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b(but)c(deleting)f(no)0 4750 y(other)28 b(edges\).)41 b(In)28 b(the)g(ab)s(o)m(v)m(e)i(pro)s(of)d(implicit)e(use)j(w)m(as)h (made)f(of)h(the)f(fact)h(that)g(remo)m(ving)f(edges)h(cannot)g Fg(add)0 4908 y Fm(new)h(triangles)f(to)i(the)f(graph;)g(but)g(in)f (the)h(case)i(of)e(induced)e(squares,)i(for)g(example,)h(remo)m(ving)f (an)g(edge)h(can)0 5066 y(actually)f(mak)m(e)h(its)f(pair)f(part)h(of)h (a)g(new)f(sp)s(ecimen)f(of)h(the)h(forbidden)c(subgraph.)141 5224 y(In)h(pro)m(ving)g(the)h(Regularit)m(y)g(Lemma)g(one)g(cannot)g (do)g(a)m(w)m(a)m(y)i(completely)d(with)g(the)h(existence)g(of)g(a)g (small)1905 5532 y(14)p eop %%Page: 15 15 15 14 bop 0 274 a Fm(fraction)23 b(of)g(non-regular)f(pairs,)h(whic)m (h)f(are)h(the)g(core)h(of)f(the)g(problem)e(in)h(pro)m(ving)g(similar) f(results)g(for)i(induced)0 432 y(subgraphs,)35 b(b)s(ecause)h(of)f (the)h(ab)s(o)m(v)m(e)h(men)m(tioned)e(inabilit)m(y)e(to)j(safely)f (remo)m(v)m(e)i(edges.)57 b(The)35 b(w)m(ork)-5 b(around)34 b(in)0 590 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b(concrete)i(example)e(can)h (help)e(explain)f(this)h({)i(in)e(the)i(case)g(of)f(testing)h(for)f(a)g (graph)0 1064 y(b)s(eing)37 b(triangle)g(free,)k(a)d(lo)m(w)m(er)h(b)s (ound)d(that)j(is)e(sup)s(er-p)s(olynomial)e(in)h Fj(\017)2599 1031 y Fq(\000)p Fi(1)2732 1064 y Fm(\(but)i(in)f(this)g(case)i(is)e (of)i(course)0 1222 y(indep)s(enden)m(t)30 b(of)j(the)g(input)d(size\)) j(is)f(giv)m(en)g(using)g(a)g(graph)h(that)g(is)e(far)i(from)f(b)s (eing)f(triangle)h(free,)h(but)f(still)0 1380 y(do)s(es)39 b(not)g(con)m(tain)g(to)s(o)h(man)m(y)f(distinct)f(triangles,)i(so)f Fj(q)j Fm(queries)c(will)e(not)k(capture)f(a)g(triangle)g(with)e(high)0 1539 y(probabilit)m(y)-8 b(.)42 b(Suc)m(h)31 b(a)h(graph)f(can)h(b)s(e) f(constructed)h(using)e(the)h(n)m(um)m(b)s(er-theoretic)h(construction) f(in)f([12)q(];)j(the)0 1697 y(details)g(and)g(a)h(generalization)g (thereof)g(to)g(other)g(prop)s(erties)e(de\014ned)h(in)f(terms)i(of)g (not)g(con)m(taining)f(a)i(\014xed)0 1855 y(\(not)c(necessarily)e 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b(metho)s(d)e(that)i(can)e(serv)m(e)i(as)0 2961 y(an)e(example)g(for)g(pro)m(ving)g(lo)m(w)m(er)h(b)s(ounds.)0 3219 y Fh(Prop)s(osition)36 b(8.1)g(\([3]\))45 b Fg(The)35 b(pr)-5 b(op)g(erty)37 b(of)e(two)g(gr)-5 b(aphs)37 b(\(given)d(by)h (their)g(adjac)-5 b(ency)36 b(matric)-5 b(es\))36 b(b)-5 b(eing)34 b(iso-)0 3377 y(morphic)g(is)f(not)622 3341 y Fi(1)p 622 3356 36 4 v 622 3408 a(8)667 3377 y Fg(-testable)g(with)h (any)f(c)-5 b(onstant)34 b(numb)-5 b(er)33 b(of)g(queries.)0 3685 y Fh(Pro)s(of:)52 b Fm(W)-8 b(e)36 b(shall)e(sho)m(w)h(that)i(for) e(an)m(y)h Fj(q)i Fm(there)e(exists)f Fj(n)p Fm(,)h(suc)m(h)g(that)g Fj(q)i Fm(edge)e(queries)f(are)g(insu\016cien)m(t)f(for)0 3843 y(distinguishing)c(b)s(et)m(w)m(een)36 b(the)f(case)h(that)f(the)g (t)m(w)m(o)h(graphs)e(are)h(isomorphic,)g(and)f(the)h(case)h(that)f (all)f(v)m(ertex)0 4001 y(p)s(erm)m(utations)d(of)i(the)g(\014rst)f (graph)g(di\013er)f(from)h(the)h(second)f(graph)g(in)f(more)i(than)3003 3965 y Fi(1)p 3003 3980 V 3003 4032 a(4)3049 3931 y Fc(\000)3087 3960 y Fe(n)3090 4032 y Fi(2)3129 3931 y Fc(\001)3200 4001 y Fm(places)f(\(remem)m(b)s(er)0 4159 y(that)f(the)g(total)g(size) f(of)g(the)h(input)d(is)i(2)1382 4090 y Fc(\000)1420 4118 y Fe(n)1424 4190 y Fi(2)1463 4090 y Fc(\001)1501 4159 y Fm(\).)141 4317 y(F)-8 b(or)22 b(this)e(w)m(e)h(construct)h (distributions)17 b(o)m(v)m(er)22 b(inputs.)36 b(The)21 b(distribution)c Fj(D)2735 4331 y Fe(P)2815 4317 y Fm(is)j(constructed) h(b)m(y)g(letting)g(the)0 4475 y(input)34 b(consist)h(of)h(a)g(random)f (graph)g Fj(G)p Fm(,)j(with)c(eac)m(h)j(edge)g(b)s(eing)d(indep)s (enden)m(tly)f(c)m(hosen)j(with)e(probabilit)m(y)10 4597 y Fi(1)p 10 4612 V 10 4664 a(2)55 4633 y Fm(,)44 b(and)d(a)g(second)g (graph)g(that)h(is)e(a)h(uniformly)d(random)j(p)s(erm)m(utation)f(of)h Fj(G)p Fm(.)73 b(The)41 b(distribution)c Fj(D)3730 4647 y Fe(N)3839 4633 y Fm(is)0 4791 y(constructed)25 b(b)m(y)g(letting)g (the)g(input)e(consist)h(of)i(a)f(random)f(graph)h Fj(G)g Fm(as)g(b)s(efore,)h(and)e(another,)j(indep)s(enden)m(tly)0 4949 y(c)m(hosen,)k(random)f(graph)g Fj(G)979 4916 y Fq(0)1002 4949 y Fm(.)141 5107 y(It)j(is)f(clear)h(that)g(an)g(input)d (c)m(hosen)k(according)e(to)i Fj(D)2022 5121 y Fe(P)2114 5107 y Fm(satis\014es)e(the)h(prop)s(ert)m(y)f(with)f(probabilit)m(y)g (1.)48 b(On)0 5265 y(the)32 b(other)g(hand,)f(b)m(y)g(a)h(standard)f (large)h(deviation)e(argumen)m(t,)j(in)d(an)i(input)d(c)m(hosen)k (according)e(to)i Fj(D)3660 5279 y Fe(N)3759 5265 y Fm(an)m(y)1905 5532 y(21)p eop %%Page: 22 22 22 21 bop 0 274 a Fm(\014xed)28 b(v)m(ertex)i(p)s(erm)m(utation)d(of)i Fj(G)g Fm(will)d(with)h(probabilit)m(y)f(more)j(than)f(1)18 b Ff(\000)e Fm(2)2681 241 y Fq(\000)p Fe(n)2779 218 y Fd(2)2814 241 y Fe(=)p Fi(50)2952 274 y Fm(di\013er)28 b(in)f(more)i(than)3736 238 y Fi(1)p 3736 253 36 4 v 3736 305 a(4)3781 205 y Fc(\000)3819 234 y Fe(n)3823 305 y Fi(2)3862 205 y Fc(\001)0 432 y Fm(places)e(from)g Fj(G)548 399 y Fq(0)572 432 y Fm(.)39 b(Th)m(us,)27 b(with)f (probabilit)m(y)f(1)14 b Ff(\000)g Fj(o)p Fm(\(1\),)30 b(all)c(v)m(ertex)i(p)s(erm)m(utations)e(of)i Fj(G)f Fm(will)e(ha)m(v)m(e)k(at)f(least)f(this)0 590 y(di\013erence)j(from)g Fj(G)693 557 y Fq(0)716 590 y Fm(.)141 748 y(W)-8 b(e)26 b(no)m(w)f(sho)m(w)f(that)h(no)g(non-adaptiv)m(e)g(algorithm)e(will)f (distinguish)f(b)s(et)m(w)m(een)k(an)g(input)e(selected)i(accord-)0 906 y(ing)i(to)i Fj(D)333 920 y Fe(P)392 906 y Fm(,)g(and)e(an)h(input) e(selected)j(according)f(to)h Fj(D)1912 920 y Fe(N)2007 906 y Fm(\(that)g(is,)f(it)g(will)d(ha)m(v)m(e)30 b(nearly)d(the)h (same)h(acceptance)0 1064 y(probabilit)m(y)f(for)i(b)s(oth\);)g(w)m(e)h (recall)f(that)h(it)f(is)f(su\016cien)m(t)h(to)h(sho)m(w)f(it)g(for)g (deterministic)f(algorithms.)141 1222 y(Supp)s(ose)f(that)i(the)f (algorithm)g(has)g(queried)f(the)i(v)m(ertex)g(pairs)e Fj(p)2441 1236 y Fi(1)2480 1222 y Fj(;)15 b(:)g(:)g(:)i(;)e(p)2728 1236 y Fe(q)2766 1222 y Fm(,)30 b(some)g(inside)d(the)j(\014rst)e (graph,)0 1380 y(some)h(inside)d(the)j(second)g(one.)40 b(F)-8 b(or)29 b(an)g(input)d(c)m(hosen)k(according)e(to)h Fj(D)2537 1394 y Fe(N)2605 1380 y Fm(,)g(the)g(v)-5 b(alues)28 b(of)g(these)h(pairs)f(will)e(b)s(e)0 1539 y Fj(q)j Fm(uniformly)23 b(and)j(indep)s(enden)m(tly)d(c)m(hosen)k(random)e(v)-5 b(alues)25 b(in)g Ff(f)p Fm(0)p Fj(;)15 b Fm(1)p Ff(g)p Fm(.)42 b(W)-8 b(e)27 b(shall)d(sho)m(w)i(that)h(the)g(distribution)0 1697 y(of)k(these)f(v)-5 b(alues)30 b(is)f(also)i(nearly)e(uniform)f (and)i(indep)s(enden)m(t)e(when)i(c)m(ho)s(osing)g(an)g(input)f (according)h(to)h Fj(D)3760 1711 y Fe(P)3819 1697 y Fm(.)141 1855 y(F)-8 b(or)25 b(this,)f(let)f Fj(u)671 1869 y Fi(1)711 1855 y Fj(;)15 b(:)g(:)g(:)h(;)f(u)964 1870 y Fe(k)1031 1855 y Fm(b)s(e)23 b(all)g(v)m(ertices)h(of)g(the)g(\014rst)f(graph)g (that)h(are)g(con)m(tained)g(in)e(an)m(y)i(pairs)f(queried)f(b)m(y)0 2013 y(the)27 b(algorithm,)g(and)f(let)g Fj(v)934 2027 y Fi(1)974 2013 y Fj(;)15 b(:)g(:)g(:)h(;)f(v)1219 2028 y Fe(l)1273 2013 y Fm(b)s(e)26 b(all)f(suc)m(h)i(v)m(ertices)g(of)g (the)g(second)g(graph.)39 b(It)26 b(is)g(clear)h(that)g Fj(k)i Ff(\024)c Fm(2)p Fj(q)30 b Fm(and)0 2171 y Fj(l)i Ff(\024)d Fm(2)p Fj(q)s Fm(.)50 b(No)m(w)34 b(let)f Fj(\033)g Fm(:)d Ff(f)p Fm(1)p Fj(;)15 b(:)g(:)g(:)j(;)d(n)p Ff(g)30 b(!)g(f)p Fm(1)p Fj(;)15 b(:)g(:)g(:)i(;)e(n)p Ff(g)34 b Fm(b)s(e)e(the)i(p)s(erm)m(utation)e(used)g(in)g Fj(D)2980 2185 y Fe(P)3072 2171 y Fm(to)i(c)m(ho)s(ose)g(the)g(second)0 2329 y(graph)24 b(according)h(to)g(the)g(\014rst.)38 b(Let)26 b Fj(E)k Fm(denote)25 b(the)g(ev)m(en)m(t)h(that,)h(for)d (this)g(p)s(erm)m(utation,)h Fj(\033)s Fm(\()p Fj(u)3266 2343 y Fe(i)3295 2329 y Fm(\))g Ff(6)p Fm(=)g Fj(v)3495 2343 y Fe(j)3557 2329 y Fm(for)f(ev)m(ery)0 2487 y(1)34 b Ff(\024)f Fj(i)h Ff(\024)g Fj(k)k Fm(and)d(1)f Ff(\024)f Fj(j)40 b Ff(\024)33 b Fj(l)r Fm(.)56 b(It)35 b(is)g(clear)g(that,)j (conditioned)c(on)h(the)h(case)g(that)g Fj(E)41 b Fm(o)s(ccurs,)36 b Fj(p)3398 2501 y Fi(1)3437 2487 y Fj(;)15 b(:)g(:)g(:)i(;)e(p)3685 2501 y Fe(q)3759 2487 y Fm(will)0 2645 y(b)s(e)32 b(again)h Fj(q)j Fm(uniformly)30 b(and)j(indep)s(enden)m(tly)d(c)m(hosen)j(v)-5 b(alues)33 b(of)g Ff(f)p Fm(0)p Fj(;)15 b Fm(1)p Ff(g)p Fm(.)51 b(But)33 b(since)f(for)h Fj(n)f Fm(large)i(enough)e(\(as)0 2803 y(a)g(function)e(of)i Fj(q)s Fm(\))f(the)h(ev)m(en)m(t)h Fj(E)k Fm(o)s(ccurs)31 b(with)f(probabilit)m(y)f(close)j(to)g(1,)h(it)e (means)g(that)h(the)g(\(unconditioned\))0 2961 y(distribution)27 b(of)k Fj(p)646 2975 y Fi(1)686 2961 y Fj(;)15 b(:)g(:)g(:)h(;)f(p)933 2975 y Fe(q)1002 2961 y Fm(will)29 b(b)s(e)h(close)h(to)h(the)f (uniform)e(indep)s(enden)m(t)g(distribution)e(also)k(when)f(an)h(input) 0 3119 y(is)e(c)m(hosen)i(according)g(to)g Fj(D)978 3133 y Fe(P)1037 3119 y Fm(.)141 3277 y(The)38 b(ab)s(o)m(v)m(e)i(sho)m(ws)e (that)i(the)e(queries)g Fj(p)1602 3291 y Fi(1)1641 3277 y Fj(;)15 b(:)g(:)g(:)i(;)e(p)1889 3291 y Fe(q)1965 3277 y Fm(cannot)40 b(distinguish)34 b(b)s(et)m(w)m(een)40 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b(of)g(functions)f(that)h(ha)m(v)m(e)i(small)c(width)g (branc)m(hing)h(programs,)i Fg(Pr)-5 b(o)g(c)g(e)g(e)g(dings)31 b(of)e(the)187 432 y Fm(41)277 399 y Fe(st)373 432 y Fg(IEEE)i(F)n(OCS)f Fm(\(2000\),)j(251{258.)0 665 y([37])47 b(C.)30 b(H.)h(P)m(apadimitriou,)d Fg(Computational)36 b(c)-5 b(omplexity)p Fm(,)32 b(Addison-W)-8 b(esley)30 b(\(1994\).)0 898 y([38])47 b(M.)27 b(P)m(arnas)g(and)f(D.)h(Ron,)g(T) -8 b(esting)27 b(metric)f(prop)s(erties,)g Fg(Pr)-5 b(o)g(c)g(e)g(e)g (dings)31 b(of)e(the)h Fm(33)3000 865 y Fe(r)r(d)3104 898 y Fg(A)n(CM)e(STOC)e Fm(\(2001\),)187 1056 y(276{285.)0 1288 y([39])47 b(M.)26 b(P)m(arnas,)h(D.)g(Ron)f(and)f(R.)h(Rubinfeld,) d(T)-8 b(esting)26 b(paren)m(thesis)f(languages,)i Fg(Pr)-5 b(o)g(c)g(e)g(e)g(dings)31 b(of)d(the)h Fm(5)3682 1256 y Fe(th)3781 1288 y Fg(In-)187 1447 y(ternational)j(Workshop)g(on)e(R) -5 b(andomization)34 b(and)c(Appr)-5 b(oximation)33 b(T)-7 b(e)i(chniques)30 b(in)g(Computer)h(Scienc)-5 b(e)187 1605 y Fm(\(2001\),)33 b(261{272.)0 1837 y([40])47 b(V.)38 b(R\177)-45 b(odl)36 b(and)h(R.)h(Duk)m(e,)j(On)36 b(graphs)h(with)g (small)f(subgraphs)f(of)j(large)g(c)m(hromatic)g(n)m(um)m(b)s(er,)g Fg(Gr)-5 b(aphs)187 1996 y(and)33 b(Combinatorics)g Fm(1)d(\(1985\),)k (91{96.)0 2228 y([41])47 b(D.)21 b(Ron,)i(Prop)s(ert)m(y)e(testing)g (\(a)g(tutorial\),)i(In:)35 b Fg(Handb)-5 b(o)g(ok)26 b(of)e(R)-5 b(andomize)g(d)27 b(Computing)22 b Fm(\(S.)f(Ra)5 b(jasek)-5 b(aran,)187 2386 y(P)d(.)31 b(M.)g(P)m(ardalos,)f(J.)g(H.)h (Reif)f(and)g(J.)g(D.)h(P)-8 b(.)31 b(Rolim)e(eds\),)h(Klu)m(w)m(er)g (Press)g(\(2001\).)0 2619 y([42])47 b(R.)42 b(Rubinfeld,)g(On)f(the)i (robustness)e(of)h(functional)f(equations,)k Fg(SIAM)d(Journal)j(on)f (Computing)f Fm(28)187 2777 y(\(1999\),)33 b(1972{1997.)0 3010 y([43])47 b(R.)29 b(Rubinfeld)c(and)j(M.)h(Sudan,)f(Robust)g(c)m (haracterization)i(of)f(p)s(olynomials)d(with)h(applications)g(to)i (pro-)187 3168 y(gram)35 b(testing,)g Fg(SIAM)h(Journal)h(of)g (Computing)f Fm(25)f(\(1996\),)j(252{271)g(\(\014rst)c(app)s(eared)f (as)i(a)g(tec)m(hnical)187 3326 y(rep)s(ort,)30 b(Cornell)e(Univ)m (ersit)m(y)-8 b(,)31 b(1993\).)0 3559 y([44])47 b(R.)22 b(Rubinfeld)c(and)j(M.)h(Sudan,)g(T)-8 b(esting)22 b(p)s(olynomial)d (functions)h(e\016cien)m(tly)i(and)f(o)m(v)m(er)i(rational)e(domains,) 187 3717 y Fg(Pr)-5 b(o)g(c)g(e)g(e)g(dings)35 b(of)d(the)h Fm(3)975 3684 y Fe(r)r(d)1083 3717 y Fg(A)n(nnual)f(A)n(CM-SIAM)e(Symp) -5 b(osium)35 b(on)e(Discr)-5 b(ete)32 b(A)n(lgorithms)g Fm(\(1992\),)h(23{43.)0 3950 y([45])47 b(E.)58 b(Szemer)m(\023)-43 b(edi,)65 b(Regular)57 b(partitions)f(of)i(graphs,)64 b(In:)95 b Fg(Pr)-5 b(o)g(c.)59 b(Col)5 b(lo)-5 b(que)58 b(Inter.)g(CNRS)g Fm(No.)h(260)187 4108 y(\(J.)30 b(C.)h(Bermond,)f(J.) g(C.)g(F)-8 b(ournier,)30 b(M.)h(Las)f(V)-8 b(ergnas)32 b(and)d(D.)i(Sotteau)h(eds.\),)f(1978,)h(399{401.)0 4341 y([46])47 b(A.)27 b(C.)g(Y)-8 b(ao,)30 b(Probabilistic)24 b(computation,)k(to)m(w)m(ards)g(a)f(uni\014ed)e(measure)i(of)g (complexit)m(y)-8 b(.)28 b Fg(Pr)-5 b(o)g(c)g(e)g(e)g(dings)32 b(of)187 4499 y(the)h Fm(18)428 4466 y Fe(th)532 4499 y Fg(IEEE)e(F)n(OCS)f Fm(\(1977\),)j(222{227.)1905 5532 y(30)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF