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1055 y(n)2163 1033 y Fv(=)25 b(\()p Fp(V)5 b(;)15 b(E)5 b Fv(\))32 b(b)m(y)141 1146 y Fp(V)46 b Fv(=)25 b Fi(f)p Fv(1)p Fp(;)15 b(:)g(:)g(:)i(;)e(n)p Fi(g)p Fv(.)141 1259 y Fp(E)31 b Fv(=)25 b Fi(f)p Fv(\()p Fp(i;)15 b(j)5 b Fv(\))27 b(:)f Fi(j)p Fp(i)21 b Fi(\000)f Fp(j)5 b Fi(j)26 b(\024)f Fp(t)p Fi(g)p Fv(.)141 1372 y(Note)33 b(that)f Fp(P)631 1339 y Fn(t)618 1394 y(n)697 1372 y Fv(has)g Fi(\024)27 b Fp(tn)k Fv(edges.)45 b(Also)31 b(note)i(that)f(all)f(the)g(edges)i(are)f(en)m(tirely)f(con)m(tained)h (in)e(one)i(of)g(the)0 1484 y(follo)m(wing)d(\(non-disjoin)m(t\))g (sets:)141 1597 y Fi(f)p Fp(j)5 b(t)21 b Fv(+)f(1)p Fp(;)15 b(:)g(:)g(:)i(;)e Fv(\()p Fp(j)27 b Fv(+)20 b(2\))p Fp(t)p Fi(g)31 b Fv(as)g Fp(j)g Fv(=)25 b(0)p Fp(;)15 b Fv(1)p Fp(;)g(:)g(:)g(:)j(;)1608 1528 y Fo(\004)1657 1562 y Fn(n)p 1657 1577 43 4 v 1666 1629 a(t)1710 1528 y Fo(\005)1769 1597 y Fi(\000)i Fv(2)141 1710 y Fi(f)p Fp(n)g Fi(\000)g Fv(2)p Fp(t)h Fv(+)f(1)p Fp(;)15 b(:)g(:)g(:)i(;)e(n)p Fi(g)p Fv(.)0 1978 y 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Fv(log)17 b Fp(q)s Fv(\))35 b(=)f Fp(O)s Fv(\()p Fp(n)3271 677 y Fr(\(4)p Fh(\000)p Fr(2)p Fn(\013)p Fr(\))p Fn(\013)3540 689 y Fe(k)q Fd(\000)p Fl(1)3678 710 y Fv(log)16 b Fp(n)p Fv(\))427 846 y(pro)s(cessors)26 b(to)i(sort)f(it)f(in)g Fp(k)16 b Fi(\000)d Fv(1)27 b(rounds.)38 b(This)24 b(requires)i Fp(O)s Fv(\()p Fp(n)2590 813 y Fr(2)p Fn(\013)p Fh(\000)p Fr(3+\(4)p Fh(\000)p Fr(2)p Fn(\013)p Fr(\))p Fn(\013)3084 825 y Fe(k)q Fd(\000)p Fl(1)3222 846 y Fv(log)17 b Fp(n)p Fv(\))26 b(pro)s(cessors.)427 959 y(A)m(t)47 b(the)e(same)h(time)f(compare)h(all)e(\()p Fp(x;)15 b(y)s Fv(\))47 b(where)d Fp(x)51 b Fi(2)f Fp(W)57 b Fv(and)45 b Fp(y)53 b Fi(2)d(f)p Fp(x)3119 973 y Fr(1)3159 959 y Fp(;)15 b(:)g(:)g(:)i(;)e(x)3413 973 y Fn(n)3460 959 y Fi(g)p Fv(.)86 b(This)44 b(is)427 1072 y Fp(O)s Fv(\()p Fp(n)p Fi(j)p Fp(W)13 b Fi(j)p Fv(\))28 b(=)f Fp(O)s Fv(\()p Fp(n)1061 1039 y Fn(\013)1111 1072 y Fv(\))32 b(comparisons.)44 b(If)31 b(y)m(ou)i(set)f Fp(\013)c Fv(=)f Fp(\013)2385 1087 y Fn(k)2460 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Fm(then)p 2664 2342 83 4 v 33 w Fp(H)40 b Fm(has)33 b Fp(O)s Fv(\()p Fp(n)3104 2381 y Fr(3)p Fn(=)p Fr(2)3229 2414 y Fv(log)17 b Fp(n)p Fv(\))32 b Fm(e)-5 b(dges.)0 2623 y Fg(Pro)s(of)30 b(sk)m(etc)m(h:)144 b Fv(Assume)25 b(y)m(ou)h(ha)m(v)m(e)h(a)f(coin)f (that)i(has)e(probabilit)m(y)e Fp(p)i Fv(=)g(\002\()2794 2583 y Fr(log)13 b Fn(n)p 2794 2602 146 4 v 2816 2611 a Fh(p)p 2875 2611 43 3 v 49 x Fn(n)2950 2623 y Fv(\))26 b(of)g(b)s(eing)e(heads.)39 b(Create)0 2751 y(a)c(graph)f(on)g Fp(n)g Fv(v)m(ertices)h(as)g(follo)m(ws:)48 b(for)34 b(eac)m(h)h Fi(f)p Fp(i;)15 b(j)5 b Fi(g)p Fv(,)38 b(\015ip)33 b(the)i(coin.)52 b(Put)34 b(the)h(edge)g Fi(f)p Fp(i;)15 b(j)5 b Fi(g)36 b Fv(in)m(to)f(the)f(graph)0 2864 y(i\013)c(the)g(coin) g(is)g(heads.)41 b(The)30 b(probabilit)m(y)e(that)j(the)f(graph)g(will) e(b)s(e)i(ha)m(v)m(e)i(the)e(desired)f(prop)s(erties)g(is)h(nonzero)0 2977 y(\(actually)g(close)h(to)g(1\).)42 b(Hence)31 b(suc)m(h)f(a)h (graph)e(exists.)p 2026 2989 34 79 v 0 3275 a Fg(Theorem)34 b(3.7)h(\([9]\))45 b Fv(sort)q(\(2)p Fp(;)15 b(n)p Fv(\))26 b Fi(\024)f Fp(O)s Fv(\()p Fp(n)1496 3242 y Fr(3)p Fn(=)p Fr(2)1621 3275 y Fv(log)17 b Fp(n)p Fv(\))p Fm(.)0 3484 y Fg(Pro)s(of)42 b(sk)m(etc)m(h:)156 b Fv(Let)36 b Fp(G)f Fv(b)s(e)g(a)h(graph)f(sho)m(wn)g(to)h(exist)f(b)m(y)h(Lemma)f(3.6.)58 b(Use)36 b(this)e(graph)h(and)g(pro)s(ceed)0 3597 y(similar)28 b(to)j(Theorem)f(3.4.)p 1050 3609 V 0 3895 a Fg(Note)35 b(3.8)46 b Fv(Some)31 b(authors)g(ha)m(v)m(e)i(credited)e([7])h(or)f ([9)q(])h(with)e(the)i(result)e(sort\()p Fp(k)s(;)15 b(n)p Fv(\))28 b(=)f Fp(O)s Fv(\()p Fp(n)3213 3862 y Fr(1+1)p Fn(=k)3431 3895 y Fv(log)17 b Fp(n)p Fv(\).)44 b(This)0 4008 y(citation)31 b(is)f(incorrect)h(and)g(this)f(result)g (is)g(not)h(kno)m(wn)g(to)g(b)s(e)g(true.)42 b(Ho)m(w)m(ev)m(er,)34 b(Bollob\023)-45 b(as)62 b([6])31 b(later)h(obtained)0 4145 y(sort\()p Fp(k)s(;)15 b(n)p Fv(\))26 b(=)f Fp(O)s Fv(\()p Fp(n)651 4112 y Fr(1+1)p Fn(=k)916 4100 y Fr(\(log)13 b Fn(n)p Fr(\))1116 4077 y Fl(2)p Fd(\000)p Fl(2)p Fe(=k)p 865 4123 483 4 v 865 4184 a Fr(\(log)g(log)g Fn(n)p Fr(\))1168 4164 y Fl(1)p Fd(\000)p Fl(1)p Fe(=k)1357 4145 y Fv(\))30 b(\(see)i(Section)e(3.4\).)0 4404 y Ff(3.3)112 b(Expander)39 b(Graphs)0 4576 y Fv(Pipp)s(enger)28 b([18)q(])j(sho)m(w)m(ed)g(that)g (sort\()p Fp(k)s(;)15 b(n)p Fv(\))26 b(=)f Fp(O)s Fv(\()p Fp(n)1767 4543 y Fr(1+1)p Fn(=k)1970 4576 y Fv(\(log)17 b Fp(n)p Fv(\))2228 4543 y Fr(2)p Fh(\000)p Fr(2)p Fn(=k)2431 4576 y Fv(\))0 4785 y Fg(De\014nition)35 b(3.9)46 b Fv([18)q(])40 b(Let)f(1)h Fi(\024)f Fp(a)h Fi(\024)f Fp(n=)p Fv(2.)67 b(An)38 b Fp(a)p Fv(-expanding)g(graph)h(is)f(a)h(graph)f(in)g(whic)m (h)g(for)g(an)m(y)i(t)m(w)m(o)0 4898 y(disjoin)m(t)29 b(sets)i(of)f(v)m(ertices)h(of)g(size)f Fp(a)20 b Fv(+)g(1,)31 b(there)g(is)e(at)i(least)g(one)f(edge)h(b)s(et)m(w)m(een)g(the)g(t)m (w)m(o)h(sets.)0 5213 y Fg(Lemma)h(3.10)i(\([18)q(]\))45 b Fm(F)-7 b(or)33 b Fv(1)26 b Fi(\024)f Fp(a)g Fi(\024)g Fp(n=)p Fv(2)33 b Fm(ther)-5 b(e)34 b(exists)f(an)g Fp(a)p Fm(-exp)-5 b(anding)34 b(gr)-5 b(aph)34 b(with)g Fp(O)s Fv(\()3236 5172 y Fn(n)3279 5148 y Fl(2)3325 5172 y Fr(log)13 b Fn(n)p 3236 5191 235 4 v 3335 5244 a(a)3481 5213 y Fv(\))33 b Fm(e)-5 b(dges.)1927 5589 y Fv(5)p eop %%Page: 6 6 6 5 bop 0 91 a Fg(Pro)s(of)39 b(sk)m(etc)m(h:)153 b Fv(Assume)33 b(y)m(ou)h(ha)m(v)m(e)h(a)f(coin)f(that)h(has)f(probabilit)m(y)e Fp(p)g Fv(=)2780 56 y Fr(2)12 b(ln)g Fn(n)p 2780 71 161 4 v 2842 123 a(a)2984 91 y Fv(of)34 b(b)s(eing)e(heads.)50 b(Create)0 204 y(a)35 b(graph)f(on)g Fp(n)g Fv(v)m(ertices)h(as)g (follo)m(ws:)48 b(for)34 b(eac)m(h)h Fi(f)p Fp(i;)15 b(j)5 b Fi(g)p Fv(,)38 b(\015ip)33 b(the)i(coin.)52 b(Put)34 b(the)h(edge)g Fi(f)p Fp(i;)15 b(j)5 b Fi(g)36 b Fv(in)m(to)f(the)f (graph)0 331 y(i\013)e(the)h(coin)g(is)f(heads.)48 b(The)32 b(probabilit)m(y)f(that)i(the)g(graph)g(will)d(b)s(e)i(an)h Fp(a)p Fv(-expander)g(graph)f(with)g Fp(O)s Fv(\()3620 291 y Fn(n)3663 267 y Fl(2)3709 291 y Fr(log)13 b Fn(n)p 3620 310 235 4 v 3718 363 a(a)3865 331 y Fv(\))0 444 y(edges)31 b(is)e(nonzero)i(\(actually)f(close)h(to)g(1\).)42 b(Hence)31 b(suc)m(h)f(a)h(graph)e(exists.)p 2699 456 34 79 v 0 732 a Fg(Lemma)k(3.11)i(\([18)q(]\))45 b Fm(If)38 b Fp(n)g Fm(elements)h(ar)-5 b(e)40 b(c)-5 b(omp)g(ar)g(e)g(d)42 b(ac)-5 b(c)g(or)g(ding)40 b(to)f(the)g(e)-5 b(dges)39 b(of)g(an)g Fp(a)p Fm(-exp)-5 b(ander)40 b(gr)-5 b(aph,)0 845 y(then)33 b(ther)-5 b(e)34 b(wil)5 b(l)33 b(b)-5 b(e)32 b(at)h(most)h Fp(O)s Fv(\()p Fp(a)15 b Fv(log)i Fp(n)p Fv(\))32 b Fm(c)-5 b(andidates)35 b(r)-5 b(emaining)34 b(for)f(any)g(given)f(r)-5 b(ank.)141 1058 y Fv(F)d(rom)31 b(Lemma)g(3.11)g(it)f(is)g(easy)h(to)g(pro)m(v)m(e)g(the)g(follo)m (wing:)0 1245 y Fg(Lemma)i(3.12)i(\([18)q(]\))45 b Fm(If)38 b Fp(n)g Fm(elements)h(ar)-5 b(e)40 b(c)-5 b(omp)g(ar)g(e)g(d)42 b(ac)-5 b(c)g(or)g(ding)40 b(to)f(the)g(e)-5 b(dges)39 b(of)g(an)g Fp(a)p Fm(-exp)-5 b(ander)40 b(gr)-5 b(aph,)0 1358 y(then)33 b(they)g(c)-5 b(an)33 b(b)-5 b(e)32 b(p)-5 b(artitione)g(d)35 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b(:)g(:)g(:)i(;)e(k)s Fv(\))29 b(Using)f(Lemma)g(3.12)i(one)e(can)g(sho)m(w)g(that)h Fi(f)p Fp(x)2760 3121 y Fr(1)2800 3107 y Fp(;)15 b(:)g(:)g(:)i(;)e(x) 3054 3121 y Fn(n)3101 3107 y Fi(g)28 b Fv(can)h(b)s(e)e(partitioned)427 3219 y(in)m(to)21 b Fp(O)s Fv(\()796 3184 y Fn(n)p 719 3199 195 4 v 719 3251 a(a)13 b Fr(log)g Fn(n)925 3219 y Fv(\))21 b(groups)g(of)g(size)g Fp(O)s Fv(\()p Fp(a)15 b Fv(log)i Fp(n)p Fv(\))k(suc)m(h)g(that)h(all)e(comparisons)g(b)s(et)m (w)m(een)i(di\013eren)m(t)f(groups)427 3332 y(are)31 b(kno)m(wn.)40 b(Sort)30 b(the)h(groups)f(inductiv)m(ely)e(in)h Fp(k)23 b Fi(\000)d Fv(1)31 b(rounds.)39 b(This)29 b(tak)m(es)1315 3552 y Fp(O)s Fv(\()1531 3490 y Fp(n)p 1432 3531 251 4 v 1432 3614 a(a)15 b Fv(log)i Fp(n)1693 3552 y Fv(\()p Fp(a)e Fv(log)i Fp(n)p Fv(\))2014 3509 y Fr(1+)2155 3482 y Fl(1)p 2114 3494 113 4 v 2114 3535 a Fe(k)q Fd(\000)p Fl(1)2241 3552 y Fv(\(log)q(\()p Fp(a)e Fv(log)j Fp(n)p Fv(\)\))2750 3509 y Fr(2)p Fh(\000)2891 3482 y Fl(2)p 2850 3494 V 2850 3535 a Fe(k)q Fd(\000)p Fl(1)2976 3552 y Fv(\))427 3770 y(pro)s(cessors.)0 3983 y(END)31 b(OF)f(ALGORITHM)141 4195 y(T)-8 b(o)31 b(ac)m(hiev)m(e)h(the)e(result)f(set)i Fp(a)26 b Fv(=)f(\002\()1490 4160 y Fn(n)1533 4136 y Fl(1)p Fd(\000)p Fl(1)p Fe(=k)p 1428 4175 348 4 v 1428 4235 a Fr(\(ln)12 b Fn(n)p Fr(\))1596 4215 y Fl(1)p Fd(\000)p Fl(2)p Fe(=k)1785 4195 y Fv(\))p Fp(:)p 1950 4207 34 79 v 0 4512 a Ff(3.4)112 b(Sup)s(er)38 b(Expander)h(Graphs)0 4684 y Fv(Alon)32 b(and)h(Azar)h([3])g(sho)m(w)m(ed)f(that)h(sort\(2)p Fp(;)15 b(n)p Fv(\))31 b(=)f Fp(O)s Fv(\()p Fp(n)1926 4651 y Fr(3)p Fn(=)p Fr(2)2135 4643 y(log)13 b Fn(n)p 2046 4663 325 4 v 2046 4673 a Fi(p)p 2121 4673 249 4 v 2121 4736 a Fr(log)g(log)g Fn(n)2380 4684 y Fv(\).)49 b(Bollob\023)-45 b(as)66 b([6)q(])33 b(extended)g(this)f(to)i(sho)m(w)0 4874 y(that)d(sort\()p Fp(k)s(;)15 b(n)p Fv(\))26 b(=)f Fp(O)s Fv(\()p Fp(n)848 4841 y Fr(1+1)p Fn(=k)1113 4830 y Fr(\(log)13 b Fn(n)p Fr(\))1313 4806 y Fl(2)p Fd(\000)p Fl(2)p Fe(=k)p 1061 4853 483 4 v 1061 4913 a Fr(\(log)h(log)f Fn(n)p Fr(\))1365 4894 y Fl(1)p Fd(\000)p Fl(1)p Fe(=k)1554 4874 y Fv(\))141 5001 y(All)22 b(these)i(results)f(use)g(graphs)g (similar)e(to)k(the)e Fp(a)p Fv(-expander)h(graphs)f(discussed)e(in)i (Section)g(3.3.)40 b(W)-8 b(e)25 b(sk)m(etc)m(h)0 5114 y(the)31 b(algorithm)e(of)h(Alon)g(and)g(Azar)h(and)f(then)g(mak)m(e)h (some)g(brief)e(commen)m(ts)i(ab)s(out)f([6)q(].)141 5227 y(W)-8 b(e)23 b(de\014ne)d(a)i(subset)f(of)g Fp(a)p Fv(-expander)g(graphs)g(that)h(has)f(additional)f(expanding)f(prop)s (erties.)37 b(The)20 b(follo)m(wing)0 5340 y(de\014nition)28 b(is)h(implicit)f(in)h([3)q(].)1927 5589 y(6)p eop %%Page: 7 7 7 6 bop 0 91 a Fg(De\014nition)35 b(3.14)47 b Fv(Let)33 b Fp(a;)15 b(n)29 b Fi(2)f Fu(N)33 b Fv(and)f Fp(a)d Fv(=)g(\012\(log)17 b Fp(n)p Fv(\).)47 b(A)33 b(graph)f Fp(G)g Fv(on)g Fp(n)g Fv(v)m(ertices)i(is)d(an)i Fp(a)p Fm(-sup)-5 b(er-exp)g(ander)0 204 y Fv(if)29 b(the)i(follo)m(wing)e (hold.)111 392 y(1.)46 b(If)31 b Fp(A)f Fv(and)h Fp(B)k Fv(are)c(disjoin)m(t)e(subsets)h(of)h(v)m(ertices)h(with)d Fp(a)i Fv(v)m(ertices)g(eac)m(h)h(then)f(some)g Fp(v)e Fi(2)c Fp(B)36 b Fv(has)30 b(at)i(least)227 505 y(log)345 527 y Fr(2)399 505 y Fp(n)e Fv(neigh)m(b)s(ors)f(in)g Fp(A)p Fv(.)111 714 y(2.)46 b(Let)36 b Fp(x)d Fi(\024)g Fp(a=e)719 617 y Fi(p)p 796 617 181 4 v 796 677 a Fr(log)887 694 y Fl(2)933 677 y Fn(n)980 714 y Fv(.)56 b(If)34 b Fp(A)i Fv(and)e Fp(B)40 b Fv(are)35 b(disjoin)m(t)f(sets)i(suc)m(h)f (that)g Fi(j)p Fp(A)p Fi(j)f Fv(=)f Fp(x)j Fv(and)e Fi(j)p Fp(B)5 b Fi(j)33 b Fv(=)g Fp(x)p Fv(\(log)3620 736 y Fr(2)3675 714 y Fp(n)p Fv(\))3765 681 y Fr(1)p Fn(=)p Fr(4)3875 714 y Fv(,)227 827 y(eac)m(h)f Fp(v)c Fi(2)d Fp(A)31 b Fv(that)g(has)f(at)h(least)g(log)1489 849 y Fr(2)1544 827 y Fp(n)e Fv(neigh)m(b)s(ors)g(in)g Fp(B)5 b Fv(.)141 1014 y(The)35 b(follo)m(wing)e(lemma)h(asserts)i(that)f (there)g(exists)g(small)e Fp(a)p Fv(-sup)s(er-expander)g(graphs.)54 b(It)35 b(is)f(similar)e(to)0 1127 y(Lemma)27 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Fv(\()p Fp(n)1548 2726 y Fr(3)p Fn(=)p Fr(2)1757 2719 y(log)13 b Fn(n)p 1668 2738 325 4 v 1668 2749 a Fi(p)p 1744 2749 249 4 v 62 x Fr(log)g(log)g Fn(n)2003 2759 y Fv(\))0 3007 y Fg(Pro)s(of)44 b(sk)m(etc)m(h:)160 b Fv(This)35 b(is)i(similar)e(to)j (the)f Fp(k)k Fv(=)36 b(2)i(case)h(of)e(Theorem)g(3.13.)64 b(The)37 b(v)-5 b(alue)37 b(of)g Fp(a)h Fv(needed)f(is)0 3120 y Fp(a)25 b Fv(=)g(\002\()275 3052 y Fi(p)p 351 3052 390 4 v 68 x Fp(n)15 b Fv(log)h(log)h Fp(n)p Fv(\).)p 916 3132 34 79 v 0 3445 a Fg(Theorem)34 b(3.18)i(\([6]\))45 b Fv(sort\()p Fp(k)s(;)15 b(n)p Fv(\))26 b(=)f Fp(O)s Fv(\()p Fp(n)1553 3412 y Fr(1+1)p Fn(=k)1818 3401 y Fr(\(log)13 b Fn(n)p Fr(\))2018 3378 y Fl(2)p Fd(\000)p Fl(2)p Fe(=k)p 1766 3424 483 4 v 1766 3485 a Fr(\(log)h(log)f Fn(n)p Fr(\))2070 3465 y Fl(1)p Fd(\000)p Fl(1)p Fe(=k)2259 3445 y Fv(\))0 3672 y Fg(Pro)s(of)52 b(sk)m(etc)m(h:)175 b Fv(A)44 b(rather)h(complicated)f(t)m(yp)s(e)h(of)g(graph)f(is)g (de\014ned)g(whic)m(h)f(will,)j(if)e(used)f(to)j(guide)0 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1533 y(By)f(lo)s(oking)e(at)j(the)e(a)m(v)m(erage)j(degrees) f(of)e(v)m(ertices)h(one)g(can)g(sho)m(w)f(that)i(there)e(is)g(a)h(set) g(of)g Fp(n=)p Fv(2)g(no)s(des)e(suc)m(h)0 1645 y(that)36 b(the)g(the)g(induced)e(subgraph)g Fp(G)1342 1612 y Fh(0)1400 1645 y Fv(on)i(them)g(is)e Fp(s)p Fv(-colorable)i(where)f Fp(s)f Fv(=)g(2)2824 1612 y Fh(\000)p Fn(k)r Fh(\000)p Fr(2)3012 1645 y Fp(n)3067 1612 y Fr(1)p Fn(=k)3180 1645 y Fv(.)56 b(Color)36 b Fp(G)3587 1612 y Fh(0)3645 1645 y Fv(with)e Fp(s)0 1758 y Fv(colors.)39 b(Let)27 b Fp(V)504 1772 y Fr(1)544 1758 y Fp(;)15 b(:)g(:)g(:)h(;)f(V)798 1772 y Fn(s)862 1758 y Fv(b)s(e)26 b(the)g(color)h(classes.)39 b(Orien)m(t)26 b Fp(G)2030 1725 y Fh(0)2079 1758 y Fv(as)g(follo)m(ws:) 38 b(F)-8 b(or)27 b(all)f(1)f Fi(\024)g Fp(i)h(<)f(j)31 b Fi(\024)25 b Fp(s)p Fv(,)i(for)f(all)f Fp(v)k Fi(2)c Fp(V)3847 1772 y Fn(i)3875 1758 y Fv(,)0 1871 y(for)32 b(all)f Fp(u)e Fi(2)f Fp(V)492 1885 y Fn(j)529 1871 y Fv(,)33 b(if)e(\()p Fp(v)s(;)15 b(u)p Fv(\))34 b(is)d(an)h(edge)h(then) f(set)h Fp(v)f(<)c(u)p Fv(.)47 b(Orien)m(t)31 b Fp(G)h Fv(so)h(that)g(y)m(ou)f(use)g(this)g(orien)m(tation)g(on)g Fp(G)3877 1838 y Fh(0)0 1984 y Fv(and)27 b(all)g(the)g(v)m(ertices)i (not)f(in)e Fp(G)1112 1951 y Fh(0)1162 1984 y Fv(are)i(less)f(than)h (all)e(the)i(v)m(ertices)h(in)d Fp(G)2465 1951 y Fh(0)2488 1984 y Fv(.)39 b(Once)28 b(the)g(transitiv)m(e)f(closure)g(of)h Fp(G)f Fv(is)0 2097 y(tak)m(en)h(one)f(still)d(needs)j(to)g(sort)g(eac) m(h)h Fp(V)1365 2111 y Fn(i)1419 2097 y Fv(in)e Fp(k)16 b Fi(\000)d Fv(1)26 b(rounds)f(\(one)j(also)e(needs)h(to)g(sort)g (those)g(elemen)m(ts)g(not)g(in)e Fp(G)3877 2064 y Fh(0)0 2210 y Fv(but)j(this)g(is)g(not)h(needed)f(for)h(the)g(lo)m(w)m(er)g(b) s(ound\).)38 b(One)29 b(can)g(sho)m(w)g(the)g(lo)m(w)m(er)g(b)s(ound)d (b)m(y)j(using)e(the)i(inductiv)m(e)0 2323 y(lo)m(w)m(er)i(b)s(ound)d (on)i(eac)m(h)i Fp(V)902 2337 y Fn(i)960 2323 y Fv(and)e(some)h (algebra.)p 1793 2335 34 79 v 0 2626 a Fg(Theorem)j(8.2)h(\([9]\))45 b Fm(F)-7 b(or)34 b(al)5 b(l)33 b Fp(c)26 b(<)1310 2553 y Fo(p)p 1393 2553 137 4 v 73 x Fv(3)p Fp(=)p Fv(2)q Fm(,)33 b Fv(sort\()p Fp(k)s(;)15 b(n)p Fv(\))26 b Fi(\025)f Fp(cn)2173 2593 y Fr(1+1)p Fn(=k)0 2838 y Fg(Pro)s(of)50 b(sk)m(etc)m(h:)171 b Fv(This)41 b(pro)s(of)h(is)g(similar)e(to)j(that) h(of)f(Theorem)f(8.1)i(except)g(that)f(the)g Fp(V)3404 2852 y Fr(1)3444 2838 y Fp(;)15 b(:)g(:)g(:)i(;)e(V)3699 2852 y Fn(s)3779 2838 y Fv(are)0 2951 y(obtained)28 b(b)m(y)g(a)h (greedy)f(coloring)g(and)g(more)g(care)h(is)f(tak)m(en)h(in)e(sho)m (wing)h(the)g(largest)h(v)-5 b(alue)28 b(of)3354 2887 y Fo(P)3442 2913 y Fn(s)3442 2974 y(i)p Fr(=1)3575 2951 y Fv(sort)q(\()p Fp(k)19 b Fi(\000)0 3064 y Fv(1)p Fp(;)c Fi(j)p Fp(V)163 3078 y Fn(i)192 3064 y Fi(j)p Fv(\).)42 b(Lagrange)31 b(m)m(ultipliers)c(are)k(used.)p 1636 3076 34 79 v 141 3248 a(The)42 b(k)m(ey)i(to)f(the)g(pro)s(ofs)f(of)h (Theorems)g(8.1)g(and)g(8.2)g(is)f(that)i(w)m(e)f(still)d(need)j(to)g (sort)g(eac)m(h)h Fp(V)3612 3262 y Fn(i)3641 3248 y Fv(.)77 b(The)0 3361 y(pro)s(of)37 b(do)s(es)g(not)h(use)f(the)h(fact)h(that)f (y)m(ou)g(migh)m(t)f(ha)m(v)m(e)i(to)f(mak)m(e)h(some)f(comparisons)f (b)s(et)m(w)m(een)h(v)m(ertices)g(in)0 3473 y(di\013eren)m(t)27 b Fp(V)409 3487 y Fn(i)437 3473 y Fv('s.)40 b(T)-8 b(o)28 b(impro)m(v)m(e)g(this)e(lo)m(w)m(er)i(b)s(ound)d(Alon)i(and)g(Azar)i (sho)m(w)m(ed)e(that)i(y)m(ou)f(will)d(ha)m(v)m(e)j(to)h(mak)m(e)f(suc) m(h)0 3586 y(comparisons.)0 3799 y Fg(Lemma)33 b(8.3)46 b Fm(L)-5 b(et)34 b Fp(G)f Fm(b)-5 b(e)33 b(a)h(gr)-5 b(aph)35 b(with)g Fp(n)e Fm(vertic)-5 b(es)33 b(and)i Fp(dn)e Fm(e)-5 b(dges.)45 b(Ther)-5 b(e)35 b(exists)e(an)h(induc)-5 b(e)g(d)35 b(sub)-5 b(gr)g(aph)35 b(on)10 3876 y Fn(n)p 10 3891 43 4 v 14 3943 a Fr(4)95 3912 y Fm(vertic)-5 b(es)33 b(such)f(that)i(\(1\))f Fp(G)1041 3879 y Fh(0)1096 3912 y Fm(has)h(de)-5 b(gr)g(e)g(e)33 b Fp(<)25 b Fv(4)p Fp(d)p Fm(,)33 b(and)g(\(2\))g(ther)-5 b(e)34 b(is)e(a)h Fv(4)p Fp(d)p Fm(-c)-5 b(oloring)34 b(of)f Fp(G)3159 3879 y Fh(0)3214 3912 y Fm(with)g(c)-5 b(olor)34 b(classes)0 4034 y Fp(V)53 4048 y Fr(1)92 4034 y Fp(;)15 b(:)g(:)g(:)i(;)e(V)347 4049 y Fr(4)p Fn(d)454 4034 y Fm(such)32 b(that)g(for)g(al)5 b(l)32 b Fv(1)25 b Fi(\024)g Fp(i;)15 b(j)32 b Fi(\024)25 b Fv(4)p Fp(d)p Fm(,)32 b(for)g(al)5 b(l)31 b Fp(x)25 b Fi(2)g Fp(V)2159 4048 y Fn(i)2187 4034 y Fm(,)32 b(ther)-5 b(e)32 b(ar)-5 b(e)32 b(at)g(most)g Fv(2)2997 4001 y Fh(j)p Fn(i)p Fh(\000)p Fn(j)t Fh(j)p Fr(+1)3273 4034 y Fm(neighb)-5 b(ors)33 b(in)e Fp(V)3836 4048 y Fn(j)3872 4034 y Fm(.)0 4247 y Fg(Pro)s(of)39 b(sk)m(etc)m(h:)150 b Fv(Remo)m(v)m(e)35 b(successiv)m(ely)d(the)h(highest)f(degree)h(v)m (ertex)2643 4211 y Fn(n)p 2643 4226 V 2647 4278 a Fr(2)2729 4247 y Fv(times.)47 b(Let)33 b Fp(G)3251 4214 y Fh(0)3307 4247 y Fv(b)s(e)f(the)h(induced)0 4360 y(subgraph)39 b(on)i(the)h(remaining)1155 4324 y Fn(n)p 1155 4339 V 1159 4391 a Fr(2)1249 4360 y Fv(v)m(ertices.)73 b(One)41 b(can)h(sho)m(w)f(that)g Fp(G)2541 4327 y Fh(0)2605 4360 y Fv(has)g(degree)h Fp(<)h Fv(4)p Fp(d)p Fv(.)73 b(Clearly)40 b Fp(G)3775 4327 y Fh(0)3839 4360 y Fv(is)0 4473 y(4)p Fp(d)p Fv(-colorable.)g(Let)28 b Fp(U)763 4487 y Fr(1)802 4473 y Fp(;)15 b(:)g(:)g(:)i(;)e(U)1066 4488 y Fr(4)p Fn(d)1169 4473 y Fv(b)s(e)26 b(the)h(color)g(classes.)39 b(A)27 b(probabilistic)c(argumen)m(t)k(sho)m(ws)g(that)g(there)g (exists)0 4586 y(a)k(p)s(erm)m(utation)e(of)i Fi(f)p Fv(1)p Fp(;)15 b(:)g(:)g(:)i(;)e Fv(4)p Fp(d)p Fi(g)32 b Fv(that)f(satis\014es)f(the)h(prop)s(erties)d(needed.)p 2690 4598 34 79 v 0 4874 a Fg(Lemma)33 b(8.4)46 b Fm(L)-5 b(et)34 b Fp(d)h Fm(b)-5 b(e)34 b(such)h(that)g Fp(d)29 b Fv(=)f Fp(o)p Fv(\()p Fp(n)p Fv(\))35 b Fm(and)g Fp(d)29 b Fv(=)f(\012\(log)17 b Fp(n)p Fv(\))p Fm(.)47 b(L)-5 b(et)34 b Fp(G)g Fm(b)-5 b(e)34 b(a)h(gr)-5 b(aph)36 b(with)g Fp(n)d Fm(vertic)-5 b(es)35 b(and)0 4987 y Fp(dn)g Fm(e)-5 b(dges.)52 b(Ther)-5 b(e)36 b(exists)g(an)h(orientation)g(of)f Fp(G)f Fm(such)h(that)h(the)f(c)-5 b(omplement)38 b(of)e(its)f(tr)-5 b(ansitive)37 b(closur)-5 b(e)37 b(has)0 5100 y(at)c(le)-5 b(ast)34 b Fv(\012\()427 5064 y Fn(n)470 5040 y Fl(2)p 427 5079 78 4 v 447 5131 a Fn(d)529 5100 y Fv(log)r(\()692 5064 y Fn(n)p 692 5079 43 4 v 695 5131 a(d)745 5100 y Fv(\)\))f Fm(e)-5 b(dges.)1905 5589 y Fv(15)p eop %%Page: 16 16 16 15 bop 0 91 a Fg(Pro)s(of)43 b(sk)m(etc)m(h:)159 b Fv(Use)38 b(Lemma)f(8.3)h(to)g(obtain)e Fp(G)1912 58 y Fh(0)1972 91 y Fv(and)g Fp(V)2208 105 y Fr(1)2248 91 y Fp(;)15 b(:)g(:)g(:)h(;)f(V)2502 106 y Fr(4)p Fn(d)2615 91 y Fv(as)38 b(sp)s(eci\014ed)d(there.)61 b(Orien)m(t)36 b Fp(G)3759 58 y Fh(0)3819 91 y Fv(as)0 204 y(follo)m(ws:)49 b(F)-8 b(or)36 b(all)e(1)f Fi(\024)g Fp(i)g(<)g(j)39 b Fi(\024)32 b Fp(m)p Fv(,)37 b(for)d(all)g Fp(v)j Fi(2)32 b Fp(V)1814 218 y Fn(i)1843 204 y Fv(,)k(for)f(all)f Fp(u)f Fi(2)f Fp(V)2410 218 y Fn(j)2447 204 y Fv(,)k(if)e(\()p Fp(v)s(;)15 b(u)p Fv(\))37 b(is)d(an)h(edge)g(then)g(set)h Fp(v)g(<)d(u)p Fv(.)0 317 y(Orien)m(t)e Fp(G)f Fv(so)i(that)g(y)m(ou)f (use)g(this)f(orien)m(tation)i(on)f Fp(G)1871 284 y Fh(0)1925 317 y Fv(and)f(all)h(the)g(v)m(ertices)h(not)f(in)f Fp(G)3058 284 y Fh(0)3112 317 y Fv(are)i(less)f(than)g(all)f(the)0 430 y(v)m(ertices)k(in)e Fp(G)513 397 y Fh(0)536 430 y Fv(.)50 b(The)33 b(complemen)m(t)h(of)g(the)g(transitiv)m(e)f (closure)g(will)d(not)k(con)m(tain)g(an)m(y)g(edges)g(within)d(an)j Fp(V)3847 444 y Fn(i)3875 430 y Fv(.)0 543 y(In)29 b(addition,)g(b)s (ecause)h(of)g(the)g(limit)e(on)i(ho)m(w)g(man)m(y)g(edges)g(can)h(go)g (from)e(an)h Fp(V)2794 557 y Fn(i)2852 543 y Fv(to)h(an)f Fp(V)3142 557 y Fn(j)3178 543 y Fv(,)h(one)f(can)g(estimate)0 656 y(additional)c(lo)m(w)m(er)j(b)s(ounds)d(on)i(the)g(n)m(um)m(b)s (er)f(of)i(edges)g(in)d(the)j(complemen)m(t)f(of)h(the)f(transitiv)m(e) g(closure.)39 b(\(This)0 769 y(is)29 b(highly)g(non-trivial.\))p 959 781 34 79 v 0 1072 a Fg(Theorem)34 b(8.5)h(\([3]\))45 b Fm(F)-7 b(or)34 b Fp(k)29 b Fi(\025)c Fv(2)p Fm(,)32 b Fv(sort)q(\()p Fp(k)s(;)15 b(n)p Fv(\))26 b Fi(\025)f Fv(\012\()p Fp(n)1942 1039 y Fr(1+1)p Fn(=k)2145 1072 y Fv(\(log)17 b Fp(n)p Fv(\))2403 1039 y Fr(1)p Fn(=k)2516 1072 y Fv(\))p Fm(.)0 1284 y Fg(Pro)s(of)36 b(sk)m(etc)m(h:)141 1397 y Fv(W)-8 b(e)34 b(pro)m(v)m(e)f(this)f(b)m(y)h(induction.)45 b(Note)34 b(that)f Fp(k)f Fv(=)d(2)k(is)f(the)h(base)f(case)i(and)e(is) g(non-trivial.)45 b(Assume)32 b(that)0 1510 y(there)k(is)g(an)g (algorithm)f(that)i(sorts)f Fp(n)g Fv(elemen)m(ts)g(in)f(2)i(rounds)e (and)g(tak)m(es)j Fp(dn)e Fv(pro)s(cessors.)58 b(W)-8 b(e)37 b(kno)m(w)f(that)0 1623 y Fp(d)d Fv(=)g(\012\()p Fp(n)340 1590 y Fr(1)p Fn(=)p Fr(3)450 1623 y Fv(\))i(b)m(y)g(Theorem)g (8.1.)56 b(W)-8 b(e)36 b(can)g(assume)e Fp(d)g Fv(=)f Fp(o)p Fv(\()p Fp(n)2206 1590 y Fr(2)p Fn(=)p Fr(3)2316 1623 y Fv(\))i(since)f(if)h(it)f(is)g(not)i(then)e(the)i(theorem)f(for) 0 1736 y Fp(k)c Fv(=)c(2)32 b(is)f(already)g(true.)45 b(Let)32 b Fp(G)g Fv(b)s(e)f(the)h(graph)f(represen)m(ting)g(the)h (\014rst)f(round.)43 b(Because)34 b(of)e(the)f(b)s(ounds)f(on)0 1849 y Fp(d)h Fv(w)m(e)h(can)f(apply)f(Lemma)h(8.4)h(to)g(the)f(graph)g (to)g(obtain)g(an)g(orien)m(tation)g(suc)m(h)g(that)g(the)g(complimen)m (t)g(of)g(the)0 1962 y(transitiv)m(e)h(closure)g(has)h(\012\()989 1926 y Fn(n)1032 1902 y Fl(2)p 989 1941 78 4 v 1009 1993 a Fn(d)1091 1962 y Fv(log)r(\()1254 1926 y Fn(n)p 1254 1941 43 4 v 1257 1993 a(d)1307 1962 y Fv(\)\))g(edges.)49 b(Hence)33 b(the)g(second)g(round)e(needs)i(\012\()3050 1926 y Fn(n)3093 1902 y Fl(2)p 3050 1941 78 4 v 3070 1993 a Fn(d)3152 1962 y Fv(log)r(\()3315 1926 y Fn(n)p 3315 1941 43 4 v 3318 1993 a(d)3368 1962 y Fv(\)\))g(pro)s(cessors.)0 2075 y(Algebra)d(sho)m(ws)g(that)h Fp(d)26 b Fv(=)f(\012\()1069 2006 y Fi(p)p 1145 2006 257 4 v 69 x Fp(n)15 b Fv(log)h Fp(n)o Fv(\).)141 2187 y(W)-8 b(e)35 b(no)m(w)f(sk)m(etc)m(h)h(the)f (induction)e(step.)51 b(It)34 b(will)d(b)s(e)i Fm(e)-5 b(asier)35 b Fv(than)e(the)h Fp(k)g Fv(=)d(2)j(case)h(since)e(it)h(do)s (es)f(not)h(use)0 2300 y(Lemma)j(8.4.)61 b(Assume)36 b(the)h(lo)m(w)m(er)g(b)s(ound)e(for)h Fp(k)28 b Fi(\000)c Fv(1)37 b(where)f Fp(k)j Fi(\025)d Fv(3.)60 b(Assume)36 b(there)h(is)f(an)g(algorithm)g(for)0 2413 y(sorting)31 b(in)g Fp(k)k Fv(rounds.)44 b(Let)33 b Fp(G)e Fv(b)s(e)g(the)i(graph)e (represen)m(ting)g(the)h(\014rst)g(round)e(of)i(the)g(algorithm.)45 b(Assume)32 b Fp(G)0 2526 y Fv(has)27 b Fp(dn)f Fv(edges.)41 b(By)27 b(T)-8 b(uran's)26 b(theorem)h(\(see)h([5)q(])f(for)g(a)g(nice) f(probabilistic)e(pro)s(of)7 b(\))27 b(a)g(graph)f(with)g Fp(dn)h Fv(edges)g(has)0 2639 y(an)j(indep)s(enden)m(t)d(set)j(of)g (size)1125 2603 y Fn(n)p 1066 2618 162 4 v 1066 2670 a Fr(2)p Fn(d)p Fr(+1)1238 2639 y Fv(.)40 b(By)30 b(rep)s(eated)g (application)e(of)i(T)-8 b(uran's)29 b(theorem)h(w)m(e)g(can)g(\014nd)e Fp(s)d Fv(=)g(\012\()p Fp(d)p Fv(\))0 2752 y(pairwise)37 b(disjoin)m(t)g(indep)s(enden)m(t)f(sets)j(of)g(size)g(\012\()1849 2716 y Fn(n)p 1808 2731 127 4 v 1808 2783 a Fr(1+)p Fn(d)1944 2752 y Fv(\))g(whic)m(h)e(w)m(e)j(denote)f 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