(\).)j(Th)o(us)13 b(when)0 982 y FB(n)h FH(is)h(indeed)g(prime,)f
(no)g FB(x)g FH(will)i(qualify)f(as)g(a)g(witness.)21 b(When)14
b FB(n)h FH(is)f(comp)q(osite,)i(Solo)o(v)m(a)o(y)f(and)f(Strassen)0
1055 y(sho)o(w)o(ed)f(that)g(the)g(set)g(of)g(false)h(witnesses)g(|)e(the)h
(n)o(um)o(b)q(ers)g(in)g Fx(f)p FH(1)p FB(;)8 b FH(.)g(.)g(.)g
FB(;)g(n)t Fx(\000)t FH(1)p Fx(g)13 b FH(that)h(violate)g(conditions)0
1127 y(\(1\))h(and)f(\(2\),)h(i.e.,)e(gcd)q(\()p FB(x;)8 b(n)p
FH(\))13 b(=)h(1)h(and)f FB(x)770 1091 y Fq(\()p Fv(n)p Fl(\000)p
Fq(1\))p 770 1101 83 2 v 804 1121 a(2)873 1127 y Fx(\021)g
FH(\()945 1109 y Fw(x)p 945 1116 22 2 v 945 1139 a(n)968 1127
y FH(\))50 b(\(mo)q(d)17 b FB(n)p FH(\))e(|)f(forms)g(a)h(prop)q(er)f
(subgroup)i(of)f FB(Z)1917 1109 y Ft(\003)1913 1139 y Fw(n)1936
1127 y FH(.)0 1199 y(Hence)f(the)h(cardinalit)o(y)j(of)d(this)i(set)e(can)h
(b)q(e)f(at)h(most)g(\()p FB(n)10 b Fx(\000)f FH(1\))p FB(=)p
FH(2.)22 b(Once)15 b(again,)i(using)g(the)e(prop)q(erties)0
1271 y(of)k(quadratic)g(residues)g(mo)q(dulo)h FB(n)p FH(,)f(the)f(witnesses)
h(for)g(comp)q(ositeness)h(are)e(de\014ned)g(in)h(suc)o(h)g(a)f(w)o(a)o(y)0
1343 y(that)f(they)e(are)i(b)q(oth)g(easily)g(c)o(hec)o(k)m(able)f(and)h
(abundan)o(t.)73 1446 y(An)22 b(in)o(teresting)h(comparison)h(of)e(the)g
(Miller-Rabin)j(and)e(Solo)o(v)m(a)o(y-Strassen)h(primalit)o(y)f(testing)0
1518 y(algorithms)f(is)e(giv)o(en)g(in)g([Mon80],)g(where)f(it)h(is)h(sho)o
(wn)f(that)g(the)f(former)g(is)h(alw)o(a)o(ys)h(more)e(e\016cien)o(t)0
1590 y(than)13 b(the)e(latter.)21 b(These)12 b(t)o(w)o(o)f(algorithms)k(are)d
(of)g(the)g(Mon)o(te)f(Carlo)j(v)m(ariet)o(y)e(b)q(ecause)g(when)g
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b(algorithm)i(\015ip-\015ops)951 2828 y(25)p eop
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b(os)21 b(and)g(Sp)q(encer)f([ES74)q(],)g(again)i(illustrates)h(random)e
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y Fu(2.3)70 b(Net)n(w)n(orks)22 b(without)h(Large)h(Hierarc)n(hies)0
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1221 y(p)q(ortfolios)h(in)f(his)f(kingdom.)36 b(As)20 b(usual,)j(Nadir)e(w)o
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1294 y(organize,)17 b(rev)o(olt,)f(and)h(\014nally)h(usurp)f(the)f(throne.)21
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b(turns)h(out)g(that)h(Nadir's)f(problem)0 2539 y(falls)i(in)f(the)f
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b(particular,)i(he)d(w)o(an)o(ts)h(a)g(net)o(w)o(ork)f FB(T)1855
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b(If)19 b(a)g(large)h(hierarc)o(h)o(y)f(is)h(found,)g(another)f
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1802 y(to)i(the)f(hierarc)o(h)o(y)g(on)h(no)q(des)h Fx(f)p
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y FH(The)i(theorem)g(is)h(pro)o(v)o(ed)f(b)o(y)f(sho)o(wing)j(that)f(there)e
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(all)g(net)o(w)o(orks)951 2828 y(27)p eop
bop 638 467 4 43 v 638 491 V 638 516 V 639 540 V 639 564 V
639 589 V 639 613 V 639 638 V 639 662 V 640 687 V 640 711 V
640 715 4 4 v 641 715 a Fk(?)670 429 y(b)710 453 y(b)749 477
y(b)789 501 y(b)828 525 y(b)868 549 y(b)907 573 y(b)947 597
y(b)986 621 y(b)1026 646 y(b)1065 670 y(b)1105 694 y(b)1144
718 y(b)1184 742 y(b)1212 745 y FG(.)1215 746 y(.)-26 b(.)1218
748 y(.)1189 745 y Fk(j)p 670 383 557 4 v 1185 381 a(-)664
357 y(\010)699 340 y(\010)735 323 y(\010)770 306 y(\010)806
289 y(\010)841 272 y(\010)876 256 y(\010)905 238 y FG(.)908
236 y(.)880 254 y Fk(*)908 291 y(\024)893 317 y(\024)878 344
y(\024)863 371 y(\024)847 398 y(\024)832 425 y(\024)817 451
y(\024)801 478 y(\024)786 505 y(\024)771 532 y(\024)755 559
y(\024)740 585 y(\024)725 612 y(\024)710 639 y(\024)694 666
y(\024)679 693 y(\024)664 719 y(\024)651 722 y FG(.)649 726
y(.)g(.)647 729 y(.)646 731 y(.)658 728 y Fk(\035)979 252 y(H)1015
270 y(H)1051 289 y(H)1087 307 y(H)1123 326 y(H)1158 345 y(H)1194
363 y(H)1223 366 y FG(.)1225 367 y(.)g(.)1228 369 y(.)1199
366 y Fk(j)961 293 y(T)977 319 y(T)993 345 y(T)1010 371 y(T)1026
397 y(T)1043 424 y(T)1059 450 y(T)1075 476 y(T)1092 502 y(T)1108
528 y(T)1124 554 y(T)1141 580 y(T)1157 607 y(T)1173 633 y(T)1190
659 y(T)1206 685 y(T)1223 711 y(T)1235 714 y FG(.)1237 717
y(.)g(.)1238 719 y(.)1240 722 y(.)1225 719 y Fk(^)669 744 y(")709
720 y(")750 696 y(")791 673 y(")832 649 y(")873 625 y(")914
601 y(")954 578 y(")995 554 y(")1036 530 y(")1077 506 y(")1118
482 y(")1158 459 y(")1199 435 y(")1228 413 y FG(.)1230 412
y(.)g(.)1233 410 y(.)1204 428 y Fk(*)p 670 766 559 4 v 1187
764 a(-)1255 425 y FG(.)1255 429 y(.)1255 433 y(.)1255 438
y(.)1255 442 y(.)1255 446 y(.)1255 450 y(.)1255 454 y(.)1255
458 y(.)1255 462 y(.)1255 467 y(.)1255 471 y(.)1255 475 y(.)1255
479 y(.)1256 483 y(.)1256 487 y(.)1256 492 y(.)1256 496 y(.)1256
500 y(.)1256 504 y(.)1256 508 y(.)1256 512 y(.)1256 517 y(.)1256
521 y(.)1256 525 y(.)1256 529 y(.)1256 533 y(.)1256 537 y(.)1256
541 y(.)1256 546 y(.)1256 550 y(.)1256 554 y(.)1256 558 y(.)1256
562 y(.)1256 566 y(.)1256 571 y(.)1256 575 y(.)1256 579 y(.)1256
583 y(.)1256 587 y(.)1256 591 y(.)1256 596 y(.)1256 600 y(.)1256
604 y(.)1256 608 y(.)1256 612 y(.)1256 616 y(.)1256 621 y(.)1256
625 y(.)1256 629 y(.)1256 633 y(.)1256 637 y(.)1256 641 y(.)1256
645 y(.)1256 650 y(.)1256 654 y(.)1256 658 y(.)1257 662 y(.)1257
666 y(.)1257 670 y(.)1257 675 y(.)1257 679 y(.)1257 683 y(.)1257
687 y(.)1257 691 y(.)1257 695 y(.)1257 700 y(.)1257 704 y(.)1257
708 y(.)1257 712 y(.)1257 716 y(.)p 1268 717 4 4 v 1270 717
a Fk(?)660 810 y(Q)697 834 y(Q)733 857 y(Q)770 880 y(Q)807
903 y(Q)843 927 y(Q)880 950 y(Q)909 953 y FG(.)912 955 y(.)883
952 y Fk(j)p 949 917 4 43 v 949 893 V 948 870 V 948 846 V 948
822 V 947 799 V 947 775 V 947 751 V 947 728 V 946 704 V 946
680 V 946 657 V 946 633 V 945 609 V 945 586 V 945 562 V 945
538 V 944 515 V 944 491 V 944 467 V 943 444 V 943 420 V 943
396 V 943 373 V 942 349 V 942 325 V 942 302 V 942 259 4 4 v
944 297 a(6)1201 805 y(")1163 828 y(")1126 850 y(")1088 872
y(")1051 895 y(")1013 917 y(")976 939 y(")963 942 y FG(.)961
943 y(.)g(.)958 945 y(.)971 942 y Fk(\031)1228 458 y(\024)1214
483 y(\024)1200 508 y(\024)1186 534 y(\024)1172 559 y(\024)1158
585 y(\024)1144 610 y(\024)1130 635 y(\024)1116 661 y(\024)1102
686 y(\024)1088 712 y(\024)1074 737 y(\024)1060 762 y(\024)1046
788 y(\024)1032 813 y(\024)1018 839 y(\024)1004 864 y(\024)990
889 y(\024)976 915 y(\024)963 918 y FG(.)961 922 y(.)g(.)960
924 y(.)958 926 y(.)971 924 y Fk(\035)658 458 y(A)672 483 y(A)686
509 y(A)700 535 y(A)714 561 y(A)728 586 y(A)741 612 y(A)755
638 y(A)769 664 y(A)783 689 y(A)797 715 y(A)811 741 y(A)824
767 y(A)838 792 y(A)852 818 y(A)866 844 y(A)880 870 y(A)894
895 y(A)907 921 y(A)915 924 y FG(.)917 928 y(.)g(.)919 930
y(.)921 932 y(.)906 930 y Fk(^)949 948 y Fj(w)1255 749 y(w)1255
383 y(w)949 217 y(w)642 383 y(w)642 749 y(w)-89 b Fz(1)586
383 y(2)702 b(4)1311 749 y(5)949 1015 y(6)945 125 y(3)8 1212
y FH(Figure)16 b(3:)22 b(A)16 b(net)o(w)o(ork)f(with)i(a)g(hierarc)o(h)o(y)f
(on)h(\014v)o(e)e(Pla)o(y)o(ers)h(with)h FB(\031)f FH(:)d Fx(f)p
FH(1)p FB(;)8 b FH(2)p FB(;)g FH(3)p FB(;)g FH(4)p FB(;)g FH(5)p
Fx(g)15 b(!)f(f)p FH(2)p FB(;)8 b FH(3)p FB(;)g FH(1)p FB(;)g
FH(4)p FB(;)g FH(5)p Fx(g)p FH(.)0 1349 y(on)21 b FB(n)f FH(no)q(des)h(and)f
(let)h(\000)465 1331 y Ft(0)465 1362 y Fw(n)509 1349 y FH(b)q(e)f(the)g
(class)h(of)f(all)i(net)o(w)o(orks)e(that)g(ha)o(v)o(e)f(a)i(hierarc)o(h)o(y)
e(on)i(1)14 b(+)g Fx(b)p FH(2)8 b(log)j FB(n)p Fx(c)0 1421
y FH(no)q(des.)22 b(W)l(e)16 b(sho)o(w)h(that)g(there)e(are)i(more)e(net)o(w)
o(orks)h(in)h(\000)1086 1428 y Fw(n)1126 1421 y FH(than)g(in)g(\000)1327
1403 y Ft(0)1327 1434 y Fw(n)1351 1421 y FH(.)73 1524 y(W)l(e)g(\014rst)g
(coun)o(t)g(the)g(n)o(um)o(b)q(er)g(of)g(net)o(w)o(orks)g(in)g(\000)1007
1531 y Fw(n)1031 1524 y FH(.)24 b(Eac)o(h)17 b(net)o(w)o(ork)g(in)h(\000)1466
1531 y Fw(n)1507 1524 y FH(consists)g(of)g FB(n)f FH(v)o(ertices)0
1596 y(and)g(\()114 1578 y Fw(n)114 1608 y FI(2)137 1596 y
FH(\))g(edges,)e(eac)o(h)h(of)h(whic)o(h)f(can)h(tak)o(e)e(t)o(w)o(o)h(p)q
(ossible)j(directions.)j(Th)o(us,)850 1718 y Fx(j)p FH(\000)894
1725 y Fw(n)918 1718 y Fx(j)27 b FH(=)h(2)1049 1697 y FI(\()1063
1686 y Fv(n)1063 1708 y Fq(2)1084 1697 y FI(\))1888 1718 y
FH(\(2\))73 1870 y(Coun)o(ting)21 b(the)e(n)o(um)o(b)q(er)g(of)h(net)o(w)o
(orks)f(in)h(\000)914 1852 y Ft(0)914 1882 y Fw(n)957 1870
y FH(is)g(a)g(bit)g(more)f(in)o(v)o(olv)o(ed.)31 b(Since)20
b(eac)o(h)f(net)o(w)o(ork)f(in)0 1942 y(\000)30 1924 y Ft(0)30
1955 y Fw(n)71 1942 y FH(has)h(a)e(hierarc)o(h)o(y)g(on)h FB(')e
FH(=)g(1)c(+)g Fx(b)p FH(2)c(log)j FB(n)p Fx(c)17 b FH(no)q(des,)i(w)o(e)e
(\014rst)g(select)g(the)h FB(')f FH(no)q(des)h(and)g(assign)i(them)0
2014 y(a)g(p)q(erm)o(utation,)h(whic)o(h)e(will)j(uniquely)e(determine)f(a)g
(hierarc)o(h)o(y)g(on)h(these)f(no)q(des.)32 b(The)20 b(remaining)0
2087 y(edges)g(in)g(the)g(graph)h(consisting)h(of)e(\()p FB(n)14
b Fx(\000)f FB(')p FH(\))20 b(no)q(des)g(can)g(b)q(e)g(assigned)i
(arbitrarily)l(.)34 b(W)l(e)20 b(coun)o(t)f(the)0 2159 y(n)o(um)o(b)q(er)12
b(of)i(net)o(w)o(orks)e(for)i(all)g(the)f(\()664 2141 y Fw(n)664
2171 y(')689 2159 y FH(\))g(p)q(ossible)j(c)o(hoices)d(of)g
FB(')g FH(no)q(des)h(and)g(all)g(the)f FB(')p FH(!)g(w)o(a)o(ys)g(of)g
(assigning)0 2231 y(them)j(a)g(p)q(erm)o(utation.)22 b(F)l(ormally)l(,)803
2303 y(\000)833 2283 y Ft(0)833 2316 y Fw(n)884 2303 y FH(=)950
2262 y Fr([)960 2354 y Fw(A)1004 2262 y Fr([)1017 2349 y Fw(\031)1059
2303 y FB(T)1088 2310 y Fw(A;\031)1888 2303 y FH(\(3\))0 2426
y(where)17 b FB(A)g FH(is)h(a)g(subset)f(of)h FB(n)f FH(no)q(des)h(suc)o(h)g
(that)f Fx(j)p FB(A)p Fx(j)e FH(=)h FB(')p FH(,)h FB(\031)i
FH(is)f(a)f(p)q(erm)o(utation)i(of)e(the)g FB(')g FH(mem)o(b)q(ers)g(of)0
2499 y FB(A)p FH(,)d(and)h FB(T)187 2506 y Fw(A;\031)261 2499
y FH(is)g(the)f(set)h(of)g(net)o(w)o(orks)f(on)h FB(n)f FH(no)q(des)i
(consisten)o(t)f(with)g(the)f(hierarc)o(h)o(y)g(on)h FB(A)g
FH(determined)0 2571 y(b)o(y)j FB(\031)r FH(.)26 b(That)19
b(is,)g(eac)o(h)e(net)o(w)o(ork)h(in)g FB(T)715 2578 y Fw(A;\031)793
2571 y FH(will)i(con)o(tain)f(a)f(hierarc)o(h)o(y)g(on)h FB(A)e
FH(uniquely)i(determined)f(b)o(y)0 2643 y FB(\031)r FH(.)23
b(The)18 b(structure)f(of)g(the)g(net)o(w)o(ork)g(on)g(the)g(remaining)i
FB(n)12 b Fx(\000)f FB(')17 b FH(no)q(des,)h(ho)o(w)o(ev)o(er,)e(is)i(unsp)q
(eci\014ed.)24 b(In)951 2828 y(28)p eop
bop 0 195 a FH(particular,)17 b(the)e(direction)i(of)f(\()598
177 y Fw(n)598 207 y FI(2)621 195 y FH(\))10 b Fx(\000)f FH(\()717
171 y Fw(')717 206 y FI(2)742 195 y FH(\))16 b(edges)g(b)q(et)o(w)o(een)e
(these)h FB(n)10 b Fx(\000)g FB(')15 b FH(no)q(des)i(is)f(unsp)q(eci\014ed.)
22 b(Hence,)794 301 y Fx(j)p FB(T)837 308 y Fw(A;\031)896 301
y Fx(j)28 b FH(=)f(2)1027 280 y FI(\()1041 269 y Fv(n)1041
291 y Fq(2)1063 280 y FI(\))p Ft(\000)p FI(\()1118 264 y Fv(')1118
292 y Fq(2)1140 280 y FI(\))1888 301 y FH(\(4\))0 407 y(Therefore,)15
b(the)h(total)i(n)o(um)o(b)q(er)d(of)i(net)o(w)o(orks)f(in)h(\000)960
389 y Ft(0)960 419 y Fw(n)1000 407 y FH(is)g(b)q(ounded)g(b)o(y)360
544 y Fx(j)p FH(\000)404 523 y Ft(0)404 556 y Fw(n)428 544
y Fx(j)27 b FB(<)535 502 y Fr(X)552 594 y Fw(A)603 502 y Fr(X)622
590 y Fw(\031)671 544 y Fx(j)p FB(T)714 551 y Fw(A;\031)774
544 y Fx(j)g FH(=)881 458 y Fr(0)881 533 y(@)939 510 y FB(n)938
582 y(')990 458 y Fr(1)990 533 y(A)1035 544 y FB(')p FH(!)13
b(2)1118 523 y FI(\()1132 512 y Fv(n)1132 534 y Fq(2)1154 523
y FI(\))p Ft(\000)p FI(\()1209 507 y Fv(')1209 535 y Fq(2)1231
523 y FI(\))1274 544 y FB(<)28 b FH(2)1364 523 y FI(\()1378
512 y Fv(n)1378 534 y Fq(2)1399 523 y FI(\))1443 544 y FH(=)f
Fx(j)p FH(\000)1552 551 y Fw(n)1576 544 y Fx(j)298 b FH(\(5\))0
680 y(This)15 b(implies)g(that)g(\000)406 687 y Fw(n)436 680
y Fx(\000)6 b FH(\000)511 662 y Ft(0)511 693 y Fw(n)549 680
y FH(is)14 b(non-empt)o(y)g(and)g(there)g(exists)g FB(T)20
b Fx(2)14 b FH(\000)1310 687 y Fw(n)1341 680 y Fx(\000)6 b
FH(\000)1416 662 y Ft(0)1416 693 y Fw(n)1453 680 y FH(con)o(taining)16
b(no)f(hierarc)o(h)o(y)0 753 y(on)i FB(')d FH(=)f(1)f(+)f Fx(b)p
FH(2)d(log)j FB(n)p Fx(c)16 b FH(no)q(des.)1336 b Fi(2)73 957
y FH(The)17 b(ab)q(o)o(v)o(e)f(theorem)g(establishes)i(an)f(upp)q(er)g(b)q
(ound)h(on)f(the)f(largest)i(in)o(teger)e FB(\037)g FH(suc)o(h)h(that)g(ev)o
(ery)0 1029 y(net)o(w)o(ork)e(on)g FB(n)h FH(no)q(des)g(con)o(tains)g(a)g
(hierarc)o(h)o(y)f(on)g FB(\037)p FH(\()p FB(n)p FH(\))g(no)q(des.)22
b(It)15 b(can)g(also)i(b)q(e)e(pro)o(v)o(ed,)g(b)o(y)f(induction)0
1102 y(on)e FB(n)p FH(,)h(that)f FB(\037)p FH(\()p FB(n)p FH(\))i
Fx(\025)f FH(1)r(+)r Fx(b)p FH(log)e FB(n)p Fx(c)p FH(.)20
b(Clearly)l(,)13 b(if)g(it)f(w)o(ere)f(the)h(case)g(in)g(Nadir's)h(p)q
(olitics)h(that)e(no)g(hierarc)o(hies)0 1174 y(b)q(e)i(formed)f(on)h
FB(m)g(<)g FH(1)6 b(+)g Fx(b)p FH(log)k FB(n)p Fx(c)k FH(no)q(des,)h(then)e
(ev)o(ery)f(assignmen)o(t)j(of)f(the)f(is-a-b)q(oss-)q(of)k(relation)e(w)o
(ould)0 1246 y(violate)21 b(Nadir's)g(requiremen)o(t)e(and)i(he)f(should)h
(mak)o(e)e(arrangemen)o(ts)i(for)f(a)h(hast)o(y)f(departure.)33
b(On)0 1318 y(the)20 b(other)g(hand,)h(for)f(v)m(alues)h(of)f
FB(m)g FH(sligh)o(tly)i(greater)e(than)g(the)g(upp)q(er)g(b)q(ound)h(of)g
(Theorem)e(1,)i(the)0 1390 y(probabilit)o(y)j(that)f(a)g(randomly)h(selected)
e(graph)h(con)o(tains)g(a)g(large)h(hierarc)o(h)o(y)e(is)h(min)o(uscule.)40
b(F)l(or)0 1463 y FB(m)14 b Fx(\025)f FH(1)f(+)f(2)p Fx(b)p
FH(log)f FB(n)p Fx(c)17 b FH(this)g(probabilit)o(y)h(is)f(b)q(ounded)h(b)o(y)
715 1566 y Fx(j)p FH(\000)759 1548 y Ft(0)759 1579 y Fw(n)782
1566 y Fx(j)p 715 1588 82 2 v 715 1634 a(j)p FH(\000)759 1641
y Fw(n)782 1634 y Fx(j)829 1600 y FB(<)895 1515 y Fr(0)895
1589 y(@)958 1566 y FB(n)952 1638 y(m)1015 1515 y Fr(1)1015
1589 y(A)1060 1600 y FB(m)p FH(!)13 b(2)1154 1579 y Ft(\000)p
FI(\()1195 1568 y Fv(m)1195 1590 y Fq(2)1225 1579 y FI(\))1888
1600 y FH(\(6\))0 1735 y(Therefore,)i(if)g(Nadir)h(w)o(ere)f(to)h(construct)f
(a)h(random)g(net)o(w)o(ork)e(on)i(1000)h(no)q(des,)f(the)f(probabilit)o(y)j
(that)0 1807 y(it)f(will)i(ha)o(v)o(e)d(a)i(hierarc)o(h)o(y)e(on)h(an)o(y)g
(subset)h(of)f(25)g(no)q(des)h(is)g(less)g(than)f(0.0000000000)q(00)q(000)q
(4.)26 b(Th)o(us)18 b(a)0 1879 y(v)o(ery)c(promising)i(strategy)g(for)f
(Nadir)g(is)h(to)f(toss)h(a)f(coin)g(to)h(determine)e(the)g(direction)i(of)f
(eac)o(h)g(edge)f(in)0 1951 y(the)i(net)o(w)o(ork;)f(the)h(o)q(dds)i(are)e
(less)h(than)g(4)f(in)h(10)913 1933 y FI(16)968 1951 y FH(that)f(he)g(will)i
(construct)f(a)f(bad)h(net)o(w)o(ork.)73 2054 y(The)f(preceding)h
(discussion,)h(unfortunately)l(,)f(lea)o(v)o(es)e(a)i(\\gra)o(y)g(area")g(in)
g(the)f(solution)i(space:)j(it)c(is)0 2126 y(not)j(clear)g(ho)o(w)f(to)h
(solv)o(e)g(the)f FF(NetHier)n(ar)n(chy)g FH(problem)h(for)g(v)m(alues)g(of)g
FB(m)f FH(b)q(et)o(w)o(een)f(1)c(+)f Fx(b)p FH(log)e FB(n)p
Fx(c)19 b FH(and)0 2198 y(1)14 b(+)f(2)p Fx(b)p FH(log)e FB(n)p
Fx(c)p FH(.)33 b(F)l(or)20 b(v)m(alues)g(of)h FB(m)e FH(less)i(than)f(the)g
(lo)o(w)o(er)g(b)q(ound)h(on)f FB(\037)p FH(,)g(the)f(solution)k(is)d
(immediate;)0 2270 y(for)h(v)m(alues)h(sligh)o(tly)h(greater)e(than)g(the)g
(upp)q(er)g(b)q(ound,)i(Theorem)d(1)i(immediately)f(yields)h(a)f(trivial)0
2342 y(probabilistic)j(algorithm)e(as)f(basic)g(coun)o(ting)h(pro)q(cedures)f
(rev)o(eal)f(that)h(there)e(is)j(an)f(abundance)g(of)0 2415
y(solutions)e(in)e(this)g(region.)23 b(Ho)o(w)o(ev)o(er,)14
b(for)j(the)f(gra)o(y)g(area)h(in)g(b)q(et)o(w)o(een)e(the)i(upp)q(er)f(and)h
(lo)o(w)o(er)g(b)q(ounds)0 2487 y(on)i FB(\037)g FH(|)f(whic)o(h)h(can)g(p)q
(ossibly)i(b)q(e)e(shrunk)g(b)o(y)f(making)i(the)e(b)q(ounds)i(tigh)o(ter)f
(|)g(exhaustiv)o(e)f(searc)o(h)0 2559 y(seems)c(to)g(b)q(e)h(the)e(only)i(w)o
(a)o(y)f(for)h(solving)h(this)f(problem.)21 b(The)14 b(latter)h(is)g
(prohibitiv)o(ely)g(exp)q(ensiv)o(e)f(ev)o(en)0 2631 y(for)i(mo)q(derate)f(v)
m(alues)h(of)g FB(n)f FH(and)h FB(m)p FH(.)21 b(F)l(or)15 b(example,)g(if)h
(Nadir)g(required)f(that)h(there)e(b)q(e)i(no)f(hierarc)o(hies)0
2704 y(on)i(18)g(no)q(des,)f(\()302 2686 y FI(1000)302 2716
y(18)375 2704 y FH(\))g(subnet)o(w)o(orks)h(m)o(ust)e(b)q(e)i(tested.)951
2828 y(29)p eop
bop 0 195 a Fu(2.4)70 b(Probabilistic)24 b(Hashing)0 330 y
FH(Man)o(y)14 b(problems)h(require)f(main)o(taining)i(a)f(table)g(of)f(v)m
(alues,)h(or)g(k)o(eys,)e(and)i(p)q(erforming)g(insert,)g(searc)o(h,)0
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b Fn(S)1695 1735 y FI(\()p Fw(i)p FI(\))1692 1772 y Fw(j)95
1844 y Fp(g)95 1928 y Fz(\()p Fn(x;)8 b(y)r Fz(\))22 b Fo(:=)i(nearest)f
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FI(\))744 1941 y Fw(j)785 1928 y Fn(;)8 b(y)830 1904 y FI(\()p
Fw(i)p FI(\))828 1941 y Fw(j)871 1928 y Fz(\))p Fp(j)p Fz(1)k
Fp(\024)h Fn(i)f Fp(\024)h Fz(4)p Fn(;)8 b Fz(1)j Fp(\024)i
Fn(j)i Fp(\024)e Fn(k)1293 1935 y Fw(i)1307 1928 y Fp(g)0 2013
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(satisfy)i FF(pr)n(ob)n(abilistic)e FH(notions)j(of)e(soundness)h(and)f
(complete-)0 731 y(ness:)0 874 y FF(Completeness.)26 b FH(if)16
b FB(w)f Fx(2)f FB(L)i FH(then,)g(with)g(v)o(ery)f(high)i(probabilit)o(y)h
(the)e(in)o(teraction)g(b)q(et)o(w)o(een)f(the)h(pro)o(v)o(er)122
947 y(and)h(the)f(v)o(eri\014er)f(m)o(ust)h(result)h(in)g(the)f(v)o
(eri\014er)f(concluding)j(that)f FB(w)g FH(is)h(indeed)e(in)h
FB(L)p FH(;)0 1060 y FF(Soundness.)26 b FH(if)16 b FB(w)f Fx(62)f
FB(L)i FH(then,)f(with)i(v)o(ery)d(high)j(probabilit)o(y)l(,)g(at)f(the)g
(end)f(of)h(the)g(proto)q(col)h(the)f(v)o(eri\014er)122 1133
y(m)o(ust)g(conclude)g(that)h FB(w)g FH(is)g(not)g(in)g FB(L)p
FH(.)0 1276 y(The)i(pro)q(of)h(m)o(ust)f(b)q(e)g(sound)h(ev)o(en)e(if)i(the)e
(pro)o(v)o(er)h(acts)g(maliciously)i(and)f(delib)q(erately)g(tries)g(to)f(fo)
q(ol)0 1349 y(the)j(v)o(eri\014er.)37 b(Sev)o(eral)22 b(prop)q(erties)h(of)f
(in)o(teractiv)o(e)f(pro)q(of)i(systems)f(concerning)h(completeness)e(and)0
1421 y(soundness,)15 b(and)f(metho)q(ds)g(for)f(constructing)i(them)e(are)g
(in)o(v)o(estigated)i(in)f([F)o(GM)1524 1403 y FI(+)1552 1421
y FH(89)q(].)20 b(Clearly)l(,)14 b(ruling)0 1493 y(b)o(y)e(probabilisti)q(c)j
(evidence)e(means)g(relaxing)h(the)f(completeness)f(and)i(correctness)e
(criteria.)21 b(Ho)o(w)o(ev)o(er,)0 1565 y(it)c(do)q(es)g(lead)g(to)f(in)o
(teresting)h(applications)j(suc)o(h)c(as)h(program)g(testing)g([BR88,)e(BK89)
q(,)g(BLR90)q(].)73 1668 y(F)l(or)f(an)g(example)f(of)h(ho)o(w)f(of)h(an)g
(in)o(teractiv)o(e)f(pro)q(of)i(system)e(|)g(in)h(particular,)h(the)e(v)o
(eri\014er)g(comp)q(o-)0 1740 y(nen)o(t)j(of)g(the)g(pro)q(of)i(|)e(can)g(b)q
(e)h(used)f(to)h(test)f(the)g(correctness)g(of)g(a)h(program,)g(consider)g
(the)f(problem)0 1812 y(of)22 b FF(gr)n(aph)g(isomorphism)t
FH(.)37 b(The)22 b(reader)g(should)h(recall)g(that)f(the)g(exact)g(complexit)
o(y)f(of)i(graph)g(iso-)0 1884 y(morphism)17 b(is)h(not)f(kno)o(wn:)23
b(while,)17 b(to)g(date,)g(no)g(p)q(olynomial-ti)q(me)i(algorithm)f(for)f
(this)h(problem)f(has)0 1956 y(b)q(een)22 b(disco)o(v)o(ered,)g(a)g(pro)q(of)
i(that)e(it)g(is)h(NP-complete)f(has)g(b)q(een)g(equally)h(elusiv)o(e.)38
b(The)22 b(follo)o(wing)0 2029 y(e\016cien)o(t)13 b(pro)q(cedure)h(for)g(c)o
(hec)o(king)g(the)f(v)m(alidit)o(y)j(of)e(a)h(graph)g(isomorphism)h(program)e
(is)h(due)f(to)h(Blum,)0 2101 y(Ragha)o(v)m(an,)i(and)g(Kannan)g([BR88,)f
(BK89].)21 b(It)16 b(is)h(based)g(on)g(an)f(in)o(teractiv)o(e)g(pro)q(of)i
(system)e(for)g(graph)0 2173 y(non-isomorphism)j(b)o(y)d(Goldreic)o(h,)g
(Micali)i(and)f(Wigderson)h([GMW91].)73 2275 y(Giv)o(en)23
b(a)g(program)h FG(P)e FH(that)h(purp)q(ortedly)h(solv)o(es)f(the)f(graph)i
(isomorphism)h(problem)e(and)g(t)o(w)o(o)0 2348 y(graphs)f
FG(G)f FH(and)g FG(H)p FH(,)f(the)h(v)o(eri\014er)f(wishes)i(to)f(determine)g
(whether)f FG(P)h FH(in)o(v)o(ok)o(ed)f(on)h FG(G)g FH(and)g
FG(H)g FH(\(denoted)0 2420 y FG(P\(G,H\))p FH(\))d(giv)o(es)j(the)f(correct)g
(result.)35 b(The)20 b(v)o(eri\014er)g FF(GI-V)l(erify)t FH(,)h(whose)h
(pseudo)q(co)q(de)f(is)g(no)o(w)g(giv)o(en,)0 2492 y(op)q(erates)c(in)g(a)g
(randomized)f(and)h(in)o(teractiv)o(e)f(manner.)0 2661 y Fy(GI-V)m(erify)27
b Fp(f)p Fo(\(*)c(Inputs:)47 b(a)24 b(program)f(P)h(and)f(graphs)g(G)h(and)f
(H)h(*\))951 2828 y FH(43)p eop
bop 143 213 a Fo(IF)23 b(P\(G,H\))g(=)h(true)f(THEN)p Fp(f)285
298 y Fo(attempt)g(to)h(establish)e(the)i(isomorphism)285 382
y(IF)g(successful)e(THEN)i(RETURN)f("P)g(is)h(correct")285
467 y(ELSE)g(RETURN)f("P)g(is)h(buggy")p Fp(g)143 552 y Fo(ELSE)p
Fp(f)285 636 y Fo(REPEAT)f Fn(k)i Fo(times)p Fp(f)428 721 y
Fo(toss)e(a)h(fair)f(coin)428 806 y(IF)h(coin)f(=)h(head)f(THEN)p
Fp(f)571 890 y Fo(generate)f(a)i(random)f(permutation)g(G)1311
874 y Ft(0)1346 890 y Fo(of)h(G)571 975 y(IF)f(P\(G,)h(G)786
959 y Ft(0)797 975 y Fo(\))g(=)g(false)f(THEN)g(RETURN)g("P)h(is)f(buggy")p
Fp(g)428 1060 y Fo(ELSE)p Fp(f)571 1144 y Fo(generate)f(a)i(random)f
(permutation)g(H)1311 1128 y Ft(0)1346 1144 y Fo(of)h(H)571
1229 y(IF)f(P\(G,)h(H)786 1213 y Ft(0)797 1229 y Fo(\))g(=)g(true)f(THEN)g
(RETURN)g("P)h(is)f(buggy")p Fp(g)285 1314 y(g)h Fo(\(*)g(end)f(REPEAT)g(*\))
285 1398 y(RETURN)g("P)h(is)g(correct")p Fp(g)143 1483 y(g)73
1724 y FF(GI-V)l(erify)e FH(starts)h(b)o(y)f(in)o(v)o(oking)h
FG(P\(G,H\))p FH(.)d(If)h FG(P)h FH(pronounces)h FG(G)f FH(and)h
FG(H)e FH(to)i(b)q(e)f(isomorphic)i(\(i.e.,)0 1796 y FG(P\(G,H\))f(=)j(true)p
FH(\),)15 b(the)i(v)o(eri\014er's)f(task)h(is)h(simple.)24
b(It)17 b(attempts)g(to)g(determine)f(the)h(corresp)q(ondence)0
1868 y(b)q(et)o(w)o(een)j(the)g(v)o(ertices)f(of)i FG(G)f FH(and)h
FG(H)f FH(\(ho)o(w)h(this)g(is)h(done)e(will)j(b)q(e)d(describ)q(ed)h
(shortly\),)h(and)f(returns)0 1941 y FG(correct)e FH(or)j FG(buggy)e
FH(accordingly)l(.)40 b(If,)22 b(on)h(the)e(other)h(hand,)i
FG(P)d FH(pronounces)i FG(G)e FH(and)i FG(H)e FH(to)h(b)q(e)g(non-)0
2013 y(isomorphic)d(\(i.e.,)c FG(P\(G,H\))24 b(=)h(false)p
FH(\),)15 b FG(V)h FH(will)j(put)e FG(P)g FH(through)h(a)f(series)g(of)h
(tests.)23 b(Should)18 b FG(P)f FH(fail)h(an)o(y)0 2085 y(one)g(of)g(these)f
(tests,)h FG(V)f FH(can)h(conclude)g(that)g FG(P)f FH(is)h
FG(buggy)p FH(.)24 b(Otherwise,)18 b FG(V)f FH(can)h(conclude,)f(with)i(a)f
(high)0 2157 y(degree)e(of)g(con\014dence,)g(that)g FG(P)g
FH(is)h FG(correct)p FH(.)73 2260 y(Consider)d(the)e(case)g
FG(P\(G,H\))23 b(=)i(true)11 b FH(\014rst.)20 b(The)13 b(v)o(eri\014er)e(can)
h(establish)i(a)f(1-to-1)h(corresp)q(ondence)0 2332 y(b)q(et)o(w)o(een)19
b(the)g(v)o(ertices)f(of)i FG(G)f FH(and)h FG(H)p FH(,)f(assuming)i(that)f
FG(P)f FH(is)h(correct)f(in)h(pronouncing)i FG(G)d FH(and)h
FG(H)f FH(to)h(b)q(e)0 2404 y(isomorphic,)k(as)f(follo)o(ws.)39
b(Starting)23 b(with)g FG(G)p FH(,)e(arbitrarily)j(n)o(um)o(b)q(er)d(the)g(v)
o(ertices)g(of)h FG(G)f FH(and)i FG(H)e FH(from)0 2476 y(1)g(to)f
FB(n)p FH(.)34 b(A)o(ttac)o(h)19 b(a)i(clique)f(of)h FB(n)14
b FH(+)g(1)21 b(v)o(ertices)e(to)i(no)q(de)g(n)o(um)o(b)q(er)e(1)i(of)f
FG(G)g FH(to)h(obtain)h(the)e(graph)h FG(G1)p FH(.)0 2549 y(Successiv)o(ely)l
(,)15 b(attac)o(h)h(a)g(similar)i(clique)e(to)g(eac)o(h)f(no)q(de)i
FB(i)e FH(in)i FG(H)e FH(to)h(obtain)i FG(H)p FB(i)p FH(,)c(and)j(test)e(if)i
FG(P\(G1,H)p FB(i)p FG(\))22 b(=)0 2621 y(true)p FH(.)h(Clearly)l(,)18
b(if)g FG(G)e FH(and)i FG(H)f FH(are)h(isomorphic,)g(and)g(if)g(no)q(de)g(1)f
(in)h FG(G)f FH(can)g(b)q(e)h(mapp)q(ed)g(to)f(no)q(de)h FB(i)f
FH(in)h FG(H)p FH(,)0 2693 y(then)e FG(P\(G1,H)p FB(i)p FG(\))d
FH(will)18 b(return)d FG(true)p FH(.)20 b(Th)o(us,)c(if)g FG(P)g
FH(returns)g FG(false)e FH(for)i(all)i FB(i)p FH(,)d FG(P)h
FH(is)g(buggy)l(.)22 b(On)16 b(the)g(other)951 2828 y(44)p
eop
bop 0 195 a FH(hand,)15 b(if)h FG(P\(G1,H)p FB(i)p FG(\))23
b(=)i(true)13 b FH(for)i(some)g FB(i)p FH(,)g(map)g(no)q(de)g(1)h(of)f
FG(G)f FH(to)i(no)q(de)f FB(i)g FH(of)g FG(H)p FH(.)f(Rep)q(eat)h(this)h(pro)
q(cedure)0 267 y(for)g(eac)o(h)f(no)q(de)i FB(j)f Fx(2)e FH([1)8
b(.)g(.)g(.)g FB(n)p FH(])16 b(of)g FG(G)p FH(.)e(A)o(t)h(an)o(y)h(p)q(oin)o
(t,)g(the)g(inabilit)o(y)i(to)e(\014nd)g(a)g(corresp)q(onding)i(no)q(de)e(in)
g FG(H)0 340 y FH(re\015ects)g(an)h(error)f(in)i(program)f
FG(P)p FH(.)f(On)g(the)h(other)f(hand,)h(if)g(all)h(the)e(v)o(ertices)g(in)h
FG(G)f FH(can)h(b)q(e)g(mapp)q(ed)g(to)0 412 y(those)g(in)h
FG(H)e FH(then)h(the)g(v)o(eri\014er)f(can)h(easily)h(test)e(if)i(the)e
(mapping)j(is)e(an)h(isomorphism)g(and)g(determine)0 484 y(if)f(the)f
(original)j(answ)o(er)d FG(P\(G,)25 b(H\))f(=)i(true)14 b FH(w)o(as)j
(correct.)73 586 y(Consider)f(the)e(case)g FG(P\(G,H\))23 b(=)j(false)12
b FH(next,)i(i.e.,)f FG(P)h FH(declares)h(that)f FG(G)g FH(and)h
FG(H)f FH(are)g(not)h(isomorphic.)0 658 y(In)h(this)h(case,)e(the)h(v)o
(eri\014er)g(relies)g(on)h(simple)g(random)f(c)o(hoice)g(and)h(input)g
(randomization)h(as)f(follo)o(ws.)0 731 y(It)h(puts)h FG(P)f
FH(through)h(a)g(series)f(of)h(tests)g(or)f FF(r)n(ounds)t
FH(.)27 b(In)18 b(eac)o(h)g(round,)h FG(V)f FH(tosses)h(a)g(fair)g(t)o(w)o
(o-sided)g(coin)0 803 y(to)g(randomly)g(c)o(ho)q(ose)g(b)q(et)o(w)o(een)f
FG(G)g FH(and)h FG(H)p FH(;)f(randomly)h(p)q(erm)o(utes)f(the)g(names)h(of)f
(the)h(v)o(ertices)e(in)i(the)0 875 y(selected)13 b(graph)i(to)f(obtain)h(a)f
(graph)h FG(K)e FH(that)h(is)h(isomorphic)g(to)f(the)g(selected)f(graph;)i
(and)f(then)g(in)o(v)o(ok)o(es)0 947 y FG(P\(G,K\))p FH(.)g(W)l(e)i(will)i
(refer)e(to)g FG(K)g FH(as)h FG(G)660 929 y Ft(0)672 947 y
FH(,)e(if)i(the)f(selected)g(graph)i(is)f FG(G)p FH(,)e(and)i(as)g
FG(H)1436 929 y Ft(0)1464 947 y FH(is)g(the)f(selected)g(graph)h(is)0
1020 y FG(H)p FH(.)73 1122 y(There)h(are)g(t)o(w)o(o)g(cases)g(dep)q(ending)h
(on)f(whether)g(or)g(not)h FG(P)e FH(is)i(correct.)25 b(If)18
b(it)g(is,)h(i.e.,)e FG(G)h FH(and)g FG(H)g FH(are)0 1194 y(actually)23
b(non-isomorphic,)h(then)e(in)g(eac)o(h)e(round)j(w)o(e)d(should)j(ha)o(v)o
(e)e FG(P\(G,)j(G)1500 1176 y Ft(0)1511 1194 y FG(\))i(=)f(true)20
b FH(when)h FG(G)g FH(is)0 1266 y(selected,)e(and)g FG(P\(G,)25
b(H)451 1248 y Ft(0)462 1266 y FG(\))g(=)g(false)18 b FH(when)h
FG(H)f FH(is)i(selected.)29 b(Th)o(us,)20 b(in)g(just)f(a)g(v)o(ery)f(small)i
(n)o(um)o(b)q(er)f(of)0 1339 y(rounds,)j(the)e(v)o(eri\014er)g(can)g(gain)i
(a)f(high)g(degree)f(of)h(con\014dence)f(in)h(the)f(correctness)g(of)h
FG(P)f FH(should)h(it)0 1411 y(resp)q(ond)c(correctly)f(in)h(eac)o(h)f
(round.)73 1513 y(On)j(the)g(other)g(hand,)g(if)h FG(P)e FH(is)i(buggy)l(,)g
(i.e.,)e FG(G)h FH(and)g FG(H)g FH(are)f(isomorphic,)j(it)f(has)f(no)h(w)o(a)
o(y)e(of)h(distin-)0 1585 y(guishing)j(b)q(et)o(w)o(een)c FG(G)413
1567 y Ft(0)444 1585 y FH(and)h FG(H)567 1567 y Ft(0)579 1585
y FH(.)30 b(This)20 b(is)h(b)q(ecause)e FG(G)999 1567 y Ft(0)1030
1585 y FH(and)g FG(H)1153 1567 y Ft(0)1184 1585 y FH(are)g(isomorphic)i(and)f
(are)f(b)q(oth)i(dra)o(wn)0 1657 y(from)d(the)g(same)h(distribution)i
(\(essen)o(tially)f(they)d(are)i(random)g(p)q(erm)o(utations)g(of)g(the)f
(same)h(graph\).)0 1730 y(Since)14 b FG(P)g FH(do)q(es)h(not)f(kno)o(w)g
(whether)g FG(G)693 1712 y Ft(0)718 1730 y FH(or)h FG(H)802
1712 y Ft(0)827 1730 y FH(is)g(b)q(eing)g(passed)g(as)g(the)f(second)g
(argumen)o(t,)g(the)g(only)g(w)o(a)o(y)0 1802 y(it)21 b(can)f(distinguish)j
(them)d(is)h(b)o(y)f(c)o(hance.)32 b(The)21 b(probabilit)o(y)h(therefore)e
(of)g FG(P)g FH(resp)q(onding)i(correctly)0 1874 y(\(i.e.,)17
b(\\y)o(es")g(to)h FG(P\(G,)25 b(G)460 1856 y Ft(0)471 1874
y FG(\))17 b FH(and)h(\\no")h(to)f FG(P\(G,)24 b(H)943 1856
y Ft(0)955 1874 y FG(\))p FH(\))17 b FB(k)i FH(straigh)o(t)g(times)f(is)g
(only)h(2)1558 1856 y Ft(\000)p Fw(k)1607 1874 y FH(.)25 b(Therefore,)17
b(the)0 1946 y(v)o(eri\014er)f(should)h(only)g(need)f(a)h(few)f(rounds)h(to)g
(determine)e(that)i FG(P)f FH(is)h(buggy)l(.)73 2049 y(The)f(v)o(eri\014er)g
(mak)o(es)g(use)g(of)h(randomization)h(to)f(its)g(adv)m(an)o(tage)g(at)g(t)o
(w)o(o)f(crucial)h(junctures)f(in)h(the)0 2121 y(ab)q(o)o(v)o(e)12
b(algorithm.)21 b(First,)13 b(it)f(generates)g(random)g(p)q(erm)o(utations)h
FG(G)1250 2103 y Ft(0)1273 2121 y FH(and)f FG(H)1389 2103 y
Ft(0)1400 2121 y FH(.)20 b(If)11 b FG(G)g FH(and)h FG(H)g FH(are)f
(isomorphic)0 2193 y(there)19 b(is)i(no)g(w)o(a)o(y)e(of)h(telling)i
FG(G)593 2175 y Ft(0)624 2193 y FH(and)f FG(H)749 2175 y Ft(0)780
2193 y FH(apart.)33 b(In)20 b(addition,)j(it)d(randomly)h(passes)g
FG(G)1667 2175 y Ft(0)1698 2193 y FH(or)f FG(H)1787 2175 y
Ft(0)1819 2193 y FH(as)g(the)0 2265 y(second)g(argumen)o(t)f(in)h(eac)o(h)f
(iteration)i(thereb)o(y)e(taxing)h(the)g(claimed)g(abilit)o(y)h(of)e
FG(P)h FH(that)f(it)h(can)g(test)0 2338 y(for)d(graph)h(isomorphism.)25
b(The)17 b(tric)o(k)g(is)h(so)f(e\013ectiv)o(e)f(that)i(it)f(will)i(catc)o(h)
d FG(P)h FH(ev)o(en)f(if)h(it)h(is)g(maliciously)0 2410 y(co)q(ded)f(and)f
(is)h(designed)h(sp)q(eci\014cally)f(to)g(fo)q(ol)h(the)e(v)o(eri\014er.)73
2512 y(The)g(ab)q(o)o(v)o(e)g(example)g(illustrates)i(the)e(p)q(o)o(w)o(er)g
(of)g(input)h(randomization)h(in)f(program)g(testing)f(and)0
2584 y(in)o(teractiv)o(e)21 b(pro)q(of)h(systems.)37 b(The)21
b(reader)g(is)h(referred)f(to)g([BR88,)g(BK89])g(for)h(more)f(probabilistic)0
2656 y(c)o(hec)o(k)o(ers)15 b(for)j(problems)g(suc)o(h)g(as)g(matrix)g(m)o
(ultiplication,)i(sorting)f(and)f(sev)o(eral)g(problems)g(in)g(group)951
2828 y(45)p eop
bop 0 195 a FH(theory)l(.)73 297 y(It)21 b(is)g(in)o(teresting)h(to)f(note)g
(that)g(in)g(the)g(ab)q(o)o(v)o(e)g(example,)g FF(GI-V)l(erify)k
FH(w)o(as)c(able)h(to)f(do)g(its)h(task)0 370 y(without)f(ha)o(ving)f(to)f
(solv)o(e)h(the)f(graph)h(isomorphism)h(problem)f(in)g(an)o(y)f(sense.)30
b(Also,)20 b(if)g(the)f(graphs)0 442 y(are)e(isomorphic,)h(then)f(the)f(v)o
(eri\014er)g(can)h(construct)g(the)g(1-1)h(map)f(b)q(et)o(w)o(een)f(the)g(v)o
(ertices)g(of)i(the)e(t)o(w)o(o)0 514 y(graphs)i(\(i.e.,)d(it)i(gains)h(more)
f(information)h(than)f(a)g(simple)h(y)o(es/no)e(answ)o(er)h(ab)q(out)h(the)e
(isomorphism)0 586 y(question\).)25 b(Ho)o(w)o(ev)o(er,)15
b(if)j(they)f(are)g(non-isomorphic,)j(the)d(v)o(eri\014er)g(gains)i(no)f
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2828 y(49)p eop
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v 476 491 a FB(P)507 498 y FI(2)p 591 537 V 412 539 181 2 v
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300 V 884 302 181 2 v 1357 419 V 1357 537 2 119 v 1421 491
a FB(P)1452 498 y FI(4)p 1535 537 V 1357 539 181 2 v 1179 891
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y(B)594 767 y(B)580 726 y(B)567 684 y(B)553 643 y(B)539 601
y(B)531 579 y(B)620 727 y FH(fork[1])531 419 y Fg(\010)573
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263 y(H)1105 373 y FH(fork[3])1299 892 y Fg(\002)1313 850 y(\002)1327
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1547 y(trying)g(:=)h(FALSE)f(\(*)p 999 1522 V 24 w Fn(s)h Fo(=)g(complement)e
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Fn(i)8 b Fp(\000)j Fn(s)p Fo(])24 b(:=)f(TRUE)273 1716 y Fp(g)273
1801 y Fo(\(*)g(eating)g(section)g(*\))273 1886 y(fork-available[)p
Fn(i)7 b Fp(\000)k Fz(1)p Fo(],)23 b(fork-available[i])f(:=)h(TRUE)136
1970 y Fp(g)0 2055 y(g)73 2296 y FH(Algorithm)e FF(DinPhil)f
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2368 y(neigh)o(b)q(or)j FB(P)236 2375 y Fw(i)p FI(+1)317 2368
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2828 y(51)p eop
bop 73 195 a FH(Lehmann)15 b(and)h(Rabin's)f(randomized)h(algorithm)g(w)o(as)
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y Fu(3.2)70 b(Comm)n(unication)24 b(Guard)g(Sc)n(heduling)0
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b FH(,)0 1871 y(ha)o(ving)17 b(the)f(follo)o(wing)j(syn)o(tax:)73
1974 y([)p FB(G)125 1981 y FI(1)159 1974 y FH(=)-8 b Fx(\))13
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2546 y Fw(i)1335 2539 y FH(talks)i(to)f FB(P)1535 2546 y Fw(i)p
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2684 y(the)12 b(end)h(of)g(the)f(election,)i FF(al)r(l)g FH(pro)q(cesses)f(m)
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b(Additionally)l(,)951 2828 y(55)p eop
bop 0 195 a FH(an)17 b(election)g(algorithm)h(m)o(ust)e(guaran)o(tee)g
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370 y(the)i(exten)o(t)f(that)i(individual)i(pro)q(cesses)e(ha)o(v)o(e)f
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442 y(ordered)g(set)f(of)h(names.)37 b(The)22 b(problem)g(of)g(leader)g
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514 y(pic)o(king)17 b(the)f(pro)q(cess)h(with)g(the)f(smallest,)h(or)g
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b(F)l(or)0 977 y(example,)14 b(the)h(adv)o(ersary)g(sc)o(heduler)f(can)h
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(their)i(next)g(computation)h(step.)38 b(Eac)o(h)22 b(pro)q(cess)g
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Fw(i)1022 2681 y Fo(in)h(a)g(ring)f(*\))951 2828 y FH(56)p
eop
bop 143 213 a Fo(\(*)23 b(s)h(:)48 b(a)23 b(list)h(of)f(names)g(*\))143
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2 206 v 983 558 258 2 v 854 659 a Fg(#)883 636 y(#)912 613
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FB(w)q FH(,)0 1371 y(the)c(algorithm)j Fx(A)8 b FH(\()p FB(I)t(;)g(w)q
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b(A)12 b(string)0 1762 y(of)21 b(bits)g(is)g(said)h(to)f(b)q(e)g
FB(k)r FH(-)p FF(wise)h(indep)n(endent)g FH(if)f(an)o(y)f FB(k)j
FH(of)e(the)f(bits)h(in)g(the)f(sequence)g(are)g(m)o(utually)0
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(then)0 1979 y(a)i(sample)g(space)f(of)h(size)f FB(O)q FH(\()p
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2586 y(clude)c(problems)h(in)g(com)o(binatorial)i(optimization)f([Rag88)q(])e
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eop
bop 0 195 a Fu(On)23 b(the)f(F)-6 b(uture)23 b(of)h(Randomized)f(Algorithms)0
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1734 y(the)d(pap)q(er,)i(including)h(Olivier)e(Devillers,)i(Martin)e
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1879 y(and)g(Mark)f(W)l(eiss.)0 2121 y FK(References)0 2263
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2335 y(ized)23 b(parallel)i(comparison-sorting)h(algorithms.)42
b FF(SIAM)24 b(Journal)f(on)h(Computing)p FH(,)301 2408 y(17:1178{119)q(2,)h
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2552 y(b)q(eha)o(vior)f(of)e(their)h(deterministic)g(coun)o(terparts.)951
2828 y(84)p eop
bop 0 195 a FH([AA)o(G)125 177 y FI(+)153 195 y FH(89])86 b(K.)19
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(Kirkpatric)o(k.)29 b(The)301 267 y(bit)24 b(complexit)o(y)e(of)h(randomized)
g(leader)g(election)g(on)g(a)g(ring.)41 b FF(SIAM)23 b(Journal)h(on)301
340 y(Computing)p FH(,)c(18\(1\):12{29,)i(F)l(eb)c(1989.)30
b(Under)18 b(v)m(arious)j(assumptions)g(ab)q(out)f(global)301
412 y(kno)o(wledge,)k(the)f(bit)g(complexit)o(y)f(of)h(leader)f(election)i
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(studied.)0 598 y([AAK90])113 b(A.)18 b(Aggarw)o(al,)j(R.)e(J.)f(Anderson,)h
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670 y(general)i(directed)f(graphs.)30 b FF(SIAM)20 b(Journal)g(on)g
(Computing)p FH(,)f(19\(2\):397{409)q(,)j(1990.)301 742 y(This)e(pap)q(er)f
(giv)o(es)f(the)g(\014rst)h(randomized)g FF(NC)f FH(algorithm)i(for)f
(depth-\014rst)f(searc)o(h)g(in)301 814 y(a)f(general)g(directed)f(graph.)0
928 y([AASS90])97 b(P)l(.)23 b(K.)g(Agarw)o(al,)j(B.)c(Arono)o(v,)i(M.)f
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1000 y(the)g(plane.)43 b(In)24 b FF(Pr)n(o)n(c.)f(Sixth)i(A)o(nn.)g(A)o(CM)e
(Symp.)h(on)h(Computational)f(Ge)n(ometry)p FH(,)301 1073 y(pages)19
b(321{331,)i(Berk)o(eley)l(,)16 b(CA,)h(June)h(1990.)28 b(The)18
b(authors)i(presen)o(t)d(a)h(randomized)301 1145 y(algorithm)i(for)d
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1909 y(graph)j(with)g FB(n)e FH(no)q(des)i(and)f Fx(j)p FB(E)s
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(algorithm)h(for)f(factoring)g(the)301 2600 y(in)o(tegers.)951
2828 y(85)p eop
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340 y(on)f(Computational)f(Ge)n(ometry)p FH(,)f(pages)g(203{210,)i(Berk)o
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(Sp)q(encer.)23 b FF(The)18 b(Pr)n(ob)n(abilistic)g(Metho)n(d)p
FH(.)23 b(John)17 b(Wiley)301 812 y(and)i(Sons,)f(1992.)27
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(as)g(dev)o(elop)q(ed)301 885 y(b)o(y)g(P)o(aul)h(Erd})-24
b(os)20 b(and)f(its)g(applications)i(in)e(Discrete)f(Mathematics)h(and)g
(Theoretical)301 957 y(Computer)e(Science.)0 1068 y([Aga90a])116
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38 b FF(Discr)n(ete)22 b(Computational)h(Ge)n(ometry)p FH(,)e(5:449{483,)k
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(h)f(of)h(whic)o(h)g(meets)301 1430 y(at)k(most)f FB(O)q FH(\()p
FB(n=r)q FH(\))i(lines)g(of)e Fx(L)p FH(.)g(He)g(sho)o(ws)h(that)f(the)g
(algorithm)j(is)e(optimal)h(up)e(to)h(a)301 1502 y(p)q(olylog)e(factor.)0
1614 y([Aga90b])113 b(P)l(.)18 b(K.)g(Agarw)o(al.)29 b(P)o(artitioning)21
b(arrangemen)o(ts)d(of)h(lines)g(I)q(I:)e(Applications.)30
b FF(Discr)n(ete)301 1686 y(Computational)20 b(Ge)n(ometry)p
FH(,)d(5:533{574,)k(1990.)28 b(Agarw)o(al)19 b(uses)f(his)h(partitioning)i
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1909 y(to)18 b(n)o(umerical)g(problems.)26 b(In)17 b Fj(Pr)n(o)n(c.)h(22nd)g
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301 2311 y(T)l(e)n(chnolo)n(gy)e(in)f(the)g(Information)g(A)n(ge)p
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2456 y(tosystem)j(to)f(toss)h(coins)g(and)g(exc)o(hange)f(results)h(in)g(a)f
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951 2828 y(96)p eop
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Fk(,)301 523 y(36:254{276,)j(1988.)28 b(The)18 b(pro)q(of)h(system)f(is)h
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(applications.)301 996 y(In)f Fj(A)n(dvanc)n(es)h(in)h(Cryptolo)n(gy{CR)l
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1068 y(V)l(ol.)c(435,)i(pages)f(547{559.)h(Springer-V)l(erlag,)g(1989.)h
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1469 y(482{493,)18 b(1990.)j(This)16 b(pap)q(er)f(con)o(tains)i(the)d
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1725 y([BMS86])118 b(E.)22 b(Bac)o(h,)h(G.)f(Miller,)i(and)f(J.)f(Shallit.)41
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1797 y(factoring.)41 b Fj(SIAM)23 b(Journal)g(on)g(Computing)p
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2270 y(CA,)j(Ma)o(y)g(1993.)42 b(A)22 b Fj(slightly)i(r)n(andom)e(sour)n(c)n
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2631 y(authors)j(sho)o(w)f(that)g(for)f(ev)o(ery)f(\014xed)h
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(w)g(that)g(there)951 2828 y(97)p eop
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267 y(for)f Fh(n)f Fk(pla)o(y)o(ers)g(that)h(tolerates)g Fh(\017n)f
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b(V)l(alian)o(t's)c([V)l(al84a)q(])e(algorithm)j(is)301 1357
y(sho)o(wn)d(to)f(b)q(e)f(the)h(b)q(est)g(p)q(ossible.)h(Also,)f(an)g
Fh(O)q Fk(\()p Fh(k)1215 1339 y Fl(4)p Ff(:)p Fl(3)1263 1357
y Fh(n)8 b Fk(log)j Fh(n)p Fk(\))j(algorithm)h(for)f(computing)301
1430 y(the)i Fh(k)r Fk(th)h(threshold)g(function)g(of)g Fh(n)f
Fk(v)m(ariables)i(is)f(giv)o(en.)0 1541 y([BP92])157 b(M.)15
b(Bellare)h(and)g(E.)g(P)o(etrank.)j(Making)e(zero-kno)o(wledge)f(pro)o(v)o
(ers)f(e\016cien)o(t.)k(In)c Fj(Pr)n(o)n(c.)301 1614 y(24th)24
b(A)o(nn.)g(A)o(CM)f(Symp.)g(on)h(The)n(ory)e(of)h(Computing)p
Fk(,)i(pages)f(711{722,)i(Victoria,)301 1686 y(B.C.,)14 b(Canada,)i(Ma)o(y)e
(1992.)20 b(They)14 b(pro)o(v)o(e)g(that)h(if)g(a)g(language)i(p)q(ossesses)f
(a)f(statistical)301 1758 y(zero-kno)o(wledge)i(pro)q(of)h(then)f(it)g(also)h
(p)q(ossesses)g(a)f(statistical)i(zero-kno)o(wledge)e(pro)q(of)301
1830 y(in)h(whic)o(h)f(the)g(pro)o(v)o(er)f(runs)h(in)g(probabilisti)q(c)i(p)
q(olynomial)h(time)d(with)g(an)h Fj(NP)f Fk(oracle.)301 1902
y(Previously)l(,)g(this)f(w)o(as)g(only)g(kno)o(wn)f(giv)o(en)h(the)f
(existence)f(of)i(one-w)o(a)o(y)f(p)q(erm)o(utations.)0 2014
y([BR88])154 b(M.)14 b(Blum)f(and)i(P)l(.)e(Ragha)o(v)m(an.)19
b(Program)c(correctness:)20 b(Can)14 b(one)g(test)g(for)g(it?)26
b(T)l(ec)o(hni-)301 2086 y(cal)13 b(Rep)q(ort)f(R)o(C)g(14038)i(\(#62902\),)f
(IBM)e(T.J.)g(W)l(atson)i(Researc)o(h)f(Cen)o(ter,)f(Septem)o(b)q(er)301
2159 y(1988.)24 b(They)17 b(presen)o(t)f(\\program)i(c)o(hec)o(k)o(ers")d
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2231 y(based)g(on)g(in)o(teractiv)o(e)f(pro)q(ofs.)0 2342 y([BR89a])130
b(L.)27 b(Babai)g(and)g(L.)g(R\023)-24 b(on)o(y)o(ai.)52 b(Computing)27
b(irreducible)h(represen)o(tations)f(of)g(\014nite)301 2415
y(groups.)36 b(In)21 b Fj(Pr)n(o)n(c.)f(30th)i(A)o(nn.)g(IEEE)f(Symp.)h(on)g
(F)l(oundations)g(of)f(Computer)h(Sci-)301 2487 y(enc)n(e)p
Fk(,)h(pages)e(93{98,)j(Researc)o(h)19 b(T)l(riangle)k(P)o(ark,)e(NC,)f
(Octob)q(er)h(1989.)h(IEEE)e(Com-)301 2559 y(puter)15 b(So)q(ciet)o(y)g
(Press.)k(In)14 b(this)i(pap)q(er,)f(the)f(authors)i(giv)o(e)f(a)g
(randomized)g(\(Las)h(V)l(egas\))301 2631 y(p)q(olynomial)k(time)d(algorithm)
i(for)f(decomp)q(osing)g(a)g(giv)o(en)f(represen)o(tation)h(of)f(a)h
(\014nite)301 2704 y(group)g(o)o(v)o(er)d(an)i(algebraic)h(n)o(um)o(b)q(er)d
(\014eld)i(in)o(to)g(absolutely)h(irreducible)f(constituen)o(ts.)951
2828 y(99)p eop
bop 0 195 a Fk([BR89b])127 b(B.)19 b(Berger)g(and)h(J.)f(Romp)q(el.)31
b(Sim)o(ulating)21 b(\(log)1224 174 y Ff(c)1250 195 y Fh(n)p
Fk(\)-wise)f(indep)q(endence)g(in)g(NC.)30 b(In)301 267 y Fj(Pr)n(o)n(c.)17
b(30th)g(A)o(nn.)h(IEEE)f(Symp.)g(on)h(F)l(oundations)g(of)f(Computer)g
(Scienc)n(e)p Fk(,)h(Researc)o(h)301 340 y(T)l(riangle)d(P)o(ark,)e(NC,)g
(Oct)f(1989.)j(IEEE)e(Computer)g(Science)f(Press.)17 b(A)12
b(general)i(frame-)301 412 y(w)o(ork)21 b(for)g(the)g(derandomization)i(of)e
(randomized)g Fj(NC)g Fk(algorithms)i(whose)f(analysis)301
484 y(uses)e(only)g(p)q(olylogari)q(thmic)i(indep)q(endence)d(is)h(presen)o
(ted.)f(This)h(framew)o(ork)f(allo)o(ws)301 556 y(the)f(deriv)m(ation)h(of)f
Fj(NC)f Fk(algorithms)j(for)d(man)o(y)g(problems)i(that)e(w)o(ere)g(not)h
(previously)301 628 y(kno)o(wn)f(to)f(b)q(e)h(in)g Fj(NC)7
b Fk(.)0 742 y([Bra85])147 b(G.)16 b(Brac)o(ha.)j(An)d Fh(O)q
Fk(\(log)10 b Fh(n)p Fk(\))16 b(exp)q(ected)e(rounds)j(randomized)f(Byzan)o
(tine)f(generals)h(pro-)301 814 y(to)q(col.)e(In)d Fj(Pr)n(o)n(c.)h(17th)g(A)
o(nn.)h(A)o(CM)g(Symp.)f(on)h(The)n(ory)e(of)i(Computing)p
Fk(,)f(pages)g(316{326,)301 887 y(1985.)k(Brac)o(ha)c(sho)o(ws)h(ho)o(w)f(to)
g(partition)i(a)f(set)f(of)g Fh(n)g Fk(sync)o(hronous)h(pro)q(cesses)f(\(of)h
(whic)o(h)301 959 y(at)k(most)g(a)g(third)g(are)g(fault)o(y\))g(in)o(to)g(o)o
(v)o(erlapping)h Fj(gr)n(oups)e Fk(of)h(pro)q(cesses)g(suc)o(h)f(that)h(the)
301 1031 y(n)o(um)o(b)q(er)c(of)h(fault)o(y)g(groups)g(is)g(at)g(most)g(the)f
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1103 y(Ben-Or's)j(algorithm)i(for)f(Byzan)o(tine)f(agreemen)o(t)f(\(see)h
(Section)h(3.5\))g(is)g(then)f(used)h(to)301 1176 y(obtain)h(an)f
Fh(O)q Fk(\(log)11 b Fh(n)p Fk(\))16 b(proto)q(col.)0 1289
y([Bro85])147 b(A.)20 b(Z.)h(Bro)q(der.)35 b(A)20 b(pro)o(v)m(ably)i(secure)e
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1362 y(distributed)e(lottery)f(problem)g(\(extended)e(abstract\).)26
b(In)17 b Fj(Pr)n(o)n(c.)h(F)l(ourth)g(A)o(nn.)h(A)o(CM)301
1434 y(Symp.)f(on)g(Principles)h(of)e(Distribute)n(d)h(Computing)p
Fk(,)f(pages)g(136{148,)i(1985.)24 b(Rabin's)301 1506 y(classic)14
b(Byzan)o(tine)e(agreemen)o(t)f(algorithm)k([Rab83)q(])c(uses)i(a)g
(coin-toss)h(whose)f(outcome)301 1578 y(is)19 b(a)o(v)m(ailable)h(to)e(all)h
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(to)e(reac)o(h)301 1650 y(Byzan)o(tine)23 b(agreemen)o(t)e(in)j(constan)o(t)f
(time.)f(Bro)q(der)h(demonstrates)g(a)g(p)q(olynomial-)301
1723 y(time)g(distributed)i(mec)o(hanism)d(to)i(implemen)o(t)e(suc)o(h)h(a)h
(coin)g(toss)f(in)h(a)g(Byzan)o(tine)301 1795 y(en)o(vironmen)o(t.)0
1909 y([Bro86])147 b(A.)19 b(Z.)g(Bro)q(der.)30 b(Ho)o(w)19
b(hard)h(is)g(it)g(to)g(marry)f(at)h(random?)32 b(\(On)19 b(the)g(appro)o
(ximation)301 1981 y(of)i(the)f(p)q(ermanen)o(t\).)32 b(In)19
b Fj(Pr)n(o)n(c.)h(18th)h(A)o(nn.)h(A)o(CM)e(Symp.)h(on)g(The)n(ory)f(of)h
(Comput-)301 2053 y(ing)p Fk(,)26 b(pages)f(50{58,)i(1986.)44
b(This)25 b(pap)q(er)f(pro)o(vides)g(a)g Fj(ful)r(l-p)n(olynomial)i(r)n
(andomize)n(d)301 2125 y(appr)n(oximation)21 b(scheme)h(\(fpr)n(as\))e
Fk(for)h(appro)o(ximating)h(the)e(p)q(ermanen)o(t.)f(Ev)m(aluating)301
2198 y(the)e(p)q(ermanen)o(t)g(of)h(a)g Fh(n)12 b Fg(\002)f
Fh(n)17 b Fk(matrix)h(is)g(equiv)m(alen)o(t)g(to)f(coun)o(ting)i(p)q(erfect)d
(matc)o(hings)301 2270 y(in)i(an)f(asso)q(ciated)i(bipartite)f(graph.)g(The)e
(problem)i(of)f(appro)o(ximately)g(coun)o(ting)h(the)301 2342
y(p)q(erfect)d(matc)o(hings)h(in)g(a)g(graph)g(is)g(reduced)e(to)i(that)g(of)
f(generating)i(them)e(uniformly)l(.)301 2414 y(See)j([JS89)q(])f(for)i(the)f
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(erratum)301 2487 y(can)13 b(b)q(e)g(found)h(in)f Fj(Pr)n(o)n(c.)g(20th)i(A)o
(nn.)g(A)o(CM)e(Symp.)h(on)h(The)n(ory)e(of)h(Computing)t Fk(,)g(1988.\).)0
2600 y([Bro89])147 b(A.)18 b(Z.)f(Bro)q(der.)28 b(Generating)19
b(random)g(spanning)h(trees.)27 b(In)18 b Fj(Pr)n(o)n(c.)g(30th)h(A)o(nn.)h
(IEEE)301 2672 y(Symp.)g(on)f(F)l(oundations)h(of)g(Computer)f(Scienc)n(e)p
Fk(,)h(pages)g(442{453,)h(Oct)d(1989.)29 b(This)938 2828 y(100)p
eop
bop 301 195 a Fk(pap)q(er)18 b(solv)o(es)g(the)f(problem)h(of)f(generating)i
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y(rected)d(graph)i Fh(G)f Fk(whic)o(h)g(as)h(the)e(follo)o(wing)k(sp)q(ecial)
e(prop)q(ert)o(y:)21 b(it)15 b(is)g(c)o(hosen)g(uniformly)301
340 y(at)j(random)h(from)e(all)i(p)q(ossible)h(spanning)g(trees)d(of)h
Fh(G)p Fk(.)g(The)g(exp)q(ected)e(running)j(time)301 412 y(of)i(the)e
(probabilisti)q(c)j(algorithm)g(is)e Fh(O)q Fk(\()p Fh(n)8
b Fk(log)k Fh(n)p Fk(\))20 b(p)q(er)g(generated)f(tree)g(for)i(almost)g(all)
301 484 y(graphs.)c(It)f(can)h(b)q(e)f Fh(O)q Fk(\()p Fh(n)767
466 y Fl(3)787 484 y Fk(\))h(p)q(er)f(generated)g(tree)g(in)h(the)f(w)o(orst)
g(case.)0 596 y([BRS91a])103 b(R.)22 b(Beigel,)h(N.)e(Reingold,)j(and)f(D.)e
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668 y(In)g Fj(Pr)n(o)n(c.)f(23r)n(d)g(A)o(nn.)i(A)o(CM)f(Symp.)g(on)h(The)n
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b(LA,)f(Ma)o(y)g(1991.)23 b(The)17 b(randomized)g(complexit)o(y)f(class)h
Fj(PP)g Fk(is)g(sho)o(wn)g(to)g(b)q(e)301 812 y(closed)g(under)f(in)o
(tersection)h(and)g(union.)0 924 y([BRS91b])100 b(A.)17 b(Blum,)h(P)l(.)g
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b(in)e(unfamiliar)i(geometric)301 996 y(terrain.)g(In)15 b
Fj(Pr)n(o)n(c.)g(23r)n(d)f(A)o(nn.)j(A)o(CM)f(Symp.)g(on)g(The)n(ory)f(of)h
(Computing)p Fk(,)g(pages)f(494{)301 1068 y(504,)i(New)f(Orleans,)g(LA,)g(Ma)
o(y)f(1991.)23 b(They)16 b(consider)g(the)g(problem)h(of)f(a)g(rob)q(ot)i
(that)301 1141 y(has)f(to)f(tra)o(v)o(el)g(from)g(a)g(start)h(lo)q(cation)h
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1213 y(obstacles)25 b(that)g(lie)f(in)g(its)h(w)o(a)o(y)l(.)e(The)h(rob)q(ot)
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1285 y(p)q(osition)d(and)d(that)g(of)g(the)g(target.)g(It)f(do)q(es)h(not,)g
(ho)o(w)o(ev)o(er,)e(kno)o(w)i(the)g(p)q(ositions)i(and)301
1357 y(exten)o(ts)11 b(of)h(the)g(obstacles)h(in)g(adv)m(ance;)g(it)f
(\014nds)h(out)f(ab)q(out)h(obstacles)h(as)e(it)g(encoun)o(ters)301
1430 y(them.)20 b(They)h(presen)o(t)f(an)h(optimal)i(randomized)e(algorithm)i
(for)e(scenes)f(con)o(taining)301 1502 y(arbitrary)e(p)q(olygonal)h
(obstacles.)0 1614 y([BS83])163 b(G.)18 b(N.)e(Buc)o(kley)g(and)i(A.)e(Silb)q
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(gen-)301 1686 y(eralized)j(input-output)g(construct)f(of)g(CSP.)29
b Fj(A)o(CM)19 b(T)l(r)n(ans.)g(on)h(Pr)n(o)n(gr)n(amming)e(L)n(an-)301
1758 y(guages)f(and)e(Systems)p Fk(,)f(5\(2\),)h(1983.)j(They)c(presen)o(t)f
(a)h(distributed)h(algorithm)h(for)d(CSP)301 1830 y(output)23
b(guards)f(based)h(on)f(priorit)o(y)g(ordering)h(of)f(pro)q(cesses.)g(Their)g
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1975 y(comm)o(unication)c(with)g(a)f(third)h(pro)q(cess)f(will)i(comm)o
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b(J.-D.)22 b(Boissonnat)i(and)f(M.)e(T)l(eillaud.)41 b(On)22
b(the)g(randomized)h(construction)g(of)f(the)301 2159 y(Delauna)o(y)c(tree.)k
Fj(The)n(or)n(etic)n(al)c(Computer)g(Scienc)n(e)p Fk(,)g(112:339{354)q(,)h
(1993.)25 b(An)16 b(on-line)301 2231 y(randomized)23 b(algorithm)g(whic)o(h)f
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2375 y(the)i(algorithm)h(is)g(optimal)g(pro)o(vided)f(that)g(the)f(p)q(oin)o
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2559 y([BV93])153 b(E.)26 b(Bernstein,)j(,)f(and)f(U.)e(V)l(azirani.)52
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2631 y(25th)e(A)o(nn.)g(A)o(CM)e(Symp.)h(on)h(The)n(ory)e(of)h(Computing)p
Fk(,)i(pages)e(11{20,)j(San)d(Diego,)301 2704 y(CA,)i(Ma)o(y)g(1993.)49
b(A)24 b Fj(quantum)j(T)l(uring)f(Machine)p Fk(,)i(as)e(originally)i(form)o
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bop 301 195 a Fk(Deutsc)o(h)23 b([Deu85],)h(ma)o(y)e(b)q(e)h(though)o(t)h(of)
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267 y(probabilisti)q(c)g(T)l(uring)e(Mac)o(hine:)30 b(it)22
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(existence)f(of)301 484 y(a)h Fj(universal)g Fk(quan)o(tum)e(T)l(uring)i(Mac)
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556 y(ally)f(b)q(ounded.)e(They)g(also)i(presen)o(t)e(the)g(\014rst)g
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628 y(more)19 b(p)q(o)o(w)o(erful)g(than)h(classical)h(probabilistic)h(TMs.)d
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y(there)g(is)h(an)g(oracle)g(relativ)o(e)g(to)g(whic)o(h)g(there)e(is)j(a)e
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Fh(n)1720 755 y Ff(o)p Fl(\(log)6 b Ff(n)p Fl(\))1855 773 y
Fk(time)301 845 y(b)o(y)16 b(a)h(b)q(ounded-error)g(probabilisti)q(c)i(TM.)0
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Fl(1)604 1031 y Fg(\021)i Fh(p)p Fk(.)44 b Fj(A)o(meric)n(an)24
b(Mathematic)n(al)g(Monthly)p Fk(,)h(19:22{27,)i(1912.)45 b(Let)301
1103 y Fh(n)38 b Fk(=)g(\005)481 1085 y Ff(i)p Fl(=)p Ff(m)481
1116 y(i)p Fl(=1)553 1103 y Fh(p)577 1081 y Ff(\027)594 1086
y Fe(i)577 1115 y Ff(i)640 1103 y Fk(b)q(e)30 b(the)g(unique)h(prime)f
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Fh(n)p Fk(\))38 b(=)301 1176 y(lcm)p Fg(f)p Fh(p)426 1154 y
Ff(\027)443 1159 y Fb(1)461 1154 y Fd(\000)p Fl(1)426 1186
y(1)508 1176 y Fk(\()p Fh(p)551 1183 y Fl(1)583 1176 y Fg(\000)10
b Fk(1\))p Fh(;)e Fk(.)g(.)g(.)g Fh(;)g(p)809 1158 y Ff(\027)826
1162 y Fe(m)856 1158 y Fd(\000)p Fl(1)809 1188 y Ff(m)903 1176
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1141 y Fh(X)t Fk(-)p Fh(Y)30 b Fk(separator)20 b(in)f(a)f(digraph,)i(and)f
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1325 y([CHM92])108 b(Z.J.)12 b(Czec)o(h,)f(G.)h(Ha)o(v)m(as,)h(and)g(B.S.)e
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1397 y(minimal)23 b(p)q(erfect)e(hash)h(functions.)37 b Fj(Information)22
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1469 y(264,)j(Oct)e(1992.)42 b(The)23 b(authors)g(describ)q(e)g(a)h
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1975 y(time.)k(The)g(metho)q(d)g(strongly)h(impro)o(v)o(es)f(on)g(the)g
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2159 y([Cic80])155 b(R.)16 b(Cic)o(helli.)21 b(Minimal)c(p)q(erfect)e(hash)i
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2231 y(the)f(A)o(CM)p Fk(,)c(23\(1\):17{19,)k(Jan)e(1980.)j(A)c(heuristic)h
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2487 y([CL89])160 b(A.)23 b(Condon)h(and)g(R.)f(Lipton.)43
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2631 y(pages)g(462{467,)h(1989.)k(In)o(teractiv)o(e)13 b(pro)q(of)j(systems)e
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eop
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267 y(on)21 b(the)g(p)q(o)o(w)o(er)f(of)h(IP)f(systems)g(that)h(halt)g(with)h
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b(I)t(M)e(E)s Fk(\(2)1770 394 y Fl(2)1788 382 y Fe(O)q Fb(\()p
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y([CPV91])120 b(P)l(.)12 b(Caspi,)i(J.)d(Piotro)o(wski,)k(and)d(R.)g(V)l
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1396 y(of)17 b(signature)g(analysis)h(e\016ciency)l(.)h Fj(IEEE)e(T)l(r)n
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y(Sept)i(1991.)25 b(This)19 b(pap)q(er)e(presen)o(ts)g(an)g(in)o(teresting)h
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1685 y(register)h(used)f(for)h(compression)g(is)g(c)o(hosen)f(at)h(random.)0
1797 y([CR79])154 b(E.)16 b(Chang)h(and)g(R.)e(Rob)q(erts.)21
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1869 y(\014nding)h(in)f(circular)h(con\014gurations)g(of)f(pro)q(cessors.)21
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2270 y([CR93])154 b(R.)19 b(Canetti)h(and)f(T.)g(Rabin.)30
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2342 y(timal)i(resilience.)36 b(In)20 b Fj(Pr)n(o)n(c.)h(25th)g(A)o(nn.)i(A)o
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y(pages)f(42{51,)i(San)d(Diego,)i(CA,)d(Ma)o(y)h(1993.)34 b(The)20
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2486 y(maxim)o(um)g(n)o(um)o(b)q(er)f(of)h(faults)h(in)g(the)f(presence)f(of)
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2611 y Ff(n)p 1431 2619 22 2 v 1433 2648 a Fl(3)1458 2631 y
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bop 301 195 a Fk(authors)20 b(sho)o(w)f(that)f(there)g(exists)h(a)f
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212 a Fl(3)1183 195 y Fg(e)13 b(\000)f Fk(1\)-resilien)o(t)20
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412 y(time.)0 523 y([CRS93])127 b(S.)15 b(Chari,)h(P)l(.)e(Rohatgi,)i(and)g
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596 y(isolation)q(,)h(with)d(applications)j(to)e(p)q(erfect)e(matc)o(hing)h
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b(A)o(nn.)f(A)o(CM)g(Symp.)g(on)h(The)n(ory)e(of)h(Computing)p
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812 y(p)q(erfect)14 b(matc)o(hing)g(that)h(uses)f Fh(O)q Fk(\(log)d
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1068 y(putational)f(geometry)l(,)c(I)q(I.)25 b Fj(Discr)n(ete)19
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1141 y(E\016cien)o(t)g(probabilisti)q(c)i(algorithms)h(are)d(presen)o(ted)f
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1758 y(Ninth)i(A)o(nn.)h(A)o(CM)e(Symp.)g(on)g(The)n(ory)g(of)g(Computing)t
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957 y(problems)d(is)g(an)g(op)q(en)g(problem.)0 1068 y([DFK91])118
b(M.)20 b(Dy)o(er,)g(A.)g(F)l(rieze,)g(and)h(R.)f(Kannan.)35
b(A)20 b(random)h(p)q(olynomial)i(time)d(algorithm)301 1141
y(for)f(appro)o(ximating)i(the)d(v)o(olume)g(of)h(a)g(con)o(v)o(ex)e(b)q(o)q
(dy)l(.)29 b Fj(Journal)19 b(of)g(the)h(A)o(CM)p Fk(,)e(38:1{)301
1213 y(17,)25 b(1991.)40 b(A)22 b(constan)o(t)h(time)f(oracle)h(is)g(assumed)
g(for)f(determining)h(if)g(a)g(p)q(oin)o(t)g(in)301 1285 y(space)f(is)g
(inside)h(or)f(outside)g(a)g(con)o(v)o(ex)e(b)q(o)q(dy)i(in)h(n-dimensional)h
(Euclidean)e(space.)301 1357 y(The)c(algorithm)i(runs)e(in)h(time)e(b)q
(ounded)i(b)o(y)e(a)h(p)q(olynomial)j(in)d Fh(n)p Fk(,)g(the)g(dimension)h
(of)301 1430 y(the)e(b)q(o)q(dy)l(,)h(and)f(1)p Fh(=\017)p
Fk(,)h(where)e Fh(\017)h Fk(is)h(the)f(relativ)o(e)g(error)g(b)q(ound.)g
(With)h(probabilit)o(y)h(3)p Fh(=)p Fk(4,)301 1502 y(it)e(\014nds)g(an)g
(appro)o(ximation)h(satisfying)g(the)e(error)g(b)q(ound.)0
1614 y([DGMP92])72 b(M.)13 b(Dietzfelbinger,)h(J.)f(Gil,)i(Y.)e(Matias,)h
(and)g(N.)f(Pipp)q(enger.)k(P)o(olynomial)f(hash)e(func-)301
1686 y(tions)f(are)e(reliable.)j(In)d Fj(Pr)n(o)n(c.)h(19th)g(Int'l.)i(Col)r
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1758 y(gr)n(amming,)19 b Fk(Lecture)f(Notes)g(in)g(Computer)g(Science,)f(V)l
(ol.)h(623,)h(pages)g(235{246,)i(Vi-)301 1830 y(enna,)14 b(Austria,)f(July)g
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301 1902 y(sho)o(ws)h(ho)o(w)f(to)g(construct)g(a)g(p)q(erfect)f(hash)i
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1975 y(for)j(real-time)g(applications)i(\(Theorems)d(6.1)h(and)g(7.1\).)0
2086 y([Dij71])160 b(E.)11 b(W.)g(Dijkstra.)i(Hierarc)o(hical)f(ordering)g
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Fk(,)301 2159 y(1\(2\):115{138,)j(1971.)f(Reprin)o(ted)d(in)g
Fj(Op)n(er)n(ating)i(Systems)g(T)l(e)n(chniques)t Fk(,)g(C.A.R.)c(Hoare)301
2231 y(and)j(R.H.)d(P)o(errot,)i(Eds.,)g(Academic)f(Press,)i(1972,)g(pp.)f
(72{93.)h(This)g(pap)q(er)f(in)o(tro)q(duces)301 2303 y(the)k(classical)j
(sync)o(hronization)f(problem)e(of)h(Dining)h(Philosophers.)0
2415 y([DKM)134 2397 y Fl(+)163 2415 y Fk(88])76 b(M.)15 b(Dietzfelbinger,)h
(A.)f(Karlin,)i(K.)d(Mehlhorn,)i(F.)f(Mey)o(er)f(auf)i(der)f(Heide,)g(H.)f
(Rohn-)301 2487 y(ert,)h(and)h(R.)f(E.)g(T)l(arjan.)20 b(Dynamic)c(p)q
(erfect)e(hashing:)23 b(Upp)q(er)15 b(and)h(lo)o(w)o(er)f(b)q(ounds.)21
b(In)301 2559 y Fj(Pr)n(o)n(c.)13 b(29th)g(A)o(nn.)i(IEEE)e(Symp.)g(on)h(F)l
(oundations)g(of)g(Computer)f(Scienc)n(e)p Fk(,)i(pages)d(524{)301
2631 y(531,)17 b(White)f(Plains,)i(NY,)c(Oct)i(1988.)21 b(A)16
b(randomized)g(algorithm)i(for)e(the)g(dictionary)301 2704
y(problem)h(based)g(on)g(p)q(erfect)e(hashing)j(is)f(presen)o(ted.)938
2828 y(108)p eop
bop 0 195 a Fk([DKS88])123 b(C.)13 b(Dw)o(ork,)g(P)l(.)f(C.)g(Kanellakis,)j
(and)e(L.)g(J.)f(Sto)q(c)o(kmey)o(er.)h(P)o(arallel)h(algorithms)h(for)e
(term)301 267 y(matc)o(hing.)23 b Fj(SIAM)17 b(Journal)h(on)g(Computing)p
Fk(,)f(17\(4\):711{731,)i(1988.)k(In)17 b(the)f(con)o(text)301
340 y(of)i(a)f(parallel)i(algorithm)g(for)e(the)g(term)f(matc)o(hing)h
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b(can)d(b)q(e)h(used)f(to)h(reduce)e(the)h(initial)j(pro)q(cessor)e
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y Fl(5)408 484 y Fk(\))f(to)h Fh(O)q Fk(\()p Fh(M)5 b Fk(\()p
Fh(n)p Fk(\)\),)23 b(where)e Fh(M)5 b Fk(\()p Fh(n)p Fk(\))21
b(is)h(the)f(pro)q(cessor)i(complexit)o(y)e(of)g(m)o(ultiplying)301
556 y(t)o(w)o(o)c Fh(n)11 b Fg(\002)g Fh(n)16 b Fk(matrices.)0
670 y([DLMV88])76 b(P)l(.)12 b(Dagum,)h(M.)e(Lub)o(y)l(,)i(M.)e(Mihail,)j
(and)f(U.V.)d(V)l(azirani.)15 b(P)o(olytop)q(es,)f(p)q(ermanen)o(ts)d(and)301
742 y(graphs)22 b(with)f(large)g(factors.)34 b(In)20 b Fj(Pr)n(o)n(c.)g(29th)
h(A)o(nn.)h(IEEE)f(Symp.)g(on)g(F)l(oundations)301 814 y(of)h(Computer)f
(Scienc)n(e)p Fk(,)i(pages)e(412{421,)j(1988.)35 b(Randomized)21
b(algorithms)i(for)d(ap-)301 887 y(pro)o(ximating)c(the)e(n)o(um)o(b)q(er)g
(of)h(p)q(erfect)e(matc)o(hings)i(in)g(a)g(graph)h(based)e(on)h(a)g
(geometric)301 959 y(reasoning)j(are)f(presen)o(ted.)0 1073
y([dlVKS93])83 b(F.)16 b(de)f(la)i(V)l(ega,)e(S.)h(Kannan,)g(and)h(M.)e(San)o
(tha.)21 b(Tw)o(o)16 b(probabilisti)q(c)i(results)f(on)f(merg-)301
1145 y(ing.)38 b Fj(SIAM)22 b(Journal)h(on)f(Computing)p Fk(,)h
(22\(2\):261{271,)j(1993.)38 b(Tw)o(o)22 b(probabilistic)301
1217 y(algorithms)e(for)d(merging)h(t)o(w)o(o)g(sorted)f(lists)i(are)e
(presen)o(ted.)f(When)i Fh(m)d(<)g(n)p Fk(,)j(the)f(\014rst)301
1289 y(algorithm)k(has)f(a)g(w)o(orst-case)g(time)e(b)q(etter)h(than)h(an)o
(y)f(deterministic)h(algorithm)h(for)301 1362 y(1)p Fh(:)p
Fk(618)16 b Fh(<)e(n=m)g(<)g Fk(3.)i(The)h(algorithm)h(is)f(extended)e(to)i
(p)q(erform)g(w)o(ell)g(for)f(an)o(y)h(v)m(alue)g(of)301 1434
y Fh(n=m)p Fk(.)0 1548 y([DMadH90])55 b(M.)11 b(Dietzfelbinger)h(and)f(F.)g
(Mey)o(er)e(auf)j(der)f(Heide.)h(Ho)o(w)f(to)g(distribute)h(a)g(dictionary)g
(in)301 1620 y(a)h(complete)e(net)o(w)o(ork.)j(In)e Fj(Pr)n(o)n(c.)g(22nd)i
(A)o(nn.)g(A)o(CM)f(Symp.)g(on)h(The)n(ory)e(of)i(Computing)p
Fk(,)301 1692 y(pages)f(117{127,)i(Baltimore,)d(MD,)f(Ma)o(y)h(1990.)j(A)c
(randomized)h(algorithm)i(is)e(giv)o(en)g(for)301 1764 y(implemen)o(ting)17
b(a)f(distributed)h(dictionary)g(on)g(a)f(complete)g(net)o(w)o(ork)f(of)h
Fh(p)g Fk(pro)q(cessors.)301 1836 y(The)g(algorithm)i(is)e(based)g(on)g
(hashing)i(and)e(uses)g Fh(O)q Fk(\()p Fh(n=p)p Fk(\))h(exp)q(ected)e(time)g
(to)h(execute)301 1909 y Fh(n)k Fk(arbitrary)h(instructions)h(\(insert,)f
(delete,)f(lo)q(okup\).)h(The)f(resp)q(onse)h(time)e(for)i(eac)o(h)301
1981 y(lo)q(okup)d(is)f(exp)q(ected)f(constan)o(t.)0 2095 y([DMadH92])55
b(M.)11 b(Dietzfelbinger)g(and)h(F.)e(Mey)o(er)f(auf)j(der)e(Heide.)i
(Dynamic)f(hashing)i(in)f(real)f(time.)h(In)301 2167 y Fj(Informatik)17
b Fg(\001)f Fj(F)l(estschrift)h(zum)g(60.)f(Geburtstag)i(von)f(G)q(\177)-26
b(unter)17 b(Holtz,)h Fk(T)l(eubner-T)l(exte)301 2239 y(zur)j(Informatik,)g
(Band)f(1,)i(pages)f(95{119.)i(B.)c(G.)h(T)l(eubner,)h(Stuttgart,)h(German)o
(y)l(,)301 2311 y(1992.)36 b(The)20 b(FKS)g(probabilisti)q(c)j(pro)q(cedure)d
(is)i(extended)d(to)i(real-time.)f(See)g(Theo-)301 2384 y(rems)c(6.1)h(and)g
(7.1)g(in)g([DGMP92)q(].)f(A)g(preliminary)h(v)o(ersion)g(of)g(this)g(pap)q
(er)g(app)q(eared)301 2456 y(as)25 b(\\A)e(new)h(univ)o(ersal)h(class)g(of)f
(hash)g(functions)h(and,)h(dynamic)d(hashing)j(in)e(real)301
2528 y(time,")d Fj(Pr)n(o)n(c.)f(17th)h(Int'l.)h(Col)r(lo)n(q.)g(on)f(A)o
(utomata,)h(L)n(anguages)g(and)f(Pr)n(o)n(gr)n(amming)t Fk(,)301
2600 y(1990,)d(pp.)e(6{19.)938 2828 y(109)p eop
bop 0 195 a Fk([DMT92])108 b(O.)11 b(Devillers,)i(S.)e(Meiser,)h(and)g(M.)f
(T)l(eillaud.)k(F)l(ully)d(dynamic)f(Delauna)o(y)i(triangulation)301
267 y(in)h(logarithmic)i(exp)q(ected)c(time)h(p)q(er)h(op)q(eration.)k
Fj(Computational)d(Ge)n(ometry:)20 b(The)n(ory)301 340 y(and)f(Applic)n
(ations)p Fk(,)f(2\(2\):55{80,)h(1992.)25 b(This)19 b(pap)q(er)e(extends)g
(the)g(results)h(of)f([BT93])301 412 y(b)o(y)f(considering)i(the)f(deletion)g
(of)g(p)q(oin)o(ts.)g(The)f(Delauna)o(y)i(triangulation)h(of)e
Fh(n)g Fk(p)q(oin)o(ts)301 484 y(is)i(up)q(dated)f(in)g Fh(O)q
Fk(\(log)11 b Fh(n)p Fk(\))17 b(exp)q(ected)g(time)g(p)q(er)g(insertion)i
(and)f Fh(O)q Fk(\(log)11 b(log)g Fh(n)p Fk(\))17 b(exp)q(ected)301
556 y(time)e(p)q(er)g(deletion.)g(The)g(insertion)h(sequence)d(is)j(assumed)f
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(assumed)g(to)g(concern)f(an)o(y)g(curren)o(tly)g(presen)o(t)g(p)q(oin)o(t)i
(with)f(the)301 701 y(same)17 b(probabilit)o(y)l(.)0 814 y([DoD83])127
b(DoD)21 b(\(United)f(States)h(Dept.)e(of)i(Defense\).)32 b
Fj(R)n(efer)n(enc)n(e)21 b(Manual)g(for)g(the)g(A)n(da)g(Pr)n(o-)301
887 y(gr)n(amming)c(L)n(anguage,)g Fk(MIL-STD)e(1815A,)h(F)l(ebruary)f(1983.)
21 b(Section)15 b(3.2)h(of)f(our)g(sur-)301 959 y(v)o(ey)i(discusses)i(a)f
(randomized)h(distributed)g(algorithm)g(for)g(the)e(sc)o(heduling)i(of)f
(input)301 1031 y(and)h(output)f(guards.)g(The)g(designers)g(of)g(Ada)g(c)o
(hose)f(only)h(to)g(allo)o(w)i(nondeterminis-)301 1103 y(tic)f(c)o(hoice)f
(among)i(the)e Fi(accept)g Fk(alternativ)o(es)h(of)g(a)g Fi(select)f
Fk(statemen)o(t.)f(This)j(design)301 1176 y(decision)d(mak)o(es)e(the)g
(guard)h(sc)o(heduling)h(problem)e(in)h(Ada)g(m)o(uc)o(h)e(easier)i(and,)f
(in)h(par-)301 1248 y(ticular,)h(obliviates)i(the)d(need)g(for)g
(randomization.)0 1362 y([Dol82])151 b(D.)12 b(Dolev.)j(The)d(Byzan)o(tine)f
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Fk(,)g(3\(1\):14{)301 1434 y(30,)j(1982.)21 b(This)16 b(is)g(the)f(in)o(tro)q
(ductory)h(pap)q(er)g(on)g(Byzan)o(tine)e(Generals.)i(Dolev)g(pro)o(v)o(es)
301 1506 y(that)i(Byzan)o(tine)e(agreemen)o(t)g(is)h(ac)o(hiev)m(able)h(in)f
Fj(any)g Fk(distributed)h(system)f(if)g(and)g(only)301 1578
y(if)i(the)f(n)o(um)o(b)q(er)g(of)h(fault)o(y)f(pro)q(cessors)i(in)f(the)f
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y(the)d(total)h(n)o(um)o(b)q(er)e(of)h(pro)q(cessors;)h(and)g(\(2\))f(less)h
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h(deterministic)301 1795 y(algorithms)d(for)d(obtaining)j(it)e(are)f(giv)o
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(er.)h(The)g(time)e(complexit)o(y)h(gap)h(for)g(2-w)o(a)o(y)f(probabilistic)
301 1981 y(\014nite-state)20 b(automata.)32 b Fj(SIAM)20 b(Journal)g(on)g
(Computing)p Fk(,)g(19\(6\):1011{10)q(23,)j(1990.)301 2053
y(Among)17 b(other)f(results,)h(this)h(pap)q(er)f(sho)o(ws)g(that)g(an)o(y)f
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b(recognizing)g(a)e(non-regular)j(language)f(m)o(ust)e(use)g(exp)q(onen)o
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g(regular)g(language)i(can)e(b)q(e)g(recognized)f(in)i(linear)301
2270 y(time,)e(a)g(time-complexit)o(y)f(gap)i(is)f(established.)h(Similar)g
(results)f(w)o(ere)f(published)i(in)301 2342 y(the)f(pap)q(er)g(en)o(titled)h
(\\On)f(the)f(P)o(o)o(w)o(er)h(of)g(2-W)l(a)o(y)g(Probabilisti)q(c)i(Finite)f
(Automata,")301 2414 y(whic)o(h)c(app)q(eared)f(in)h Fj(Pr)n(o)n(c.)f(30th)h
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2487 y(Scienc)n(e)t Fk(,)i(1989.)0 2600 y([DSMP87])83 b(A.)23
b(De)h(San)o(tis,)i(S.)d(Micali,)j(and)e(G.)g(P)o(ersiano.)44
b(Non-in)o(teractiv)o(e)24 b(zero-kno)o(wledge)301 2672 y(pro)q(of-systems.)
45 b(In)24 b Fj(A)n(dvanc)n(es)g(in)h(Cryptolo)n(gy{CR)l(YPTO)f(87,)h
Fk(Lecture)f(Notes)f(in)938 2828 y(110)p eop
bop 301 195 a Fk(Computer)23 b(Science,)g(V)l(ol.)g(293,)i(pages)e(52{72.)h
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b(Non-in)o(teractiv)o(e)24 b(zero-kno)o(wledge)301 523 y(pro)q(of-systems)d
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(Science,)g(V)l(ol.)g(403,)h(pages)g(269{283.)i(Springer-V)l(erlag,)301
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y(prepro)q(cessing)23 b(stage)f(then)e(an)o(y)h(su\016cien)o(tly)g(short)h
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p Fk(,)g(19\(2\):472{499,)j(1990.)g(An)12 b(e\016cien)o(t)g(random-)301
1068 y(ized)18 b(proto)q(col)i(is)f(presen)o(ted)e(that)h(tolerates)h(up)f
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1213 y(toss.)g(Also)f(giv)o(en)g(is)h(a)f(Byzan)o(tine)f(Generals)i
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1285 y(ures)18 b(and)g(runs)g(in)g(constan)o(t)g(exp)q(ected)f(n)o(um)o(b)q
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y(of)k(this)f(pap)q(er)h(app)q(eared)g(in)f Fj(Pr)n(o)n(c.)g(27th)g(A)o(nn.)i
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b(Scienc)n(e)t Fk(,)f(1986.)0 1541 y([DSY90])124 b(A.)23 b(De)g(San)o(tis)h
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1686 y(90,)d Fk(Lecture)e(Notes)g(in)h(Computer)f(Science,)f(V)l(ol.)i(537.)g
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(non-in)o(teractiv)o(ely)h(in)g(zero-kno)o(wledge.)0 1942 y([ER60])156
b(P)l(.)15 b(Erd})-24 b(os)15 b(and)h(A.)e(R)o(\023)-23 b(en)o(yi.)18
b(On)d(the)f(ev)o(olution)i(of)f(random)g(graphs.)20 b Fj(Publ.)d(Math.)f
(Inst.)301 2014 y(Hung.)e(A)n(c)n(ad.)e(Sci.)p Fk(,)g(5:17{61,)i(1960.)g(A)d
(seminal)h(pap)q(er)f(on)h(random)g(graphs.)f(Reprin)o(ted)301
2086 y(in)22 b Fj(Paul)g(Er)n(d})-25 b(os:)30 b(The)22 b(A)o(rt)g(of)f
(Counting.)i(Sele)n(cte)n(d)h(Writings)t Fk(,)e(J.H.)d(Sp)q(encer,)j(Ed.,)301
2159 y(V)l(ol.)c(5)f(of)h(the)f(series)g Fj(Mathematicians)i(of)g(Our)f(Time)
t Fk(,)f(MIT)g(Press,)g(1973,)h(pp.)f(574{)301 2231 y(617.)0
2342 y([ES74])165 b(P)l(.)18 b(Erd})-24 b(os)19 b(and)f(J.)g(Sp)q(encer.)25
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Fk(.)26 b(Academic)301 2415 y(Press,)16 b(New)f(Y)l(ork)h(and)g(London,)h
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301 2487 y(small,)h(p)q(o)o(w)o(er)f(pac)o(k)o(ed)f(monograph)i(on)f
(non-constructiv)o(e)g(probabilisti)q(c)i(metho)q(ds)e(in)301
2559 y(com)o(binatorics.)j(Our)e(algorithm)i(for)f(net)o(w)o(orks)f(without)i
(large)f(hierarc)o(hies)g(is)h(based)301 2631 y(on)13 b(the)f(discussion)i
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bop 0 195 a Fk([F)o(CDH91])85 b(E.)22 b(F)l(o)o(x,)h(Q.F.)e(Chen,)i(A.)e
(Daoud,)k(and)d(L.S.)g(Heath.)38 b(Order)22 b(preserving)h(minimal)301
267 y(p)q(erfect)14 b(hash)i(functions)f(and)g(information)i(retriev)m(al.)i
Fj(A)o(CM)c(T)l(r)n(ans.)g(on)h(Information)301 340 y(Systems)p
Fk(,)h(9\(2\):281{308,)i(July)e(1991.)23 b(This)18 b(algorithm)g(com)o(bines)
f(the)f(tec)o(hniques)g(of)301 412 y(em)o(b)q(edding)22 b(the)g(k)o(eys)f(in)
o(to)h(an)g Fh(r)q Fg(\000)p Fk(graph)h(and)f(t)o(w)o(o-lev)o(el)g(hashing)h
(to)f(design)h(hash)301 484 y(functions)16 b(that)f(are)f(optimal)i(in)f
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556 y(algorithm)22 b(to)f(generate)f(the)g(hash)h(functions)g(uses)f
(near-optimal)j(space)d(and)h(time.)301 628 y(An)o(y)16 b(desired)g(order)h
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b(P)l(.)g(F)l(eynman.)34 b(Sim)o(ulating)23 b(ph)o(ysics)f(with)g(computers.)
35 b Fj(International)23 b(Journal)301 814 y(of)c(The)n(or)n(etic)n(al)g
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b(p)q(oin)o(ts)i(out)g(the)e(curi-)301 887 y(ous)i(problem)g(that)f(it)h(app)
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(tum)301 959 y(ph)o(ysical)25 b(system)e(on)g(a)h(probabilisti)q(c)i(T)l
(uring)e(Mac)o(hine)f(without)i(an)f(exp)q(onen)o(tial)301
1031 y(slo)o(wdo)o(wn,)h(ev)o(en)20 b(if)i(the)g(quan)o(tum)f(ph)o(ysical)i
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y(\(lik)o(e)17 b(some)f(kind)h(of)f(quan)o(tum)g(cellular)i(automaton\).)0
1217 y([FFS87])134 b(U.)15 b(F)l(eige,)h(A.)f(Fiat,)h(and)g(A.)f(Shamir.)22
b(Zero-kno)o(wledge)16 b(pro)q(ofs)h(of)f(iden)o(tit)o(y)l(.)21
b(In)15 b Fj(Pr)n(o)n(c.)301 1289 y(19th)j(A)o(nn.)h(A)o(CM)e(Symp.)h(on)g
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1362 y(kno)o(wledge)19 b(pro)q(ofs,)h(in)f(the)g(traditional)i(sense,)d(rev)o
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(viz.)h Fh(w)g Fg(2)f Fh(L)g Fk(or)h Fh(w)g Fg(62)f Fh(L)p
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1506 y(kno)o(wledge")f(pro)q(ofs)h(where)e(the)g(pro)o(v)o(er)f(con)o(vinces)
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Fh(w)h Fk(is)f(or)g(is)h(not)f(in)g Fh(L)p Fk(,)g(without)h(rev)o(ealing)f
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(presen)o(ted.)0 1836 y([F)o(GM)128 1818 y Fl(+)157 1836 y
Fk(89])82 b(M.)22 b(F)q(\177)-26 b(urer,)24 b(O.)e(Goldreic)o(h,)j(Y.)c
(Mansour,)k(M.)c(Sipser,)k(and)e(S.)f(Zac)o(hos.)40 b(On)22
b(com-)301 1909 y(pleteness)h(and)g(soundness)h(in)f(in)o(teractiv)o(e)g(pro)
q(of)g(systems.)40 b(In)22 b(S.)h(Micali,)i(editor,)301 1981
y Fj(A)n(dvanc)n(es)g(in)g(Computing)f(R)n(ese)n(ar)n(ch)f(5:)36
b(R)n(andomness)24 b(and)g(Computation)p Fk(,)h(pages)301 2053
y(429{442.)20 b(JAI)d(Press,)h(Green)o(wic)o(h,)f(CT,)g(1989.)27
b(An)17 b(in)o(teractiv)o(e)g(pro)q(of)i(system)e(for)h(a)301
2125 y(language)j Fh(L)d Fk(is)h(said)h(to)e(ha)o(v)o(e)g Fj(p)n(erfe)n(ct)h
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Fk(accepts)301 2198 y Fh(w)d Fk(if)g Fh(w)e Fg(2)g Fh(L)p Fk(.)h(This)h(pap)q
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(teractiv)o(e,)g(p)q(os-)301 2270 y(sibly)g(un)o(b)q(ounded,)e(pro)q(of)i
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2342 y Fj(NP)g Fk(ha)o(v)o(e)e(in)o(teractiv)o(e)g(pro)q(ofs)i(with)g(p)q
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2414 y(under)g(the)f(title)h(\\In)o(teractiv)o(e)e(pro)q(of)j(system:)j(pro)o
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y(selection,")e(authored)f(b)o(y)e(O.)h(Goldreic)o(h,)h(Y.)f(Mansour)h(and)f
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b(IEEE)e(Symp.)g(on)h(F)l(oundations)g(of)f(Computer)h(Scienc)n(e)p
Fk(,)f(1987,)h(pp.)e(449{461.)938 2828 y(112)p eop
bop 0 195 a Fk([FH84])156 b(V.)15 b(A.)g(F)l(eldman)g(and)h(D.)g(Harel.)k(A)
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267 y(puter)i(and)f(System)h(Scienc)n(es)p Fk(,)f(28\(2\):193{215)q(,)h
(1984.)25 b(This)18 b(pap)q(er)g(de\014nes)f(a)g(formal)301
340 y(logic)i Fi(Pr)p Fk(DL)g(to)e(reason)i(ab)q(out)f(probabilisti)q(c)i
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526 y([FHCD92])84 b(E.)14 b(F)l(o)o(x,)g(L.S.)f(Heath,)h(Q.F.)f(Chen,)h(and)g
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598 y(functions)22 b(for)e(large)h(databases.)35 b Fj(Communic)n(ations)21
b(of)g(the)h(A)o(CM)p Fk(,)d(35\(1\):105{121,)301 670 y(Jan)o(uary)k(1992.)41
b(This)23 b(pap)q(er)g(presen)o(ts)f(t)o(w)o(o)g(randomized)h(algorithm)h
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(designed)h(for)g(use)f(with)h(data)g(bases)g(with)g(as)301
814 y(man)o(y)d(as)g(a)h(million)h(k)o(eys.)c(The)i(algorithms)i(ha)o(v)o(e)e
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(and)h(the)e(second)h(generates)g(functions)g(that)g(approac)o(h)h(the)e
(theoretical)301 1031 y(lo)o(w)o(er)h(b)q(ound)g(of)g(\012\()p
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(predecessor)f(of)h([F)o(CDH91)o(].)0 1145 y([FKS82])128 b(M.)12
b(L.)h(F)l(redman,)g(J.)f(Koml\023)-24 b(os,)14 b(and)f(E.)f(Szemeredi.)j
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1217 y(w)o(orst)21 b(case)e(access)h(time.)32 b(In)19 b Fj(Pr)n(o)n(c.)h(23r)
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1289 y(Computer)15 b(Scienc)n(e)p Fk(,)g(pages)f(165{169,)i(1982.)h(This)e
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y(results)i(that)f(are)g(essen)o(tial)h(for)f(constructing)h(a)f(p)q(erfect)f
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1548 y([FL82])163 b(M.)15 b(J.)h(Fisc)o(her)f(and)i(N.)e(Lync)o(h.)20
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b(pro)o(v)o(e)301 1692 y(that)20 b(no)g(deterministic)g(solution)i(to)e(the)f
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y(agreemen)o(t)14 b(in)h(less)h(than)f Fh(t)8 b Fk(+)g(1)14
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b(Appro)o(ximate)21 b(coun)o(ting:)34 b(A)21 b(detailed)i(analysis.)40
b Fj(BIT)p Fk(,)21 b(25:113{134,)301 1950 y(1985.)g(In)15 b(1978,)i(R.)d
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2095 y(sen)o(ted)h(a)h(randomized)g(algorithm,)i(kno)o(wn)d(as)i
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2167 y(one)f(to)f(appro)o(ximately)h(main)o(tain)h(a)e(coun)o(ter)g(whose)h
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y(v)m(al)g(1)g(to)f Fh(M)18 b Fk(using)d(only)e(ab)q(out)i(log)10
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Fh(O)q Fk(\(2)767 2582 y Fd(\000)p Ff(\016)q(=)p Fl(2)850 2600
y Fk(\).)938 2828 y(113)p eop
bop 0 195 a Fk([Fla90])156 b(P)l(.)19 b(Fla)s(jolet.)31 b(On)20
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1176 y(pro)o(v)o(es)e(that)i(ev)o(ery)d(completely)h(async)o(hronous,)i
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b(er's)23 b(conjecture)g(ab)q(out)i(the)f(spacing)h(of)f(prime)938
2828 y(117)p eop
bop 301 195 a Fk(n)o(um)o(b)q(ers)22 b(is)h(true)f(than)g(there)g(exists)g(a)
g(random)h(p)q(olynomial)i(time)d(algorithm)i(for)301 267 y(primalit)o(y)18
b(testing.)0 381 y([GKS92])122 b(L.)15 b(J.)g(Guibas,)h(D.)e(E.)h(Kn)o(uth,)f
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453 y(tion)23 b(of)g(Delauna)o(y)f(and)h(V)l(oronoi)g(diagrams.)40
b Fj(A)o(lgorithmic)n(a)p Fk(,)23 b(7\(4\):381{413,)j(1992.)301
526 y(They)c(giv)o(e)g(a)g(new)g(randomized)g(incremen)o(tal)g(algorithm)h
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(and)f(Delauna)o(y)g(triangulations.)i(Their)d(algorithm)i(tak)o(es)301
670 y(exp)q(ected)j(time)g Fh(O)q Fk(\()p Fh(n=)8 b Fk(log)k
Fh(n)p Fk(\))23 b(and)g(space)g Fh(O)q Fk(\()p Fh(n)p Fk(\),)h(is)g(v)o(ery)d
(practical)j(to)f(implemen)o(t,)301 742 y(and)16 b(along)h(with)g(the)e
(algorithm)i(of)f([BT93],)e(is)i(more)f(\\on-line")j(than)e(earlier)g
(similar)301 814 y(metho)q(ds.)0 928 y([GKS93])122 b(W.)16
b(Go)q(ddard,)i(V.)d(King,)i(and)g(L.)f(Sc)o(h)o(ulman.)21
b(Optimal)c(randomized)g(algorithms)i(for)301 1000 y(lo)q(cal)j(sorting)f
(and)f(set-maxima.)31 b Fj(SIAM)21 b(Journal)f(on)h(Computing)p
Fk(,)f(22\(2\):272{283,)301 1073 y(April)d(1993.)k(Nearly)16
b(optimal)h(randomized)f(algorithms)i(are)d(presen)o(ted)g(for)h(the)f(lo)q
(cal)301 1145 y(sorting)k(problem)f(\(i.e.,)e(determining)h(the)g(relativ)o
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y(v)o(ertices)12 b(in)i(a)f(graph)h(in)g(whic)o(h)f(eac)o(h)f(v)o(ertex)g(is)
h(assigned)i(an)e(elemen)o(t)f(of)h(a)h(total)g(order\))301
1289 y(and)j(the)e(set-maxima)i(problem)f(\(i.e.,)f(determining)h(the)g
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h(of)e(sets)h(whose)g(elemen)o(ts)e(are)h(dra)o(wn)h(from)f(a)h(total)g
(order\).)0 1475 y([GL89])157 b(R.)21 b(I.)f(Green)o(b)q(erg)h(and)h(C.)e(E.)
h(Leiserson.)37 b(Randomized)22 b(routing)g(on)g(fat-trees.)35
b(In)301 1548 y(S.)16 b(Micali,)g(editor,)g Fj(A)n(dvanc)n(es)i(in)f
(Computing)h(R)n(ese)n(ar)n(ch)d(5:)22 b(R)n(andomness)17 b(and)g(Com-)301
1620 y(putation)p Fk(,)23 b(pages)f(345{374.)i(JAI)c(Press,)j(Green)o(wic)o
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y(class)d(of)e(routing)i(net)o(w)o(orks)e(in)g(parallel)j(computation.)e(Giv)
o(en)f(a)g(set)h(of)f(messages)h(to)301 1764 y(send,)j(the)e(c)o(hoice)g(is)i
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(at)h(what)301 1836 y(time.)15 b(This)h(approac)o(h)f(is)h(di\013eren)o(t)f
(from)g(that)g(of)g([V)l(al82)q(].)f(See)h(also)h Fj(Pr)n(o)n(c.)f(17th)h(A)o
(nn.)301 1909 y(A)o(CM)h(Symp.)h(on)f(The)n(ory)g(of)g(Computing)t
Fk(,)g(1985,)g(pp.)f(241{249.)0 2022 y([GM84])142 b(S.)17 b(Goldw)o(asser)i
(and)f(S.)f(Macali.)25 b(Probabilisti)q(c)19 b(encryption.)25
b Fj(Journal)18 b(of)g(Computer)301 2095 y(and)c(System)g(Scienc)n(es)p
Fk(,)h(28\(2\):270{299,)h(1984.)g(This)d(pap)q(er)g(in)o(tro)q(duces)g(a)g
(new)f(proba-)301 2167 y(bilistic)i(encryption)e(tec)o(hnique.)f(It)g(also)j
(con)o(tains)f(an)f(excellen)o(t)f(in)o(tro)q(duction)j(to)e(other)301
2239 y(public)h(k)o(ey)e(cryptosystems)g(with)i(discussion)h(on)e(ob)s
(jections)h(to)f(cryptosystems)g(based)301 2311 y(on)17 b(trap)q(do)q(or)h
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(and)e(R.)f(Riv)o(est.)35 b(A)20 b(digital)j(signature)f(sc)o(heme)e(secure)
301 2497 y(against)13 b(adaptiv)o(e)f(c)o(hosen-message)f(attac)o(k.)i
Fj(SIAM)f(Journal)h(on)g(Computing)p Fk(,)f(17:281{)301 2570
y(308,)17 b(1988.)23 b(This)17 b(is)h(a)e(companion)i(pap)q(er)f(of)f
([KPU88].)938 2828 y(118)p eop
bop 0 195 a Fk([GMR89])106 b(S.)15 b(Goldw)o(asser,)h(S.)f(Macali,)h(and)f
(C.)g(Rac)o(k)o(o\013.)k(The)c(kno)o(wledge)g(complexit)o(y)g(of)g(in)o(ter-)
301 267 y(activ)o(e)e(pro)q(of)h(systems.)i Fj(SIAM)e(Journal)g(on)h
(Computing)p Fk(,)f(pages)g(186{208,)i(1989.)h(This)301 340
y(pap)q(er)j(\014rst)f(app)q(eared)g(in)h Fj(Pr)n(o)n(c.)e(17th)i(A)o(nn.)h
(A)o(CM)e(Symp.)g(on)h(The)n(ory)f(of)h(Comput-)301 412 y(ing)p
Fk(,)e(1985,)h(pp.)d(291{304.)k(It)d(in)o(tro)q(duces)g(the)g(imp)q(ortan)o
(t)h(notion)h(of)f(zero-kno)o(wledge)301 484 y(in)o(teractiv)o(e)g(pro)q
(ofs.)h(The)f(authors)h(sho)o(w)g(that)f(it)h(is)f(p)q(ossible)j(to)d(pro)o
(v)o(e)f(that)i(certain)301 556 y(theorems)d(are)h(true)f(without)h
(divulging)i(wh)o(y)d(this)h(is)g(so.)0 668 y([GMV91])105 b(J.)24
b(Gil,)j(Y.)c(Matias,)j(and)f(U.)e(Vishkin.)44 b(T)l(o)o(w)o(ards)25
b(a)f(theory)g(of)g(nearly)g(constan)o(t-)301 740 y(time)17
b(parallel)h(algorithms.)24 b(In)16 b Fj(Pr)n(o)n(c.)h(32nd)h(A)o(nn.)g(IEEE)
g(Symp.)f(on)h(F)l(oundations)g(of)301 812 y(Computer)h(Scienc)n(e)p
Fk(,)g(pages)f(698{710,)j(1991.)26 b(This)19 b(pap)q(er)f(presen)o(ts)f(a)h
(paradigm)h(for)301 885 y(obtaining)f Fh(O)q Fk(\(log)637 863
y Fd(\003)665 885 y Fh(n)p Fk(\))d(running)h(time)e(for)h(problems)g(suc)o(h)
g(as)g(directory)g(main)o(tenance,)301 957 y(load)j(balancing)h(and)d
(hashing)j(using)e Fh(n=)8 b Fk(log)1153 935 y Fd(\003)1181
957 y Fh(n)16 b Fk(pro)q(cessors.)0 1068 y([GMW87])92 b(O.)21
b(Goldreic)o(h,)h(S.)e(Micali,)j(and)e(A.)f(Wigderson.)36 b(Ho)o(w)20
b(to)i(pla)o(y)f(an)o(y)f(men)o(tal)h(game)301 1141 y(or)e(a)f(completeness)g
(theorem)f(for)h(proto)q(cols)i(with)f(honest)f(ma)s(jorit)o(y)l(.)26
b(In)17 b Fj(Pr)n(o)n(c.)h(19th)301 1213 y(A)o(nn.)d(A)o(CM)f(Symp.)h(on)f
(The)n(ory)f(of)i(Computing)p Fk(,)f(pages)f(218{229,)j(1987.)h(Goldreic)o(h)
d(et)301 1285 y(al.)i(demonstrate)f(the)g(use)g(of)g(zero-kno)o(wledge)h(pro)
q(ofs)g(on)f(pro)o(ving)h(the)f(completeness)301 1357 y(theorem)h(for)h
(proto)q(cols)h(with)f(honest)g(ma)s(jorit)o(y)l(.)0 1469 y([GMW91])92
b(O.)13 b(Goldreic)o(h,)h(S.)f(Micali,)h(and)g(A.)e(Wigderson.)17
b(Pro)q(ofs)e(that)e(yield)h(nothing)h(but)e(their)301 1541
y(v)m(alidit)o(y)k(or)e(all)h(languages)h(in)f(NP)e(ha)o(v)o(e)g(zero-kno)o
(wledge)i(pro)q(of)g(systems.)i Fj(Journal)e(of)301 1614 y(the)k(A)o(CM)p
Fk(,)d(38\(1\):691{729,)k(1991.)28 b(They)17 b(sho)o(w)i(that)f(for)g(a)h
(language)h Fh(L)e Fk(in)g Fj(NP)g Fk(and)301 1686 y(a)i(string)h
Fh(w)g Fk(in)f Fh(L)p Fk(,)h(there)e(exists)h(a)g(probabilistic)j(in)o
(teractiv)o(e)c(pro)q(of)i(that)f(e\016cien)o(tly)301 1758
y(demonstrates)14 b(mem)o(b)q(ership)f(of)g Fh(x)g Fk(in)h
Fh(L)f Fk(without)i(con)o(v)o(eying)d(additional)17 b(information.)301
1830 y(Previously)l(,)23 b(zero-kno)o(wledge)f(pro)q(ofs)h(w)o(ere)d(kno)o
(wn)h(only)h(for)g(some)f(problems)h(that)301 1902 y(w)o(ere)f(in)g(b)q(oth)h
Fj(NP)f Fk(and)h Fj(c)n(o-NP)5 b Fk(.)21 b(A)f(preliminary)i(v)o(ersion)g(of)
f(this)h(pap)q(er)f(app)q(eared)301 1975 y(in)e Fj(Pr)n(o)n(c.)f(27th)h(A)o
(nn.)h(IEEE)f(Symp.)g(on)h(F)l(oundations)g(of)f(Computer)g(Scienc)n(e)p
Fk(,)h(1986,)301 2047 y(under)13 b(the)f(title)h(\\Pro)q(ofs)h(that)f(yield)g
(nothing)h(but)f(their)g(v)m(alidit)o(y)g(and)g(a)g(metho)q(dology)301
2119 y(of)k(cryptographic)g(proto)q(col)h(design.".)0 2231
y([Gol92])150 b(M.)18 b(Goldwurm.)28 b(Probabilistic)21 b(estimation)e(of)g
(the)f(n)o(um)o(b)q(er)f(of)h(pre\014xes)g(of)g(a)h(trace.)301
2303 y Fj(The)n(or)n(etic)n(al)g(Computer)g(Scienc)n(e)p Fk(,)i(92:249{268,)g
(1992.)29 b(The)19 b(author)g(uses)f(the)h(result)301 2375
y(to)e(determine)f(the)g(b)q(eha)o(vior)h(of)f(sev)o(eral)g(algorithms)j
(relating)f(to)e(trace)g(languages.)0 2487 y([Gon84])136 b(G.H.)16
b(Gonnet.)25 b(Determining)18 b(the)f(equiv)m(alence)g(of)h(expressions)g(in)
f(random)h(p)q(olyno-)301 2559 y(mial)j(time.)29 b(In)19 b
Fj(Pr)n(o)n(c.)g(16th)h(A)o(nn.)h(A)o(CM)f(Symp.)g(on)g(The)n(ory)f(of)h
(Computing)p Fk(,)g(pages)301 2631 y(334{341,)g(1984.)26 b(Hashing)19
b(functions)f(are)g(used)f(to)h(determine)e(algebraic)j(expression)301
2704 y(equiv)m(alence)f(with)g(a)f(small)i(probabilit)o(y)g(of)e(error.)g
(The)g(probabilit)o(y)i(of)f(error)f(can)g(b)q(e)938 2828 y(119)p
eop
bop 301 195 a Fk(made)12 b(arbitrarily)h(small,)g(dep)q(ending)g(on)f(the)f
(n)o(um)o(b)q(er)g(of)g(iterations)j(of)d(the)h(algorithm.)301
267 y(See)k([Gon86)q(])g(for)h(some)f(related)g(w)o(ork.)0
379 y([Gon86])136 b(G.H.)14 b(Gonnet.)20 b(New)14 b(results)i(for)f(random)h
(determination)g(of)f(equiv)m(alence)g(of)g(expres-)301 451
y(sions.)27 b(In)17 b(B.W.)g(Char,)h(editor,)g Fj(ISSA)o(C)h('86:)24
b(Pr)n(o)n(c.)18 b(Int'l.)h(Symp.)g(on)g(Symb)n(olic)g(and)301
523 y(A)o(lgebr)n(aic)i(Computation)p Fk(,)e(pages)h(127{131.)h(A)o(CM)d
(Press,)i(1986.)30 b(Some)19 b(op)q(en)g(prob-)301 596 y(lems)h(in)g(the)g
(same)f(general)i(area)f(as)g(that)g(co)o(v)o(ered)e(b)o(y)h([Gon84)q(])g
(are)h(solv)o(ed)g(in)g(this)301 668 y(pap)q(er.)0 780 y([GRSS93])97
b(M.)22 b(Golin,)k(R.)c(Raman,)i(C.)e(Sc)o(h)o(w)o(arz,)h(and)g(M.)f(Smid.)40
b(Randomized)23 b(data)h(struc-)301 852 y(tures)15 b(for)f(the)h(dynamic)f
(closest-pair)j(problem.)h(In)c Fj(Pr)n(o)n(c.)h(F)l(ourth)h(A)o(nn.)g(A)o
(CM-SIAM)301 924 y(Symp.)f(on)f(Discr)n(ete)h(A)o(lgorithms)p
Fk(,)e(pages)h(301{310,)i(Austin,)d(TX,)f(Jan)o(uary)i(1993.)j(The)301
996 y(authors)22 b(describ)q(e)e(a)h(new)f(randomized)h(data)g(structure,)f
(the)g Fj(sp)n(arse)g(p)n(artition)t Fk(,)g(for)301 1068 y(solving)f(the)d
(dynamic)g(closest-pair)j(problem.)d(Using)h(this)g(data)g(structure,)f(the)g
(clos-)301 1141 y(est)k(pair)h(of)f(a)g(set)g(of)g Fh(n)g Fk(p)q(oin)o(ts)h
(in)g Fh(k)r Fk(-dimensional)h(space,)e(for)g(an)o(y)g(\014xed)g
Fh(k)r Fk(,)g(can)g(b)q(e)301 1213 y(found)e(in)f(constan)o(t)h(time.)e(If)h
(the)f(p)q(oin)o(ts)j(are)d(c)o(hosen)h(from)g(a)g(\014nite)h(univ)o(erse,)e
(and)i(if)301 1285 y(the)13 b(\015o)q(or)i(function)f(is)f(a)o(v)m(ailable)j
(at)d(unit-cost,)i(then)e(the)g(data)h(structure)f(supp)q(orts)h(in-)301
1357 y(sertions)h(in)o(to)g(and)f(deletions)h(from)f(the)f(set)h(in)h(exp)q
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1430 y(exp)q(ected)k Fh(O)q Fk(\()p Fh(n)p Fk(\))h(space.)f(Here,)f(it)i(is)g
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1502 y(adv)o(ersary)c(who)h(do)q(es)g(not)g(kno)o(w)f(the)g(random)g(c)o
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b(S.)22 b(Goldw)o(asser)i(and)f(M.)e(Sipser.)40 b(Priv)m(ate)23
b(coins)g(v)o(ersus)f(public)h(coins)g(in)g(in)o(terac-)301
1686 y(tiv)o(e)g(pro)q(of)i(systems.)41 b Fj(A)n(dvanc)n(es)25
b(in)f(Computing)h(R)n(ese)n(ar)n(ch)d(5:)35 b(R)n(andomness)24
b(and)301 1758 y(Computation)p Fk(,)f(1989.)38 b(This)23 b(w)o(ork)f
(establishes)h(equiv)m(alence)e(b)q(et)o(w)o(een)g(the)g(notions)301
1830 y(of)c(in)o(teractiv)o(e)f(pro)q(ofs)i(in)o(tro)q(duced)e(in)h([GMR89)q
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(eared)f(in)g Fj(Pr)n(o)n(c.)f(18th)i(A)o(nn.)g(A)o(CM)f(Symp.)g(on)h(The)n
(ory)f(of)g(Computing)p Fk(,)g(1986,)301 1975 y(pp.)h(59{68\).)0
2086 y([Gup93])133 b(R.)16 b(Gupta.)23 b(\010-test:)f(P)o(erfect)15
b(hashed)i(index)f(test)h(for)f(resp)q(onse)h(v)m(alidation.)25
b(In)16 b Fj(Pr)n(o)n(c.)301 2159 y(1993)k(IEEE)f(Int'l.)i(Conf.)f(on)g
(Computer)f(Design)p Fk(,)h(Cam)o(bridge,)f(MA,)f(Oct)g(1993.)30
b(A)301 2231 y(sc)o(heme)23 b(for)h(c)o(hec)o(king)f(the)h(\014delit)o(y)g
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2303 y(tailored)f(sequence)e(of)h(test)f(inputs)i(is)f(describ)q(ed.)f
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h(p)q(erfect)f(hashing)h(function)g Fh(h)p Fk(\()p Fh(x)p Fk(\))e(that)i(map)
f(the)f(exp)q(ected)g(test)301 2447 y(outcomes)j(to)f(the)g(sequence)g([1)8
b(.)g(.)g(.)f Fh(m)p Fk(].)20 b(This)h(sequence)e(is)i(c)o(hec)o(k)o(ed)d(b)o
(y)i(a)h(hardw)o(are)301 2520 y(implemen)o(tation)d(of)e Fh(h)p
Fk(\()p Fh(x)p Fk(\))g(and)h(an)g(up-coun)o(ter.)0 2631 y([GW86])137
b(A.)20 b(G.)g(Green)o(b)q(erg)g(and)h(A.)e(W)l(eiss.)34 b(A)20
b(lo)o(w)o(er)g(b)q(ound)i(for)e(probabilisti)q(c)j(algorithms)301
2704 y(for)15 b(\014nite)g(state)g(mac)o(hines.)j Fj(Journal)d(of)h(Computer)
g(and)g(System)g(Scienc)n(es)p Fk(,)g(33\(1\):88,)938 2828
y(120)p eop
bop 301 196 a Fk(August)16 b(1986.)22 b(A)15 b(pro)q(of)i(that)f(the)g
(running)h(time)e(cannot)h(b)q(e)g(b)q(etter)f(than)i(\012\(2)1827
164 y Fe(n)p 1827 170 20 2 v 1829 191 a Fb(2)1853 196 y Fh(n)p
Fk(\))f(is)301 269 y(presen)o(ted.)0 380 y([GY89])150 b(R.)12
b(Graham)g(and)g(A.)f(Y)l(ao.)j(On)e(the)f(improbabilit)o(y)j(of)e(reac)o
(hing)h(Byzan)o(tine)e(consensus.)301 452 y(In)h Fj(Pr)n(o)n(c.)h(21st)g(A)o
(nn.)i(A)o(CM)e(Symp.)g(on)h(The)n(ory)f(of)h(Computing)p Fk(,)f(pages)g
(467{478,)i(1989.)301 524 y(The)i(maxim)o(um)f(probabilit)o(y)j
Fh(\014)908 531 y Ff(n;t)970 524 y Fk(of)e(obtaining)i(consensus)e(is)g
(attac)o(k)o(ed)f(for)h Fh(t)d Fg(\025)g Fh(n=)p Fk(3)301 597
y(\(F)l(or)19 b(smaller)h(v)m(alues,)f(deterministic)g(algorithms)i(are)e(a)o
(v)m(ailable,)i(so)e Fh(\014)1672 604 y Ff(n;t)1735 597 y Fk(=)f(1.\))g(The)
301 669 y(smallest)i(non-trivial)g(case,)f Fh(\014)880 676
y Fl(3)p Ff(;)p Fl(1)926 669 y Fk(,)f(is)h(sho)o(wn)g(to)g(b)q(e)f(\()1308
619 y Fc(q)p 1349 619 19 2 v 1349 669 a Fk(\(5\))13 b Fg(\000)f
Fk(1\))p Fh(=)p Fk(2,)19 b(the)f(recipro)q(cal)i(of)301 741
y(the)c(golden)i(ratio.)f(In)f(a)h(restricted)f(mo)q(del,)h(it)f(is)i(sho)o
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Fk(1,)j(if)301 813 y Fh(t=n)d(>)g Fk(1)e Fg(\000)556 792 y
Fl(1)p Fd(\000)p Fl(log)5 b(1)p Fd(\000)p Ff(\017)712 780 y
Fb(1)p Fe(=)p Fb(2)p 528 802 260 2 v 528 832 a Fl(log)h(\(1)p
Fd(\000)p Fl(\(1)p Fd(\000)p Ff(\017)p Fl(\))726 822 y Fb(1)p
Fe(=)p Fb(2)774 832 y Fl(\))793 813 y Fk(,)15 b(then)h Fh(\014)961
820 y Ff(n;t)1021 813 y Fh(<)e(\017)p Fk(.)0 925 y([Had86])137
b(V.)18 b(Hadzilacos.)29 b(Ben-Or's)18 b(randomized)h(proto)q(col)h(for)f
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301 1069 y(Science,)h(Univ)o(ersit)o(y)f(of)h(T)l(oron)o(to,)i(Octob)q(er)e
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1141 y(rectness)d(of)g(Ben-Or's)g([BO83)o(])g(probabilistic)j(algorithm)f
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301 1397 y(Symp.)g(on)g(The)n(ory)e(of)i(Computing)p Fk(,)f(pages)g(299{306,)
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1831 y(randomized)h(parallel)h(algorithms)g(for)e(the)g(c)o(hain)g(sorting)h
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Fj(A)o(lgorithmics:)22 b(The)15 b(Spirit)g(of)g(Computing)p
Fk(.)i(Addison-W)l(esley)l(,)e(1987.)j(This)301 2087 y(b)q(o)q(ok)26
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2270 y([Has90])145 b(J.)19 b(Hastad.)31 b(Pseudo-random)20
b(generators)g(under)g(uniform)f(assumptions.)32 b(In)19 b
Fj(Pr)n(o)n(c.)301 2342 y(22nd)g(A)o(nn.)h(A)o(CM)e(Symp.)h(on)g(The)n(ory)f
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2415 y(MD,)h(Ma)o(y)f(1990.)39 b(Hastad)23 b(pro)o(v)o(es)e(that)h(giv)o(en)g
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2487 y(the)d(uniform)g(mo)q(del)g(\(i.e.,)e(cannot)i(b)q(e)g(in)o(v)o(erted)e
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y(p)q(olynomial)d(time)d(statistical)j(tests.)938 2828 y(121)p
eop
bop 0 195 a Fk([Her92])147 b(T.)16 b(Herman.)j(Self-stabilizatio)q(n:)24
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267 y(puting)p Fk(,)24 b(6\(2\):95{98,)g(1992.)38 b(Herman)20
b(uses)i(randomization)h(to)f(con)o(v)o(ert)e(Dijkstra's)301
340 y(k-state)f(m)o(utual)h(exclusion)g(proto)q(col)g(for)f(unidirectional)j
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(unreliable)i(threshold)f(circuits)g(are)301 959 y(considered.)0
1073 y([Hoa74])140 b(C.)19 b(A.)f(R.)g(Hoare.)28 b(Monitors:)g(An)18
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1145 y(munic)n(ations)19 b(of)f(the)h(A)o(CM)p Fk(,)d(17\(2\):549{557,)k
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2353 y(V)l(erlag,)e(1981.)22 b(This)c(w)o(ork)e(is)h(an)g(early)f(surv)o(ey)g
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670 y(a)g(function)g Fh(\033)g Fk(tak)o(es)f(less)g(than)h
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1145 y Fh(F)7 b Fk(\()p Fh(x)p Fk(\))15 b(is)h(one)f(that)h(is)g(easily)h
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Fh(h)g Fk(is)h(guaran)o(teed,)301 1830 y(the)g(sc)o(heme)e(su\013ers)j(from)e
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301 1902 y(of)j(the)f(searc)o(h)g(for)g Fh(h)p Fk(.)0 2014
y([JKS84])135 b(J.)22 b(Ja'Ja',)g(V.)f(K.)g(Prasanna)j(Kumar,)e(and)h(J.)e
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g(sets)g(of)g(proto)q(cols.)31 b Fj(SIAM)20 b(Journal)g(on)g(Computing)p
Fk(,)g(13\(4\):840{849,)301 2159 y(No)o(v)o(em)o(b)q(er)e(1984.)34
b(This)21 b(pap)q(er)f(is)h(a)f(study)g(of)g(the)g(comm)o(unication)g
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b(computation)g(mo)q(dels.)e(It)f(is)i(widely)g(conjectured)e(that)h
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b Fh(P)7 b(P)301 2375 y Fk(for)21 b(p)q(olynomial)i(time)d(complexit)o(y)g
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2447 y(gaps)d(exist)e(among)h(the)f(corresp)q(onding)j(comm)o(unication)e
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b(J.)h(v)m(an)g(Leeu)o(w)o(en,)g(editor,)301 2631 y Fj(Handb)n(o)n(ok)d(of)g
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(67{161.)h(Elsevier)e(and)g(The)g(MIT)f(Press)i(\(co-publishers\),)938
2828 y(124)p eop
bop 301 195 a Fk(1990.)52 b(Johnson)27 b(presen)o(ts)e(an)h(extensiv)o(e)f
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b Fj(SIAM)d(Journal)301 526 y(on)25 b(Computing)p Fk(,)h(18\(6\):1149{117)q
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814 y(are)j(matc)o(hings)g(in)g(the)f(graph.)g(F)l(or)h(this)g(sc)o(heme)e
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1073 y([JVV86])126 b(M.)12 b(R.)f(Jerrum,)h(L.)g(G.)g(V)l(alian)o(t,)h(and)g
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1145 y(natorial)i(structures)d(from)g(a)h(uniform)g(distribution.)18
b Fj(The)n(or)n(etic)n(al)13 b(Computer)h(Scienc)n(e)p Fk(,)301
1217 y(43:169{188,)i(1986.)g(This)d(pap)q(er)g(considers)g(the)f(class)h(of)g
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b(generation)i(of)e(com)o(binatorial)i(structures)e(from)g(a)g(uniform)h
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Fk(uniform)f(generation)h(of)f(`e\016cien)o(tly)e(v)o(eri\014able')i(com)o
(binatorial)301 1434 y(structures)f(is)g(reducible)f(to)h(appro)o(ximate)g
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1692 y(Ma)o(y)k(1992.)27 b(A)17 b(randomized)h(v)m(arian)o(t)g(of)g(the)f
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Fk(constrain)o(ts,)g(uses)f(an)h(exp)q(ected)301 1836 y(sub)q(exp)q(onen)o
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2022 y(pro)q(ducts.)36 b Fj(Journal)22 b(of)f(the)h(A)o(CM)p
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Fk(\()476 2292 y Fl(1)p 446 2300 76 2 v 446 2329 a Ff(n)467
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b Fj(Communic)n(ations)301 2497 y(of)22 b(the)h(A)o(CM)p Fk(,)e
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2642 y(putational)e(complexit)o(y)d(theory)l(.)g(It)f(discusses)j
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eop
bop 301 195 a Fk(abilistic)23 b(analysis)f(of)f(appro)o(ximation)h
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267 y(problems.)0 381 y([Kar90])144 b(R.)21 b(M.)f(Karp.)34
b(An)21 b(in)o(tro)q(duction)h(to)f(randomized)h(algorithms.)36
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(algorithm)301 856 y(in)o(v)o(ests)e(an)f(amoun)o(t)h(of)g(w)o(ork)f
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928 y Fk(\()p Fh(x)p Fk(\))p Fh(;)8 b(h)465 935 y Fl(2)484
928 y Fk(\()p Fh(x)p Fk(\))p Fh(;)g Fg(\001)g(\001)g(\001)g
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1000 y Fh(h)329 1007 y Ff(i)360 1000 y Fk(are)k(random)g(v)m(ariables)h(|)f
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1362 y(and)17 b(presen)o(ts)f(a)h(n)o(um)o(b)q(er)e(of)h(t)o(ypical)h
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Fh(n)301 1836 y Fk(v)o(ertices)15 b(and)i Fh(m)e Fk(\(p)q(ossibly)k(w)o(eigh)
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1909 y(the)f(v)o(ertices)e(in)o(to)i(t)o(w)o(o)g(non-empt)o(y)f(sets)g
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eop
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1758 y(Symp.)g(on)f(F)l(oundations)h(of)g(Computer)f(Scienc)n(e)t
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301 2191 y(an)d(exp)q(onen)o(tial)h(reduction)e(in)h(the)f(size)h(of)f(the)g
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2704 y(lems.)38 b Fj(Information)23 b(Pr)n(o)n(c)n(essing)f(L)n(etters)p
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1878 y(Conf.)20 b(on)f(Pr)n(ob)n(abilistic)g(Computational)h(Complexity)p
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(k)f(for)h(o)q(ccurrences)h(of)g(the)f(pattern)h(string)h(in)f(the)g(text)301
556 y(string.)g(The)f(running)h(time)e(of)h(the)g(algorithm)i(is)e
Fh(O)q Fk(\(\()p Fh(n)t Fg(\000)t Fh(m)t Fk(+)t(1\))p Fh(m)p
Fk(\))f(in)i(the)e(w)o(orst)h(case,)301 628 y(where)18 b(the)g(text)g(is)h
(of)g(length)g Fh(n)f Fk(and)h(the)f(pattern)h(is)g(of)g(length)g
Fh(m)p Fk(,)f(but)g(can)h(b)q(e)f(ex-)301 701 y(p)q(ected)13
b(to)h(run)f(in)h(time)f Fh(O)q Fk(\()p Fh(n)5 b Fk(+)g Fh(m)p
Fk(\))13 b(in)h(practice.)f(The)g(probabilit)o(y)i(that)f(the)f(algorithm)301
773 y(rep)q(orts)21 b(a)g(false)g(matc)o(h)f(is)h(1)p Fh(=n)p
Fk(.)g(Tw)o(o-dimensional)i(patterns)e(are)f(also)i(considered.)301
845 y(An)13 b(earlier)g(v)o(ersion)g(of)g(this)h(pap)q(er)f(app)q(eared)g(as)
g(T)l(ec)o(hnical)h(Rep)q(ort)f(TR-31-81,)i(Aik)o(en)301 917
y(Computation)j(Lab,)f(Harv)m(ard)g(Univ)o(ersit)o(y)l(,)e(1981.)0
1031 y([KR88])151 b(H.)13 b(Karlo\013)h(and)g(P)l(.)f(Ragha)o(v)m(an.)k
(Randomized)c(algorithms)i(and)f(pseudorandom)g(n)o(um-)301
1103 y(b)q(ers.)i(In)c Fj(Pr)n(o)n(c.)g(20th)i(A)o(nn.)h(A)o(CM)e(Symp.)h(on)
g(The)n(ory)f(of)g(Computing)p Fk(,)h(pages)f(310{321,)301
1176 y(1988.)20 b(F)l(ollo)o(wing)c(up)f(on)f(Bac)o(h's)f(w)o(ork)h([Bac91],)
g(this)h(pap)q(er)g(studies)g(pseudo-random)301 1248 y(substitutes)k(\(with)g
(small)g(seeds\))f(for)g(purely)g(random)h(c)o(hoices)e(in)i(sorting,)g
(selection)301 1320 y(and)j(oblivious)j(message)c(routing.)i(An)e(in)o
(teresting)h(result)g(is)h(that)f(the)f(linear)i(con-)301 1392
y(gruence)f(pseudo-random)i(n)o(um)o(b)q(er)e(generator)h(prop)q(osed)h(b)o
(y)e(Kn)o(uth)g([Kn)o(u73])g(can)301 1465 y(in)o(teract)17
b(with)g(some)f(quic)o(ksort)g(algorithms.)0 1578 y([Kro85])144
b(L.)16 b(Kronsjo.)21 b Fj(Computational)c(Complexity)g(of)g(Se)n(quential)i
(and)d(Par)n(al)r(lel)i(A)o(lgorithms)p Fk(.)301 1650 y(John)c(Wiley)f(and)g
(Sons,)h(New)e(Y)l(ork,)g(1985.)17 b(Chapter)c(5,)g(Section)g(5.3,)g
(addresses)g(prob-)301 1723 y(abilistic)20 b(algorithms.)g(Rabin's)e
(algorithms)i(for)e(primalit)o(y)g(and)g(the)f(Nearest)g(Neigh-)301
1795 y(b)q(ors)h(problem)e(are)h(describ)q(ed.)0 1909 y([KRR91])115
b(H.)17 b(Karlo\013,)i(Y.)d(Rabani,)j(and)f(Y.)e(Ra)o(vid.)25
b(Lo)o(w)o(er)17 b(b)q(ounds)i(for)f(randomized)g Fh(k)r Fk(-serv)o(er)301
1981 y(and)e(motion)h(planning.)22 b(In)15 b Fj(Pr)n(o)n(c.)h(23r)n(d)f(A)o
(nn.)j(A)o(CM)e(Symp.)h(on)g(The)n(ory)e(of)i(Comput-)301 2053
y(ing)p Fk(,)k(pages)f(278{288,)i(New)d(Orleans,)h(LA,)f(Ma)o(y)g(1991.)32
b(Lo)o(w)o(er)19 b(b)q(ounds)h(are)g(pro)o(v)o(ed)301 2125
y(on)k(the)f(comp)q(etitiv)o(e)g(ratio)i(of)e(randomized)h(algorithms)h(for)f
(the)f(on-line)i Fh(k)r Fk(-serv)o(er)301 2198 y(problem)17
b(and)g(an)g(on-line)h(motion-planning)h(problem.)0 2311 y([KR)l(T93])120
b(M.-Y.)13 b(Kao,)i(J.)f(H.)f(Reif,)i(and)f(S.)g(R.)g(T)l(ate.)k(Searc)o
(hing)d(in)g(an)g(unkno)o(wn)f(en)o(vironmen)o(t:)301 2384
y(An)i(optimal)i(randomized)e(algorithm)i(for)f(the)e(co)o(w-path)i(problem.)
k(In)16 b Fj(Pr)n(o)n(c.)g(F)l(ourth)301 2456 y(A)o(nn.)i(A)o(CM-SIAM)e
(Symp.)g(on)h(Discr)n(ete)g(A)o(lgorithms)p Fk(,)e(pages)h(441{447,)h
(Austin,)f(TX,)301 2528 y(Jan)o(uary)h(1993.)24 b(The)17 b(\014rst)g
(randomized)g(algorithm)i(for)e(the)f Fh(w)q Fk(-lane)i(co)o(w-path)f(prob-)
301 2600 y(lem,)h(a)h(problem)g(of)f(searc)o(hing)h(in)g(an)f(unkno)o(wn)h
(en)o(vironmen)o(t,)e(is)i(giv)o(en.)f(The)g(algo-)938 2828
y(131)p eop
bop 301 195 a Fk(rithm)14 b(is)g(optimal)g(for)g Fh(w)h Fk(=)f(2)f(and)h
(evidence)e(is)i(supplied)h(that)e(it)h(is)g(optimal)h(for)e(larger)301
267 y(v)m(alues)18 b(of)e Fh(w)q Fk(.)0 379 y([KS92])160 b(P)l(.)18
b(N.)e(Klein)j(and)f(S.)f(Sairam.)27 b(A)17 b(parallel)j(randomized)e(appro)o
(ximation)h(sc)o(heme)e(for)301 451 y(shortest)23 b(paths.)40
b(In)22 b Fj(Pr)n(o)n(c.)g(24th)h(A)o(nn.)h(A)o(CM)e(Symp.)h(on)g(The)n(ory)f
(of)h(Computing)p Fk(,)301 523 y(pages)d(750{758,)i(Victoria,)e(B.C.,)d
(Canada,)k(Ma)o(y)d(1992.)30 b(A)19 b(randomized)g(algorithm)301
596 y(is)c(giv)o(en)g(for)f(appro)o(ximate)h(shortest)g(path)g(computation)g
(in)g(an)g(undirected)f(w)o(eigh)o(ted)301 668 y(graph.)0 780
y([KS93])160 b(D.)18 b(R.)e(Karger)i(and)g(C.)f(Stein.)25 b(An)1002
767 y(~)990 780 y Fh(O)r Fk(\()p Fh(n)1077 761 y Fl(2)1097
780 y Fk(\))17 b(algorithm)j(for)d(minim)o(um)g(cuts.)25 b(In)17
b Fj(Pr)n(o)n(c.)301 852 y(25th)j(A)o(nn.)f(A)o(CM)g(Symp.)g(on)h(The)n(ory)e
(of)h(Computing)p Fk(,)g(pages)g(757{765,)h(San)f(Diego,)301
924 y(CA,)g(Ma)o(y)f(1993.)31 b(A)18 b Fj(minimum)i(cut)g Fk(is)g(a)f(set)g
(of)g(edges)g(of)g(minim)o(um)g(w)o(eigh)o(t)g(whose)301 996
y(remo)o(v)m(al)12 b(disconnects)f(a)h(giv)o(en)f(graph.)g(Karger)h(and)f
(Stein)h(giv)o(e)f(a)g(strongly)h(p)q(olynomial)301 1068 y(randomized)23
b(algorithm)g(whic)o(h)f(\014nds)h(a)f(minim)o(um)g(cut)f(with)i(high)g
(probabilit)o(y)g(in)301 1141 y Fh(O)q Fk(\()p Fh(n)387 1123
y Fl(2)416 1141 y Fk(log)479 1119 y Fl(3)507 1141 y Fh(n)p
Fk(\))f(time.)f(Their)i(algorithm)h(can)e(b)q(e)g(implemen)o(ted)f(in)h
Fj(RNC)g Fk(using)h(only)301 1213 y Fh(n)330 1195 y Fl(2)371
1213 y Fk(pro)q(cessors,)f(and)f(is)h(th)o(us)e(the)h(\014rst)f(e\016cien)o
(t)g Fj(RNC)g Fk(algorithm)j(for)d(the)h(min-cut)301 1285 y(problem.)0
1397 y([KST90])125 b(P)l(.)13 b(Klein,)h(C.)f(Stein,)g(and)h(E.)e(T)l(ardos.)
17 b(Leigh)o(ton-Rao)e(migh)o(t)e(b)q(e)h(practical:)20 b(F)l(aster)13
b(ap-)301 1469 y(pro)o(ximation)19 b(algorithms)g(for)e(concurren)o(t)f
(\015o)o(w)i(with)f(uniform)h(capacities.)24 b(In)17 b Fj(Pr)n(o)n(c.)301
1541 y(22nd)i(A)o(nn.)h(A)o(CM)e(Symp.)h(on)g(The)n(ory)f(of)g(Computing)p
Fk(,)h(pages)f(310{321,)j(Baltimore,)301 1614 y(MD,)f(Ma)o(y)g(1990.)34
b(They)20 b(giv)o(e)g(an)h Fh(O)q Fk(\()p Fh(m)1081 1595 y
Fl(2)1109 1614 y Fk(log)10 b Fh(m)p Fk(\))20 b(exp)q(ected-time)f(randomized)
i(algo-)301 1686 y(rithm)15 b(for)g(appro)o(ximately)g(solving)i(the)d
(concurren)o(t)g(m)o(ulticommo)q(dit)o(y)h(\015o)o(w)g(problem)301
1758 y(with)i(uniform)g(capacities.)0 1870 y([Kur87])141 b(S.)12
b(A.)e(Kurtz.)j(A)e(note)h(on)g(random)g(p)q(olynomial)i(time.)f
Fj(SIAM)g(Journal)g(on)h(Computing)p Fk(,)301 1942 y(16\(5\):852{853)q(,)j
(Octob)q(er)d(1987.)20 b(Sho)o(ws)15 b(that)g Fh(P)1229 1924
y Ff(A)1264 1942 y Fg(\\)7 b Fh(P)1342 1924 y Ff(B)1387 1942
y Fk(=)14 b Fh(B)s(P)7 b(P)21 b Fk(with)15 b(probabilit)o(y)i(1)301
2014 y(for)24 b(indep)q(enden)o(t)f(random)g(sets)g Fh(A)g
Fk(and)g Fh(B)s Fk(.)f(Here,)h Fh(A)g Fk(and)g Fh(B)j Fk(are)d(sets)g
(consisting)301 2086 y(of)h(strings)h(c)o(hosen)f(at)g(random,)h(and)f
Fh(P)1101 2068 y Ff(A)1154 2086 y Fk(and)g Fh(P)1294 2068 y
Ff(B)1348 2086 y Fk(are)g(relativized)g(to)g Fh(A)f Fk(and)h
Fh(B)301 2159 y Fk(resp)q(ectiv)o(ely)l(.)16 b(See)f([Gil77)r(])h(for)g
(additional)k(notation.)0 2270 y([KUW86])100 b(R.)15 b(M.)g(Karp,)g(E.)g
(Upfal,)h(and)g(A.)e(Wigderson.)21 b(Constructing)c(a)f(p)q(erfect)f(matc)o
(hing)g(in)301 2342 y(Random)d(NC.)h Fj(Combinatoric)n(a)p
Fk(,)e(6:35{48,)i(1986.)i(P)o(erfect)10 b(matc)o(hing)h(is)h(a)f(fundamen)o
(tal)301 2415 y(problem)17 b(that)g(is)h(not)f(kno)o(wn)g(to)g(b)q(e)g(solv)m
(able)h(b)o(y)e(an)h Fj(NC)g Fk(algorithm,)h(i.e.,)e(a)h(parallel)301
2487 y(algorithm)c(running)g(in)f(time)f(p)q(olynomial)j(in)d(log)g
Fh(n)g Fk(and)h(using)g(a)g(n)o(um)o(b)q(er)e(of)i(pro)q(cessors)301
2559 y(p)q(olynomial)23 b(in)e Fh(n)p Fk(.)e(This)j(pap)q(er)e(pro)o(v)o(es)g
(that)g(p)q(erfect)f(matc)o(hing)i(is)g(in)f(random)h Fj(NC)301
2631 y Fk(and)15 b(giv)o(es)f(a)h(fast,)g(parallel,)h(randomized)f(algorithm)
h(for)e(\014nding)i(a)e(p)q(erfect)g(matc)o(hing)301 2704 y(in)j(a)g(simple)g
(graph.)938 2828 y(132)p eop
bop 0 195 a Fk([KVV90])113 b(R.)20 b(M.)f(Karp,)h(U.)f(V.)g(V)l(azirani,)i
(and)f(V.)f(V.)g(V)l(azirani.)32 b(An)20 b(optimal)h(algorithm)h(for)301
267 y(on-line)i(bipartite)f(matc)o(hing.)39 b(In)21 b Fj(Pr)n(o)n(c.)h(22nd)h
(A)o(nn.)g(A)o(CM)f(Symp.)h(on)g(The)n(ory)e(of)301 340 y(Computing)p
Fk(,)d(pages)g(352{358,)i(Baltimore,)e(MD,)f(Ma)o(y)g(1990.)26
b(An)17 b(on-line)i(algorithm)301 412 y(receiv)o(es)14 b(a)h(sequence)f(of)h
(requests)f(and)i(m)o(ust)e(resp)q(ond)i(to)f(eac)o(h)f(request)g(as)i(so)q
(on)g(as)f(it)301 484 y(is)j(receiv)o(ed.)d(In)h(con)o(trast,)g(an)i
(o\013-line)g(algorithm)h(ma)o(y)d(w)o(ait)h(un)o(til)h(all)g(requests)e(ha)o
(v)o(e)301 556 y(b)q(een)23 b(receiv)o(ed)e(b)q(efore)i(determining)h(its)f
(resp)q(onses.)g(The)g(authors)h(giv)o(e)f(a)g(simple,)301
628 y(randomized,)17 b(optimal,)g(on-line)h(algorithm)g(for)f(bipartite)g
(matc)o(hing.)0 740 y([KW85])137 b(R.)22 b(M.)f(Karp)h(and)g(A.)f(Wigderson.)
39 b(A)22 b(fast)g(parallel)i(algorithm)g(for)e(the)f(maximal)301
812 y(indep)q(enden)o(t)h(set)g(problem.)39 b Fj(Journal)22
b(of)h(the)g(A)o(CM)p Fk(,)e(32\(4\):762{773,)26 b(1985.)40
b(This)301 885 y(imp)q(ortan)o(t)14 b(pap)q(er)g(sho)o(w)o(ed)f(that)g(the)g
(maximal)h(indep)q(enden)o(t)f(set)f(problem)i(for)f(graphs)301
957 y(can)i(b)q(e)g(solv)o(ed)g(in)g(p)q(olylogarithmic)j(time)c(using)i(a)f
(p)q(olynomial)i(n)o(um)o(b)q(er)d(of)g(pro)q(cesses)301 1029
y(on)i(a)g(PRAM)e(in)i(whic)o(h)f(concurren)o(t)g(reads)g(and)h(writes)g(are)
f(disallo)o(w)o(ed.)i(They)e(deriv)o(e)301 1101 y(their)23
b(algorithm)h(from)d(a)i(randomized)f(one)g(using)i(a)e(tec)o(hnique)f(that)i
(has)f(b)q(ecome)301 1174 y(kno)o(wn)17 b(as)g(derandomization)h(via)f
Fh(k)r Fk(-wise)g(indep)q(endence.)0 1285 y([KZ88])157 b(R.)11
b(M.)f(Karp)i(and)g(Y.)e(Zhang.)k(A)c(randomized)i(parallel)h(branc)o(h)e
(and)h(b)q(ound)g(pro)q(cedure.)301 1357 y(In)24 b Fj(Pr)n(o)n(c.)f(20th)h(A)
o(nn.)h(A)o(CM)e(Symp.)h(on)h(The)n(ory)e(of)h(Computing)p
Fk(,)h(pages)g(290{300,)301 1430 y(1988.)h(A)16 b(general)i(tec)o(hnique)e
(assuming)j(no)f(sp)q(ecial)g(comm)o(unication)g(capabilities)i(is)301
1502 y(presen)o(ted.)0 1614 y([Lak90])145 b(Y.)20 b(N.)f(Lakshman.)34
b(On)21 b(the)f(complexit)o(y)g(of)g(computing)h(a)g(Gr\177)-24
b(obner)21 b(basis)h(for)e(the)301 1686 y(radical)15 b(of)e(a)h(zero)e
(dimensional)k(ideal.)h(In)12 b Fj(Pr)n(o)n(c.)i(22nd)g(A)o(nn.)h(A)o(CM)f
(Symp.)g(on)h(The)n(ory)301 1758 y(of)i(Computing)p Fk(,)e(pages)h(555{562,)i
(Baltimore,)e(MD,)f(Ma)o(y)f(1990.)21 b(Lakshmanan)c(sho)o(ws)301
1830 y(that)h(if)g(a)g(system)f(of)g(p)q(olynomials)k Fh(f)1012
1837 y Fl(1)1032 1830 y Fk(,)c Fh(f)1087 1837 y Fl(2)1106 1830
y Fh(;)8 b Fk(.)g(.)g(.)g Fh(;)17 b(f)1249 1837 y Ff(r)1285
1830 y Fk(in)h Fh(n)g Fk(v)m(ariables)h(with)f Fh(deg)r Fk(\()p
Fh(f)1821 1837 y Ff(i)1835 1830 y Fk(\))e Fg(\024)g Fh(d)301
1902 y Fk(o)o(v)o(er)22 b(the)h(rational)i(n)o(um)o(b)q(ers)d(has)h(only)h
(\014nitely)f(man)o(y)f(a\016ne)h(zeros,)h(then)e(all)i(the)301
1975 y(a\016ne)17 b(zeros)h(can)f(b)q(e)g(determined)g(in)h(time)f(p)q
(olynomial)j(in)e Fh(d)1475 1957 y Ff(n)1516 1975 y Fk(b)o(y)e(a)i(Las)g(V)l
(egas)g(t)o(yp)q(e)301 2047 y(randomized)f(algorithm.)0 2159
y([LC88])160 b(T.G.)19 b(Lewis)h(and)f(C.R.)f(Co)q(ok.)30 b(Hashing)21
b(for)e(dynamic)g(and)g(static)h(in)o(ternal)g(tables.)301
2231 y Fj(Computer)p Fk(,)d(21:45{56,)i(1988.)24 b(The)17 b(authors)h(surv)o
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(tec)o(hniques)301 2375 y(exc)o(hange)d(sp)q(eed)g(for)g(memory)l(.)e(It)i
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2447 y(topics,)e(dynamic)f(and)g(static)g(hash)h(tables,)f(p)q(erfect)f
(hashing,)i(and)g(minimal)g(p)q(erfect)301 2520 y(hashing.)0
2631 y([Leh27])146 b(D.)21 b(H.)e(Lehmer.)33 b Fj(Bul)r(letin)24
b(of)d(the)g(A)o(meric)n(an)h(Mathematic)n(al)f(So)n(ciety)p
Fk(,)g(33:327{340,)301 2704 y(1927.)i(This)17 b(pap)q(er)g(presen)o(ts)f(the)
g(Lucas-Lehmer)h(heuristic)g(for)g(primalit)o(y)g(testing.)938
2828 y(133)p eop
bop 0 195 a Fk([Leh82])146 b(D.)19 b(Lehmann.)28 b(On)19 b(primalit)o(y)h
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(y)301 267 y(1982.)h(Lehmann)15 b(presen)o(ts)f(t)o(w)o(o)g(algorithms)j(for)
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y(24th)17 b(A)o(nn.)f(A)o(CM)g(Symp.)g(on)h(The)n(ory)e(of)h(Computing)p
Fk(,)f(pages)h(77{96,)h(Victoria,)f(B.C.,)301 668 y(Canada,)i(Ma)o(y)e(1992.)
22 b(This)c(surv)o(ey)d(includes)i(the)f(topic)h(of)g(randomized)g(wiring.)0
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852 y(teractiv)o(e)e(pro)q(of)h(systems.)30 b(In)18 b Fj(Pr)n(o)n(c.)h(31st)h
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y(Computer)h(Scienc)n(e)p Fk(,)j(pages)d(2{10,)i(1990.)38 b(The)22
b(authors)g(presen)o(t)f(a)h(new)f(algebraic)301 996 y(tec)o(hnique)e(for)h
(constructing)h(IP)f(systems)f(and)i(pro)o(v)o(e)e(that)h(ev)o(ery)e
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1252 y([LLM90])120 b(F.)27 b(T.)f(Leigh)o(ton,)31 b(D.)c(Lisinski,)k(and)d
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1325 y(randomly-wired)d(m)o(ultistage)g(net)o(w)o(orks.)45
b(In)24 b Fj(Pr)n(o)n(c.)g(1990)g(IEEE)h(Int'l.)h(Conf.)f(on)301
1397 y(Computer)d(Design)p Fk(,)g(pages)f(380{385,)j(1990.)36
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(Randomly-wired)301 1614 y(m)o(ultibutter\015ies)j(p)q(erform)g(b)q(etter)f
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2270 y(and)i(graph)g(connectivit)o(y)l(.)k Fj(Combinatoric)n(a)p
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2342 y(algorithms)24 b(for)d(connectivit)o(y)g(computation,)j(b)q(oth)e(of)f
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(Symp.)h(on)301 2704 y(F)l(oundations)d(of)e(Computer)g(Scienc)n(e)t
Fk(,)h(1986,)f(pp.)f(39{53.\).)938 2828 y(134)p eop
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301 340 y(Symp.)f(on)g(F)l(oundations)h(of)f(Computer)g(Scienc)n(e)p
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670 y(m)o(ultibutter\015ies)g(and)g(randomly-wired)h(splitter)g(net)o(w)o
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Fk(,)k(41\(5\):578{587,)i(Ma)o(y)c(1992.)33 b(This)21 b(pap)q(er)g(describ)q
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1145 y(in)o(terconnection)c(net)o(w)o(orks.)k Fj(Information)17
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b(surv)o(ey)301 1217 y(of)22 b(recen)o(t)d(researc)o(h)i(on)g(randomly)h
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1289 y(b)q(een)16 b(found)h(to)f(b)q(e)g(exceptionally)h(fault-toleran)o(t)h
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y(routing)h(and)f(circuit-switc)o(hing)h(applications.)0 1475
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2022 y(editor,)23 b Fj(F)l(undamentals)g(of)e(Computing)h(The)n(ory)p
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b(\(fairly\))h(simple)h(circuit)f(that)f(\(usually\))301 2353
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(1990.)34 b(A)20 b Fh(k)r Fk(-round)h(tournamen)o(t)f(o)o(v)o(er)f
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2570 y(in)k(a)g(sorting)h(net)o(w)o(ork)d(and)i(t)o(w)o(o)f(randomized)h
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301 267 y(p)q(erm)o(utation)e(net)o(w)o(orks.)i Fj(IEEE)e(T)l(r)n(ans.)g(on)g
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y(1981.)36 b(This)22 b(pap)q(er)f(presen)o(ts)f(deterministic)h(algorithms)i
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y(parallel)19 b(time.)0 598 y([LR81])159 b(D.)15 b(Lehmann)h(and)f(M.)g(O.)f
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1981 y(enc)n(e)p Fk(,)g(pages)g(482{492,)i(1992.)25 b(The)17
b(authors)i(presen)o(t)d(an)i Fh(O)q Fk(\()p Fh(n)1499 1963
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Fk(\))g(algorithm)h(to)301 2053 y(appro)o(ximate)i(the)g(v)o(olume)f(of)h(a)f
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Fk(in)h Fh(E)s(P)7 b(S)s(P)g(AC)t(E)s Fk(,)16 b Fh(N)5 b(P)1217
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2582 y Ff(A)1518 2600 y Fk(\(2)1561 2582 y Ff(n)1585 2600 y
Fh(=n)p Fk(\).)938 2828 y(136)p eop
bop 0 195 a Fk([LSP82])135 b(L.)22 b(Lamp)q(ort,)i(R.)d(Shostak,)j(and)e(M.)f
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1289 y(771{782,)e(Victoria,)e(B.C.,)e(Canada,)i(Ma)o(y)f(1992.)25
b(E\016cien)o(t)17 b(randomized)h(and)f(deter-)301 1362 y(ministic)e
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b(probabilistic)h(b)o(y)d(determining)g(algebraic)301 1692
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(Com-)301 1909 y(puter)e(Scienc)n(e)t Fk(,)h(1985,)f(pp.)e(65{73.)j(It)d(is)h
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301 2672 y(allel,)i(distributed)f(and)g(sequen)o(tial)g(algorithms)i(are)d
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267 y(sion)20 b(of)e(the)f(Omega)h(net)o(w)o(ork.)26 b Fj(Journal)19
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340 y(is)19 b(an)g(extension)f(of)g(V)l(alian)o(t)h(and)g(Aleliunas')g
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Fh(x)916 821 y Ff(i)947 814 y Fk(and)i Fh(x)1071 821 y Ff(j)1089
814 y Fk(,)f Fh(x)1148 821 y Ff(i)1161 814 y Fk(\()p Fh(modp)p
Fk(\))f Fg(6)p Fk(=)g Fh(x)1412 821 y Ff(j)1430 814 y Fk(\()p
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1217 y(ume)22 b(1)g(of)g Fj(EA)l(TCS)i(Mono)n(gr)n(aphs)d(on)i(The)n(or)n
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1289 y(V)l(erlag,)27 b(1984.)47 b(V)l(olume)25 b(1)g(of)g(this)g(three-v)o
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1692 y(Mono)n(gr)n(aphs)d(on)i(The)n(or)n(etic)n(al)e(Computer)i(Scienc)n(e)p
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301 2239 y(geometry)l(.)0 2353 y([Mig80])143 b(M.)17 b(Mignotte.)26
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267 y(for)15 b(primalit)o(y)g(testing.)g(The)g(\014rst)f(one)h(runs)f(in)h
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340 y(is)23 b(actually)g(a)f(p)q(olynomial)i(time)d(algorithm)j(\()p
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1758 y(ized)i(parallel)h(algorithm)g(for)f(subtree)f(isomorphism)i(that)f
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1830 y Fh(O)q Fk(\()p Fh(n=)8 b Fk(log)k Fh(n)p Fk(\))21 b(pro)q(cessors.)h
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(the)301 1902 y(problem.)16 b(See)f(also)i(the)e(related)g(pap)q(er)h(\\P)o
(arallel)i(tree)d(con)o(traction)h(and)g(its)g(applica-)301
1975 y(tions,")h(in)e Fj(Pr)n(o)n(c.)g(26th)h(A)o(nn.)h(IEEE)f(Symp.)g(on)h
(F)l(oundations)g(of)f(Computer)g(Scienc)n(e)t Fk(,)301 2047
y(1985,)i(pp.)e(478{489;)i(and)f(the)f(companion)i(pap)q(er)f([MR91)o(].)0
2159 y([MR91])144 b(G.)21 b(L.)f(Miller)i(and)f(J.)f(H.)f(Reif.)34
b(P)o(arallel)23 b(tree)c(con)o(traction,)j(P)o(art)f(2:)30
b(F)l(urther)20 b(Ap-)301 2231 y(plications.)38 b Fj(SIAM)22
b(Journal)g(on)g(Computing)p Fk(,)g(20\(6\):1128{11)q(47,)j(Decem)o(b)q(er)20
b(1991.)301 2303 y(In)h(this)g(follo)o(w-up)i(of)e([MR89],)g(the)f(authors)i
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y(\\parallel)h(tree)c(con)o(traction)i(tec)o(hnique,")e(including)j
(algorithms)g(for)f(sub)q(expression)301 2447 y(ev)m(aluation,)d(tree)d(and)h
(graph)g(isomorphism,)h(and)f(building)i(cononical)f(forms)f(of)f(trees)301
2520 y(and)j(planar)h(graphs.)0 2631 y([MS82])153 b(K.)17 b(Mehlhorn)g(and)g
(E.)g(Sc)o(hmidt.)23 b(Las)17 b(V)l(egas)g(is)h(b)q(etter)f(than)g
(determinism)g(in)g(VLSI)301 2704 y(and)23 b(distributed)g(computing.)39
b(In)21 b Fj(Pr)n(o)n(c.)h(14th)g(A)o(nn.)h(A)o(CM)g(Symp.)f(on)h(The)n(ory)e
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bop 301 195 a Fj(Computing)p Fk(,)19 b(pages)g(330{337,)h(1982.)28
b(This)19 b(pap)q(er)f(demonstrates)h(a)f(problem)g(where)301
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Fk(.)g(.)g(.)g Fh(x)1328 347 y Ff(n)1351 340 y Fk(\),)18 b
Fh(Y)28 b Fk(=)16 b(\()p Fh(y)1555 347 y Fl(1)1575 340 y Fh(;)8
b(y)1621 347 y Fl(2)1640 340 y Fh(;)g Fk(.)g(.)g(.)f Fh(y)1751
347 y Ff(n)1775 340 y Fk(\),)17 b(where)301 412 y Fh(x)329
419 y Ff(i)355 412 y Fk(and)c Fh(y)470 419 y Ff(i)496 412 y
Fk(are)f(in)o(tegers)g(b)q(et)o(w)o(een)g(0)g(and)h(2)1085
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491 y Ff(i)1847 484 y Fk(=)i Fh(y)1923 491 y Ff(i)1936 484
y Fk(,)301 556 y(and)f(0)f(otherwise|is)h(to)f(b)q(e)g(computed)g(with)g
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Fh(n)605 610 y Fl(2)640 628 y Fk(message)g(bits)h(in)f(the)g(deterministic)g
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Fk(log)j Fh(n)p Fk(\))301 701 y(a)o(v)o(erage)16 b(running-time)i
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b(S.)25 b(Micali)h(and)f(A.)f(Shamir.)47 b(An)24 b(impro)o(v)o(emen)o(t)f(of)
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1029 y(zero-kno)o(wledge)e(based)f(iden)o(ti\014cation)i(and)e(digital)j
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(complexit)o(y)f(unc)o(hanged.)0 1357 y([MS92])153 b(B.)14
b(M.)f(Maggs)i(and)g(R.)f(K.)f(Sitaraman.)20 b(Simple)14 b(algorithms)j(for)d
(routing)i(on)e(butter\015y)301 1430 y(net)o(w)o(orks)h(with)g(b)q(ounded)h
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(B.C.,)e(Canada,)i(Ma)o(y)e(1992.)30 b(The)19 b(authors)301
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Fh(N)5 b Fk(-input)18 b(butter\015y)e(with)h(b)q(ounded-size)h(queues)e(in)h
Fh(O)q Fk(\(log)11 b Fh(N)5 b Fk(\))17 b(steps,)301 1719 y(with)g
(high-probabil)q(i)q(t)o(y)l(.)0 1830 y([MSV85])116 b(F.)16
b(Ma\016oli,)h(M.)f(G.)g(Sp)q(eranza,)h(and)g(C.)f(V)l(ercellis.)21
b(Randomized)c(algorithms.)24 b Fj(Com-)301 1902 y(binatorial)19
b(Optimization|A)o(nnotate)n(d)g(Biblio)n(gr)n(aphies)p Fk(,)e(pages)h
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b(J.)22 b(Matousek,)i(M.)d(Sharir,)k(and)d(E.)g(W)l(elzl.)40
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2159 y(programming.)17 b(In)12 b Fj(Pr)n(o)n(c.)h(Eighth)h(A)o(nn.)h(A)o(CM)f
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b(1{8,)h(Berlin,)f(German)o(y)l(,)e(June)h(1992.)34 b(They)20
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Fk(\()p Fh(nde)639 2355 y Fl(4)657 2322 y Fg(p)p 698 2322 148
2 v 698 2355 a Ff(d)6 b Fl(ln\()p Ff(n)p Fl(+1\))848 2375 y
Fk(\))16 b(time)g(in)h(the)f(unit)h(cost)g(mo)q(del.)0 2487
y([MT85])145 b(U.)21 b(Man)o(b)q(er)g(and)h(M.)e(T)l(ompa.)37
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2559 y(decision)h(trees.)36 b Fj(Journal)22 b(of)f(the)i(A)o(CM)p
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e(Decision)938 2828 y(141)p eop
bop 301 195 a Fk(trees)18 b(that)g(allo)o(w)h(in)o(ternal)g(randomization)g
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267 y(of)14 b(error)g(are)g(sho)o(wn)g(to)h(run)e(no)i(faster)f
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y(for)h(a)g(collection)g(of)g(problems.)f(An)g(earlier)h(v)o(ersion)g(of)g
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y(algorithms)25 b(for)e(problems)g(in)g(computational)h(geometry)l(.)e(The)g
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Fh(k)r Fk(,)e(and)i(\(3\))301 1574 y(hidden)e(surface)f(remo)o(v)m(al)h(for)f
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1902 y(131,)e(North)g(Con)o(w)o(a)o(y)l(,)f(NH,)e(June)i(1991.)21
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1975 y(called)j(dynamic)f(sampling,)h(that)f(can)f(b)q(e)h(used)g(to)g
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2231 y([Mul91b])113 b(K.)24 b(Mulm)o(uley)l(.)44 b(Randomized)24
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b(results)301 2303 y(in)g(dynamic)f(sampling.)40 b(In)21 b
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b(balancing)h(and)301 2631 y(dynamic)i(sh)o(u\017ing.)23 b(In)16
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(Com-)301 2704 y(puter)e(Scienc)n(e)p Fk(,)g(pages)e(180{196,)j(1991.)i(This)
c(pap)q(er)f(presen)o(ts)g(a)g(general)g(randomized)938 2828
y(142)p eop
bop 301 195 a Fk(algorithm)20 b(for)e(problems)h(suc)o(h)f(as)g(the)g
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267 y(Hulls)g(and)e(V)l(oronoi)i(Diagrams.)0 381 y([Mul92])140
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670 y(substituted)14 b(for)e(the)g(random)h(source)g(in)g(man)o(y)f
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b(In)301 1073 y Fj(Pr)n(o)n(c.)c(F)l(ourth)g(Int'l.)h(Conf.)g(on)g
(Combinatorial)f(Pattern)i(Matching,)f Fk(Lecture)e(Notes)301
1145 y(in)h(Computer)f(Science,)f(V)l(ol.)h(684,)h(pages)g(164{178,)h(P)o
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16 b Fj(Pr)n(o)n(c.)g(23r)n(d)g(A)o(nn.)i(A)o(CM)f(Symp.)301
1981 y(on)i(The)n(ory)e(of)i(Computing)p Fk(,)f(pages)g(307{316,)i(New)c
(Orleans,)j(LA,)d(Ma)o(y)h(1991.)26 b(Ran-)301 2053 y(domized)15
b(parallel)j(algorithms)f(are)e(giv)o(en)g(for)g(constructing)h(a)f(p)q
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2198 y(p)q(olylogari)q(thmic)k(time.)0 2311 y([MVN93])106 b(Y.)13
b(Matias,)i(J.)f(S.)f(Vitter,)h(and)g(W.-C.)f(Ni.)18 b(Dynamic)c(generation)h
(of)f(discrete)f(random)301 2384 y(v)m(ariables.)32 b(In)19
b Fj(Pr)n(o)n(c.)g(F)l(ourth)h(A)o(nn.)h(A)o(CM-SIAM)f(Symp.)g(on)g(Discr)n
(ete)g(A)o(lgorithms)p Fk(,)301 2456 y(pages)13 b(361{370,)i(Austin,)e(TX,)e
(Jan)o(uary)h(1993.)k(E\016cien)o(t)11 b(randomized)i(algorithms)h(are)301
2528 y(giv)o(en)h(to)g(generate)g(a)g(random)g(v)m(ariate)h(distributed)g
(according)g(to)f(a)g(dynamically)h(set)301 2600 y(of)h(w)o(eigh)o(ts.)g(The)
f(base)h(v)o(ersion)g(of)g(eac)o(h)f(algorithm)j(generates)e(the)f(discrete)g
(random)301 2672 y(v)m(ariate)d(in)g Fh(O)q Fk(\(log)631 2651
y Fd(\003)659 2672 y Fh(N)5 b Fk(\))12 b(exp)q(ected)f(time)h(and)g(up)q
(dates)h(a)f(w)o(eigh)o(t)g(in)g Fh(O)q Fk(\(2)1628 2654 y
Fl(log)1675 2642 y Fa(\003)1699 2654 y Ff(N)1732 2672 y Fk(\))g(exp)q(ected)
938 2828 y(143)p eop
bop 301 195 a Fk(time)13 b(in)h(the)f(w)o(orst)g(case.)g(It)g(is)g(sho)o(wn)h
(ho)o(w)g(to)f(reduce)f(the)h(up)q(date)h(time)f(to)g Fh(O)q
Fk(\(log)1859 174 y Fd(\003)1887 195 y Fh(N)5 b Fk(\))301 267
y(amortized)17 b(exp)q(ected)e(time.)0 381 y([MV)o(O91])106
b(A.)13 b(Menezes,)g(S.)g(V)l(anstone,)h(and)h(T.)e(Ok)m(amoto.)18
b(Reducing)c(elliptic)h(curv)o(e)e(logarithms)301 453 y(to)19
b(logarithms)h(in)e(a)g(\014nite)g(\014eld.)27 b(In)17 b Fj(Pr)n(o)n(c.)h
(23r)n(d)g(A)o(nn.)h(A)o(CM)g(Symp.)f(on)h(The)n(ory)f(of)301
526 y(Computing)p Fk(,)g(pages)g(80{89,)g(New)f(Orleans,)g(LA,)g(Ma)o(y)g
(1991.)25 b(They)16 b(presen)o(t)h(a)g(prob-)301 598 y(abilistic)22
b(p)q(olynomial-ti)q(me)f(algorithm)g(for)f(the)f(elliptic)i(curv)o(e)d
(logarithm)j(problem,)301 670 y(the)16 b(\014rst)h(sub)q(exp)q(onen)o
(tial-time,)h(general-purp)q(ose)g(algorithm)g(for)f(the)f(problem.)0
784 y([MVV87])106 b(K.)11 b(Mulm)o(uley)l(,)h(U.)e(V.)h(V)l(azirani,)i(and)e
(V.)g(V.)f(V)l(azirani.)15 b(Matc)o(hing)c(is)h(as)g(easy)g(as)g(matrix)301
856 y(in)o(v)o(ersion.)31 b Fj(Combinatoric)n(a)p Fk(,)19 b(7:105{113,)j
(1987.)31 b(An)18 b(elegan)o(t)i(parallel,)h(randomized)301
928 y(algorithm)g(for)e(\014nding)i(a)e(p)q(erfect)f(matc)o(hing)h(in)h(a)f
(simple)h(graph)g(based)f(on)g(T)l(utte's)301 1000 y(matrix)e(is)g(presen)o
(ted.)e(The)h(algorithm,)i(whic)o(h)e(is)h(made)f(p)q(ossible)j(b)o(y)c(a)i
(probabilistic)301 1073 y(lemma)e(called)g(the)f(isolation)k(lemma,)c
(requires)g(in)o(v)o(ersion)h(of)g(a)g(single)g(in)o(teger)g(matrix)301
1145 y(whic)o(h)i(can)f(b)q(e)h(parallelized.)0 1259 y([MW90])130
b(B.)11 b(McKa)o(y)g(and)h(N.)f(W)l(ormald.)j(Uniform)e(generation)h(of)f
(random)g(graphs)h(of)f(mo)q(derate)301 1331 y(degree.)35 b
Fj(Journal)22 b(of)g(A)o(lgorithms)p Fk(,)f(11:52{67,)j(1990.)37
b(A)21 b(randomized)g(algorithm)i(is)301 1403 y(giv)o(en)15
b(for)g(generating)g Fh(k)r Fk(-regular)h(graphs)g(on)f Fh(n)g
Fk(v)o(ertices,)e(uniformly)j(at)e(random.)h(The)301 1475 y(exp)q(ected)d
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Fh(O)q Fk(\()p Fh(n)1600 1443 y Fb(1)p 1600 1449 16 2 v 1600
1470 a(3)1623 1475 y Fk(\).)e(Sp)q(ecial)h(cases,)301 1548
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(are)i(considered.)0 1661 y([MWHC93])58 b(B.S.)21 b(Ma)s(jewski,)i(N.C.)d(W)l
(ormald,)k(G.)e(Ha)o(v)m(as,)h(and)f(Z.J.)f(Czec)o(h.)36 b(Graphs,)24
b(h)o(yp)q(er-)301 1733 y(graphs)19 b(and)f(hashing.)26 b(In)17
b Fj(Pr)n(o)n(c.)g(19th)h(Int'l.)h(Workshop)f(on)g(Gr)n(aph-The)n(or)n(etic)f
(Con-)301 1806 y(c)n(epts)22 b(in)g(Computer)f(Scienc)n(e)i(\(WG'93\))p
Fk(,)d(Utrec)o(h)o(t,)g(The)g(Netherlands,)h(June)g(1993.)301
1878 y(The)i(authors)h(generalize)g(the)e(metho)q(d)h(presen)o(ted)f(in)i
([CHM92])e(b)o(y)g(mapping)i(the)301 1950 y(input)c(set)f(in)o(to)g(a)g(h)o
(yp)q(ergraph)g(rather)g(than)h(a)f(graph.)g(This)h(mo)q(di\014cation)h(allo)
o(ws)g(a)301 2022 y(reduction)f(in)g(the)f(size)g(of)h(the)f(program,)h
(while)g(main)o(taining)i(all)e(other)g(features)f(of)301 2095
y(the)d(metho)q(d.)g(Also,)h(the)f(hash)h(function)g(generation)h(time)e(is)h
(reduced.)0 2208 y([MZ86])150 b(N.)25 b(Megiddo)h(and)g(E.)f(Zemel.)48
b(An)25 b Fh(O)q Fk(\()p Fh(n)8 b Fk(log)j Fh(n)p Fk(\))25
b(randomizing)i(algorithm)h(for)d(the)301 2281 y(w)o(eigh)o(ted)c(Euclidean)i
(1-cen)o(ter)d(problem.)36 b Fj(Journal)22 b(of)g(A)o(lgorithms)p
Fk(,)f(7\(3\):358{368,)301 2353 y(Sep)12 b(1986.)i(A)d(set)g(of)g(p)q(oin)o
(ts)i Fh(p)848 2360 y Ff(i)876 2353 y Fk(=)h(\()p Fh(x)975
2360 y Ff(i)989 2353 y Fh(;)8 b(y)1035 2360 y Ff(i)1049 2353
y Fk(\))j(and)h(their)f(w)o(eigh)o(ts)h Fh(w)1484 2360 y Ff(i)1498
2353 y Fk(,)23 b(1)14 b Fg(\024)g Fh(i)f Fg(\024)h Fh(n)d Fk(are)g(giv)o(en.)
301 2425 y(It)17 b(is)i(required)e(to)h(\014nd)f(a)h(p)q(oin)o(t)h
Fh(p)f Fk(that)f(minimizes)i(the)e(maxim)o(um)g(\014rst)g(momen)o(t)g(of)301
2497 y(the)d(w)o(eigh)o(ts)h(of)f(the)g Fh(p)714 2504 y Ff(i)742
2497 y Fk(s,)g(i.e.,)g(the)g Fh(p)g Fk(that)g(minimizes)h Fh(H)t
Fk(\()p Fh(p)p Fk(\))g(=)e Fh(M)5 b(AX)1630 2504 y Fl(1)p Fd(\024)p
Ff(i)p Fd(\024)p Ff(n)1739 2497 y Fh(w)1774 2504 y Ff(i)1802
2497 y Fh(d)p Fk(\()p Fh(p;)j(p)1916 2504 y Ff(i)1931 2497
y Fk(\))301 2570 y(where)k Fh(d)p Fk(\()p Fh(p;)c(p)552 2577
y Ff(i)568 2570 y Fk(\))k(is)h(the)f(magnitude)h(of)f(the)g(distance)h(b)q
(et)o(w)o(een)f Fh(p)g Fk(and)h Fh(p)1614 2577 y Ff(i)1629
2570 y Fk(.)f(A)f(randomized)301 2642 y(algorithm)19 b(that)d(do)q(es)h(this)
g(with)g(a)g(small)h(probabilit)o(y)g(of)f(error)f(is)h(presen)o(ted.)938
2828 y(144)p eop
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(distributions.)301 267 y Fj(SIAM)d(Journal)g(on)g(Computing)p
Fk(,)f(21\(3\):438{449,)j(June)c(1992.)j(Natara)s(jan)e(generalizes)301
340 y(the)k(mo)q(del)h(of)g(probably)g(appro)o(ximate)g(learning)h(prop)q
(osed)g(b)o(y)e(V)l(alian)o(t)h([V)l(al84b)q(].)0 453 y([Nis90])156
b(N.)24 b(Nisan.)45 b(Pseudorandom)26 b(generators)f(for)f(space-b)q(ounded)i
(computations.)46 b(In)301 526 y Fj(Pr)n(o)n(c.)16 b(22nd)g(A)o(nn.)i(A)o(CM)
e(Symp.)g(on)h(The)n(ory)e(of)h(Computing)p Fk(,)g(pages)g(204{212,)i(Balti-)
301 598 y(more,)d(MD,)g(ma)o(y)f(1990.)21 b(Pseudorandom)c(generators)e(are)h
(constructed)f(that)g(con)o(v)o(ert)301 670 y Fh(O)q Fk(\()p
Fh(S)d Fk(log)e Fh(R)p Fk(\))17 b(truly)g(random)f(bits)i(to)e
Fh(R)h Fk(bits)g(that)f(app)q(ear)i(random)e(to)h(an)o(y)f(algorithm)301
742 y(that)h(runs)f(in)h Fj(SP)l(A)o(CE\(S\))r Fk(.)f(In)g(particular,)i(an)o
(y)e(randomized)g(p)q(olynomial)j(time)d(algo-)301 814 y(rithm)21
b(that)h(runs)f(in)g(space)g Fh(S)j Fk(can)c(b)q(e)h(sim)o(ulated)h(using)g
(only)f Fh(O)q Fk(\()p Fh(S)12 b Fk(log)f Fh(n)p Fk(\))20 b(random)301
887 y(bits.)e(Applications)i(are)e(giv)o(en)g(for)f(\\deterministic)i
(ampli\014cation,")h(the)e(problem)g(of)301 959 y(reducing)f(the)f
(probabilit)o(y)j(of)d(error)g(of)h(randomized)g(algorithms.)0
1073 y([Nis93])156 b(N.)17 b(Nisan.)27 b(On)17 b(read-once)h(vs.)f(m)o
(ultiple)i(access)f(to)g(randomness)g(in)g(logspace.)27 b Fj(The-)301
1145 y(or)n(etic)n(al)21 b(Computer)f(Scienc)n(e)p Fk(,)i(107:135{144,)h
(1993.)32 b(This)21 b(pap)q(er)f(sho)o(ws)h(that)f(ev)o(ery)301
1217 y(language)h(accepted)d(with)i(b)q(ounded)g(t)o(w)o(o-sided)g(error)f(b)
o(y)f(a)h(read-once)g(randomized)301 1289 y(logspace)i(mac)o(hine)d(can)h(b)q
(e)g(accepted)f(with)i(zero)e(error)h(b)o(y)f(a)h(randomized)h(logspace)301
1362 y(mac)o(hine)c(with)h(m)o(ultiple)g(access)f(to)g(the)g(random)g(bits.)g
(Also,)h(the)e(class)j(of)e(languages)301 1434 y(accepted)h(with)g(t)o(w)o
(o-sided)h(error)f(b)o(y)f(a)h(randomized)h(logspace)g(mac)o(hine)f(with)g(m)
o(ulti-)301 1506 y(ple)f(access)f(to)g(the)g(random)h(bits)g(is)g(sho)o(wn)f
(to)h(b)q(e)f(the)g(class)h(of)f(languages)j(that)d(are)g(in)301
1578 y(logspace)j(relativ)o(e)e(to)h(almost)h(ev)o(ery)c(oracle.)0
1692 y([NN90])151 b(J.)19 b(Naor)h(and)g(M.)e(Naor.)31 b(Small-bias)22
b(probabilit)o(y)f(spaces:)28 b(E\016cien)o(t)19 b(constructions)301
1764 y(and)f(applications.)28 b(In)17 b Fj(Pr)n(o)n(c.)h(22nd)h(A)o(nn.)g(A)o
(CM)f(Symp.)g(on)h(The)n(ory)e(of)i(Computing)p Fk(,)301 1836
y(pages)25 b(213{223,)k(Baltimore,)e(MD,)c(Ma)o(y)h(1990.)47
b(This)25 b(pap)q(er)g(sho)o(ws)g(an)g(e\016cien)o(t)301 1909
y(construction)20 b(of)f(a)g(small)h(probabilit)o(y)g(space)f(on)g
Fh(n)f Fk(binary)i(random)f(v)m(ariables)h(suc)o(h)301 1981
y(that)k(for)g(ev)o(ery)e(subset,)j(its)f(parit)o(y)g(is)h(either)e(zero)g
(or)h(one)g(with)g(\\almost")i(equal)301 2053 y(probabilit)o(y)l(.)18
b(Applications)h(are)e(sho)o(wn)g(in)g(problems)g(suc)o(h)g(as)g(the)f
(derandomization)301 2125 y(of)23 b(algorithms)h(and)e(reducing)h(the)f(n)o
(um)o(b)q(er)f(of)h(random)h(bits)f(required)g(b)o(y)g(certain)301
2198 y(randomized)17 b(algorithms.)0 2311 y([NS93])161 b(M.)18
b(Naor)g(and)h(L.)f(Sto)q(c)o(kmey)o(er.)24 b(What)19 b(can)f(b)q(e)g
(computed)g(lo)q(cally?)38 b(In)18 b Fj(Pr)n(o)n(c.)g(25th)301
2384 y(A)o(nn.)j(A)o(CM)e(Symp.)g(on)h(The)n(ory)f(of)h(Computing)p
Fk(,)f(pages)h(184{193,)h(San)f(Diego,)g(CA,)301 2456 y(Ma)o(y)i(1993.)42
b(In)22 b(the)g(con)o(text)g(of)g(a)h(distributed)h(net)o(w)o(ork,)f(Naor)g
(and)g(Sto)q(c)o(kmey)o(er)301 2528 y(in)o(v)o(estigate)15
b Fj(L)n(o)n(c)n(al)r(ly)g(Che)n(ckable)j(L)n(ab)n(eling)c
Fk(\()p Fj(LCL)p Fk(\))g(problems,)g(where)g(the)g(legalit)o(y)i(of)f(a)301
2600 y(lab)q(eling)20 b(\(e.g.,)c(coloring\))k(can)d(b)q(e)g(c)o(hec)o(k)o
(ed)e(lo)q(cally;)20 b(i.e.,)c(within)j(time)e(\(or)g(distance\))301
2672 y(indep)q(enden)o(t)j(of)g(the)g(size)f(of)h(the)g(net)o(w)o(ork.)e
(Among)i(their)g(results)g(they)f(sho)o(w)i(that)938 2828 y(145)p
eop
bop 301 195 a Fk(randomization)24 b(cannot)d(mak)o(e)g(an)g(LCL)h(problem)g
(lo)q(cal;)j(i.e.,)c(if)h(a)f(problem)h(has)g(a)301 267 y(lo)q(cal)d
(randomized)d(algorithm)j(then)d(it)g(has)h(a)g(lo)q(cal)h(deterministic)f
(algorithm.)0 381 y([NY90])151 b(M.)23 b(Naor)h(and)h(M.)e(Y)l(ung.)43
b(Public-k)o(ey)24 b(cryptosystems)f(pro)o(v)m(ably)i(secure)e(against)301
453 y(c)o(hosen)g(cypher-text)e(attac)o(k.)40 b(In)22 b Fj(Pr)n(o)n(c.)g
(22nd)i(A)o(nn.)g(A)o(CM)f(Symp.)g(on)g(The)n(ory)f(of)301
526 y(Computing)p Fk(,)14 b(pages)g(427{437,)h(1990.)i(The)c(authors)h(sho)o
(w)f(ho)o(w)h(to)f(construct)g(a)g(public-)301 598 y(k)o(ey)18
b(cryptosystem)f(secure)h(against)j Fj(chosen)f(ciphertest)h(attacks)t
Fk(,)e(giv)o(en)g(a)g(publi-k)o(ey)301 670 y(cryptosystem)d(secure)f(against)
j(passiv)o(e)e(ea)o(v)o(esdropping)g(and)h(a)f(nonin)o(teractiv)o(e)g(zero-)
301 742 y(kno)o(wledge)h(pro)q(of)g(system)f(in)h(the)f(shared)h(string)g(mo)
q(del.)0 856 y([NZ93])158 b(N.)19 b(Nisan)h(and)g(D.)f(Zuc)o(k)o(erman.)29
b(More)20 b(deterministic)g(sim)o(ulation)h(in)f(logspace.)32
b(In)301 928 y Fj(Pr)n(o)n(c.)20 b(25th)h(A)o(nn.)h(A)o(CM)f(Symp.)g(on)g
(The)n(ory)f(of)h(Computing)p Fk(,)h(pages)f(235{244,)i(San)301
1000 y(Diego,)13 b(CA,)e(Ma)o(y)g(1993.)k(It)10 b(is)i(sho)o(wn)g(that)g(an)o
(y)f(randomized)h Fj(sp)n(ac)n(e\(S\))f Fk(algorithm)j(that)301
1073 y(uses)j(only)g Fj(p)n(oly\(S\))f Fk(random)h(bits)g(can)g(b)q(e)f(sim)o
(ulated)i(deterministically)g(in)f Fj(sp)n(ac)n(e\(S\))r Fk(,)301
1145 y(for)g Fh(S)s Fk(\()p Fh(n)p Fk(\))d Fg(\025)f Fk(log)e
Fh(n)p Fk(.)0 1259 y([Ore87])146 b(Y.)24 b(Oren.)46 b(On)24
b(the)g(cunning)i(p)q(o)o(w)o(er)e(of)h(c)o(heating)g(v)o(eri\014ers:)38
b(Some)24 b(observ)m(ations)301 1331 y(ab)q(out)f(zero)d(kno)o(wledge)i(pro)q
(ofs.)36 b(In)21 b Fj(Pr)n(o)n(c.)f(28th)i(A)o(nn.)g(IEEE)g(Symp.)f(on)h(F)l
(ounda-)301 1403 y(tions)d(of)e(Computer)h(Scienc)n(e)p Fk(,)h(pages)e
(462{471,)i(1987.)25 b(Oren)16 b(di\013eren)o(tiates)i(b)q(et)o(w)o(een)301
1475 y Fj(auxiliary-input)30 b Fk(zero-kno)o(wledge)25 b(and)g
Fj(blackb)n(ox-simulation)31 b Fk(zero-kno)o(wledge.)25 b(He)301
1548 y(sho)o(ws)18 b(that)f(all)h(kno)o(wn)f(zero-kno)o(wledge)g(pro)q(ofs)i
(are)d(in)i(the)e(latter)h(category)l(.)g(In)f(ad-)301 1620
y(dition,)h(it)f(is)h(pro)o(v)o(ed)e(that)h Fj(blackb)n(ox-simulation)22
b Fk(zero-kno)o(wledge)16 b(implies)h Fj(auxiliary-)301 1692
y(input)28 b Fk(kno)o(wledge,)c(and)f(that)g(the)f(latter)h(corresp)q(onds)h
(to)f(the)f(original)j(de\014nition)301 1764 y(giv)o(en)17
b(in)g([GMR89].)0 1878 y([P)o(ac87])147 b(J.)12 b(P)o(ac)o(hl.)i(A)d(lo)o(w)o
(er)h(b)q(ound)h(for)f(probabilistic)j(distributed)e(algorithms.)j
Fj(Journal)d(of)g(A)o(l-)301 1950 y(gorithms)p Fk(,)g(8\(1\):53{65,)i(1987.)h
(The)c(minim)o(um)h(n)o(um)o(b)q(er)e(of)i(messages)g(required)f(to)h(\014nd)
301 2022 y(the)19 b(extremal)f(v)m(alue)h(of)f(no)q(de)i(ids)f(in)g(an)g
(async)o(hronous)g(net)o(w)o(ork)f(deterministically)301 2095
y(is)h(\002\()p Fh(n)27 b Fk(log)10 b Fh(n)p Fk(\).)18 b(This)i(pap)q(er)e
(sho)o(ws)h(that)g(this)g(b)q(ound)h(holds)f(ev)o(en)e(for)i(probabilistic)
301 2167 y(algorithms.)0 2281 y([P)o(az71])147 b(A.)18 b(P)o(az.)29
b Fj(Intr)n(o)n(duction)19 b(to)h(Pr)n(ob)n(abilistic)g(A)o(utomata)p
Fk(.)29 b(Academic)18 b(Press,)h(1971.)30 b(P)o(az)301 2353
y(dev)o(elops)17 b(a)f(theory)h(of)f(equiv)m(alence)g(among)i(probabilistic)h
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814 y(Berlin,)24 b(German)o(y)l(,)e(June)g(1992.)40 b(Giv)o(en)22
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1620 y([P)o(er85])152 b(K.)16 b(P)o(erry)l(.)k(Randomized)c(Byzan)o(tine)f
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(enden)o(t)f(of)h Fh(n)301 1909 y Fk(and)k Fh(t)p Fk(.)e(As)h(usual,)i
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301 2456 y(for)15 b(\014nding)g(the)f(largest)h(of)f(a)h(set)f(of)g
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301 2528 y(algorithm)g(requires)e Fh(O)q Fk(\()p Fh(n)8 b Fk(log)j
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y(the)18 b(A)o(CM)p Fk(,)d(29\(1\):261{266)q(,)j(Jan)o(uary)f(1982.)22
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y(imating)17 b(pac)o(king)e(in)o(teger)g(problems.)k Fj(Journal)d(of)g
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1430 y(computational)g(geometry)l(,)c(graph)i(theory)l(,)e(VLSI,)h(and)g(net)
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b(pap)q(er)g(presen)o(ts)301 1902 y(randomized)c(algorithms)i(for)d
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y(on)k(mesh)e(connected)h(computers.)f(The)h(time)g(b)q(ounds)h(of)f(these)g
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y([Ra)s(j91b])120 b(S.)13 b(Ra)s(jasek)m(aran.)k(Randomized)d(algorithms)h
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2231 y(nical)26 b(Rep)q(ort)f(MS-CIS-91-92,)j(Dept.)d(of)f(Computer)h(and)g
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2447 y(on)e(a)g(mesh)g(connected)f(computer)g(are)g(surv)o(ey)o(ed.)f(The)h
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1942 y(Journal)i(on)h(Computing)p Fk(,)f(14\(2\):396{409)q(,)i(Ma)o(y)d
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523 y(parallel)i(sorting)e(algorithm.)22 b Fj(SIAM)16 b(Journal)h(on)f
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1397 y(238,)f(1984.)36 b(They)20 b(presen)o(t)h(probabilistic)i(distributed)f
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1614 y(nizing)j(In)o(terpro)q(cess)d(Comm)o(unication)i(in)g(Real)f(Time,")g
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2487 y(21\(3\):466{485)q(,)h(June)d(1992.)19 b(An)13 b(optimal)j(parallel)g
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bop 301 195 a Fk(a)17 b(new)f(random)h(sampling)h(tec)o(hnique)d(for)h
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267 y(A)o(nn.)j(A)o(CM)d(Symp.)i(on)f(The)n(ory)g(of)g(Computing)t
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(analysis)301 1259 y(of)c(deterministic)g(algorithms)h(with)f(random)g
(inputs,)g(i.e.,)f(algorithms)i(for)f(whic)o(h)f(the)301 1331
y(space)j(of)h(all)g(inputs)g(has)f(a)h(kno)o(wn)f(probabilit)o(y)i
(distribution.)g(It)d(also)j(presen)o(ts)d(t)o(w)o(o)301 1403
y(probabilisti)q(c)i(algorithms)f(with)f(real)g(time)f(resp)q(onse)h(for)g
(the)f(problem)g(of)h(comm)o(uni-)301 1475 y(cation)h(guard)f(sc)o(heduling.)
0 1589 y([Spr77])152 b(R.)22 b(Sprugnoli.)43 b(P)o(erfect)21
b(hash)j(functions:)35 b(A)22 b(single)i(prob)q(e)f(retriev)m(al)h(metho)q(d)
f(for)301 1661 y(static)c(sets.)26 b Fj(Communic)n(ations)18
b(of)h(the)g(A)o(CM)p Fk(,)e(20:841{850,)k(1977.)27 b(This)19
b(is)f(the)g(\014rst)301 1733 y(discussion)c(on)e(p)q(erfect)f(hashing;)j
(describ)q(es)e(heuristics)g(for)g(constructing)g(p)q(erfect)f(hash)301
1806 y(functions.)0 1919 y([SS77])171 b(R.)21 b(Solo)o(v)m(a)o(y)i(and)f(V.)f
(Strassen.)38 b(A)21 b(fast)h(Mon)o(te-Carlo)h(test)f(for)g(primalit)o(y)l(.)
38 b Fj(SIAM)301 1992 y(Journal)21 b(on)h(Computing)p Fk(,)f(6\(1\):84{85,)j
(Marc)o(h)19 b(1977.)35 b(Another)20 b(test)g(for)h(primalit)o(y)301
2064 y(based)d(on)f(the)g(abundance)g(of)g(witnesses)h(to)f(comp)q(ositeness)
i(of)e Fh(n)g Fk(is)g(presen)o(ted.)f(The)301 2136 y(test)23
b(en)o(tails)h(pic)o(king)f(a)g(random)g(n)o(um)o(b)q(er)f
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Fh(")g Fk(=)301 2208 y Fh(a)327 2190 y Fl(\()p Ff(n)p Fd(\000)p
Fl(1\))p Ff(=)p Fl(2)458 2208 y Fk(\()50 b(\(mo)q(d)17 b Fh(n)p
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Fk(=)e(\()p Fh(a=n)p Fk(\))301 2281 y(equals)h Fh(")f Fk(then)g
Fh(n)g Fk(is)h(prime,)f(else,)g(if)g(either)g Fh(g)r(cd)p Fk(\()p
Fh(a;)8 b(n)p Fk(\))15 b Fh(>)g Fk(1)j(or)f Fh(\016)g Fg(6)p
Fk(=)e Fh(")p Fk(,)h(decide)h Fh(n)g Fk(to)h(b)q(e)301 2353
y(comp)q(osite.)f(The)f(second)h(decision)g(has)g(less)g(than)1282
2333 y Fl(1)p 1282 2341 18 2 v 1282 2370 a(2)1321 2353 y Fk(probabilit)o(y)h
(of)f(b)q(eing)g(wrong.)0 2467 y([SS78])171 b(R.)17 b(Solo)o(v)m(a)o(y)h(and)
f(V.)g(Strassen.)24 b(Erratum:)f(A)16 b(fast)i(Mon)o(te-Carlo)g(test)f(for)g
(primalit)o(y)l(.)301 2539 y Fj(SIAM)f(Journal)f(on)g(Computing)p
Fk(,)g(7\(1\),)f(F)l(eb.)f(1978.)19 b(A)14 b(minor)g(correction)g(in)h(the)e
(anal-)301 2611 y(ysis)i(presen)o(ted)e(in)i([SS77])f(is)g(rep)q(orted)g(b)o
(y)g(the)f(authors.)i(The)f(basic)h(results)f(of)g([SS77)q(],)301
2683 y(ho)o(w)o(ev)o(er,)h(still)j(hold.)938 2828 y(156)p eop
bop 0 195 a Fk([SS90])171 b(J.)15 b(P)l(.)f(Sc)o(hmidt)h(and)g(A.)f(Siegel.)
20 b(The)15 b(spatial)i(complexit)o(y)d(of)h(oblivious)j Fh(k)r
Fk(-prob)q(e)d(hash)301 267 y(functions.)41 b Fj(SIAM)23 b(Journal)g(on)h
(Computing)p Fk(,)g(19\(5\):775{786,)j(1990.)41 b(This)23 b(pap)q(er)301
340 y(giv)o(es,)d(among)g(other)g(results,)g(a)g(lo)o(w)o(er)f(b)q(ound)i
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y(program)g(for)f(oblivious)j Fh(k)r Fk(-prob)q(e)e(hash)g(function.)f(A)g
(probabilistic)j(construction)e(of)301 484 y(a)g(family)g(of)f(oblivious)j
Fh(k)r Fk(-prob)q(e)e(hash)g(function)g(that)f(nearly)h(matc)o(h)e(this)i(b)q
(ound)g(is)301 556 y(also)e(giv)o(en.)0 670 y([SSS93])144 b(J.)13
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b(Cherno\013-Ho)q(e\013ding)e(b)q(ounds)f(for)e(ap-)301 742
y(plications)h(with)f(limited)g(indep)q(endence.)g(In)e Fj(Pr)n(o)n(c.)h(F)l
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b(Cherno\013-)301 887 y(Ho)q(e\013ding)15 b(b)q(ounds)f(are)f(frequen)o(tly)g
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959 y(algorithms)21 b(to)e(b)q(ound)h(the)f(tail)h(probabilities)i(of)d(the)f
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1176 y Fk(among)k(the)e(random)g(v)m(ariables.)0 1289 y([Sto85])155
b(L.)14 b(Sto)q(c)o(kmey)o(er.)g(On)f(appro)o(ximation)j(algorithms)f(for)f
(#P.)i Fj(SIAM)e(Journal)h(on)g(Com-)301 1362 y(puting)p Fk(,)j
(14\(4\):849{861,)i(1985.)k(The)17 b(author)g(explores)g(the)g(e\013ect)f(of)
h(appro)o(ximation)301 1434 y(and)23 b(randomization)h(on)e(the)f(complexit)o
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1506 y(#P)17 b(whic)o(h)f(has)h(problems)h(suc)o(h)e(as)h(coun)o(ting)g(the)f
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1578 y(a)g(graph,)g(the)f(size)g(of)h(bac)o(ktrac)o(k)e(searc)o(h)h(trees,)f
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(azirani.)17 b(Generating)e(quasi-random)f(sequences)f(from)g(semi-)301
1764 y(random)g(sources.)i Fj(Journal)f(of)f(Computer)h(and)g(System)g
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(ariable)301 1981 y(bias.)j(The)g(adv)o(ersary)f(can)g(lo)q(ok)i(at)e(the)g
(previously)h(output)g(bits,)g(and)g(use)f(them)g(to)301 2053
y(set)g(the)f(bias)h(in)g(the)f(coin.)h(The)f(bias,)h(whic)o(h)g(helps)g(mo)q
(del)g(correlation)g(among)h(bits,)301 2125 y(is)e(constrained)h(to)e(b)q(e)h
(b)q(et)o(w)o(een)e(t)o(w)o(o)h(limits.)0 2239 y([TN91])153
b(T.)19 b(T)l(okuy)o(ama)f(and)h(J.)g(Nak)m(ano.)28 b(Geometric)19
b(algorithms)h(for)f(a)g(minim)o(um)f(cost)h(as-)301 2311 y(signmen)o(t)24
b(problem.)45 b(In)23 b Fj(Pr)n(o)n(c.)g(Seventh)k(A)o(nn.)e(A)o(CM)f(Symp.)g
(on)g(Computational)301 2384 y(Ge)n(ometry)p Fk(,)c(pages)g(262{271,)j(North)
d(Con)o(w)o(a)o(y)l(,)g(NH,)e(June)i(1991.)33 b(An)20 b(e\016cien)o(t)e(ran-)
301 2456 y(domized)13 b(algorithm)g(is)g(giv)o(en)f(for)g(the)g(minim)o(um)g
(cost)g Fh(\025)p Fk(-assignmen)o(t)i(problem,)f(whic)o(h)301
2528 y(is)h(equiv)m(alen)o(t)g(to)g(the)f(minim)o(um)g(w)o(eigh)o(t)g
(one-to-man)o(y)h(matc)o(hing)g(problem)f(in)h(a)g(com-)301
2600 y(plete)h(bipartite)i(graph)f(\000)e(=)g(\()p Fh(A;)8
b(B)s Fk(\).)13 b(If)i Fh(A)g Fk(and)g Fh(B)j Fk(ha)o(v)o(e)c
Fh(n)h Fk(and)h Fh(k)h Fk(no)q(des)f(resp)q(ectiv)o(ely)l(,)301
2672 y(then)g(the)g(algorithm)j(requires)d Fh(O)q Fk(\()p Fh(k)r(n)c
Fk(+)f Fh(k)1103 2654 y Fl(3)p Ff(:)p Fl(5)1150 2672 y Fh(n)1179
2654 y Fl(0)p Ff(:)p Fl(5)1226 2672 y Fk(\))16 b(exp)q(ected)g(time.)938
2828 y(157)p eop
bop 0 195 a Fk([TO92])152 b(S.)27 b(T)l(o)q(da)h(and)g(M.)e(Ogiw)o(ara.)55
b(Coun)o(ting)28 b(classes)g(are)f(at)h(least)f(as)h(hard)g(as)f(the)301
267 y(p)q(olynomial-ti)q(me)14 b(hierarc)o(h)o(y)l(.)f Fj(SIAM)h(Journal)f
(on)h(Computing)p Fk(,)f(21\(2\):316{328,)i(April)301 340 y(1992.)24
b(Man)o(y)17 b(coun)o(ting)g(classes)h(are)f(sho)o(wn)g(to)g(b)q(e)g
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(hierarc)o(h)o(y)l(,)e(under)i(a)f(notion)i(of)f(randomized)g(reducibilit)o
(y)l(,)g(unless)301 484 y(the)d(p)q(olynomial-ti)q(me)i(hierarc)o(h)o(y)e
(collapses.)0 596 y([T)l(ut47])148 b(W.)13 b(T.)g(T)l(utte.)j(The)c
(factorization)j(of)f(linear)g(graphs.)j Fj(Journal)d(of)h(the)f(L)n(ondon)g
(Mathe-)301 668 y(matic)n(al)j(So)n(ciety)p Fk(,)d(22:107{111,)k(1947.)i(Let)
15 b Fh(G)p Fk(\()p Fh(V)s(;)8 b(E)s Fk(\))15 b(b)q(e)g(a)g(giv)o(en)f
(simple)i(graph)f(where)301 740 y Fh(V)30 b Fk(=)18 b Fg(f)p
Fk(1)p Fh(;)8 b Fk(2)p Fh(;)g Fk(.)g(.)g(.)g Fh(n)p Fg(g)p
Fk(.)18 b(Asso)q(ciate)i(a)f(v)m(ariable)h Fh(x)1160 747 y
Ff(ij)1209 740 y Fk(with)f(eac)o(h)g(edge)f Fh(e)1571 747 y
Ff(ij)1619 740 y Fg(2)g Fh(E)k Fk(and)d(de\014ne)301 812 y(the)d
Fh(n)10 b Fg(\002)f Fh(n)16 b Fk(matrix)g Fh(B)g Fk(=)e([)p
Fh(b)815 819 y Ff(ij)845 812 y Fk(])h(as)h(follo)o(ws.)h(If)e(there)g(is)i
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885 y(v)o(ertex)j Fh(j)k Fk(them)c Fh(b)642 892 y Ff(ij)691
885 y Fk(=)g(0.)h(Otherwise,)h Fh(b)1073 892 y Ff(ij)1121 885
y Fk(=)e Fh(x)1205 892 y Ff(ij)1254 885 y Fk(if)i Fh(i)e(>)g(j)k
Fk(and)e Fh(b)1554 892 y Ff(ij)1602 885 y Fk(=)f Fg(\000)p
Fh(x)1726 892 y Ff(ij)1775 885 y Fk(if)g Fh(i)f(<)g(j)s Fk(.)301
957 y(This)g(pap)q(er)f(pro)o(v)o(es)e(that)i Fh(G)g Fk(has)g(a)f(p)q(erfect)
g(matc)o(hing)h(if)f(and)h(only)g(if)g(det\()p Fh(B)s Fk(\))c
Fg(6)p Fk(=)h(0.)0 1068 y([TW87])140 b(M.)19 b(T)l(ompa)g(and)h(H.)e(W)l
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(ter-)301 1141 y(activ)o(e)h(pro)q(ofs)i(of)e(p)q(ossession)i(of)f
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1213 y(on)e(F)l(oundations)f(of)g(Computer)g(Scienc)n(e)p Fk(,)h(pages)e
(472{482,)j(1987.)k(T)l(ompa)18 b(and)f(W)l(oll)301 1285 y(presen)o(t)e(a)g
(general)h(theory)l(,)e(of)h(whic)o(h)h(IP)e(pro)q(ofs)j(for)e(graph)h
(isomorphism,)g(quadratic)301 1357 y(residuosit)o(y)i(and)f(kno)o(wledge)f
(of)h(discrete)f(logarithms)i(are)f(sp)q(ecial)h(cases.)0 1469
y([Tze89])146 b(W.)12 b(G.)g(Tzeng.)j(The)d(equiv)m(alence)h(and)g(learning)g
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y(30th)18 b(A)o(nn.)g(IEEE)f(Symp.)g(on)h(F)l(oundations)g(of)f(Computer)g
(Scienc)n(e)p Fk(,)h(pages)f(268{273,)301 1614 y(1989.)31 b(The)18
b(equiv)m(alence)h(problem)h(of)f(probabilistic)i(automata)f(is)g(solv)m
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y Fl(1)435 1686 y Fk(+)8 b Fh(n)510 1693 y Fl(2)529 1686 y
Fk(\))548 1668 y Fl(4)568 1686 y Fk(\),)15 b(where)f Fh(n)784
1693 y Fl(1)819 1686 y Fk(and)h Fh(n)941 1693 y Fl(2)975 1686
y Fk(are)g(the)g(n)o(um)o(b)q(er)f(of)h(states)g(in)g(the)g(t)o(w)o(o)f
(automata.)301 1758 y(The)25 b(problem)f(of)h(learning)g(probabilisti)q(c)i
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y(p)q(olynomial)19 b(time)d(is)i(also)f(presen)o(ted.)0 1942
y([Upf89])146 b(E.)21 b(Upfal.)37 b(An)20 b Fh(O)q Fk(\(log)11
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b(In)21 b Fj(Pr)n(o)n(c.)g(21st)301 2014 y(A)o(nn.)d(A)o(CM)f(Symp.)f(on)h
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b(pap)q(er)301 2086 y(presen)o(ts)22 b(the)f(\014rst)h(deterministic)h
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2303 y(can)g(b)q(e)f(routed)h(in)g Fh(O)q Fk(\(log)10 b Fh(N)17
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2487 y(algorithms.)29 b Fj(SIAM)19 b(Journal)g(on)h(Computing)p
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2559 y(transitiv)o(e)d(closure)f(algorithms)i(are)e(presen)o(ted)f(for)h(the)
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b(A)19 b(sc)o(heme)e(for)i(fast)g(parallel)i(comm)o(unication.)29
b Fj(SIAM)20 b(Journal)g(on)301 267 y(Computing)p Fk(,)c(11\(2\):350{361)q(,)
i(Ma)o(y)d(1982.)21 b(V)l(alian)o(t)c(giv)o(es)f(a)g(distributed)g
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(unique)i(sources)f(to)g(unique)g(destinations)i(in)301 412
y(an)17 b Fh(n)p Fk(-dimensional)i(binary)e(cub)q(e)f(in)h
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userdict /end-hook known{end-hook}if
%%EOF