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b(or)g(3\))g(is)g(well-solv)o (ed,)h(so)g(the)f(main)g(issue)g(is)g(that)f(of)h(dealing)h(with)f(a) -152 4948 y Fk(Permission)h(to)f(mak)o(e)h(digital)i(or)d(hard)g (copies)i(of)e(all)h(or)f(part)g(of)g(this)h(w)o(ork)f(for)-152 5022 y(personal)f(or)f(classroom)g(use)f(is)g(granted)j(without)e(fee)h (pro)o(vided)g(that)f(copies)h(are)-152 5097 y(not)c(made)f(or)g (distrib)o(uted)i(for)e(pro\002t)g(or)g(commercial)i(adv)n(antage)h (and)d(that)h(copies)-152 5172 y(bear)e(this)f(notice)i(and)e(the)g (full)h(citation)h(on)e(the)g(\002rst)g(page.)25 b(T)-5 b(o)15 b(cop)o(y)j(otherwise,)g(to)-152 5246 y(republish,)g(to)e(post)f (on)h(serv)o(ers)g(or)g(to)g(redistrib)o(ute)i(to)e(lists,)g(requires)i (prior)e(speci\002c)-152 5321 y(permission)i(and/or)h(a)e(fee.)-152 5396 y(Cop)o(yright)j(200X)d(A)m(CM)g(X\255XXXXX\255XX\255X/XX/XX)g (...)p Fr($)p Fk(5.00.)2040 1524 y Fr(lar)o(ge)24 b(number)h(of)f (dimensions,)i(the)e(so-called)g(\223curse)h(of)e(dimensional-)2040 1611 y(ity\224.)36 b(Despite)23 b(decades)i(of)e(intensi)n(v)o(e)h(ef)n (fort,)g(the)f(current)h(solutions)f(are)2040 1698 y(not)d(entirely)g (satisf)o(actory;)g(in)f(f)o(act,)h(for)f(lar)o(ge)h(enough)h Fq(d)p Fr(,)e(in)h(theory)g(or)f(in)2040 1785 y(practice,)24 b(the)o(y)g(often)g(pro)o(vide)g(little)e(impro)o(v)o(ement)i(o)o(v)o (er)g(a)f(linear)g(algo-)2040 1872 y(rithm)e(which)g(compares)h(a)f (query)g(to)g(each)h(point)f(from)g(the)g(database.)29 b(In)2040 1960 y(particular)m(,)17 b(it)g(w)o(as)g(sho)n(wn)i(in)e ([28])g(\(both)h(empirically)f(and)h(theoretically\))2040 2047 y(that)25 b Fm(all)g Fr(current)h(inde)o(xing)g(techniques)h (\(based)f(on)g(space)g(partitioning\))2040 2134 y(de)o(grade)20 b(to)f(linear)g(search)g(for)g(suf)n(\002ciently)g(high)h(dimensions.) 2115 2221 y(In)27 b(recent)g(years,)i(se)n(v)o(eral)e(researchers)g (proposed)i(to)e(a)o(v)o(oid)f(the)h(run-)2040 2308 y(ning)d(time)e (bottleneck)i(by)f(using)h Fm(appr)m(oximation)g Fr(\(e.g.,)f([3,)g (22,)h(19,)g(24,)2040 2396 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b(functions)g(for)g(the)f (case)h(when)g(the)f(points)h(li)n(v)o(e)f(in)h(binary)g(Hamming)2040 4052 y(space)k Fl(f)p Fn(0)p Fq(;)14 b Fn(1)p Fl(g)2413 4020 y Fp(d)2450 4052 y Fr(.)27 b(The)o(y)21 b(sho)n(wed)h(e)o (xperimentally)f(that)f(the)h(data)f(structure)2040 4139 y(achie)n(v)o(es)d(lar)o(ge)e(speedup)i(o)o(v)o(er)f(se)n(v)o(eral)g (tree-based)g(data)g(structures)g(when)2040 4226 y(the)21 b(data)g(is)f(stored)h(on)g(disk.)29 b(In)21 b(addition,)g(since)g(the) g(LSH)e(is)i(a)f(hashing-)2040 4313 y(based)i(scheme,)g(it)f(can)g(be)h (naturally)f(e)o(xtended)i(to)e(the)g Fm(dynamic)h Fr(setting,)2040 4400 y(i.e.,)29 b(when)g(insertion)f(and)h(deletion)f(operations)i (also)e(need)h(to)f(be)g(sup-)2040 4488 y(ported.)35 b(This)22 b(a)o(v)o(oids)h(the)g(comple)o(xity)g(of)g(dealing)g(with)g (tree)f(structures)2040 4575 y(when)e(the)f(data)g(is)f(dynamic.)2115 4662 y(The)30 b(LSH)e(algorithm)i(has)h(been)f(since)g(used)h(in)e (numerous)j(applied)2040 4749 y(settings,)27 b(e.g.,)g(see)f([14,)h (10,)h(16,)f(27,)h(5,)f(7,)g(29,)h(6,)f(26,)g(13].)44 b(Ho)n(we)n(v)o(er)m(,)2040 4836 y(it)22 b(suf)n(fers)g(from)g(a)g (fundamental)i(dra)o(wback:)30 b(it)22 b(is)g(f)o(ast)g(and)h(simple)f (only)2040 4924 y(when)j(the)f(input)g(points)g(li)n(v)o(e)g(in)g(the)g (Hamming)g(space)h(\(indeed,)g(almost)2040 5011 y(all)d(of)h(the)g(abo) o(v)o(e)g(applications)h(in)m(v)o(olv)o(ed)f(binary)g(data\).)35 b(As)22 b(mentioned)2040 5098 y(in)d([19,)g(14],)h(it)e(is)h(possible)h (to)f(e)o(xtend)h(the)f(algorithm)h(to)f(the)g Fq(l)3709 5106 y Fo(2)3762 5098 y Fr(norm,)h(by)2040 5185 y(embedding)h Fq(l)2414 5193 y Fo(2)2467 5185 y Fr(space)e(into)g Fq(l)2809 5193 y Fo(1)2862 5185 y Fr(space,)g(and)g(then)h Fq(l)3362 5193 y Fo(1)3415 5185 y Fr(space)f(into)g(Hamming)2040 5272 y(space.)27 b(Ho)n(we)n(v)o(er)m(,)20 b(it)f(increases)h(the)g (query)h(time)e(and/or)i(error)f(by)g(a)g(lar)o(ge)2040 5359 y(f)o(actor)f(and)h(complicates)f(the)g(algorithm.)p eop %%Page: 2 2 2 1 bop -77 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b(algorithm)g(also)h(inherits)f(tw)o(o)g(v)o(ery)g (con)m(v)o(enient)i(properties)f(of)-152 2059 y(LSH)26 b(schemes.)47 b(The)27 b(\002rst)f(one)h(is)f(that)h(it)f(w)o(orks)h (well)f(on)i(data)e(that)h(is)-152 2146 y(e)o(xtremely)h (high-dimensional)g(b)o(ut)f(sparse.)47 b(Speci\002cally)-5 b(,)28 b(the)f(running)-152 2233 y(time)20 b(bound)h(remains)f (unchanged)j(if)c Fq(d)g Fr(denotes)i(the)f(maximum)h(number)-152 2320 y(of)j(non-zero)h(elements)f(in)f(v)o(ectors.)38 b(T)-6 b(o)23 b(our)h(kno)n(wledge,)i(this)e(property)-152 2407 y(is)29 b(not)h(shared)g(by)g(other)f(kno)n(wn)i(spatial)e(data)g (structures.)55 b(Thanks)30 b(to)-152 2494 y(this)c(property)-5 b(,)29 b(we)d(were)g(able)h(to)f(use)g(our)h(ne)n(w)g(LSH)e(scheme)i (\(specif-)-152 2582 y(ically)-5 b(,)30 b(the)e Fq(l)200 2590 y Fo(1)262 2582 y Fr(norm)h(v)o(ersion\))f(for)g(f)o(ast)g(color)o (-based)h(image)f(similarity)-152 2669 y(search)19 b([20].)24 b(In)18 b(that)g(conte)o(xt,)h(each)h(image)e(w)o(as)h(represented)h 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b(papers)g([9,)i(21,)g(23].)55 b(LSH)28 b(solv)o(es)i(an)g(easier)f(problem)h(\(near)-152 4063 y(neighbor)18 b(under)f Fq(l)339 4071 y Fo(2)390 4063 y Fr(norm\),)g(while)f(w)o(orking)h(under)g(weak)o(er)g (assumptions)-152 4151 y(about)g(the)e(gro)n(wth)h(function.)23 b(It)15 b(is)g(also)h(some)n(what)g(f)o(aster)m(,)g(due)h(to)e(the)h(f) o(act)-152 4238 y(that)h(the)h Fn(log)d Fq(n)i Fr(f)o(actor)g(in)g(the) h(query)g(time)f(of)g(the)g(earlier)g(schemes)h(is)f Fm(mul-)-152 4325 y(tiplied)i Fr(by)g(a)g(function)h(of)f Fq(b)p Fr(,)f(while)h(in)g(our)g(case)g(this)g(f)o(actor)g(is)g(additi) n(v)o(e.)-77 4412 y(W)-6 b(e)18 b(complement)i(our)e(theoretical)h (analysis)g(with)f(e)o(xperimental)h(e)n(v)n(al-)-152 4499 y(uation)k(of)e(the)h(algorithm)g(on)h(data)f(with)f(wide)h(range) h(of)e(parameters.)33 b(In)-152 4587 y(particular)m(,)20 b(we)f(compare)h(our)g(algorithm)g(to)f(an)h(approximate)h(v)o(ersion)f (of)-152 4674 y(the)j Fq(k)r(d)p Fr(-tree)g(algorithm)h([2].)35 b(W)-6 b(e)23 b(performed)h(the)f(e)o(xperiments)h(on)f(syn-)-152 4761 y(thetic)i(data)h(sets)f(containing)h(\223planted\224)g(near)g (neighbor)h(\(see)e(section)g(5)-152 4848 y(for)e(more)g(details\);)h (similar)e(model)i(w)o(as)f(earlier)f(used)h(in)g([30].)35 b(Our)23 b(e)o(x-)-152 4935 y(periments)16 b(indicate)g(that)f(the)g (ne)n(w)h(LSH)e(scheme)i(achie)n(v)o(es)h(query)f(time)f(of)-152 5022 y(up)20 b(to)f(40)g(times)g(better)f(than)i(the)f(query)g(time)g (of)g(the)g Fq(k)r(d)p Fr(-tree)f(algorithm.)-152 5172 y Fs(1.1)99 b(Notations)25 b(and)g(pr)n(oblem)h(de\002nitions)-77 5280 y Fr(W)-6 b(e)25 b(use)h Fq(l)195 5248 y Fp(d)194 5293 y(p)257 5280 y Fr(to)f(denote)i(the)f(space)g Fl(<)928 5248 y Fp(d)990 5280 y Fr(under)g(the)g Fq(l)1325 5288 y Fp(p)1386 5280 y Fr(norm.)44 b(F)o(or)25 b(an)o(y)-152 5378 y(point)20 b Fh(v)j Fl(2)f(<)216 5346 y Fp(d)253 5378 y Fr(,)c(we)h(denote)h(by)f Fl(jj)747 5377 y Fq(~)749 5378 y Fh(v)s Fl(jj)838 5386 y Fp(p)894 5378 y Fr(the)g Fq(l)1027 5386 y Fp(p)1081 5378 y Fr(norm)h(of)f(the)g(v)o(ector)1651 5377 y Fq(~)1653 5378 y Fh(v)r Fr(.)k(Let)2040 -69 y Fl(M)e Fn(=)g(\()p Fq(X)r(;)13 b(d)p Fn(\))18 b Fr(be)h(an)o(y)f (metric)g(space,)h(and)g Fq(v)24 b Fl(2)e Fq(X)6 b Fr(.)23 b(The)18 b Fm(ball)g Fr(of)g(radius)h Fq(r)2040 19 y Fr(centered)h(at)e Fq(v)k Fr(is)d(de\002ned)g(as)g Fq(B)t Fn(\()p Fq(v)s(;)13 b(r)r Fn(\))21 b(=)g Fl(f)p Fq(q)k Fl(2)c Fq(X)28 b Fl(j)22 b Fq(d)p Fn(\()p Fq(v)s(;)12 b(q)s Fn(\))22 b Fl(\024)f Fq(r)r Fl(g)p Fr(.)2115 106 y(Let)i Fq(c)30 b Fn(=)f(1)21 b(+)f Fq(\017)p Fr(.)37 b(In)23 b(this)g(paper)h(we)g(focus)g(on)g(the)f Fn(\()p Fq(R)q(;)13 b(c)p Fn(\))p Fr(-NN)23 b(prob-)2040 193 y(lem.)44 b(Observ)o(e)27 b(that)e Fn(\()p Fq(R)q(;)13 b(c)p Fn(\))p Fr(-NN)26 b(is)f(simply)h(a)g(decision)h(v)o(ersion)f(of) g(the)2040 280 y(Approximate)j(Nearest)g(Neighbor)g(problem.)52 b(Although)29 b(in)f(man)o(y)h(ap-)2040 367 y(plications)i(solving)g (the)f(decision)h(v)o(ersion)g(is)f(good)h(enough,)k(one)c(can)2040 454 y(also)25 b(reduce)h(the)e(approximate)i(NN)f(problem)g(to)g (approximate)h(NN)e(via)2040 542 y(binary-search-lik)o(e)c(approach.)25 b(In)18 b(particular)m(,)h(it)f(is)g(kno)n(wn)i([19,)f(15])g(that)2040 629 y(the)k Fq(c)p Fr(-approximate)h(NN)e(problem)i(reduces)g(to)e Fq(O)r Fn(\(log)s(\()p Fq(n=\017)p Fn(\)\))h Fr(instances)2040 716 y(of)28 b Fn(\()p Fq(R)q(;)13 b(c)p Fn(\))p Fr(-NN.)27 b(Then,)j(the)e(comple)o(xity)h(of)f Fq(c)p Fr(-approximate)h(NN)f(is)f (the)2040 803 y(same)19 b(\(within)g(log)g(f)o(actor\))g(as)g(that)g (of)g(the)g Fn(\()p Fq(R)q(;)12 b(c)p Fn(\))p Fr(-NN)19 b(problem.)2040 999 y Fs(2.)99 b(LOCALITY)-9 b(\255SENSITIVE)27 b(HASHING)2115 1107 y Fr(An)19 b(important)h(technique)g(from)f([19],)g (to)h(solv)o(e)f(the)g Fn(\()p Fq(R)q(;)13 b(c)p Fn(\))p Fr(-NN)19 b(prob-)2040 1194 y(lem)28 b(is)f(locality)g(sensiti)n(v)o(e) h(hashing)h(or)f(LSH.)e(F)o(or)h(a)h(domain)g Fq(S)k Fr(of)27 b(the)2040 1282 y(points)19 b(set)g(with)g(distance)g(measure) h Fq(D)r Fr(,)f(an)g(LSH)f(f)o(amily)h(is)f(de\002ned)i(as:)2125 1422 y(D)t Fg(E)t(FI)t(N)t(I)t(T)t(I)t(O)t(N)40 b Fr(1.)e Fm(A)14 b(family)g Fl(H)21 b Fn(=)g Fl(f)p Fq(h)h Fn(:)g Fq(S)k Fl(!)21 b Fq(U)8 b Fl(g)15 b Fm(is)f(called)g Fn(\()p Fq(r)3825 1430 y Fo(1)3860 1422 y Fq(;)f(r)3929 1430 y Fo(2)3963 1422 y Fq(;)g(p)4036 1430 y Fo(1)4070 1422 y Fq(;)g(p)4143 1430 y Fo(2)4177 1422 y Fn(\))p Fr(-)2040 1510 y(sensiti)n(v)o(e)19 b Fm(for)g Fq(D)i Fm(if)e(for)f(any)i Fq(v)s(;)13 b(q)24 b Fl(2)e Fq(S)2151 1591 y Fl(\017)38 b Fm(if)18 b Fq(v)24 b Fl(2)e Fq(B)t Fn(\()p Fq(q)s(;)13 b(r)2619 1599 y Fo(1)2653 1591 y Fn(\))19 b Fm(then)g Fn(Pr)2931 1599 y Ff(H)2988 1591 y Fn([)p Fq(h)p Fn(\()p Fq(q)s Fn(\))j(=)f Fq(h)p Fn(\()p Fq(v)s Fn(\)])g Fl(\025)h Fq(p)3560 1599 y Fo(1)3594 1591 y Fm(,)2151 1734 y Fl(\017)38 b Fm(if)18 b Fq(v)33 b(=)-47 b Fl(2)22 b Fq(B)t Fn(\()p Fq(q)s(;)13 b(r)2619 1742 y Fo(2)2653 1734 y Fn(\))19 b Fm(then)g Fn(Pr)2931 1742 y Ff(H)2988 1734 y Fn([)p Fq(h)p Fn(\()p Fq(q)s Fn(\))j(=)f Fq(h)p Fn(\()p Fq(v)s Fn(\)])g Fl(\024)h Fq(p)3560 1742 y Fo(2)3594 1734 y Fm(.)2040 1875 y Fr(In)h(order)g(for) g(a)g(locality-sensiti)n(v)o(e)h(hash)f(\(LSH\))f(f)o(amily)h(to)g(be)g (useful,)h(it)2040 1962 y(has)19 b(to)g(satisfy)g(inequalities)g Fq(p)2851 1970 y Fo(1)2907 1962 y Fq(>)i(p)3027 1970 y Fo(2)3080 1962 y Fr(and)e Fq(r)3241 1970 y Fo(1)3297 1962 y Fq(<)i(r)3413 1970 y Fo(2)3447 1962 y Fr(.)2115 2050 y(W)-6 b(e)25 b(will)g(brie\003y)g(describe,)j(from)e([19],)h(ho)n (w)f(a)g(LSH)e(f)o(amily)i(can)g(be)2040 2137 y(used)k(to)g(solv)o(e)f (the)h Fn(\()p Fq(R)q(;)13 b(c)p Fn(\))p Fr(-NN)29 b(problem:)45 b(W)-6 b(e)29 b(choose)h Fq(r)3660 2145 y Fo(1)3736 2137 y Fn(=)41 b Fq(R)30 b Fr(and)2040 2224 y Fq(r)2075 2232 y Fo(2)2149 2224 y Fn(=)39 b Fq(c)24 b Fl(\001)h Fq(R)q Fr(.)51 b(Gi)n(v)o(en)29 b(a)f(f)o(amily)g Fl(H)h Fr(of)f(hash)i (functions)f(with)f(parame-)2040 2311 y(ters)19 b Fn(\()p Fq(r)2232 2319 y Fo(1)2266 2311 y Fq(;)13 b(r)2335 2319 y Fo(2)2370 2311 y Fq(;)g(p)2443 2319 y Fo(1)2477 2311 y Fq(;)g(p)2550 2319 y Fo(2)2584 2311 y Fn(\))19 b Fr(as)g(in)h (De\002nition)f(1,)g(we)g(amplify)h(the)f(gap)h(between)2040 2398 y(the)32 b(\223high\224)i(probability)e Fq(p)2793 2406 y Fo(1)2860 2398 y Fr(and)g(\223lo)n(w\224)h(probability)g Fq(p)3607 2406 y Fo(2)3673 2398 y Fr(by)g(concate-)2040 2486 y(nating)d(se)n(v)o(eral)h(functions.)56 b(In)30 b(particular)m(,)i(for)e Fq(k)h Fr(speci\002ed)g(later)m(,)g(de-)2040 2573 y(\002ne)e(a)g(function)g(f)o(amily)g Fl(G)45 b Fn(=)40 b Fl(f)p Fq(g)j Fn(:)d Fq(S)45 b Fl(!)40 b Fq(U)3382 2541 y Fp(k)3420 2573 y Fl(g)29 b Fr(such)h(that)f Fq(g)s Fn(\()p Fq(v)s Fn(\))39 b(=)2040 2660 y(\()p Fq(h)2114 2668 y Fo(1)2149 2660 y Fn(\()p Fq(v)s Fn(\))p Fq(;)13 b(:)g(:)g(:)26 b(;)13 b(h)2476 2669 y Fp(k)2514 2660 y Fn(\()p Fq(v)s Fn(\)\))p Fr(,)23 b(where)h Fq(h)2936 2668 y Fp(i)2992 2660 y Fl(2)29 b(H)p Fr(.)36 b(F)o(or)22 b(an)i(inte)o(ger)f Fq(L)g Fr(we)g(choose)2040 2747 y Fq(L)28 b Fr(functions)i Fq(g)2464 2755 y Fo(1)2498 2747 y Fq(;)13 b(:)g(:)g(:)26 b(;)13 b(g)2718 2755 y Fp(L)2792 2747 y Fr(from)28 b Fl(G)t Fr(,)i(independently)h(and)e(uniformly)g(at) 2040 2834 y(random.)35 b(During)23 b(preprocessing,)i(we)e(store)f (each)h Fq(v)32 b Fl(2)c Fq(P)33 b Fr(\(input)23 b(point)2040 2921 y(set\))h(in)h(the)g(b)o(uck)o(et)g Fq(g)2629 2929 y Fp(j)2662 2921 y Fn(\()p Fq(v)s Fn(\))p Fr(,)g(for)g Fq(j)37 b Fn(=)32 b(1)p Fq(;)13 b(:)g(:)h(:)25 b(;)14 b(L)p Fr(.)40 b(Since)25 b(the)g(total)f(num-)2040 3009 y(ber)h(of)f(b)o(uck)o(ets)i(may)f(be)g(lar)o(ge,)g(we)g(retain)f(only) h(the)g(non-empty)h(b)o(uck-)2040 3096 y(ets)h(by)h(resorting)f(to)h (hashing.)49 b(T)-6 b(o)27 b(process)h(a)f(query)h Fq(q)s Fr(,)h(we)e(search)h(all)2040 3183 y(b)o(uck)o(ets)e Fq(g)2328 3191 y Fo(1)2362 3183 y Fn(\()p Fq(q)s Fn(\))p Fq(;)13 b(:)g(:)g(:)26 b(;)13 b(g)2679 3191 y Fp(L)2725 3183 y Fn(\()p Fq(q)s Fn(\))p Fr(;)28 b(as)d(it)g(is)f(possible)i (\(though)h(unlik)o(ely\))f(that)2040 3270 y(the)e(total)f(number)i(of) f(points)g(stored)g(in)g(those)g(b)o(uck)o(ets)h(is)e(lar)o(ge,)i(we)f (in-)2040 3357 y(terrupt)j(search)g(after)f(\002nding)h(\002rst)f Fn(3)p Fq(L)h Fr(points)g(\(including)g(duplicates\).)2040 3444 y(Let)d Fq(v)2201 3452 y Fo(1)2236 3444 y Fq(;)13 b(:)g(:)h(:)25 b(;)14 b(v)2457 3452 y Fp(t)2509 3444 y Fr(be)25 b(the)g(points)g(encountered)i(therein.)41 b(F)o(or)24 b(each)h Fq(v)3909 3452 y Fp(j)3942 3444 y Fr(,)h(if)2040 3532 y Fq(v)2077 3540 y Fp(j)2131 3532 y Fl(2)c Fq(B)t Fn(\()p Fq(q)s(;)13 b(r)2402 3540 y Fo(2)2436 3532 y Fn(\))18 b Fr(then)i(we)f(return)i Fg(Y)t(E)t(S)e Fr(and)h Fq(v)3241 3540 y Fp(j)3274 3532 y Fr(,)e(else)h(we)g(return)i Fg(N)t(O)r Fr(.)2115 3619 y(The)j(parameters)h Fq(k)h Fr(and)f Fq(L)f Fr(are)h(chosen)g(so)g(as)f(to)g(ensure)h(that)g(with)e (a)2040 3706 y(constant)d(probability)f(the)g(follo)n(wing)h(tw)o(o)f (properties)g(hold:)2133 3847 y(1.)38 b(If)20 b(there)h(e)o(xists)g Fq(v)2697 3815 y Ff(\003)2758 3847 y Fl(2)k Fq(B)t Fn(\()p Fq(q)s(;)13 b(r)3032 3855 y Fo(1)3066 3847 y Fn(\))21 b Fr(then)g Fq(g)3303 3855 y Fp(j)3335 3847 y Fn(\()p Fq(v)3405 3815 y Ff(\003)3441 3847 y Fn(\))k(=)g Fq(g)3618 3855 y Fp(j)3650 3847 y Fn(\()p Fq(q)s Fn(\))20 b Fr(for)h(some)2227 3934 y Fq(j)26 b Fn(=)21 b(1)13 b Fq(:)g(:)g(:)h(L)p Fr(,)k(and)2133 4078 y(2.)38 b(The)28 b(total)f(number)i(of)f (collisions)g(of)g Fq(q)j Fr(with)c(points)h(from)g Fq(P)35 b Fl(\000)2227 4165 y Fq(B)t Fn(\()p Fq(q)s(;)13 b(r)2425 4173 y Fo(2)2459 4165 y Fn(\))19 b Fr(is)f(less)h(than)g Fn(3)p Fq(L)p Fr(,)g(i.e.)2520 4283 y Fp(L)2485 4305 y Fj(X)2488 4463 y Fp(j)s Fo(=1)2609 4376 y Fl(j)p Fn(\()p Fq(P)28 b Fl(\000)17 b Fq(B)t Fn(\()p Fq(q)s(;)c(r)3012 4384 y Fo(2)3046 4376 y Fn(\)\))k Fl(\\)g Fq(g)3231 4340 y Ff(\000)p Fo(1)3228 4393 y Fp(j)3313 4376 y Fn(\()p Fq(g)3380 4384 y Fp(j)3412 4376 y Fn(\()p Fq(q)s Fn(\)\))p Fl(j)k Fq(<)g Fn(3)p Fq(L:)2115 4617 y Fr(Observ)o(e)27 b(that)f(if)h(\(1\))f(and)h(\(2\))g(hold,)i(then)e(the)f(algorithm)h 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5298 y Fo(1)3459 5290 y Fq(;)g(p)3532 5298 y Fo(2)3566 5290 y Fn(\))p Fm(-sensitive)18 b(fam-)2040 5378 y(ily)26 b Fl(H)h Fm(for)g(a)f(distance)i(measur)m(e)g Fq(D)r Fm(.)46 b(Then)27 b(ther)m(e)g(e)o(xists)f(an)h(algorithm)p eop %%Page: 3 3 3 2 bop -152 -69 a Fm(for)19 b Fn(\()p Fq(R)q(;)13 b(c)p Fn(\))p Fm(-NN)19 b(under)h(measur)m(e)g Fq(D)h Fm(whic)o(h)f(uses)f Fq(O)r Fn(\()p Fq(dn)f Fn(+)f Fq(n)1491 -100 y Fo(1+)p Fp(\032)1604 -69 y Fn(\))i Fm(space)o(,)-152 19 y(with)24 b(query)h(time)f(dominated)i(by)e Fq(O)r Fn(\()p Fq(n)918 -13 y Fp(\032)955 19 y Fn(\))g Fm(distance)h(computations,)i(and)-152 106 y Fq(O)r Fn(\()p Fq(n)-16 74 y Fp(\032)34 106 y Fn(log)133 124 y Fo(1)p Fp(=p)227 134 y Fe(2)277 106 y Fq(n)p Fn(\))18 b Fm(e)o(valuations)i(of)f(hash)h(functions)g(fr)m(om)e Fl(H)p Fm(,)h(wher)m(e)g Fq(\032)i Fn(=)-142 182 y Fo(ln)10 b(1)p Fp(=p)14 192 y Fe(1)p -142 201 189 4 v -142 244 a Fo(ln)g(1)p Fp(=p)14 254 y Fe(2)57 218 y Fm(.)-152 424 y Fs(3.)100 b(OUR)24 b(LSH)h(SCHEME)-77 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Fr(for)f(all)f(v)n(alues)i(of)f Fq(c)p Fr(.)p eop %%Page: 4 4 4 3 bop 103 1254 a @beginspecial 76 @llx 199 @lly 548 @urx 589 @ury 2126 @rwi 1587 @rhi @setspecial %%BeginDocument: l1_plot.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: /afs/ir.stanford.edu/users/d/a/datar/papers/socg03/l1/l11_plot.eps %%CreationDate: 12/03/2002 20:07:06 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 76 199 548 589 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s {show newpath} bdef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray 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types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 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3 30 3 30 3 30 3 29 3 30 3 30 3 30 3 29 4 30 3 30 3 30 4 29 3 30 4 30 3 30 4 29 3 30 4 30 4 30 3 30 4 29 4 30 4 30 4 30 4 29 4 30 4 30 4 30 4 29 5 30 4 30 4 30 5 29 4 30 5 30 5 30 4 29 5 30 5 30 5 30 5 29 5 30 5 30 5 30 6 29 5 30 5 30 6 30 6 29 5 30 6 30 6 30 6 29 7 30 6 30 6 30 7 29 6 30 7 30 7 30 7 29 7 30 7 30 7 30 8 30 7 29 8 30 8 30 8 30 8 29 9 3309 3736 100 MP stroke 30 8 30 9 30 8 29 10 30 9 30 9 30 10 29 10 30 9 30 11 30 11 29 10 30 11 30 11 30 12 29 11 30 12 30 12 30 13 29 13 30 13 30 13 30 14 29 15 30 14 30 15 30 16 29 15 30 17 30 16 30 18 29 16 30 18 30 20 30 19 29 20 30 20 30 22 30 20 30 23 29 22 30 24 30 26 30 26 29 26 30 27 30 28 30 31 29 30 30 32 30 32 30 35 29 38 30 34 30 42 30 41 29 43 30 44 30 44 30 49 29 50 30 57 30 53 30 62 29 63 30 66 30 67 30 76 29 76 30 87 30 90 30 97 29 98 30 112 30 116 30 125 29 141 30 149 30 163 30 178 928 583 81 MP stroke DO 30 2 30 2 30 3 30 2 29 2 30 2 30 2 30 3 29 2 30 2 30 3 30 2 29 2 30 3 30 2 30 3 29 2 30 3 30 2 30 3 29 3 30 2 30 3 30 3 29 2 30 3 30 3 30 3 29 3 30 3 30 3 30 3 29 3 30 3 30 3 30 3 29 3 30 3 30 3 30 4 29 3 30 3 30 4 30 3 29 4 30 3 30 4 30 4 30 3 29 4 30 4 30 4 30 3 29 4 30 4 30 4 30 5 29 4 30 4 30 4 30 5 29 4 30 4 30 5 30 5 29 4 30 5 30 5 30 5 29 5 30 5 30 5 30 5 29 5 30 6 30 5 30 6 29 5 30 6 30 6 30 6 29 6 30 6 30 6 30 6 29 7 30 6 30 7 30 7 29 7 30 7 30 7 30 7 30 8 29 7 30 8 30 8 30 8 29 8 3309 3778 100 MP stroke 30 9 30 8 30 9 29 9 30 9 30 9 30 10 29 9 30 10 30 10 30 11 29 10 30 11 30 11 30 11 29 12 30 12 30 12 30 12 29 13 30 13 30 13 30 14 29 14 30 15 30 14 30 16 29 15 30 16 30 17 30 17 29 17 30 18 30 19 30 19 29 20 30 20 30 21 30 22 30 22 29 23 30 24 30 25 30 25 29 27 30 27 30 29 30 29 29 31 30 32 30 33 30 34 29 36 30 38 30 39 30 41 29 42 30 45 30 47 30 49 29 51 30 55 30 57 30 60 29 63 30 67 30 71 30 76 29 80 30 85 30 91 30 97 29 104 30 112 30 120 30 130 29 141 30 153 30 167 30 183 928 590 81 MP stroke gr DO 2985 4903 mt (Approximation factor c) s SO 1 sg 0 299 573 0 0 -299 5622 748 4 MP PP -573 0 0 299 573 0 0 -299 5622 748 5 MP stroke 4 w DO SO 6 w 0 sg 5622 748 mt 6195 748 L 5622 449 mt 6195 449 L 5622 748 mt 5622 449 L 6195 748 mt 6195 449 L 5622 748 mt 6195 748 L 5622 748 mt 5622 449 L 5622 748 mt 6195 748 L 5622 449 mt 6195 449 L 5622 748 mt 5622 449 L 6195 748 mt 6195 449 L 5955 571 mt (rho) s 5955 709 mt (1/c) s gs 5622 449 574 300 MR c np 200 0 5688 531 2 MP stroke DO 200 0 5688 669 2 MP stroke SO gr end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument @endspecial 696 1424 a Fr(\(a\))18 b(Optimal)h Fq(\032)f Fr(for)h Fq(l)1248 1432 y Fo(1)2006 1254 y @beginspecial 76 @llx 199 @lly 548 @urx 589 @ury 2126 @rwi 1587 @rhi @setspecial %%BeginDocument: l2_plot.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: /afs/ir.stanford.edu/users/d/a/datar/papers/socg03/l2/l2_plot.eps %%CreationDate: 12/03/2002 20:11:31 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 76 199 548 589 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s {show newpath} bdef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup string currentfile exch readhexstring pop /bmap xdef pop pop} bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform {bmap} image gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 76 199 548 589 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 698 274 5673 4681 MR c np 91 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6913 5186 PR 6 w 0 4226 5356 0 0 -4226 899 4615 4 MP PP -5356 0 0 4226 5356 0 0 -4226 899 4615 5 MP stroke 4 w DO SO 6 w 0 sg 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L 899 4615 mt 6255 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4561 L 899 389 mt 899 442 L %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 120 FMSR 866 4760 mt (1) s 1494 4615 mt 1494 4561 L 1494 389 mt 1494 442 L 1461 4760 mt (2) s 2089 4615 mt 2089 4561 L 2089 389 mt 2089 442 L 2056 4760 mt (3) s 2684 4615 mt 2684 4561 L 2684 389 mt 2684 442 L 2651 4760 mt (4) s 3279 4615 mt 3279 4561 L 3279 389 mt 3279 442 L 3246 4760 mt (5) s 3874 4615 mt 3874 4561 L 3874 389 mt 3874 442 L 3841 4760 mt (6) s 4469 4615 mt 4469 4561 L 4469 389 mt 4469 442 L 4436 4760 mt (7) s 5064 4615 mt 5064 4561 L 5064 389 mt 5064 442 L 5031 4760 mt (8) s 5659 4615 mt 5659 4561 L 5659 389 mt 5659 442 L 5626 4760 mt (9) s 6255 4615 mt 6255 4561 L 6255 389 mt 6255 442 L 6189 4760 mt (10) s 899 4615 mt 952 4615 L 6255 4615 mt 6201 4615 L 798 4659 mt (0) s 899 4192 mt 952 4192 L 6255 4192 mt 6201 4192 L 698 4236 mt (0.1) s 899 3769 mt 952 3769 L 6255 3769 mt 6201 3769 L 698 3813 mt (0.2) s 899 3347 mt 952 3347 L 6255 3347 mt 6201 3347 L 698 3391 mt (0.3) s 899 2924 mt 952 2924 L 6255 2924 mt 6201 2924 L 698 2968 mt (0.4) s 899 2502 mt 952 2502 L 6255 2502 mt 6201 2502 L 698 2546 mt (0.5) s 899 2079 mt 952 2079 L 6255 2079 mt 6201 2079 L 698 2123 mt (0.6) s 899 1656 mt 952 1656 L 6255 1656 mt 6201 1656 L 698 1700 mt (0.7) s 899 1234 mt 952 1234 L 6255 1234 mt 6201 1234 L 698 1278 mt (0.8) s 899 811 mt 952 811 L 6255 811 mt 6201 811 L 698 855 mt (0.9) s 899 389 mt 952 389 L 6255 389 mt 6201 389 L 798 433 mt (1) s 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L gs 899 389 5357 4227 MR c np 30 1 30 2 30 2 30 2 29 2 30 1 30 2 30 2 29 2 30 2 30 2 30 2 29 2 30 2 30 2 30 2 29 2 30 2 30 2 30 3 29 2 30 2 30 2 30 3 29 2 30 2 30 2 30 3 29 2 30 3 30 2 30 3 29 2 30 3 30 2 30 3 29 3 30 2 30 3 30 3 29 3 30 2 30 3 30 3 29 3 30 3 30 3 30 3 30 3 29 3 30 4 30 3 30 3 29 3 30 4 30 3 30 4 29 3 30 4 30 3 30 4 29 4 30 4 30 3 30 4 29 4 30 4 30 5 30 4 29 4 30 4 30 5 30 4 29 5 30 4 30 5 30 5 29 4 30 5 30 5 30 5 29 6 30 5 30 5 30 6 29 5 30 6 30 6 30 6 29 6 30 6 30 6 30 6 30 7 29 6 30 7 30 7 30 7 29 7 3309 3922 100 MP stroke 30 7 30 8 30 8 29 7 30 8 30 8 30 9 29 8 30 9 30 9 30 9 29 9 30 10 30 10 30 10 29 10 30 11 30 11 30 11 29 11 30 12 30 12 30 12 29 13 30 13 30 13 30 14 29 14 30 15 30 15 30 16 29 16 30 16 30 17 30 18 29 18 30 19 30 19 30 21 30 20 29 22 30 23 30 23 30 24 29 25 30 26 30 27 30 28 29 30 30 31 30 32 30 33 29 35 30 37 30 38 30 40 29 43 30 44 30 47 30 49 29 52 30 55 30 59 30 61 29 66 30 70 30 74 30 79 29 85 30 91 30 98 30 105 29 114 30 123 30 133 30 146 29 158 30 174 30 192 30 211 928 623 81 MP stroke DO 30 2 30 2 30 3 30 2 29 2 30 2 30 2 30 3 29 2 30 2 30 3 30 2 29 2 30 3 30 2 30 3 29 2 30 3 30 2 30 3 29 3 30 2 30 3 30 3 29 2 30 3 30 3 30 3 29 3 30 3 30 3 30 3 29 3 30 3 30 3 30 3 29 3 30 3 30 3 30 4 29 3 30 3 30 4 30 3 29 4 30 3 30 4 30 4 30 3 29 4 30 4 30 4 30 3 29 4 30 4 30 4 30 5 29 4 30 4 30 4 30 5 29 4 30 4 30 5 30 5 29 4 30 5 30 5 30 5 29 5 30 5 30 5 30 5 29 5 30 6 30 5 30 6 29 5 30 6 30 6 30 6 29 6 30 6 30 6 30 6 29 7 30 6 30 7 30 7 29 7 30 7 30 7 30 7 30 8 29 7 30 8 30 8 30 8 29 8 3309 3778 100 MP stroke 30 9 30 8 30 9 29 9 30 9 30 9 30 10 29 9 30 10 30 10 30 11 29 10 30 11 30 11 30 11 29 12 30 12 30 12 30 12 29 13 30 13 30 13 30 14 29 14 30 15 30 14 30 16 29 15 30 16 30 17 30 17 29 17 30 18 30 19 30 19 29 20 30 20 30 21 30 22 30 22 29 23 30 24 30 25 30 25 29 27 30 27 30 29 30 29 29 31 30 32 30 33 30 34 29 36 30 38 30 39 30 41 29 42 30 45 30 47 30 49 29 51 30 55 30 57 30 60 29 63 30 67 30 71 30 76 29 80 30 85 30 91 30 97 29 104 30 112 30 120 30 130 29 141 30 153 30 167 30 183 928 590 81 MP stroke gr DO 2985 4903 mt (Approximation factor c) s SO 1 sg 0 299 573 0 0 -299 5622 748 4 MP PP -573 0 0 299 573 0 0 -299 5622 748 5 MP stroke 4 w DO SO 6 w 0 sg 5622 748 mt 6195 748 L 5622 449 mt 6195 449 L 5622 748 mt 5622 449 L 6195 748 mt 6195 449 L 5622 748 mt 6195 748 L 5622 748 mt 5622 449 L 5622 748 mt 6195 748 L 5622 449 mt 6195 449 L 5622 748 mt 5622 449 L 6195 748 mt 6195 449 L 5955 571 mt (rho) s 5955 709 mt (1/c) s gs 5622 449 574 300 MR c np 200 0 5688 531 2 MP stroke DO 200 0 5688 669 2 MP stroke SO gr end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument @endspecial 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b(to)g(test)f(our)g(algorithm.)32 b(The)21 b(dimensionality)i(of)e(the)h(underlying)2040 5129 y Fq(l)2063 5137 y Fo(2)2120 5129 y Fr(space)h(w)o(as)g(v)n(aried) g(between)g Fn(20)h Fr(and)f Fn(500)p Fr(.)35 b(W)-6 b(e)22 b(considered)i(generat-)2040 5216 y(ing)g(all)g(the)g(data)h (and)f(query)h(points)g(independently)h(at)e(random.)40 b(Thus,)2040 5303 y(for)21 b(a)h(data)f(point)h(\(or)f(query)h(point\)) g(its)e(coordinate)j(along)f(e)n(v)o(ery)g(dimen-)p eop %%Page: 5 5 5 4 bop 103 -69 a gsave currentpoint currentpoint translate -90 neg rotate neg exch neg exch translate 103 -69 a @beginspecial 50 @llx 50 @lly 554 @urx 770 @ury 1587 @rwi 2126 @rhi @setspecial %%BeginDocument: r_var_l1.ps %!PS-Adobe-2.0 %%Title: r_var_l1.ps %%Creator: gnuplot 3.7 patchlevel 1 %%CreationDate: Wed Nov 27 01:28:42 2002 %%DocumentFonts: (atend) %%BoundingBox: 50 50 554 770 %%Orientation: Landscape %%Pages: (atend) %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def 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rotate neg exch neg exch translate 2006 -69 a @beginspecial 50 @llx 50 @lly 554 @urx 770 @ury 1587 @rwi 2126 @rhi @setspecial %%BeginDocument: r_var_l2.ps %!PS-Adobe-2.0 %%Title: r_var_l2.ps %%Creator: gnuplot 3.7 patchlevel 1 %%CreationDate: Wed Nov 27 00:52:31 2002 %%DocumentFonts: (atend) %%BoundingBox: 50 50 554 770 %%Orientation: Landscape %%Pages: (atend) %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end %%EndProlog %%Page: 1 1 gnudict begin gsave 50 50 translate 0.100 0.100 scale 90 rotate 0 -5040 translate 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 630 420 M 63 0 V 6269 0 R -63 0 V 546 420 M (0) Rshow 630 865 M 63 0 V 6269 0 R -63 0 V 546 865 M (0.1) Rshow 630 1310 M 63 0 V 6269 0 R 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copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 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%%EndDocument @endspecial 1426 -69 a currentpoint grestore moveto 1426 -69 a 759 1397 a Fr(\(a\))19 b Fq(\032)f Fr(vs)h Fq(c)g Fr(for)g Fq(l)1185 1405 y Fo(1)2006 -69 y gsave currentpoint currentpoint translate -90 neg rotate neg exch neg exch translate 2006 -69 a @beginspecial 50 @llx 50 @lly 554 @urx 770 @ury 1587 @rwi 2126 @rhi @setspecial %%BeginDocument: c_var_l2.ps %!PS-Adobe-2.0 %%Title: c_var_l2.ps %%Creator: gnuplot 3.7 patchlevel 1 %%CreationDate: Tue Dec 3 22:48:58 2002 %%DocumentFonts: (atend) %%BoundingBox: 50 50 554 770 %%Orientation: Landscape %%Pages: (atend) %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt 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b(Har)o(-Peled.)g(A)g(replacement)i (for)f(v)o(oronoi)h(diagrams)f(of)g(near)2202 5341 y(linear)g(size.)f Fm(Pr)m(oceedings)i(of)f(the)g(Symposium)h(on)f(F)-8 b(oundations)21 b(of)p eop %%Page: 7 7 7 6 bop 103 1254 a @beginspecial 70 @llx 196 @lly 549 @urx 589 @ury 2126 @rwi 1587 @rhi @setspecial %%BeginDocument: n_time.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: /afs/athena.mit.edu/user/m/i/mirrokni/research/pstable/mayur/test2/n_time.eps %%CreationDate: 02/18/2003 03:22:14 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 70 196 549 589 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s {show newpath} bdef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup string currentfile exch readhexstring pop /bmap xdef pop pop} bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform {bmap} image gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 70 196 549 589 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 631 274 5746 4716 MR c np 91 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} 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%%EndDocument @endspecial 703 1424 a Fr(\(a\))18 b(query)i(time)f(vs)g Fq(n)2006 1254 y @beginspecial 78 @llx 196 @lly 548 @urx 589 @ury 2126 @rwi 1587 @rhi @setspecial %%BeginDocument: n_speedup.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: /afs/athena.mit.edu/user/m/i/mirrokni/research/pstable/mayur/test2/n_speedup.eps %%CreationDate: 02/18/2003 03:22:56 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 78 196 548 589 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s {show newpath} bdef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup string currentfile exch readhexstring pop /bmap xdef pop pop} bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform {bmap} image gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 78 196 548 589 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 731 274 5640 4716 MR c np 91 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} 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bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef 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3214 -1898 1071 -578 536 -627 321 0 1113 4483 5 MP stroke gr 36 36 1113 4483 FO 36 36 1434 4483 FO 36 36 1970 3856 FO 36 36 3041 3278 FO 36 36 6255 1380 FO gs 899 389 5357 4227 MR c np gr end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument @endspecial 570 1424 a Fr(\(a\))19 b(query)h(time)e(vs)h(dimension) 2006 1254 y @beginspecial 78 @llx 213 @lly 551 @urx 589 @ury 2126 @rwi 1587 @rhi @setspecial %%BeginDocument: dim_speedup.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: /afs/athena.mit.edu/user/m/i/mirrokni/research/pstable/mayur/dim_report/dim_speedup.eps %%CreationDate: 02/18/2003 04:16:13 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 78 213 551 589 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s {show newpath} bdef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR 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Fr(Noting)17 b Fn(log)r(\(1)8 b Fl(\000)g Fq(l)q(x)p Fn(\))22 b Fq(<)f Fn(0)p Fr(,)c(the)f(claim)h (is)f(equi)n(v)n(alent)i(to)e Fq(l)e Fn(log)r(\(1)8 b Fl(\000)2040 4633 y Fq(x)p Fn(\))21 b Fl(\025)g Fn(log)r(\(1)c Fl(\000)g Fq(l)q(x)p Fn(\))p Fr(.)23 b(This)18 b(in)h(turn)g(is)g(equi) n(v)n(alent)h(to)2526 4792 y Fq(g)s Fn(\()p Fq(x)p Fn(\))g Fl(\021)h Fn(\(1)d Fl(\000)e Fq(l)q(x)p Fn(\))h Fl(\000)g Fn(\(1)g Fl(\000)g Fq(x)p Fn(\))3361 4756 y Fp(l)3406 4792 y Fl(\024)k Fn(0)p Fq(:)2040 4940 y Fr(This)f(is)f(tri)n(vially)h (true)f(for)h Fq(x)j Fn(=)g(0)p Fr(.)j(Furthermore,)20 b(taking)h(the)f(deri)n(v)n(ati)n(v)o(e,)2040 5027 y(we)i(see)g Fq(g)2306 4995 y Ff(0)2328 5027 y Fn(\()p Fq(x)p Fn(\))k(=)g Fl(\000)p Fq(l)21 b Fn(+)e Fq(l)q Fn(\(1)g Fl(\000)g Fq(x)p Fn(\))2991 4995 y Fp(l)p Ff(\000)p Fo(1)3093 5027 y Fr(,)j(which)h(is)e(non-positi)n(v)o(e)i(for)f Fq(x)k Fl(2)2040 5114 y Fn([0)p Fq(;)14 b Fn(1\))25 b Fr(and)h Fq(l)35 b Fl(\025)f Fn(1)p Fr(.)42 b(Therefore,)27 b Fq(g)h Fr(is)d(non-increasing)i(in)e(the)h(re)o(gion)f(in)2040 5201 y(which)k(we)f(are)g(interested,)j(and)e(so)g Fq(g)s Fn(\()p Fq(x)p Fn(\))38 b Fl(\024)h Fn(0)28 b Fr(for)g(all)g(v)n(alues) h(in)g(this)2040 5289 y(re)o(gion.)2115 5376 y(No)n(w)19 b(our)g(goal)h(is)e(to)h(upper)h(bound)3092 5343 y Fo(1)p Ff(\000)p Fp(p)3202 5353 y Fe(1)p 3092 5359 143 4 v 3092 5401 a Fo(1)p Ff(\000)p Fp(p)3202 5411 y Fe(2)3245 5376 y Fr(.)p eop %%Page: 9 9 9 8 bop 103 1254 a @beginspecial 70 @llx 213 @lly 549 @urx 589 @ury 2126 @rwi 1587 @rhi @setspecial %%BeginDocument: eps_time.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: /afs/athena.mit.edu/user/m/i/mirrokni/research/pstable/mayur/eps_report/eps_time.eps %%CreationDate: 02/18/2003 04:36:16 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 70 213 549 589 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s {show newpath} bdef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup string currentfile exch readhexstring pop /bmap xdef pop pop} bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform {bmap} image gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 70 213 549 589 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 631 274 5742 4514 MR c np 91 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6913 5186 PR 6 w 0 4226 5356 0 0 -4226 899 4615 4 MP PP -5356 0 0 4226 5356 0 0 -4226 899 4615 5 MP stroke 4 w DO SO 6 w 0 sg 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L 899 4615 mt 6255 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4561 L 899 389 mt 899 442 L %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 120 FMSR 816 4760 mt (0.5) s 1970 4615 mt 1970 4561 L 1970 389 mt 1970 442 L 1937 4760 mt (1) s 3041 4615 mt 3041 4561 L 3041 389 mt 3041 442 L 2958 4760 mt (1.5) s 4112 4615 mt 4112 4561 L 4112 389 mt 4112 442 L 4079 4760 mt (2) s 5183 4615 mt 5183 4561 L 5183 389 mt 5183 442 L 5100 4760 mt (2.5) s 6255 4615 mt 6255 4561 L 6255 389 mt 6255 442 L 6222 4760 mt (3) s 899 4615 mt 952 4615 L 6255 4615 mt 6201 4615 L 798 4659 mt (0) s 899 3769 mt 952 3769 L 6255 3769 mt 6201 3769 L 631 3813 mt (0.05) s 899 2924 mt 952 2924 L 6255 2924 mt 6201 2924 L 698 2968 mt (0.1) s 899 2079 mt 952 2079 L 6255 2079 mt 6201 2079 L 631 2123 mt (0.15) s 899 1234 mt 952 1234 L 6255 1234 mt 6201 1234 L 698 1278 mt (0.2) s 899 389 mt 952 389 L 6255 389 mt 6201 389 L 631 433 mt (0.25) s 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L gs 899 389 5357 4227 MR c np 2143 10 2142 38 1071 265 899 4273 4 MP stroke gr 863 4273 mt 935 4273 L 899 4237 mt 899 4309 L 1934 4538 mt 2006 4538 L 1970 4502 mt 1970 4574 L 4076 4576 mt 4148 4576 L 4112 4540 mt 4112 4612 L 6219 4586 mt 6291 4586 L 6255 4550 mt 6255 4622 L 874 4248 mt 924 4298 L 924 4248 mt 874 4298 L 1945 4513 mt 1995 4563 L 1995 4513 mt 1945 4563 L 4087 4551 mt 4137 4601 L 4137 4551 mt 4087 4601 L 6230 4561 mt 6280 4611 L 6280 4561 mt 6230 4611 L gs 899 389 5357 4227 MR c np 2143 387 2142 1015 1071 1535 899 1093 4 MP stroke gr 36 36 899 1093 FO 36 36 1970 2628 FO 36 36 4112 3643 FO 36 36 6255 4030 FO gs 899 389 5357 4227 MR c np gr end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument @endspecial 710 1424 a Fr(\(a\))19 b(query)h(time)e(vs)h Fq(\017)2006 1254 y @beginspecial 78 @llx 213 @lly 549 @urx 589 @ury 2126 @rwi 1587 @rhi @setspecial %%BeginDocument: eps_speedup.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: /afs/athena.mit.edu/user/m/i/mirrokni/research/pstable/mayur/eps_report/eps_speedup.eps %%CreationDate: 02/18/2003 04:36:40 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 78 213 549 589 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rm /rmoveto ldef /rl /rlineto ldef /s {show newpath} bdef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef /rc {rectclip} bdef /rf {rectfill} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont} bdef /reencode {exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop} bdef /isroman {findfont /CharStrings get /Agrave known} bdef /FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS} bdef /csm {1 dpi2point div -1 dpi2point div scale neg translate dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L {lineto stroke} bdef /MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef /AP {{rlineto} repeat} bdef /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /PP {closepath eofill} bdef /DP {closepath stroke} bdef /MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath} bdef /FR {MR stroke} bdef /PR {MR fill} bdef /L1i {{currentfile picstr readhexstring pop} image} bdef /tMatrix matrix def /MakeOval {newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix} bdef /FO {MakeOval stroke} bdef /PO {MakeOval fill} bdef /PD {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /FA {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef /PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef /FAn {newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef /PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath} bdef /FRR {MRR stroke } bdef /PRR {MRR fill } bdef /MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath} bdef /FlrRR {MlrRR stroke } bdef /PlrRR {MlrRR fill } bdef /MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath} bdef /FtbRR {MtbRR stroke } bdef /PtbRR {MtbRR fill } bdef /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup string currentfile exch readhexstring pop /bmap xdef pop pop} bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform {bmap} image gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup 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y Fn(1)p 2760 5188 39 4 v 2763 5255 a Fq(c)2809 5205 y Fn(\(1)h(+)g(2)p Fq(\016)3046 5169 y Ff(0)3069 5205 y Fn(\))2616 5360 y Fl(\024)2760 5312 y Fn(1)p 2760 5343 V 2763 5411 a Fq(c)2809 5360 y Fn(\(1)g(+)g Fq(\015)t Fn(\))p Fq(:)880 b Fr(\(1\))p eop %%Page: 10 10 10 9 bop -152 -69 a Fd(Case)19 b(2:)k Fq(p)d(<)h Fn(1)p Fr(.)i(F)o(or)18 b(this)f(case)i(we)f(will)f(choose)i(our)f(parameters) h(so)f(that)-152 19 y(we)k(can)g(use)g(the)g(P)o(areto)f(estimation)g (for)h(the)g(density)g(function.)32 b(Choose)-152 106 y Fq(x)-108 114 y Fo(0)-53 106 y Fr(lar)o(ge)21 b(enough)i(so)e(that)f (the)h(P)o(areto)g(estimation)g(is)f(accurate)i(to)f(within)-152 193 y(a)e(f)o(actor)g(of)g Fn(\(1)f Fl(\006)e Fq(\016)s Fn(\))j Fr(for)g Fq(x)h(>)i(x)717 201 y Fo(0)751 193 y Fr(.)h(Then)c(for)g Fq(x)h(>)h(x)1260 201 y Fo(0)1295 193 y Fr(,)-32 370 y Fq(G)p Fn(\()p Fq(x)p Fn(\))g(=)246 339 y Fo(1)p 244 353 35 4 v 244 395 a Fp(x)301 310 y Fj(R)352 330 y Fp(x)387 340 y Fe(0)337 397 y Fo(0)436 370 y Fq(tf)8 b Fn(\()p Fq(t)p Fn(\))p Fq(dt)17 b Fn(+)771 339 y Fo(1)p 768 353 V 768 395 a Fp(x)826 310 y Fj(R)877 330 y Fp(x)862 397 y(x)897 407 y Fe(0)946 370 y Fq(tf)8 b Fn(\()p Fq(t)p Fn(\))p Fq(dt)99 559 y(<)192 528 y Fo(1)p 190 542 V 190 584 a Fp(x)247 498 y Fj(R)298 519 y Fp(x)333 529 y Fe(0)283 586 y Fo(0)382 559 y Fq(tf)g Fn(\()p Fq(t)p Fn(\))p Fq(dt)17 b Fn(+)714 528 y Fo(1+)p Fp(\016)p 714 542 107 4 v 751 584 a(x)844 498 y Fj(R)895 519 y Fp(x)880 586 y(x)915 596 y Fe(0)965 559 y Fq(pC)1059 567 y Fp(p)1094 559 y Fq(t)1122 527 y Ff(\000)p Fp(p)1206 559 y Fq(dt)99 745 y Fn(=)192 715 y Fo(1)p 190 729 35 4 v 190 771 a Fp(x)247 685 y Fj(R)298 705 y Fp(x)333 715 y Fe(0)283 772 y Fo(0)382 745 y Fq(tf)8 b Fn(\()p Fq(t)p Fn(\))p Fq(dt)18 b Fn(+)155 846 y(\()236 806 y Fp(pC)312 814 y Fi(p)p 194 829 194 4 v 194 871 a Fp(x)p Fo(\(1)p Ff(\000)p Fp(p)p Fo(\))398 846 y Fq(x)442 814 y Ff(\000)p Fp(p)p Fo(+1)619 846 y Fl(\000)747 806 y Fp(pC)823 814 y Fi(p)p 706 829 V 706 871 a Fp(x)p Fo(\(1)p Ff(\000)p Fp(p)p Fo(\))909 846 y Fq(x)953 854 y Fo(0)987 814 y Ff(\000)p Fp(p)p Fo(+1)1148 846 y Fn(\)\(1)g(+)e Fq(\016)s Fn(\))99 1048 y(=)192 1017 y Fo(1)p 190 1031 35 4 v 190 1073 a Fp(x)234 1048 y Fn(\()264 988 y Fj(R)315 1008 y Fp(x)350 1018 y Fe(0)301 1075 y Fo(0)400 1048 y Fq(tf)8 b Fn(\()p Fq(t)p Fn(\))p Fq(dt)16 b Fl(\000)731 1008 y Fp(pC)807 1016 y Fi(p)842 1008 y Fo(\(1+)p Fp(\016)r Fo(\))p 731 1031 266 4 v 785 1073 a(\(1)p Ff(\000)p Fp(p)p Fo(\))1007 1048 y Fq(x)1051 1056 y Fo(0)1085 1016 y Ff(\000)p Fp(p)p Fo(+1)1247 1048 y Fn(\))i(+)184 1129 y Fo(1)p 165 1143 70 4 v 165 1185 a Fp(x)200 1171 y Fi(p)244 1160 y Fn(\()308 1120 y Fp(pC)384 1128 y Fi(p)p 284 1143 159 4 v 284 1185 a Fo(\(1)p Ff(\000)p Fp(p)p Fo(\))452 1160 y Fn(\(1)g(+)f Fq(\016)s Fn(\)\))p Fq(:)-152 1345 y Fr(Since)h Fq(x)76 1353 y Fo(0)129 1345 y Fr(is)g(a)g(constant)h(that)g(depends)h(on)e Fq(\016)s Fr(,)g(the)h(\002rst)e(term)h(decreases)i(as)-152 1432 y Fn(1)p Fq(=x)i Fr(while)g(the)f(second)i(term)f(decreases)h(as)e Fn(1)p Fq(=x)1189 1400 y Fp(p)1247 1432 y Fr(where)h Fq(p)k(<)h Fn(1)p Fr(.)k(Thus)-152 1519 y(for)18 b(e)n(v)o(ery)h Fq(\016)171 1487 y Ff(0)211 1519 y Fr(there)f(is)g(some)h Fq(x)666 1527 y Fo(1)718 1519 y Fr(such)f(that)g(for)g(all)g Fq(x)j(>)g(x)1390 1527 y Fo(1)1424 1519 y Fr(,)d(the)g(\002rst)f(term) -152 1606 y(is)22 b(at)h(most)f Fq(\016)201 1575 y Ff(0)246 1606 y Fr(times)g(the)h(second)g(term.)34 b(W)-6 b(e)22 b(choose)i Fq(\016)1350 1575 y Ff(0)1400 1606 y Fn(=)k Fq(\016)s Fr(.)34 b(Then)23 b(for)-152 1694 y Fq(x)e(>)g Fn(max)o(\()p Fq(x)211 1702 y Fo(1)245 1694 y Fq(;)13 b(x)323 1702 y Fo(0)357 1694 y Fn(\))p Fr(,)335 1881 y Fq(G)p Fn(\()p Fq(x)p Fn(\))21 b Fq(<)g Fn(\(1)c(+)g Fq(\016)s Fn(\))830 1845 y Fo(2)877 1776 y Fj(\022)1034 1832 y Fq(pC)1128 1840 y Fp(p)p 943 1864 311 4 v 943 1931 a Fn(\(1)h Fl(\000)f Fq(p)p Fn(\))p Fq(x)1219 1910 y Fp(p)1264 1776 y Fj(\023)1333 1881 y Fq(:)-152 2063 y Fr(In)i(the)g(same)g(w)o(ay)h(we)f(obtain)335 2248 y Fq(G)p Fn(\()p Fq(x)p Fn(\))i Fq(>)g Fn(\(1)c Fl(\000)g Fq(\016)s Fn(\))830 2212 y Fo(2)877 2142 y Fj(\022)1034 2199 y Fq(pC)1128 2207 y Fp(p)p 943 2231 V 943 2298 a Fn(\(1)h Fl(\000)f Fq(p)p Fn(\))p Fq(x)1219 2277 y Fp(p)1264 2142 y Fj(\023)1333 2248 y Fq(:)-152 2435 y Fr(Using)j(these)f(tw)o(o)g (bounds,)h(we)f(see)g(for)g Fq(r)k(>)e(c)13 b Fn(max)o(\()p Fq(x)1329 2443 y Fo(1)1363 2435 y Fq(;)g(x)1441 2443 y Fo(0)1475 2435 y Fn(\))p Fr(,)292 2630 y Fq(G)p Fn(\()p Fq(r)r Fn(\))p 256 2661 229 4 v 256 2729 a Fq(G)p Fn(\()p Fq(r)r(=c)p Fn(\))569 2678 y Fq(<)767 2600 y Fn(\(1)18 b(+)f Fq(\016)s Fn(\))997 2568 y Fo(2)1044 2517 y Fj(\020)1156 2560 y Fp(pC)1232 2568 y Fi(p)p 1100 2583 224 4 v 1100 2625 a Fo(\(1)p Ff(\000)p Fp(p)p Fo(\))p Fp(r)1289 2611 y Fi(p)1333 2517 y Fj(\021)p 714 2661 720 4 v 714 2756 a Fn(\(1)g Fl(\000)g Fq(\016)s Fn(\))943 2735 y Fo(2)990 2674 y Fj(\020)1156 2716 y Fp(pC)1232 2724 y Fi(p)p 1046 2739 332 4 v 1046 2782 a Fo(\(1)p Ff(\000)p Fp(p)p Fo(\)\()p Fp(r)r(=c)p Fo(\))1343 2768 y Fi(p)1387 2674 y Fj(\021)569 2910 y Fl(\024)729 2861 y Fn(1)p 714 2893 70 4 v 714 2960 a Fq(c)747 2939 y Fp(p)793 2910 y Fn(\(1)g(+)g(9)p Fq(\016)s Fn(\))569 3065 y Fl(\024)729 3017 y Fn(1)p 714 3048 V 714 3116 a Fq(c)747 3094 y Fp(p)793 3065 y Fn(\(1)g(+)g Fq(\015)t Fn(\))-152 3232 y Fr(for)i Fq(\016)24 b(<)d Fn(min)o(\()p Fq(\015)t(=)p Fn(9)p Fq(;)14 b Fn(1)p Fq(=)p Fn(2\))p Fr(.)-77 3319 y(W)-6 b(e)23 b(no)n(w)h(consider)g(the)f (special)h(cases)f(of)h Fq(p)29 b Fl(2)h(f)p Fn(1)p Fq(;)14 b Fn(2)p Fl(g)p Fr(.)36 b(F)o(or)23 b(the)g(case)-152 3406 y(of)j Fq(p)35 b Fn(=)g(1)p Fr(,)28 b(we)e(ha)o(v)o(e)g(the)g (Cauchy)h(distrib)o(ution)f(and)h(we)f(can)h(compute)-152 3514 y(directly)e Fq(G)p Fn(\()p Fq(r)r Fn(\))33 b(=)393 3478 y Fo(ln)o(\()p Fp(r)499 3458 y Fe(2)531 3478 y Fo(+1\))p 393 3497 240 4 v 478 3540 a Fp(\031)r(r)667 3514 y Fr(and)25 b Fq(F)11 b Fn(\()p Fq(r)r Fn(\))32 b(=)h(1)22 b Fl(\000)1236 3484 y Fo(2)p 1232 3498 38 4 v 1232 3540 a Fp(\031)1293 3514 y Fn(tan)1404 3483 y Ff(\000)p Fo(1)1486 3514 y Fn(\()p Fq(r)r Fn(\))p Fr(.)41 b(In)25 b(f)o(act)-152 3621 y(for)18 b(the)g(ratio)256 3585 y Fp(F)8 b Fo(\()p Fp(r)r Fo(\))p 226 3604 185 4 v 226 3647 a Fp(F)g Fo(\()p Fp(r)r(=c)p Fo(\))421 3621 y Fr(,)17 b(the)h(pre)n(vious)g(analysis)h (for)e(general)h Fq(p)f Fr(w)o(orks)i(here.)-152 3741 y(As)j(for)h(the)f(ratio)375 3705 y Fp(G)p Fo(\()p Fp(r)r Fo(\))p 345 3723 186 4 v 345 3766 a Fp(G)p Fo(\()p Fp(r)r(=c)p Fo(\))541 3741 y Fr(,)g(we)h(can)f(pro)o(v)o(e)h(the)g(upper)g(bound)h (of)1616 3710 y Fo(1)p 1616 3724 31 4 v 1617 3766 a Fp(c)1679 3741 y Fr(using)-152 3838 y(L)-7 b('Hopital)19 b(rule,)f(as)h(follo)n (ws:)240 4097 y Fn(lim)217 4142 y Fp(r)r Ff(!1)399 4049 y Fq(G)p Fn(\()p Fq(r)r Fn(\))p 393 4080 171 4 v 393 4148 a Fq(G)p Fn(\()493 4117 y Fp(r)p 492 4131 31 4 v 493 4173 a(c)533 4148 y Fn(\))647 4097 y(=)98 b(lim)782 4142 y Fp(r)r Ff(!1)1046 4049 y Fn(ln\()p Fq(r)1177 4017 y Fo(2)1228 4049 y Fn(+)17 b(1\))p 957 4080 505 4 v 957 4148 a Fq(c)c Fn(ln\(\()p Fq(r)r(=c)p Fn(\))1265 4126 y Fo(2)1317 4148 y Fn(+)j(1\))647 4314 y(=)98 b(lim)782 4359 y Fp(r)r Ff(!1)1153 4214 y Fo(2)p Fp(r)p 1090 4228 188 4 v 1090 4271 a Fo(\()p Fp(r)1145 4257 y Fe(2)1176 4271 y Fo(+1\))p 957 4296 454 4 v 957 4369 a Fq(c)p Fn(\()1170 4338 y Fo(2)p Fp(r)p 1030 4352 341 4 v 1030 4396 a(c)1058 4382 y Fe(2)1090 4396 y Fo(\()p Fp(r)1145 4382 y Fe(2)1177 4396 y Fp(=c)1237 4382 y Fe(2)1269 4396 y Fo(+1\))1380 4369 y Fn(\))647 4529 y(=)g(lim)782 4574 y Fp(r)r Ff(!1)957 4480 y Fq(c)990 4449 y Fo(2)1025 4480 y Fn(\()p Fq(r)1092 4449 y Fo(2)1126 4480 y Fq(=c)1197 4449 y Fo(2)1249 4480 y Fn(+)17 b(1\))p 957 4512 438 4 v 1027 4579 a Fq(c)p Fn(\()p Fq(r)1127 4558 y Fo(2)1179 4579 y Fn(+)g(1\))647 4719 y(=)98 b(lim)782 4764 y Fp(r)r Ff(!1)957 4670 y Fq(c)990 4639 y Fo(2)1025 4670 y Fn(\(2)p Fq(r)r(=c)1201 4639 y Fo(2)1236 4670 y Fn(\))p 957 4702 309 4 v 1057 4769 a(2)p Fq(cr)647 4877 y Fn(=)792 4829 y(1)p 792 4860 39 4 v 795 4928 a Fq(c)840 4877 y(:)-77 5039 y Fr(Also)19 b(for)g(the)h(case)f Fq(p)j Fn(=)f(2)p Fr(,)e(i.e.)24 b(the)19 b(normal)g(distrib)o(ution,)g(the)h(compu-)-152 5126 y(tation)g(is)f(straightforw)o(ard.)26 b(W)-6 b(e)19 b(use)h(the)g(f)o(act)g(that)f(for)h(this)f(case)h Fq(F)11 b Fn(\()p Fq(r)r Fn(\))22 b Fl(')-152 5275 y Fq(f)8 b Fn(\()p Fq(r)r Fn(\))p Fq(=r)24 b Fr(and)e Fq(G)p Fn(\()p Fq(r)r Fn(\))27 b(=)540 5244 y Fo(2)p 496 5258 119 4 v 496 5266 a Ff(p)p 546 5266 69 3 v 41 x Fo(2)p Fp(\031)634 5244 y Fo(1)p Ff(\000)p Fp(e)752 5196 y Fb(\000)p Fi(r)826 5180 y Fe(2)p 752 5207 107 3 v 791 5239 a(2)p 634 5258 239 4 v 738 5300 a Fp(r)883 5275 y Fr(,)21 b(where)h Fq(f)8 b Fn(\()p Fq(r)r Fn(\))22 b Fr(is)f(the)h(normal)g(den-)-152 5378 y(sity)f(function.)30 b(F)o(or)21 b(lar)o(ge)g(v)n(alues)h(of)f Fq(r)r Fr(,)f Fq(G)p Fn(\()p Fq(r)r Fn(\))h Fr(clearly)g(dominates)h Fq(F)11 b Fn(\()p Fq(r)r Fn(\))p Fr(,)2040 -69 y(because)34 b Fq(F)11 b Fn(\()p Fq(r)r Fn(\))32 b Fr(decreases)h(e)o(xponentially)h (\()p Fq(e)3315 -100 y Ff(\000)p Fp(r)3394 -121 y Fe(2)3426 -100 y Fp(=)p Fo(2)3492 -69 y Fr(\))f(while)f Fq(G)p Fn(\()p Fq(r)r Fn(\))g Fr(de-)2040 32 y(creases)26 b(as)f Fn(1)p Fq(=r)r Fr(.)42 b(Thus,)26 b(we)f(need)h(to)f(approximate)3536 -4 y Fp(G)p Fo(\()p Fp(r)r Fo(\))p 3506 14 186 4 v 3506 57 a Fp(G)p Fo(\()p Fp(r)r(=c)p Fo(\))3726 32 y Fr(as)g Fq(r)i Fr(tends)2040 140 y(to)19 b(in\002nity)-5 b(,)18 b(which)i(is)e(clearly)2867 110 y Fo(1)p 2867 124 31 4 v 2868 166 a Fp(c)2908 140 y Fr(.)2447 432 y Fn(lim)2424 477 y Fp(r)r Ff(!1)2606 384 y Fq(G)p Fn(\()p Fq(r)r Fn(\))p 2599 415 171 4 v 2599 483 a Fq(G)p Fn(\()2699 452 y Fp(r)p 2699 466 31 4 v 2700 508 a(c)2740 483 y Fn(\))2801 432 y(=)44 b(lim)2882 477 y Fp(r)r Ff(!1)3118 384 y Fn(1)17 b Fl(\000)g Fq(e)3296 329 y Fb(\000)p Fi(r)3370 312 y Fe(2)p 3296 341 107 3 v 3335 374 a(2)p 3057 415 421 4 v 3057 521 a Fq(c)p Fn(\(1)h Fl(\000)e Fq(e)3288 483 y Ff(\000)3359 464 y Fi(r)3388 452 y Fe(2)p 3346 472 88 3 v 3346 513 a(2)p Fi(c)3401 501 y Fe(2)3448 521 y Fn(\))3509 432 y(=)3600 384 y(1)p 3600 415 39 4 v 3603 483 a Fq(c)2115 649 y Fr(Notice)21 b(that)h(similar)f(to)h(the)f(pre)n (vious)i(parts,)f(we)g(can)g(\002nd)g(the)g(appro-)2040 736 y(priate)d Fq(r)r Fn(\()p Fq(c;)13 b(p;)g(\015)t Fn(\))18 b Fr(such)i(that)2824 703 y Fo(1)p Ff(\000)p Fp(p)2934 713 y Fe(1)p 2824 719 143 4 v 2824 762 a Fo(1)p Ff(\000)p Fp(p)2934 772 y Fe(2)2996 736 y Fr(is)e(at)h(most)g(most)g Fn(\(1)e(+)g Fq(\015)t Fn(\))3711 706 y Fo(1)p 3711 720 31 4 v 3712 762 a Fp(c)3751 736 y Fr(.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF