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(the)f(dag,)i(and)f(the)2049 4135 y Fl(critical-path)18 b(length)g Fu(T)2704 4143 y Fs(1)2787 4135 y Fv(is)g(the)h(length)g(of) f(a)h(longest)g(\(directed\))g(path)g(in)2049 4218 y(the)24 b(dag.)38 b(The)24 b(ratio)f Fu(T)2673 4226 y Ft(1)2708 4218 y Fu(=T)2791 4226 y Fs(1)2880 4218 y Fv(is)g(called)h(the)g Fl(av)o(erage)h(parallelism)p Fv(.)38 b(The)2049 4301 y(dag)18 b(is)f(generated)i(dynamically)f(during)g(e)o(x)o(ecution,)h (and)f(a)f(scheduler)h(op-)2049 4384 y(erates)k(in)g(an)g(on-line)h(f)o (ashion.)33 b(When)22 b(all)g(of)g(the)g(parents)h(of)f(a)g(node)h(are) 2049 4467 y(e)o(x)o(ecuted)f(we)g(say)f(that)h(the)f(node)h(is)f Fl(ready)p Fv(,)i(and)f(only)g(ready)g(nodes)g(may)2049 4550 y(be)d(e)o(x)o(ecuted.)2126 4633 y(W)-6 b(e)29 b(mak)o(e)h(tw)o(o) f(assumptions)i(related)e(to)g(the)g(structure)h(of)f(the)g(dag.)2049 4716 y(First,)g(we)g(assume)g(that)f(each)h(node)h(has)f(out-de)o(gree) g(at)g(most)f Fr(2)p Fv(.)52 b(This)2049 4799 y(assumption)19 b(is)e(consistent)i(with)e(our)h(con)m(v)o(ention)i(that)e(a)f(node)i (represents)2049 4882 y(a)c(single)h(instruction.)22 b(Second,)16 b(we)f(assume)h(that)f(the)g(dag)h(has)g(e)o(xactly)f(one) 2049 4965 y Fl(root)k Fv(node)h(with)f(in-de)o(gree)g Fr(0)g Fv(and)h(one)g Fl(\002nal)e Fv(node)h(with)g(out-de)o(gree)h Fr(0)p Fv(.)2126 5048 y(W)-6 b(e)20 b(present)i(a)f(non-blocking)j (implementation)e(of)f(the)g(w)o(ork-stealing)2049 5131 y(algorithm)c([8],)f(and)h(we)f(analyze)h(the)f(performance)h(of)g (this)f(non-blocking)2049 5214 y(w)o(ork)k(stealer)f(in)g (multiprogrammed)i(en)m(vironments.)26 b(In)19 b(this)g(implemen-)2049 5297 y(tation,)27 b(all)e(concurrent)h(data)g(structures)g(are)f (non-blocking)j([23)q(,)d(24])h(so)2049 5380 y(that)d(if)f(the)h(k)o (ernel)h(preempts)f(a)g(process,)i(it)d(does)i(not)f(hinder)g(other)h (pro-)p eop %%Page: 2 2 2 1 bop -150 -75 a Fv(cesses,)29 b(for)e(e)o(xample)g(by)h(holding)g (locks.)47 b(Moreo)o(v)o(er)m(,)30 b(this)c(implemen-)-150 8 y(tation)g(mak)o(es)g(use)h(of)e(\223yield\224)i(system)f(calls)f (that)h(constrain)g(the)g(k)o(ernel)-150 91 y(adv)o(ersary)g(in)g(a)f (manner)h(that)f(models)h(the)g(beha)o(vior)g(of)f Fk(yield)h Fv(system)-150 174 y(calls)c(found)h(in)f(current)g(multiprocessor)h (operating)g(systems.)33 b(W)-6 b(e)21 b(sho)n(w)-150 257 y(that)h(for)h(an)o(y)g(multithreaded)g(computation)h(with)f(w)o (ork)g Fu(T)1432 265 y Ft(1)1488 257 y Fv(and)h(critical-)-150 340 y(path)30 b(length)g Fu(T)269 348 y Fs(1)334 340 y Fv(,)i(the)e(non-blocking)i(w)o(ork)e(stealer)g(runs)g(in)g(e)o (xpected)-150 423 y(time)c Fu(O)r Fr(\()p Fu(T)144 431 y Ft(1)178 423 y Fu(=P)265 431 y Fq(A)338 423 y Fr(+)d Fu(T)466 431 y Fs(1)530 423 y Fu(P)6 b(=P)672 431 y Fq(A)724 423 y Fr(\))p Fv(.)43 b(This)26 b(bound)i(is)d(optimal)h(to)g(within)g (a)-150 506 y(constant)j(f)o(actor)g(and)h(achie)n(v)o(es)f(linear)g (speedup)h(\227)e(that)g(is,)j(e)o(x)o(ecution)-150 589 y(time)19 b Fu(O)r Fr(\()p Fu(T)137 597 y Ft(1)171 589 y Fu(=P)258 597 y Fq(A)309 589 y Fr(\))g Fv(\227)f(whene)n(v)o(er)i Fu(P)32 b Fr(=)21 b Fu(O)r Fr(\()p Fu(T)1054 597 y Ft(1)1089 589 y Fu(=T)1172 597 y Fs(1)1237 589 y Fr(\))p Fv(.)i(W)-6 b(e)18 b(also)h(sho)n(w)h(that)-150 672 y(for)f(an)o(y)h Fu(")i(>)f Fr(0)p Fv(,)f(with)e(probability)i(at)f(least)g Fr(1)f Fm(\000)f Fu(")p Fv(,)i(the)g(e)o(x)o(ecution)h(time)f(is)-150 755 y Fu(O)r Fr(\()p Fu(T)-15 763 y Ft(1)20 755 y Fu(=P)107 763 y Fq(A)175 755 y Fr(+)d(\()p Fu(T)326 763 y Fs(1)408 755 y Fr(+)h(lg)r(\(1)p Fu(=")p Fr(\)\))p Fu(P)6 b(=P)890 763 y Fq(A)941 755 y Fr(\))p Fv(.)-73 838 y(This)22 b(result)h(impro)o (v)o(es)h(on)f(pre)n(vious)i(results)d([8)q(])g(in)h(tw)o(o)g(w)o(ays.) 36 b(First,)-150 921 y(we)16 b(consider)h(arbitrary)e(multithreaded)i (computations)g(as)f(opposed)i(to)e(the)-150 1004 y(special)28 b(case)g(of)g(\223fully)g(strict\224)f(computations.)52 b(Second,)30 b(we)e(consider)-150 1087 y(multiprogrammed)f(en)m (vironments)g(as)e(opposed)j(to)d(dedicated)i(en)m(viron-)-150 1170 y(ments.)47 b(A)27 b(multiprogrammed)h(en)m(vironment)h(is)d(a)h (generalization)h(of)f(a)-150 1253 y(dedicated)d(en)m(vironment,)h (because)f(we)e(can)h(vie)n(w)g(a)g(dedicated)h(en)m(viron-)-150 1336 y(ment)e(as)h(a)f(multiprogrammed)h(en)m(vironment)h(in)e(which)g (the)h(k)o(ernel)g(e)o(x)o(e-)-150 1419 y(cutes)g(the)f(computation)i (on)f Fu(P)33 b Fv(dedicated)24 b(processors.)35 b(Moreo)o(v)o(er)m(,) 24 b(note)-150 1502 y(that)h(in)h(this)f(case,)i(we)f(ha)o(v)o(e)f Fu(P)703 1510 y Fq(A)787 1502 y Fr(=)34 b Fu(P)11 b Fv(,)26 b(and)g(our)g(bound)h(for)f(multipro-)-150 1585 y(grammed)g(en)m (vironments)h(specializes)e(to)g(match)g(the)g Fu(O)r Fr(\()p Fu(T)1475 1593 y Ft(1)1510 1585 y Fu(=P)33 b Fr(+)21 b Fu(T)1756 1593 y Fs(1)1821 1585 y Fr(\))-150 1668 y Fv(bound)h(established)g(earlier)e(for)h(fully)g(strict)f (computations)i(e)o(x)o(ecuting)g(in)-150 1751 y(dedicated)e(en)m (vironments.)-73 1834 y(Our)28 b(non-blocking)i(w)o(ork)f(stealer)f (has)g(been)h(implemented)h(and)e(nu-)-150 1917 y(merous)22 b(performance)g(studies)g(ha)o(v)o(e)f(been)h(conducted)h([9].)29 b(These)22 b(stud-)-150 2000 y(ies)17 b(sho)n(w)g(that)g(application)g (performance)h(conforms)g(to)f(the)g Fu(O)r Fr(\()p Fu(T)1610 2008 y Ft(1)1645 2000 y Fu(=P)1732 2008 y Fq(A)1791 2000 y Fr(+)-150 2083 y Fu(T)-105 2091 y Fs(1)-40 2083 y Fu(P)6 b(=P)102 2091 y Fq(A)153 2083 y Fr(\))25 b Fv(bound)j(and)f(that)e(the) h(constant)h(hidden)g(in)f(the)g(big-Oh)g(no-)-150 2166 y(tation)d(is)f(small,)h(roughly)h(1.)34 b(Moreo)o(v)o(er)m(,)24 b(these)f(studies)g(sho)n(w)h(that)e(non-)-150 2249 y(blocking)g(data)f (structures)g(and)g(the)g(use)g(of)g(yields)g(are)f(essential)h(in)g (prac-)-150 2332 y(tice.)29 b(If)21 b(an)o(y)g(of)g(these)h (implementation)g(mechanisms)g(are)f(omitted,)g(then)-150 2415 y(performance)f(de)o(grades)g(dramatically)g(for)e Fu(P)1076 2423 y Fq(A)1148 2415 y Fu(<)j(P)11 b Fv(.)-73 2498 y(The)16 b(remainder)h(of)f(this)f(paper)i(is)f(or)o(ganized)h(as) f(follo)n(ws.)22 b(In)16 b(Section)g(2,)-150 2581 y(we)28 b(formalize)g(our)h(model)g(of)f(multiprogrammed)i(en)m(vironments.)52 b(W)-6 b(e)-150 2664 y(also)17 b(pro)o(v)o(e)h(a)e(lo)n(wer)h(bound)i (implying)e(that)g(the)g(performance)h(of)f(the)g(non-)-150 2747 y(blocking)29 b(w)o(ork)f(stealer)g(is)f(optimal)h(to)f(within)h (a)f(constant)i(f)o(actor)l(.)49 b(W)-6 b(e)-150 2830 y(present)25 b(the)g(non-blocking)i(w)o(ork)e(stealer)g(in)f(Section)h (3,)h(and)f(we)g(pro)o(v)o(e)-150 2914 y(an)i(important)f(structural)h (lemma)f(that)g(is)g(needed)i(for)e(the)h(analysis.)46 b(In)-150 2997 y(Section)26 b(4)g(we)g(establish)h(optimal)f(upper)h (bounds)h(on)e(the)g(performance)-150 3080 y(of)f(the)g(w)o(ork)h (stealer)f(under)h(v)n(arious)f(assumptions)i(with)d(respect)i(to)f (the)-150 3163 y(k)o(ernel.)46 b(In)27 b(Section)f(5,)i(we)e(consider)h (related)g(w)o(ork.)46 b(In)26 b(Section)g(6)h(we)-150 3246 y(of)n(fer)19 b(some)g(concluding)i(remarks.)-150 3413 y Fw(2)74 b(Multipr)o(ogramming)-150 3539 y Fv(W)-6 b(e)23 b(model)h(a)g(multiprogrammed)g(en)m(vironment)h(with)f(a)f(k)o (ernel)h(that)g(be-)-150 3622 y(ha)o(v)o(es)17 b(as)f(an)h(adv)o (ersary)-5 b(.)24 b(Whereas)17 b(a)f(user)o(-le)n(v)o(el)g(scheduler)i (maps)f(threads)-150 3705 y(onto)33 b(a)f(\002x)o(ed)h(collection)f(of) h Fu(P)43 b Fv(processes,)36 b(a)c(k)o(ernel-le)n(v)o(el)i(scheduler) -150 3788 y(maps)c(processes)h(onto)f(processors.)57 b(In)29 b(this)h(section,)i(we)e(de\002ne)g(e)o(x)o(e-)-150 3872 y(cution)i(schedules,)j(and)d(we)f(pro)o(v)o(e)h(upper)h(and)f(lo) n(wer)f(bounds)i(on)f(the)-150 3955 y(length)c(of)f(e)o(x)o(ecution)h (schedules.)49 b(These)27 b(bounds)i(are)e(straightforw)o(ard)-150 4038 y(and)19 b(are)f(included)h(primarily)f(to)g(gi)n(v)o(e)g(the)g (reader)h(a)f(better)g(understanding)-150 4121 y(of)24 b(the)f(model)i(of)e(computation)i(and)g(the)e(central)h(issues)g(that) g(we)f(intend)-150 4204 y(to)29 b(address.)54 b(The)29 b(lo)n(wer)f(bound)j(demonstrates)f(the)f(optimality)g(of)g(the)-150 4287 y Fu(O)r Fr(\()p Fu(T)-15 4295 y Ft(1)20 4287 y Fu(=P)107 4295 y Fq(A)174 4287 y Fr(+)16 b Fu(T)295 4295 y Fs(1)360 4287 y Fu(P)6 b(=P)502 4295 y Fq(A)553 4287 y Fr(\))18 b Fv(upper)i(bound)g(that)e(we)h(will)f(establish)h(for)f (our)-150 4370 y(non-blocking)j(w)o(ork)f(stealer)l(.)-73 4453 y(The)k(k)o(ernel)h(operates)g(in)g(discrete)f Fl(steps)p Fv(,)i(numbered)g(from)e(1,)i(as)e(fol-)-150 4536 y(lo)n(ws.)38 b(At)23 b(each)h(step)g Fu(i)p Fv(,)h(the)f(k)o(ernel)g(chooses)h(an)o (y)f(subset)h(of)e(the)h Fu(P)35 b Fv(pro-)-150 4619 y(cesses,)29 b(and)f(then)f(these)g(chosen)h(processes)g(are)f(allo)n (wed)g(to)g(e)o(x)o(ecute)h(a)-150 4702 y(single)15 b(instruction.)22 b(W)-6 b(e)15 b(let)f Fu(p)644 4710 y Fq(i)684 4702 y Fv(denote)i(the)f(number)h(of)f(chosen)h(processes,)-150 4785 y(and)j(we)f(say)g(that)h(these)f Fu(p)539 4793 y Fq(i)583 4785 y Fv(processes)h(are)f Fl(sc)o(heduled)h Fv(at)f(step)g Fu(i)p Fv(.)23 b(The)18 b(k)o(er)o(-)-150 4868 y(nel)29 b(may)g(choose)g(to)g(schedule)h(an)o(y)f(number)g(of)g (processes)g(between)h Fr(0)-150 4951 y Fv(and)c Fu(P)11 b Fv(,)26 b(so)f Fr(0)34 b Fm(\024)f Fu(p)383 4959 y Fq(i)442 4951 y Fm(\024)g Fu(P)11 b Fv(.)42 b(W)-6 b(e)24 b(can)i(vie)n(w)f(the)h(k)o(ernel)g(as)f(producing)i(a)-150 5034 y Fl(k)n(ernel)d(sc)o(hedule)f Fv(that)h(maps)h(each)f(positi)n(v) o(e)g(inte)o(ger)h(to)e(a)h(subset)h(of)f(the)-150 5117 y(processes.)29 b(That)21 b(is,)f(a)h(k)o(ernel)g(schedule)h(maps)f (each)h(step)e Fu(i)h Fv(to)g(the)f(set)h(of)-150 5200 y(processes)h(that)e(are)h(scheduled)h(at)f(step)g Fu(i)p Fv(,)g(and)g Fu(p)1169 5208 y Fq(i)1215 5200 y Fv(is)g(the)f(size)h(of) g(that)f(set.)2049 -75 y(The)f Fl(processor)h(av)o(erage)g Fu(P)2804 -67 y Fq(A)2873 -75 y Fv(o)o(v)o(er)g Fu(T)29 b Fv(steps)19 b(is)f(de\002ned)i(as)2786 134 y Fu(P)2835 142 y Fq(A)2906 134 y Fr(=)3006 86 y(1)p 2997 117 56 4 v 2997 185 a Fu(T)3113 32 y Fq(T)3076 57 y Fj(X)3086 231 y Fq(i)p Ft(=1)3208 134 y Fu(p)3247 142 y Fq(i)3292 134 y Fu(:)650 b Fv(\(1\))2049 340 y(Though)20 b(our)g(analysis)f(is)g (based)h(on)g(this)e(step-by-step,)i(synchronous)i(e)o(x-)2049 423 y(ecution)i(model,)h(our)e(w)o(ork)h(stealer)f(is)g(asynchronous)j (and)e(does)g(not)g(de-)2049 506 y(pend)19 b(on)g(synchron)o(y)h(for)e (correctness.)24 b(The)18 b(synchronous)j(model)e(admits)2049 589 y(the)j(possibility)h(that)f(at)g(a)g(step)h Fu(i)p Fv(,)g(tw)o(o)f(or)g(more)h(processes)g(may)g(e)o(x)o(ecute)2049 672 y(instructions)18 b(that)f(reference)h(a)f(common)h(memory)g (location.)23 b(W)-6 b(e)17 b(assume)2049 755 y(that)i(the)g(ef)n(fect) g(of)g(step)g Fu(i)g Fv(is)f(equi)n(v)n(alent)j(to)e(some)g(serial)g(e) o(x)o(ecution)h(of)f(the)2049 838 y Fu(p)2088 846 y Fq(i)2138 838 y Fv(instructions)24 b(e)o(x)o(ecuted)h(by)g(the)f Fu(p)3050 846 y Fq(i)3100 838 y Fv(scheduled)h(processes,)h(where)f (the)2049 921 y(order)g(of)g(e)o(x)o(ecution)g(is)f(determined)i(in)e (some)i(arbitrary)e(manner)i(by)f(the)2049 1004 y(k)o(ernel.)2126 1087 y(Gi)n(v)o(en)30 b(a)g(k)o(ernel)h(schedule,)j(an)c Fl(ex)o(ecution)g(sc)o(hedule)g Fv(speci\002es,)j(for)2049 1170 y(each)27 b(step)f Fu(i)p Fv(,)i(the)e(particular)g(subset)h(of)f (at)g(most)g Fu(p)3451 1178 y Fq(i)3503 1170 y Fv(ready)h(nodes)g(to)f (be)2049 1253 y(e)o(x)o(ecuted)j(by)g(the)f Fu(p)2601 1261 y Fq(i)2655 1253 y Fv(scheduled)i(processes)f(at)f(step)g Fu(i)p Fv(.)51 b(W)-6 b(e)28 b(de\002ne)h(the)2049 1336 y Fl(length)c Fv(of)i(an)f(e)o(x)o(ecution)i(schedule)f(to)g(be)f(the)h (number)g(of)g(steps)f(in)h(the)2049 1419 y(schedule.)45 b(An)25 b(e)o(x)o(ecution)i(schedule)g(is)e(determined)i(both)f(by)g (the)g(user)o(-)2049 1502 y(le)n(v)o(el)17 b(scheduler)h(and)f(by)g (the)g(k)o(ernel.)23 b(Speci\002cally)-5 b(,)17 b(an)g(on-line)g(user)o (-le)n(v)o(el)2049 1585 y(scheduler)27 b(does)f(not)g(kno)n(w)g(the)g (k)o(ernel)g(schedule)h(for)e(future)h(steps.)43 b(At)2049 1668 y(each)25 b(step,)g(a)f(user)o(-le)n(v)o(el)f(scheduler)i(can)g (determine)f(what)g(instruction)h(a)2049 1751 y(process)f(should)g(e)o (x)o(ecute)g(ne)o(xt,)g(b)o(ut)f(it)g(has)g(no)h(w)o(ay)f(to)g (determine)h(when)2049 1834 y(the)18 b(k)o(ernel)h(will)e(actually)h (let)f(the)h(process)h(e)o(x)o(ecute)g(that)e(ne)o(xt)i(instruction.) 2126 1917 y(The)32 b(follo)n(wing)h(theorem)g(sho)n(ws)h(that)e Fu(T)3279 1925 y Ft(1)3313 1917 y Fu(=P)3400 1925 y Fq(A)3483 1917 y Fv(and)i Fu(T)3669 1925 y Fs(1)3733 1917 y Fu(P)6 b(=P)3875 1925 y Fq(A)3959 1917 y Fv(are)2049 2000 y(both)27 b(lo)n(wer)f(bounds)i(on)e(the)g(length)h(of)f(an)o(y)h(e)o(x)o (ecution)g(schedule.)46 b(The)2049 2083 y(lo)n(wer)32 b(bound)h(of)f Fu(T)2606 2091 y Ft(1)2640 2083 y Fu(=P)2727 2091 y Fq(A)2809 2083 y Fv(holds)h(re)o(gardless)f(of)g(the)f(k)o (ernel)i(schedule,)2049 2166 y(while)d(the)g(lo)n(wer)g(bound)i(of)e Fu(T)2918 2174 y Fs(1)2982 2166 y Fu(P)6 b(=P)3124 2174 y Fq(A)3205 2166 y Fv(holds)31 b(only)g(for)f(some)g(k)o(ernel)2049 2249 y(schedules.)f(That)20 b(is,)g(there)g(e)o(xists)h(k)o(ernel)g (schedules)g(such)h(that)e(an)o(y)h(e)o(x)o(e-)2049 2332 y(cution)c(schedule)i(has)e(length)g(at)g(least)g Fu(T)3120 2340 y Fs(1)3185 2332 y Fu(P)6 b(=P)3327 2340 y Fq(A)3378 2332 y Fv(.)22 b(Moreo)o(v)o(er)m(,)c(there)f(e)o(xist)2049 2415 y(such)22 b(k)o(ernel)g(schedules)g(with)f Fu(P)2928 2423 y Fq(A)2999 2415 y Fv(ranging)h(from)f Fu(P)32 b Fv(do)n(wn)22 b(to)f(v)n(alues)h(ar)o(-)2049 2498 y(bitrarily)17 b(close)h(to)f(0.)23 b(These)17 b(lo)n(wer)h(bounds)h(imply)e (corresponding)j(lo)n(wer)2049 2581 y(bounds)h(on)e(the)g(performance)h (of)f(an)o(y)h(user)o(-le)n(v)o(el)e(scheduler)l(.)2049 2703 y Fw(Theorem)h(1)41 b Fi(Consider)28 b(any)f(multithr)m(eaded)h (computation)g(with)e(work)2049 2786 y Fu(T)2094 2794 y Ft(1)2153 2786 y Fi(and)g(critical-path)f(length)h Fu(T)2945 2794 y Fs(1)3010 2786 y Fi(,)f(and)h(any)g(number)g Fu(P)35 b Fi(of)25 b(pr)m(ocesses.)2049 2869 y(Then)d(for)g(any)h(k)o (ernel)f(sc)o(hedule)o(,)i(e)o(very)e(e)o(xecution)h(sc)o(hedule)g(has) f(length)2049 2952 y(at)d(least)g Fu(T)2331 2960 y Ft(1)2365 2952 y Fu(=P)2452 2960 y Fq(A)2503 2952 y Fi(,)f(wher)m(e)i Fu(P)2788 2960 y Fq(A)2857 2952 y Fi(is)e(the)h(pr)m(ocessor)i(aver)o (a)o(g)o(e)f(o)o(ver)g(the)f(length)2049 3035 y(of)35 b(the)h(sc)o(hedule)o(.)74 b(In)36 b(addition,)k(for)35 b(any)i(number)f Fu(P)3614 3003 y Fs(0)3603 3050 y Fq(A)3689 3035 y Fi(of)f(the)h(form)2049 3118 y Fu(T)2094 3126 y Fs(1)2159 3118 y Fu(P)6 b(=)p Fr(\()p Fu(k)26 b Fr(+)c Fu(T)2475 3126 y Fs(1)2540 3118 y Fr(\))k Fi(wher)m(e)h Fu(k)h Fi(is)e(a)h(nonne)m(gative)h(inte)m(g)o(er)-8 b(,)29 b(ther)m(e)e(e)o(xists)f(a)2049 3201 y(k)o(ernel)j(sc)o(hedule)f (suc)o(h)h(that)f(e)o(very)g(e)o(xecution)g(sc)o(hedule)h(has)g(length) f(at)2049 3284 y(least)19 b Fu(T)2254 3292 y Fs(1)2319 3284 y Fu(P)6 b(=P)2461 3292 y Fq(A)2512 3284 y Fi(,)18 b(and)i Fu(P)2729 3292 y Fq(A)2798 3284 y Fi(is)e(in)h(the)g(r)o(ang)o (e)h Fm(b)p Fu(P)3338 3252 y Fs(0)3327 3300 y Fq(A)3378 3284 y Fm(c)h(\024)g Fu(P)3563 3292 y Fq(A)3635 3284 y Fm(\024)g Fu(P)3776 3252 y Fs(0)3765 3300 y Fq(A)3815 3284 y Fi(.)2049 3406 y(Pr)m(oof:)74 b Fv(The)18 b(processor)h(a)o(v)o (erage)f(o)o(v)o(er)g(the)f(length)h Fu(T)28 b Fv(of)18 b(the)f(schedule)i(is)2049 3489 y(de\002ned)h(by)f(Equation)h(\(1\),)e (so)h(we)g(ha)o(v)o(e)2786 3698 y Fu(T)32 b Fr(=)2984 3649 y(1)p 2954 3681 100 4 v 2954 3748 a Fu(P)3003 3756 y Fq(A)3113 3596 y(T)3076 3621 y Fj(X)3086 3795 y Fq(i)p Ft(=1)3208 3698 y Fu(p)3247 3706 y Fq(i)3292 3698 y Fu(:)650 b Fv(\(2\))2049 3926 y(F)o(or)42 b(both)i(lo)n(wer)e(bounds,)50 b(we)43 b(bound)h Fu(T)54 b Fv(by)43 b(bounding)3763 3866 y Fj(P)3850 3887 y Fq(T)3850 3953 y(i)p Ft(=1)3966 3926 y Fu(p)4005 3934 y Fq(i)4031 3926 y Fv(.)2049 4018 y(The)30 b(lo)n(wer)f(bound)i(of)f Fu(T)2745 4026 y Ft(1)2779 4018 y Fu(=P)2866 4026 y Fq(A)2946 4018 y Fv(is)f(immediate)h(from)g (the)g(lo)n(wer)f(bound)2049 4058 y Fj(P)2137 4079 y Fq(T)2137 4145 y(i)p Ft(=1)2253 4118 y Fu(p)2292 4126 y Fq(i)2358 4118 y Fm(\025)41 b Fu(T)2504 4126 y Ft(1)2538 4118 y Fv(,)31 b(which)f(follo)n(ws)f(from)g(the)g(f)o(act)g(that)g(an) o(y)h(e)o(x)o(ecution)2049 4201 y(schedule)19 b(is)e(required)h(to)f(e) o(x)o(ecute)i(all)d(of)i(the)f(nodes)i(in)e(the)h(multithreaded)2049 4284 y(computation.)23 b(F)o(or)15 b(the)h(lo)n(wer)g(bound)h(of)f Fu(T)3197 4292 y Fs(1)3261 4284 y Fu(P)6 b(=P)3403 4292 y Fq(A)3454 4284 y Fv(,)16 b(we)g(pro)o(v)o(e)g(the)g(lo)n(wer)2049 4385 y(bound)2254 4325 y Fj(P)2342 4345 y Fq(T)2342 4412 y(i)p Ft(=1)2458 4385 y Fu(p)2497 4393 y Fq(i)2545 4385 y Fm(\025)21 b Fu(T)2671 4393 y Fs(1)2735 4385 y Fu(P)11 b Fv(.)2126 4468 y(W)-6 b(e)14 b(construct)i(a)e(k)o(ernel)i(schedule)g (that)f(forces)g(e)n(v)o(ery)g(e)o(x)o(ecution)h(sched-)2049 4551 y(ule)i(to)g(satisfy)g(this)f(bound)j(as)e(follo)n(ws.)k(Let)c Fu(k)i Fv(be)e(as)g(de\002ned)g(in)g(the)g(state-)2049 4634 y(ment)g(of)f(the)g(lemma.)23 b(The)18 b(k)o(ernel)g(schedule)g (sets)g Fu(p)3432 4642 y Fq(i)3479 4634 y Fr(=)j(0)c Fv(for)h Fr(1)j Fm(\024)g Fu(i)h Fm(\024)f Fu(k)r Fv(,)2049 4717 y(sets)j Fu(p)2224 4725 y Fq(i)2281 4717 y Fr(=)31 b Fu(P)j Fv(for)24 b Fu(k)g Fr(+)c(1)32 b Fm(\024)f Fu(i)g Fm(\024)g Fu(k)23 b Fr(+)e Fu(T)3208 4725 y Fs(1)3273 4717 y Fv(,)k(and)g(sets)f Fu(p)3624 4725 y Fq(i)3681 4717 y Fr(=)31 b Fm(b)p Fu(P)3866 4685 y Fs(0)3855 4733 y Fq(A)3905 4717 y Fm(c)24 b Fv(for)2049 4800 y Fu(k)g Fr(+)d Fu(T)2239 4808 y Fs(1)2336 4800 y Fu(<)31 b(i)p Fv(.)40 b(An)o(y)25 b(e)o(x)o(ecution)h(schedule)g(has)e(length)i Fu(T)42 b Fm(\025)32 b Fu(k)23 b Fr(+)e Fu(T)3966 4808 y Fs(1)4031 4800 y Fv(,)2049 4898 y(so)h(we)f(ha)o(v)o(e)g(the)h(lo)n (wer)f(bound)2914 4838 y Fj(P)3002 4859 y Fq(T)3002 4925 y(i)p Ft(=1)3118 4898 y Fu(p)3157 4906 y Fq(i)3209 4898 y Fm(\025)26 b Fu(T)3340 4906 y Fs(1)3404 4898 y Fu(P)11 b Fv(.)30 b(It)21 b(remains)h(only)g(to)2049 4981 y(sho)n(w)d(that)f Fu(P)2402 4989 y Fq(A)2471 4981 y Fv(is)g(in)g(the)h(desired)g(range.)k (The)c(processor)g(a)o(v)o(erage)g(for)g(the)2049 5064 y(\002rst)j Fu(k)g Fr(+)e Fu(T)2375 5072 y Fs(1)2463 5064 y Fv(steps)j(is)g Fu(T)2753 5072 y Fs(1)2818 5064 y Fu(P)6 b(=)p Fr(\()p Fu(k)23 b Fr(+)d Fu(T)3129 5072 y Fs(1)3194 5064 y Fr(\))29 b(=)g Fu(P)3402 5033 y Fs(0)3391 5080 y Fq(A)3441 5064 y Fv(.)35 b(F)o(or)22 b(all)h(subsequent)2049 5147 y(steps)e Fu(i)j(>)g(k)c Fr(+)e Fu(T)2536 5155 y Fs(1)2601 5147 y Fv(,)i(we)g(ha)o(v)o(e)g Fu(p)2944 5155 y Fq(i)2994 5147 y Fr(=)k Fm(b)p Fu(P)3172 5116 y Fs(0)3161 5163 y Fq(A)3212 5147 y Fm(c)p Fv(.)j(Thus,)20 b Fu(P)3529 5155 y Fq(A)3600 5147 y Fv(f)o(alls)g(within)g(the)2049 5230 y(desired)f(range.)p 4008 5230 42 42 v 2126 5347 a(W)-6 b(e)20 b(say)h(that)g(an)g(e)o(x)o(ecution)h(schedule)g(is)e Fl(greedy)i Fv(if)e(at)h(each)g(step)g Fu(i)g Fv(the)2049 5430 y(number)f(of)e(ready)h(nodes)h(e)o(x)o(ecuted)g(is)e(equal)h(to)g (the)f(minimum)h(of)g Fu(p)3898 5438 y Fq(i)3942 5430 y Fv(and)1929 5679 y Fw(2)p eop %%Page: 3 3 3 2 bop -150 -75 a Fv(the)23 b(number)g(of)g(ready)g(nodes.)35 b(Note)23 b(that)g(an)f(on-line)h(user)o(-le)n(v)o(el)g(sched-)-150 8 y(uler)29 b(cannot)g(al)o(w)o(ays)h(produce)g(greedy)g(e)o(x)o (ecution)f(schedules,)k(because)-150 91 y(some)d(amount)h(of)e (scheduling)i(o)o(v)o(erhead)g(is)f(often)g(una)o(v)o(oidable.)56 b(The)-150 174 y(follo)n(wing)19 b(theorem)h(about)g(greedy)g(e)o(x)o (ecution)g(schedules)g(also)f(holds)h(for)-150 257 y(le)n(v)o(el-by-le) n(v)o(el)31 b(\(Brent)f([10]\))g(e)o(x)o(ecution)h(schedules,)j(with)c (only)g(tri)n(vial)-150 340 y(changes)20 b(to)f(the)g(proof.)-150 499 y Fw(Theorem)g(2)i(\(Greed)o(y)e(Sc)o(hedules\))38 b Fi(Consider)32 b(any)h(multithr)m(eaded)-150 582 y(computation)25 b(with)d(work)i Fu(T)616 590 y Ft(1)673 582 y Fi(and)g(critical-path)f (length)h Fu(T)1459 590 y Fs(1)1524 582 y Fi(,)g(any)f(num-)-150 665 y(ber)c Fu(P)30 b Fi(of)19 b(pr)m(ocesses,)g(and)h(any)g(k)o(ernel) 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f(scheduled)i(at)e(step)h Fu(i)p Fv(,)g(and)g(then)g(we)g(bound)h(the) -150 1338 y(total)k(number)i(of)f(tok)o(ens)g(collected.)35 b(Moreo)o(v)o(er)m(,)25 b(we)d(collect)h(the)g(tok)o(ens)-150 1421 y(in)18 b(tw)o(o)g(b)o(uck)o(ets:)24 b(a)18 b Fl(w)o(ork)g(b)o(uc) o(k)n(et)g Fv(and)h(an)f Fl(idle)g(b)o(uc)o(k)n(et)p Fv(.)23 b(Consider)18 b(a)g(step)h Fu(i)-150 1504 y Fv(and)k(a)g (process)g(that)g(is)f(scheduled)i(at)e(step)h Fu(i)p Fv(.)34 b(If)22 b(the)h(process)g(e)o(x)o(ecutes)h(a)-150 1587 y(node)18 b(of)f(the)h(computation,)g(then)g(it)e(puts)i(its)e (tok)o(en)i(into)f(the)h(w)o(ork)f(b)o(uck)o(et,)-150 1670 y(and)25 b(otherwise)g(we)g(say)g(that)f(the)h(process)g(is)f (idle)h(and)g(it)f(puts)h(its)f(tok)o(en)-150 1753 y(into)h(the)f(idle) g(b)o(uck)o(et.)41 b(After)24 b(the)g(last)g(step,)i(the)f(w)o(ork)g(b) o(uck)o(et)g(contains)-150 1836 y(e)o(xactly)17 b Fu(T)126 1844 y Ft(1)177 1836 y Fv(tok)o(ens)h(\227)e(one)i(tok)o(en)f(for)g (each)g(node)h(of)f(the)g(computation.)23 b(It)-150 1919 y(remains)17 b(only)g(to)f(pro)o(v)o(e)h(that)f(the)h(idle)f(b)o(uck)o (et)h(contains)h(at)e(most)g Fu(T)1629 1927 y Fs(1)1694 1919 y Fr(\()p Fu(P)i Fm(\000)-150 2002 y Fr(1\))h Fv(tok)o(ens.)-73 2085 y(Consider)g(a)g(step)h(during)f(which)h(some)f(process)h(places)g (a)f(tok)o(en)h(in)f(the)-150 2168 y(idle)i(b)o(uck)o(et.)29 b(W)-6 b(e)20 b(refer)h(to)f(such)i(a)f(step)g(as)f(an)h Fl(idle)g(step)p Fv(.)29 b(At)20 b(an)h(idle)g(step)-150 2251 y(we)e(ha)o(v)o(e)g(an)g(idle)g(process)g(and)h(since)f(the)g (schedule)h(is)f(greedy)-5 b(,)20 b(it)e(follo)n(ws)-150 2334 y(that)k(e)n(v)o(ery)g(ready)g(node)h(is)e(e)o(x)o(ecuted)i(at)f (an)g(idle)f(step.)32 b(This)21 b(observ)n(ation)-150 2417 y(leads)16 b(to)g(tw)o(o)h(further)f(observ)n(ations.)23 b(First,)16 b(at)f(e)n(v)o(ery)i(step)f(there)h(is)e(at)h(least)-150 2500 y(one)23 b(ready)h(node,)g(so)e(of)h(the)g Fu(p)682 2508 y Fq(i)730 2500 y Fv(processes)h(scheduled)g(at)e(an)h(idle)f (step)h Fu(i)p Fv(,)-150 2583 y(at)d(most)h Fu(p)129 2591 y Fq(i)173 2583 y Fm(\000)d Fr(1)24 b Fm(\024)g Fu(P)29 b Fm(\000)18 b Fr(1)i Fv(could)i(be)e(idle.)28 b(Second,)21 b(for)f(each)h(step)g Fu(i)p Fv(,)f(let)-150 2666 y Fu(G)-90 2674 y Fq(i)-41 2666 y Fv(denote)25 b(the)e(sub-dag)h (of)f(the)h(computation)g(consisting)g(of)f(just)g(those)-150 2749 y(nodes)j(that)f(ha)o(v)o(e)h(not)f(yet)g(been)h(e)o(x)o(ecuted)h (after)d(step)i Fu(i)p Fv(.)42 b(If)24 b(step)h Fu(i)h Fv(is)e(an)-150 2832 y(idle)f(step,)h(then)g(e)n(v)o(ery)g(node)g(with) f(in-de)o(gree)h Fr(0)f Fv(in)g Fu(G)1316 2840 y Fq(i)p Fs(\000)p Ft(1)1444 2832 y Fv(gets)h(e)o(x)o(ecuted)-150 2915 y(at)c(step)h Fu(i)p Fv(,)g(so)f(a)h(longest)g(path)g(in)f Fu(G)794 2923 y Fq(i)841 2915 y Fv(is)g(one)h(node)h(shorter)f(than)g (a)f(longest)-150 2998 y(path)c(in)g Fu(G)128 3006 y Fq(i)p Fs(\000)p Ft(1)233 2998 y Fv(.)21 b(Since)16 b(the)g(longest)g (path)g(in)f Fu(G)1070 3006 y Ft(0)1121 2998 y Fv(has)h(length)g Fu(T)1483 3006 y Fs(1)1548 2998 y Fv(,)f(there)h(can)-150 3081 y(be)22 b(at)f(most)h Fu(T)229 3089 y Fs(1)315 3081 y Fv(idle)g(steps.)31 b(Putting)22 b(these)f(tw)o(o)h(observ)n(ations)h (together)m(,)-150 3164 y(we)18 b(conclude)h(that)f(after)g(the)g(last) f(step,)h(the)g(idle)g(b)o(uck)o(et)g(contains)h(at)f(most)-150 3247 y Fu(T)-105 3255 y Fs(1)-40 3247 y Fr(\()p Fu(P)27 b Fm(\000)17 b Fr(1\))i Fv(tok)o(ens.)p 1809 3247 42 42 v -73 3364 a(Theorems)33 b(1)g(and)g(2)g(sho)n(w)g(that)g(for)g (some)g(k)o(ernel)g(schedules,)k(an)o(y)-150 3447 y(greedy)23 b(e)o(x)o(ecution)h(schedule)g(is)e(within)g(a)g(f)o(actor)h(of)f(tw)o (o)h(of)f(optimal.)34 b(In)-150 3530 y(addition,)16 b(though)g(we)e (shall)h(not)f(pro)o(v)o(e)i(it,)e(for)h(an)o(y)g(k)o(ernel)g (schedule,)h(some)-150 3613 y(greedy)i(e)o(x)o(ecution)g(schedule)g(is) f(optimal.)23 b(W)-6 b(e)16 b(remark)h(that)g(the)g(latter)g(f)o(act) -150 3696 y(does)29 b(not)g(imply)g(the)f(e)o(xistence)i(of)e(a)h (polynomial-time)g(algorithm)g(for)-150 3779 y(computing)21 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y(process)g(is)f(e)o(x)o(ecuting)i(a)e(thread,)h(then)g(we)f (de\002ne)h(the)f Fl(assigned)h(node)g Fv(of)2049 2302 y(the)j(process)g(to)g(be)g(the)f(currently)i(ready)f(node)h(of)e(the)h (assigned)h(thread.)2049 2385 y(Applied)d(to)f(nodes)i(instead)f(of)f (threads,)i(the)e(scheduler)i(operates)f(as)f(fol-)2049 2468 y(lo)n(ws.)51 b(If)28 b(a)g(process)h(does)g(not)g(ha)o(v)o(e)f (an)h(assigned)g(node,)j(then)c(it)g(pops)2049 2552 y(the)21 b(bottom-most)g(node)h(of)n(f)e(its)g(deque,)i(and)f(that)g(node)g (becomes)h(the)f(as-)2049 2635 y(signed)h(node.)30 b(If)21 b(the)g(deque)h(is)f(empty)-5 b(,)22 b(then)f(it)f(pops)i(the)f (top-most)h(node)2049 2718 y(of)n(f)e(the)h(deque)g(of)f(a)g(randomly)i (chosen)f(victim,)f(and)h(that)f(node)i(becomes)2049 2801 y(the)d(assigned)i(node.)k(The)20 b(process)g(then)g(e)o(x)o (ecutes)g(the)f(assigned)i(node.)k(If)2049 2884 y(e)o(x)o(ecution)g(of) g(the)f(assigned)h(node)h(enables)f(tw)o(o)f(child)h(nodes,)h(then)f (one)2049 2967 y(child)i(is)g(pushed)h(onto)f(the)g(bottom)h(of)f(the)g (deque)h(and)f(the)g(other)h(child)2049 3050 y(becomes)22 b(the)e(assigned)h(node.)29 b(If)20 b(one)h(child)f(is)g(enabled,)i (then)e(that)h(child)2049 3133 y(becomes)f(the)f(assigned)h(node.)k(If) 18 b(no)h(child)g(is)f(enabled,)i(then)f(the)g(process)2049 3216 y(has)h(no)f(assigned)i(node,)f(and)g(it)e(returns)i(to)f(its)g (deque)h(or)f(becomes)i(a)e(thief)2049 3299 y(to)g(obtain)g(an)h (assigned)g(node.)2126 3382 y(Though)27 b(each)g(node)g(of)g(the)f(dag) h(represents)g(one)g(instruction)f(and)h(is)2049 3465 y(therefore)21 b(e)o(x)o(ecuted)h(atomically)-5 b(,)21 b(for)f(the)h(case)f(of)h(a)f(node)i(whose)f(e)o(x)o(ecu-)2049 3548 y(tion)j(enables)h(tw)o(o)f(children,)h(we)f(de\002ne)g(the)g(e)o (x)o(ecution)h(to)f Fi(occur)i Fv(at)e(the)2049 3631 y(linearization)19 b(time)g(of)g(the)g(ensuing)h Fk(pushBottom)p Fv(.)2126 3714 y(If)f(the)i(e)o(x)o(ecution)g(of)f(node)h Fu(u)f Fv(enables)h(node)g Fu(v)s Fv(,)f(then)h(we)f(call)g(the)g(edge) 2049 3797 y Fr(\()p Fu(u;)13 b(v)s Fr(\))20 b Fv(an)g Fl(enabling)f(edge)p Fv(,)i(and)g(we)e(call)h Fu(u)g Fv(the)g Fl(designated)g(parent)g Fv(of)g Fu(v)s Fv(.)2049 3880 y(Note)26 b(that)f(e)n(v)o(ery)i(node)f(e)o(xcept)h(the)e(root)h (node)h(has)f(e)o(xactly)g(one)g(desig-)2049 3963 y(nated)18 b(parent,)f(so)g(the)h(subgraph)g(of)f(the)g(dag)h(consisting)g(of)f (only)h(enabling)2049 4046 y(edges)k(forms)f(a)g(rooted)h(tree)f(that)g (we)g(call)g(the)g Fl(enabling)g(tree)p Fv(.)30 b(Note)21 b(that)2049 4129 y(each)27 b(e)o(x)o(ecution)h(of)e(the)g(computation)i (may)f(ha)o(v)o(e)g(a)f(dif)n(ferent)h(enabling)2049 4212 y(tree.)34 b(If)22 b Fu(d)p Fr(\()p Fu(u)p Fr(\))g Fv(is)g(the)h(depth)g(of)g(a)f(node)i Fu(u)f Fv(in)f(the)h(enabling)g (tree,)g(then)g(its)2049 4295 y Fl(w)o(eight)15 b Fv(is)g(de\002ned)i (as)f Fu(w)r Fr(\()p Fu(u)p Fr(\))21 b(=)g Fu(T)2954 4303 y Fs(1)3024 4295 y Fm(\000)5 b Fu(d)p Fr(\()p Fu(u)p Fr(\))p Fv(.)21 b(The)15 b(root)h(of)g(the)g(dag,)g(which)2049 4378 y(is)h(also)g(the)g(root)h(of)f(the)g(enabling)i(tree,)e(has)g (weight)h Fu(T)3491 4386 y Fs(1)3556 4378 y Fv(.)k(Our)17 b(analysis)h(of)2049 4461 y(Section)h(4)g(emplo)o(ys)h(a)f(potential)g (function)h(based)g(on)f(the)g(node)h(weights.)2126 4544 y(As)29 b(illustrated)f(in)h(Figure)g(4,)j(the)d(structural)g(lemma)g (states)g(that)g(for)2049 4627 y(an)o(y)e(deque,)j(at)d(all)f(times)h (during)g(the)g(e)o(x)o(ecution)h(of)f(the)g(w)o(ork-stealing)2049 4710 y(algorithm,)e(the)f(designated)h(parents)f(of)g(the)g(nodes)h(in) e(the)h(deque)h(lie)e(on)2049 4793 y(some)k(root-to-leaf)f(path)g(of)g (the)g(enabling)h(tree.)44 b(Moreo)o(v)o(er)m(,)29 b(the)d(order)o(-) 2049 4876 y(ing)f(of)f(these)h(designated)g(parents)g(along)g(this)f (path)h(corresponds)h(to)f(the)2049 4959 y(top-to-bottom)g(ordering)h (of)e(the)h(nodes)h(in)e(the)h(deque.)41 b(As)24 b(a)g(corollary)-5 b(,)2049 5042 y(we)24 b(observ)o(e)i(that)f(the)f(weights)h(of)g(the)f 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Fu(v)776 1420 y Ft(0)811 1412 y Fu(;)13 b(x)p Fr(\))27 b Fv(and)h Fr(\()p Fu(v)1148 1420 y Ft(0)1183 1412 y Fu(;)13 b(y)s Fr(\))27 b Fv(are)g(both)i (enabling)-150 1495 y(edges,)21 b(it)f(is)g(straightforw)o(ard)g(to)h (v)o(erify)f(that)g(if)g(the)g(lemma)g(holds)h(before)-150 1579 y(the)h(node)h(e)o(x)o(ecution,)g(then)g(it)e(also)h(holds)g (after)l(.)32 b(This)22 b(case)g(gi)n(v)o(es)g(rise)g(to)-150 1662 y(the)h(possibility)h(that)f(the)g(bottom-most)h(node)h(in)e(the)g (deque)i(and)f(the)f(as-)-150 1745 y(signed)e(node)f(both)h(ha)o(v)o(e) e(the)h(same)g(designated)h(parent.)26 b(Other)20 b(cases)g(are)-150 1828 y(check)o(ed)h(in)e(a)f(similar)h(f)o(ashion.)p 1809 1828 42 42 v -150 2021 a Fw(Cor)o(ollar)q(y)f(4)41 b Fi(If)28 b Fu(v)392 2029 y Ft(0)427 2021 y Fu(;)13 b(v)498 2029 y Ft(1)533 2021 y Fu(;)g(:)g(:)g(:)h(;)f(v)741 2030 y Fq(k)808 2021 y Fi(ar)m(e)29 b(as)g(de\002ned)h(in)e(the)h (statement)g(of)-150 2104 y(Lemma)e(3,)i(then)f(we)e(have)i Fu(w)r Fr(\()p Fu(v)739 2112 y Ft(0)774 2104 y Fr(\))37 b Fm(\024)g Fu(w)r Fr(\()p Fu(v)1062 2112 y Ft(1)1097 2104 y Fr(\))g Fu(<)f(:)13 b(:)h(:)37 b(<)g(w)r Fr(\()p Fu(v)1608 2113 y Fq(k)q Fs(\000)p Ft(1)1724 2104 y Fr(\))g Fu(<)-150 2187 y(w)r Fr(\()p Fu(v)-26 2196 y Fq(k)12 2187 y Fr(\))p Fi(.)p 1809 2187 V -150 2387 a Fw(4)74 b(Anal)o(ysis)18 b(of)j(the)e(w)o(ork)h(stealer)-150 2513 y Fv(In)30 b(this)f(section)h(we)g(establish)g(optimal)g(bounds)h (on)f(the)g(running)h(time)-150 2596 y(of)22 b(the)h(non-blocking)i(w)o (ork)e(stealer)f(under)h(v)n(arious)g(assumptions)h(about)-150 2679 y(the)d(k)o(ernel.)30 b(It)20 b(should)i(be)g(emphasized)g(that)f (the)g(w)o(ork)h(stealer)e(performs)-150 2762 y(correctly)f(for)g(an)o (y)h(k)o(ernel.)25 b(W)-6 b(e)18 b(consider)i(v)n(arious)g (restrictions)f(on)h(k)o(ernel)-150 2846 y(beha)o(vior)c(in)f(order)g (to)g(demonstrate)h(en)m(vironments)h(in)e(which)g(the)h(running)-150 2929 y(time)j(of)f(the)h(w)o(ork)h(stealer)f(is)f(optimal.)-73 3012 y(The)26 b(follo)n(wing)h(de\002nitions)g(will)f(pro)o(v)o(e)h(to) g(be)g(useful)g(in)f(our)h(analy-)-150 3095 y(sis.)c(An)c(instruction)g (in)g(the)g(sequence)i(e)o(x)o(ecuted)f(by)f(some)h(process)g Fu(q)h Fv(is)e(a)-150 3178 y Fl(milestone)h Fv(if)f(and)i(only)g(if)e (one)i(of)f(the)g(follo)n(wing)h(tw)o(o)f(conditions)h(holds:)-150 3261 y(\(i\))k(e)o(x)o(ecution)h(of)g(a)f(node)i(by)e(process)i Fu(q)h Fv(occurs)e(at)f(that)h(instruction,)h(or)-150 3344 y(\(ii\))c(a)h Fk(popTop)i Fv(in)m(v)o(ocation)f(completes.)39 b(It)24 b(is)g(straightforw)o(ard)g(to)g(ar)o(gue)-150 3427 y(that)c(a)g(gi)n(v)o(en)h(process)f(may)h(e)o(x)o(ecute)g(at)e (most)h(some)h(constant)g(number)g(of)-150 3510 y(instructions)26 b(between)h(successi)n(v)o(e)g(milestones.)44 b(Throughout)27 b(this)f(sec-)-150 3593 y(tion,)d(we)f(let)g Fu(C)27 b Fv(denote)d(a)e(suf)n(\002ciently)g(lar)o(ge)h(constant)g(such)g (that)f(in)g(an)o(y)-150 3676 y(sequence)28 b(of)f Fu(C)k 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b(the)h(number)g(of)g (\223thro)n(ws\224,)i(that)-150 4593 y(is,)17 b(steal)h(attempts)g (satisfying)g(a)g(technical)h(condition)g(stated)f(belo)n(w)-5 b(.)23 b(This)-150 4676 y(goal)15 b(cannot)g(be)g(achie)n(v)o(ed)h (without)e(restricting)h(the)f(k)o(ernel,)i(so)f(in)f(addition)-150 4759 y(to)22 b(pro)o(ving)h(this)e(bound)j(on)e(e)o(x)o(ecution)h (time,)f(we)f(shall)h(state)g(and)g(justify)-150 4842 y(certain)d(k)o(ernel)h(restrictions.)-73 4925 y(One)j(fundamental)h (obstacle)g(pre)n(v)o(ents)f(us)g(from)g(pro)o(ving)h(the)f(desired) -150 5008 y(performance)39 b(bound)g(within)f(the)f(\(unrestricted\))h (multiprogramming)-150 5092 y(model)27 b(of)e(Section)h(2.)45 b(The)25 b(problem)i(is)f(that)g(the)g(k)o(ernel)g(may)h(bias)f(the) -150 5175 y(random)17 b(steal)e(attempts)h(to)n(w)o(ards)h(the)f(empty) g(deques.)23 b(In)16 b(particular)m(,)g(con-)-150 5258 y(sider)i(the)g(steal)f(attempts)h(initiated)f(within)h(some)g(\002x)o (ed)g(interv)n(al)g(of)g(steps.)2049 -75 y(The)k(adv)o(ersary)i(can)e (bias)h(these)f(steal)g(attempts)g(to)n(w)o(ards)h(the)g(empty)g(de-) 2049 8 y(ques)16 b(by)f(delaying)h(those)g(steal)f(attempts)g(that)g (choose)h(non-empty)h(deques)2049 91 y(as)i(victims)g(so)g(that)g(the)o (y)g(occur)g(after)g(the)g(end)g(of)g(the)g(interv)n(al.)2126 174 y(T)-6 b(o)16 b(address)i(this)e(issue,)h(we)g(restrict)f(the)h(k)o (ernel)g(to)g(schedule)g(in)g Fl(rounds)2049 257 y Fv(rather)k(than)h (steps.)30 b(A)21 b(process)h(that)f(is)g(scheduled)h(in)f(a)g (particular)h(round)2049 340 y(gets)i(to)h(e)o(x)o(ecute)g(between)g Fr(2)p Fu(C)k Fv(and)c Fr(3)p Fu(C)30 b Fv(instructions)25 b(without)f(preemp-)2049 423 y(tion,)19 b(where)h Fu(C)k Fv(is)18 b(the)i(constant)g(de\002ned)g(at)f(the)g(be)o(ginning)i(of)e (Section)g(4.)2049 506 y(The)24 b(precise)g(number)i(of)e(instructions) g(that)g(a)g(process)h(e)o(x)o(ecutes)g(during)2049 589 y(a)20 b(round)h(is)f(determined)h(by)g(the)f(k)o(ernel)h(in)f(an)g (arbitrary)h(manner)l(.)27 b(W)-6 b(e)20 b(as-)2049 672 y(sume)d(that)g(the)g(process)h(e)o(x)o(ecutes)g(these)f Fr(2)p Fu(C)22 b Fv(to)17 b Fr(3)p Fu(C)22 b Fv(instructions)17 b(in)g(serial)2049 755 y(order)m(,)i(b)o(ut)f(we)g(allo)n(w)g(the)h (instruction)f(streams)h(of)f(dif)n(ferent)g(processes)i(to)2049 838 y(be)27 b(interlea)o(v)o(ed)f(arbitrarily)-5 b(,)28 b(as)e(determined)h(by)g(the)f(k)o(ernel.)46 b(W)-6 b(e)26 b(claim)2049 921 y(that)e(our)g(requirement)h(that)f(processes)h(be)g (scheduled)g(in)f(rounds)h(of)f Fr(2)p Fu(C)2049 1004 y Fv(to)j Fr(3)p Fu(C)33 b Fv(instructions)28 b(is)f(a)g(reasonable)i (one.)49 b(Because)28 b(of)f(the)g(o)o(v)o(erhead)2049 1087 y(associated)d(with)f(conte)o(xt-switching,)i(practical)e(k)o (ernels)h(tend)g(to)f(assign)2049 1170 y(processes)18 b(to)f(processors)i(for)e(some)h(nontri)n(vial)f(scheduling)i(quantum.) 24 b(In)2049 1253 y(f)o(act,)29 b(a)e(typical)g(scheduling)i(quantum)f (is)f(orders)h(of)f(magnitude)h(higher)2049 1336 y(than)k(the)f(modest) h(v)n(alue)g(of)f Fu(C)37 b Fv(needed)32 b(to)f(achie)n(v)o(e)i(our)e (performance)2049 1419 y(bounds.)2126 1502 y(Recall)16 b(that)h(a)g(steal)g(attempt)g(e)o(x)o(ecuted)h(by)f(some)h(process)f Fu(q)j Fv(completes)2049 1585 y(at)25 b(a)g(milestone.)43 b(W)-6 b(e)24 b(de\002ne)i(such)g(a)f(steal)g(attempt)g(to)g(be)h(a)f Fl(thro)n(w)g Fv(if)f(it)2049 1668 y(completes)g(at)f Fu(q)s Fv(')l(s)g(second)h(milestone)g(in)f(a)g(round.)37 b(Such)24 b(a)f(thro)n(w)g(com-)2049 1751 y(pletes)h(in)g(the)g(same)g (round)h(as)f(the)g(round)h(in)f(which)g(the)g(identity)g(of)g(the)2049 1834 y(associated)30 b(random)f(victim)g(is)f(determined.)53 b(This)29 b(property)g(is)f(useful)2049 1917 y(because)20 b(it)d(ensures)i(that)f(the)h(random)g(victim)f(distrib)o(ution)g (cannot)h(be)g(bi-)2049 2000 y(ased)d(by)g(the)g(k)o(ernel.)23 b(The)15 b(follo)n(wing)i(lemma)e(bounds)j(the)d(e)o(x)o(ecution)i (time)2049 2083 y(in)i(terms)g(of)f(the)h(number)h(of)f(thro)n(ws.)2049 2206 y Fw(Lemma)h(5)41 b Fi(Consider)19 b(any)g(multithr)m(eaded)g (computation)g(with)e(work)i Fu(T)4016 2214 y Ft(1)2049 2289 y Fi(being)31 b(e)o(xecuted)f(by)g(the)g(non-bloc)o(king)i(work)e (stealer)-8 b(.)56 b(Then)30 b(the)g(e)o(xe-)2049 2372 y(cution)d(time)f(is)h(at)f(most)h Fu(O)r Fr(\()p Fu(T)2881 2380 y Ft(1)2915 2372 y Fu(=P)3002 2380 y Fq(A)3076 2372 y Fr(+)c Fu(S=P)3293 2380 y Fq(A)3343 2372 y Fr(\))p Fi(,)28 b(wher)m(e)f Fu(S)k Fi(denotes)d(the)2049 2455 y(number)20 b(of)f(thr)m(ows.)2049 2578 y(Pr)m(oof:)74 b Fv(As)15 b(in)h(the)f(proof)h(of)f(Theorem)h(2,)g(we)f(bound)i(the)e (e)o(x)o(ecution)h(time)2049 2676 y(by)23 b(using)h(Equation)f(\(2\))g (and)g(bounding)3168 2616 y Fj(P)3255 2637 y Fq(T)3255 2703 y(i)p Ft(=1)3372 2676 y Fu(p)3411 2684 y Fq(i)3436 2676 y Fv(.)35 b(At)22 b(each)h(round,)i(we)2049 2759 y(collect)j(a)f(tok)o(en)i(from)f(each)h(scheduled)g(process.)50 b(W)-6 b(e)28 b(will)f(sho)n(w)h(that)2049 2842 y(the)23 b(total)h(number)g(of)f(tok)o(ens)i(collected)f(is)f Fu(T)3285 2850 y Ft(1)3340 2842 y Fr(+)d Fu(S)t Fv(.)36 b(Since)24 b(each)g(round)2049 2926 y(consists)h(of)f(at)g(most)h Fr(3)p Fu(C)k Fv(steps,)d(this)e(bound)i(on)f(the)f(number)h(of)g(tok)o (ens)2049 3009 y(implies)19 b(the)g(desired)g(time)g(bound.)2126 3092 y(When)e(a)g(process)h Fu(q)i Fv(is)d(scheduled)h(in)f(a)g(round,) i(it)d(e)o(x)o(ecutes)i(at)f(least)g(tw)o(o)2049 3175 y(milestones,)30 b(and)f(the)f(process)h(places)g(its)e(tok)o(en)i(in)f (one)h(of)f(tw)o(o)g(b)o(uck-)2049 3258 y(ets,)g(as)e(determined)i(by)f (the)f(second)i(milestone.)46 b(There)26 b(are)h(tw)o(o)f(types)2049 3341 y(of)f(milestones.)43 b(If)25 b Fu(q)s Fv(')l(s)g(second)i (milestone)e(marks)h(the)g(occurrence)g(of)g(a)2049 3424 y(node)21 b(e)o(x)o(ecution,)f(then)h Fu(q)h Fv(places)e(its)f(tok)o (en)i(in)f(the)g Fl(w)o(ork)g(b)o(uc)o(k)n(et)p Fv(.)25 b(Clearly)2049 3507 y(there)20 b(are)f(at)g(most)h Fu(T)2611 3515 y Ft(1)2664 3507 y Fv(tok)o(ens)h(in)e(the)h(w)o(ork)g(b)o(uck)o (et.)25 b(The)19 b(second)i(type)f(of)2049 3590 y(milestone)d(marks)f (the)h(completion)g(of)f(a)h(steal)f(attempt,)g(and)h(if)f Fu(q)s Fv(')l(s)g(second)2049 3673 y(milestone)k(is)e(of)i(this)e (type,)i(then)f Fu(q)j Fv(places)e(its)f(tok)o(en)h(in)f(the)g Fl(steal)g(b)o(uc)o(k)n(et)p Fv(.)2049 3756 y(In)24 b(this)g(case,)i (we)e(observ)o(e)i(that)e(the)g(steal)g(attempt)g(is)g(a)g(thro)n(w)-5 b(,)26 b(so)f(there)2049 3839 y(are)19 b(e)o(xactly)g Fu(S)k Fv(tok)o(ens)d(in)f(the)g(steal)g(b)o(uck)o(et.)p 4008 3839 V 2049 4010 a Fw(4.2)74 b(The)19 b(potential)f(function)2049 4120 y Fv(As)28 b(ar)o(gued)g(in)g(Section)f(4.1,)j(it)d(remains)h (only)h(to)e(analyze)i(the)f(number)2049 4203 y(of)f(thro)n(ws.)49 b(W)-6 b(e)27 b(perform)g(this)g(analysis)h(using)g(an)g(amortization)g (ar)o(gu-)2049 4286 y(ment)22 b(based)i(on)e(a)h(potential)f(function)h (that)f(decreases)i(as)e(the)g(algorithm)2049 4369 y(progresses.)39 b(Our)24 b(high-le)n(v)o(el)g(strate)o(gy)g(is)g(to)f(di)n(vide)i(the)f (e)o(x)o(ecution)h(into)2049 4452 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(which)-150 399 y(has)25 b(weight)h Fu(T)248 407 y Fs(1)337 399 y Fv(and)g(is)f(assigned)h(to)f(some)g(process,)i(so)f(we)f(start)f (with)-150 482 y Fr(\010)-95 490 y Ft(0)-20 482 y Fr(=)40 b(3)118 450 y Ft(2)p Fq(T)184 458 y Fb(1)246 450 y Fs(\000)p Ft(1)328 482 y Fv(.)53 b(When)29 b(e)o(x)o(ecution)g(terminates,)i (there)e(are)g(no)g(ready)-150 565 y(nodes,)20 b(so)f(the)g(\002nal)f (potential)h(is)g Fr(0)p Fv(.)-73 648 y(Throughout)i(the)e(e)o(x)o (ecution,)i(the)e(potential)h(ne)n(v)o(er)g(increases.)25 b(That)19 b(is,)-150 731 y(for)j(each)g(round)h Fu(i)p Fv(,)f(we)g(ha)o(v)o(e)g Fr(\010)704 739 y Fq(i)p Ft(+1)834 731 y Fm(\024)27 b Fr(\010)976 739 y Fq(i)1002 731 y Fv(.)32 b(The)21 b(w)o(ork)i(stealer)e(performs)-150 814 y(only)k(tw)o(o)g(actions)g(that)f(may)h(change)h(the)f(potential,) h(and)f(both)g(of)g(them)-150 897 y(decrease)i(the)f(potential.)45 b(The)26 b(\002rst)f(action)i(that)f(changes)h(the)f(potential)-150 980 y(is)f(the)g(remo)o(v)n(al)g(of)g(a)g(node)h Fu(u)f 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Fv(becomes)f(the)f(assigned)h (node.)53 b(Thus,)32 b(the)c(potential)-150 1572 y(decreases)20 b(by)134 1708 y Fu(\036)180 1716 y Fq(i)206 1708 y Fr(\()p Fu(u)p Fr(\))d Fm(\000)g Fu(\036)450 1716 y Fq(i)p Ft(+1)553 1708 y Fr(\()p Fu(x)p Fr(\))f Fm(\000)h Fu(\036)796 1716 y Fq(i)p Ft(+1)899 1708 y Fr(\()p Fu(y)s Fr(\))217 1833 y(=)82 b(3)397 1797 y Ft(2)p Fq(w)q Ft(\()p Fq(u)p Ft(\))p Fs(\000)p Ft(1)658 1833 y Fm(\000)17 b Fr(3)773 1797 y Ft(2)p Fq(w)q Ft(\()p Fq(x)p Ft(\))954 1833 y Fm(\000)g Fr(3)1069 1797 y Ft(2)p Fq(w)q Ft(\()p Fq(y)r Ft(\))p Fs(\000)p Ft(1)217 1941 y Fr(=)82 b(3)397 1905 y Ft(2)p Fq(w)q Ft(\()p Fq(u)p Ft(\))p Fs(\000)p Ft(1)658 1941 y Fm(\000)17 b Fr(3)773 1905 y Ft(2\()p Fq(w)q Ft(\()p Fq(u)p Ft(\))p Fs(\000)p Ft(1\))1082 1941 y Fm(\000)g Fr(3)1197 1905 y Ft(2\()p Fq(w)q Ft(\()p Fq(u)p Ft(\))p Fs(\000)p Ft(1\))p Fs(\000)p Ft(1)217 2077 y Fr(=)82 b(3)397 2041 y Ft(2)p Fq(w)q Ft(\()p Fq(u)p Ft(\))p Fs(\000)p Ft(1)654 1987 y Fj(\020)704 2077 y Fr(1)17 b Fm(\000)846 2029 y Fr(1)p 846 2060 39 4 v 846 2128 a(3)911 2077 y Fm(\000)998 2029 y Fr(1)p 998 2060 V 998 2128 a(9)1047 1987 y Fj(\021)217 2263 y Fr(=)369 2215 y(5)p 369 2246 V 369 2314 a(9)418 2263 y Fu(\036)464 2271 y Fq(i)490 2263 y Fr(\()p Fu(u)p Fr(\))h Fu(;)-150 2426 y Fv(which)g(is)f(positi)n (v)o(e.)23 b(If)17 b(the)g(e)o(x)o(ecution)h(of)g Fu(u)f Fv(enables)h(fe)n(wer)g(than)g(tw)o(o)f(chil-)-150 2509 y(dren,)f(then)g(the)g(potential)g(decreases)h(e)n(v)o(en)f(more.)22 b(Thus,)16 b(the)g(e)o(x)o(ecution)h(of)-150 2592 y(a)i(node)h Fu(u)f Fv(at)f(round)i Fu(i)f Fv(decreases)h(the)f(potential)g(by)h(at) e(least)h Fr(\(5)p Fu(=)p Fr(9\))p Fu(\036)1676 2600 y Fq(i)1703 2592 y Fr(\()p Fu(u)p Fr(\))p Fv(.)-73 2675 y(T)-6 b(o)27 b(f)o(acilitate)g(the)h(analysis,)i(we)d(partition)h(the) f(potential)h(among)h(the)-150 2758 y(processes)f(and)g(we)g (separately)g(consider)g(the)f(processes)i(whose)f(deque)-150 2841 y(is)22 b(empty)i(and)f(the)g(processes)h(whose)g(deque)g(is)e (non-empty)-5 b(.)36 b(At)23 b(the)g(be-)-150 2924 y(ginning)j(of)f (round)h Fu(i)p Fv(,)h(for)e(an)o(y)g(process)h Fu(q)s Fv(,)g(let)f Fu(R)1193 2932 y Fq(i)1219 2924 y Fr(\()p Fu(q)s Fr(\))g Fv(denote)h(the)f(set)g(of)-150 3007 y(ready)20 b(nodes)g(that)f(are)h(in)f Fu(q)s Fv(')l(s)g(deque)h(along)g(with)f (the)h(ready)g(node,)g(if)e(an)o(y)-5 b(,)-150 3090 y(that)21 b(is)g(assigned)h(to)f Fu(q)s Fv(.)30 b(W)-6 b(e)21 b(say)g(that)g (each)h(node)g Fu(u)f Fv(in)g Fu(R)1395 3098 y Fq(i)1422 3090 y Fr(\()p Fu(q)s Fr(\))f Fi(belongs)j Fv(to)-150 3173 y(process)d Fu(q)s Fv(.)j(Then)c(the)g(potential)g(that)g(we)g (associate)g(with)g Fu(q)i Fv(is)481 3336 y Fr(\010)536 3344 y Fq(i)563 3336 y Fr(\()p Fu(q)s Fr(\))g(=)817 3259 y Fj(X)762 3435 y Fq(u)p Fs(2)p Fq(R)887 3446 y Fc(i)913 3435 y Ft(\()p Fq(q)r Ft(\))1004 3336 y Fu(\036)1050 3344 y Fq(i)1076 3336 y Fr(\()p Fu(u)p Fr(\))d Fu(:)-150 3568 y Fv(In)e(addition,)h(let)e Fu(A)356 3576 y Fq(i)398 3568 y Fv(denote)i(the)f(set)f(of)h(processes)h(whose)g(deque)g(is)e (empty)-150 3651 y(at)24 b(the)h(be)o(ginning)h(of)e(round)i Fu(i)p Fv(,)f(and)g(let)f Fu(D)1015 3659 y Fq(i)1066 3651 y Fv(denote)h(the)g(set)f(of)h(all)e(other)-150 3734 y(processes.)h(W)-6 b(e)19 b(partition)f(the)h(potential)h Fr(\010)1012 3742 y Fq(i)1057 3734 y Fv(into)f(tw)o(o)g(parts)463 3870 y Fr(\010)518 3878 y Fq(i)566 3870 y Fr(=)i(\010)702 3878 y Fq(i)729 3870 y Fr(\()p Fu(A)817 3878 y Fq(i)843 3870 y Fr(\))c(+)g(\010)1022 3878 y Fq(i)1048 3870 y Fr(\()p Fu(D)1141 3878 y Fq(i)1168 3870 y Fr(\))i Fu(;)-150 4006 y Fv(where)-41 4153 y Fr(\010)14 4161 y Fq(i)41 4153 y Fr(\()p Fu(A)129 4161 y Fq(i)155 4153 y Fr(\))i(=)299 4076 y Fj(X)287 4251 y Fq(q)r Fs(2)p Fq(A)405 4262 y Fc(i)444 4153 y Fr(\010)499 4161 y Fq(i)526 4153 y Fr(\()p Fu(q)s Fr(\))149 b Fv(and)h Fr(\010)1084 4161 y Fq(i)1111 4153 y Fr(\()p Fu(D)1204 4161 y Fq(i)1231 4153 y Fr(\))21 b(=)1377 4076 y Fj(X)1363 4251 y Fq(q)r Fs(2)p Fq(D)1485 4262 y Fc(i)1523 4153 y Fr(\010)1578 4161 y Fq(i)1605 4153 y Fr(\()p Fu(q)s Fr(\))e Fu(;)-150 4382 y Fv(and)h(we)e(analyze)i (the)f(tw)o(o)g(parts)g(separately)-5 b(.)-73 4465 y(W)f(e)15 b(no)n(w)h(wish)g(to)g(sho)n(w)g(that)g(whene)n(v)o(er)h Fu(P)26 b Fv(or)16 b(more)g(thro)n(ws)g(tak)o(e)g(place)-150 4548 y(o)o(v)o(er)29 b(a)f(sequence)i(of)e(rounds,)k(the)c(potential)h (decreases)g(by)g(a)f(constant)-150 4631 y(fraction)18 b(with)g(constant)h(probability)-5 b(.)24 b(This)18 b(claim)g(will)f (be)h(pro)o(v)o(ed)i(in)e(tw)o(o)-150 4714 y(stages.)42 b(First,)25 b(we)g(sho)n(w)g(that)g Fr(3)p Fu(=)p Fr(4)h Fv(of)f(the)g(potential)h Fr(\010)1389 4722 y Fq(i)1415 4714 y Fr(\()p Fu(D)1508 4722 y Fq(i)1535 4714 y Fr(\))f Fv(is)f(sitting)-150 4797 y(\223e)o(xposed\224)29 b(at)d(the)h(top)g (of)f(the)h(deques)h(where)f(it)f(is)g(accessible)h(to)g(steal)-150 4880 y(attempts.)j(Second,)22 b(we)f(use)g(a)g(\223balls)g(and)h (weighted)g(bins\224)f(ar)o(gument)h(to)-150 4963 y(sho)n(w)c(that)g Fr(1)p Fu(=)p Fr(2)h Fv(of)e(this)h(e)o(xposed)h(potential)f(is)f (stolen)h(with)f Fr(1)p Fu(=)p Fr(4)i Fv(probabil-)-150 5046 y(ity)-5 b(.)23 b(The)c(potential)g Fr(\010)436 5054 y Fq(i)463 5046 y Fr(\()p Fu(A)551 5054 y Fq(i)577 5046 y Fr(\))f Fv(is)h(considered)h(separately)-5 b(.)-150 5181 y Fw(Lemma)20 b(6)h(\(T)-6 b(op-Hea)o(vy)17 b(Deques\))39 b Fi(Consider)21 b(any)f(r)m(ound)g Fu(i)g Fi(and)g(any)-150 5264 y(pr)m(ocess)30 b Fu(q)j Fi(in)c Fu(D)321 5272 y Fq(i)348 5264 y Fi(.)55 b(The)30 b(top-most)g(node)h Fu(u)e Fi(in)h Fu(q)s Fi(')m(s)f(deque)i(contrib)o(utes)-150 5347 y(at)c(least)g Fr(3)p Fu(=)p Fr(4)g Fi(of)g(the)g(potential)g (associated)h(with)f Fu(q)s Fi(.)46 b(That)27 b(is,)h(we)f(have)-150 5430 y Fu(\036)-104 5438 y Fq(i)-78 5430 y Fr(\()p Fu(u)p Fr(\))21 b Fm(\025)g Fr(\(3)p Fu(=)p Fr(4\)\010)357 5438 y Fq(i)385 5430 y Fr(\()p Fu(q)s Fr(\))p Fi(.)2049 -75 y(Pr)m(oof:)74 b Fv(This)27 b(lemma)g(follo)n(ws)h(directly)f(from)g (the)g(Structural)g(Lemma)2049 8 y(\(Lemma)e(3\),)g(and)h(in)e (particular)h(from)g(Corollary)g(4.)40 b(Suppose)26 b(the)f(top-)2049 91 y(most)g(node)i Fu(u)e Fv(in)g Fu(q)s Fv(')l(s)g(deque)i(is)e(also)g (the)h(only)g(node)g(in)f Fu(q)s Fv(')l(s)g(deque,)j(and)2049 174 y(in)22 b(addition,)h Fu(u)f Fv(has)g(the)g(same)g(designated)i (parent)e(as)g(the)g(node)h Fu(y)h Fv(that)e(is)2049 257 y(assigned)e(to)f Fu(q)s Fv(.)k(In)c(this)f(case,)h(we)g(ha)o(v)o (e)2558 414 y Fr(\010)2613 422 y Fq(i)2640 414 y Fr(\()p Fu(q)s Fr(\))82 b(=)h Fu(\036)3008 422 y Fq(i)3034 414 y Fr(\()p Fu(u)p Fr(\))17 b(+)g Fu(\036)3278 422 y Fq(i)3304 414 y Fr(\()p Fu(y)s Fr(\))2819 539 y(=)83 b(3)3000 503 y Ft(2)p Fq(w)q Ft(\()p Fq(u)p Ft(\))3183 539 y Fr(+)16 b(3)3297 503 y Ft(2)p Fq(w)q Ft(\()p Fq(y)r Ft(\))p Fs(\000)p Ft(1)2819 647 y Fr(=)83 b(3)3000 611 y Ft(2)p Fq(w)q Ft(\()p Fq(u)p Ft(\))3183 647 y Fr(+)16 b(3)3297 611 y Ft(2)p Fq(w)q Ft(\()p Fq(u)p Ft(\))p Fs(\000)p Ft(1)2819 784 y Fr(=)2972 735 y(4)p 2972 767 V 2972 834 a(3)3020 784 y Fu(\036)3066 792 y Fq(i)3092 784 y Fr(\()p Fu(u)p Fr(\))j Fu(:)2049 968 y Fv(In)e(all)f(other)i(cases,)f Fu(u)g Fv(contrib)o(utes)g(an)h(e)n(v)o(en)f(lar)o(ger)g(fraction)g(of) g(the)g(poten-)2049 1051 y(tial)h(associated)i(with)e Fu(q)s Fv(.)p 4008 1051 42 42 v 2049 1243 a Fw(Lemma)i(7)h(\(Balls)e (and)h(W)m(eighted)e(Bins\))40 b Fi(Suppose)k(that)e Fu(P)53 b Fi(balls)2049 1326 y(ar)m(e)30 b(thr)m(own)g(independently)i (and)e(uniformly)g(at)g(r)o(andom)h(into)f Fu(P)40 b Fi(bins,)2049 1409 y(wher)m(e)20 b(for)g Fu(i)i Fr(=)h(1)p Fu(;)13 b(:)g(:)g(:)h(;)f(P)e Fi(,)19 b(bin)g Fu(i)h Fi(has)g(a)g(weight)f Fu(W)3424 1417 y Fq(i)3450 1409 y Fi(.)25 b(The)19 b(total)h(weight)f(is)2049 1507 y Fu(W)32 b Fr(=)2235 1446 y Fj(P)2322 1467 y Fq(P)2322 1534 y(i)p Ft(=1)2438 1507 y Fu(W)2511 1515 y Fq(i)2537 1507 y Fi(.)23 b(F)-8 b(or)19 b(eac)o(h)h(bin)f Fu(i)p Fi(,)f(de\002ne)j(the)e(r)o(andom)h(variable)f Fu(X)3926 1515 y Fq(i)3972 1507 y Fi(as)2390 1734 y Fu(X)2453 1742 y Fq(i)2501 1734 y Fr(=)2582 1619 y Fj(\032)2686 1691 y Fu(W)2759 1699 y Fq(i)2868 1691 y Fi(if)f(some)h(ball)g(lands)h(in)e (bin)i Fu(i)p Fi(;)2686 1774 y Fr(0)144 b Fi(otherwise)o(.)2049 1971 y(If)18 b Fu(X)28 b Fr(=)2285 1911 y Fj(P)2373 1932 y Fq(P)2373 1998 y(i)p Ft(=1)2489 1971 y Fu(X)2552 1979 y Fq(i)2579 1971 y Fi(,)18 b(then)i(for)f(any)g Fu(\014)k Fi(in)c(the)g(r)o(ang)o(e)h Fr(0)i Fu(<)f(\014)26 b(<)21 b Fr(1)p Fi(,)e(we)g(have)2049 2063 y Fr(Pr)13 b Fm(f)p Fu(X)28 b Fm(\025)21 b Fu(\014)t(W)11 b Fm(g)21 b Fu(>)g Fr(1)d Fm(\000)f Fr(1)p Fu(=)p Fr(\(\(1)h Fm(\000)e Fu(\014)t Fr(\))p Fu(e)p Fr(\))p Fi(.)2049 2221 y(Pr)m(oof:)74 b Fv(F)o(or)21 b(each)g(bin)g Fu(i)p Fv(,)g(consider)h(the)f(random)g (v)n(ariable)h Fu(W)3745 2229 y Fq(i)3789 2221 y Fm(\000)c Fu(X)3930 2229 y Fq(i)3957 2221 y Fv(.)28 b(It)2049 2304 y(tak)o(es)21 b(on)f(the)g(v)n(alue)h Fu(W)2680 2312 y Fq(i)2726 2304 y Fv(when)g(no)f(ball)g(lands)h(in)f(bin)g Fu(i)p Fv(,)g(and)h(otherwise)f(it)2049 2387 y(is)f Fr(0)p Fv(.)k(Thus,)c(we)f(ha)o(v)o(e)2498 2590 y Fr(E)13 b([)q Fu(W)2658 2598 y Fq(i)2701 2590 y Fm(\000)j Fu(X)2840 2598 y Fq(i)2867 2590 y Fr(])83 b(=)g Fu(W)3187 2598 y Fq(i)3226 2500 y Fj(\020)3275 2590 y Fr(1)18 b Fm(\000)3428 2542 y Fr(1)p 3418 2573 60 4 v 3418 2641 a Fu(P)3487 2500 y Fj(\021)3537 2514 y Fq(P)2971 2736 y Fm(\024)83 b Fu(W)3187 2744 y Fq(i)3213 2736 y Fu(=e)19 b(:)2049 2893 y Fv(It)h(follo)n(ws)h(that)g Fr(E)13 b([)p Fu(W)28 b Fm(\000)16 b Fu(X)6 b Fr(])26 b Fm(\024)f Fu(W)r(=e)p Fv(.)j(From)21 b(Mark)o(o)o(v')l(s)h(Inequality)g(we)2049 2976 y(ha)o(v)o(e)d(that)2370 3164 y Fr(Pr)13 b Fm(f)p Fu(W)28 b Fm(\000)16 b Fu(X)28 b(>)21 b Fr(\(1)c Fm(\000)g Fu(\014)t Fr(\))p Fu(W)11 b Fm(g)21 b Fu(<)3325 3115 y Fr(E)13 b([)q Fu(W)27 b Fm(\000)17 b Fu(X)6 b Fr(])p 3325 3147 355 4 v 3341 3215 a(\(1)17 b Fm(\000)g Fu(\014)t Fr(\))p Fu(W)3708 3164 y(:)2049 3372 y Fv(Thus,)i(we)g(conclude)h Fr(Pr)13 b Fm(f)q Fu(X)27 b(<)21 b(\014)t(W)11 b Fm(g)21 b Fu(<)g Fr(1)p Fu(=)p Fr(\(\(1)d Fm(\000)f Fu(\014)t Fr(\))p Fu(e)p Fr(\))p Fv(.)p 4008 3372 42 42 v 2126 3489 a(W)-6 b(e)26 b(no)n(w)i(sho)n(w)f(that)g(whene)n(v)o(er)h Fu(P)37 b Fv(or)27 b(more)g(thro)n(ws)g(occur)m(,)i(the)e(po-)2049 3572 y(tential)i(decreases)i(by)f(a)f(constant)h(fraction)g(of)f Fr(\010)3433 3580 y Fq(i)3460 3572 y Fr(\()p Fu(D)3553 3580 y Fq(i)3580 3572 y Fr(\))g Fv(with)g(constant)2049 3655 y(probability)-5 b(.)2049 3813 y Fw(Lemma)20 b(8)41 b Fi(Consider)24 b(any)g(r)m(ound)g Fu(i)f Fi(and)h(any)g(later)f(r)m (ound)h Fu(j)j Fi(suc)o(h)d(that)2049 3896 y(at)16 b(least)h Fu(P)27 b Fi(thr)m(ows)16 b(occur)i(at)e(r)m(ounds)i(fr)m(om)e Fu(i)g Fi(\(inclusive\))h(to)f Fu(j)21 b Fi(\(e)o(xclusive\).)2049 3979 y(Then)e(we)g(have)2518 4160 y Fr(Pr)2614 4069 y Fj(n)2669 4160 y Fr(\010)2724 4168 y Fq(i)2768 4160 y Fm(\000)e Fr(\010)2900 4168 y Fq(j)2954 4160 y Fm(\025)3045 4111 y Fr(1)p 3045 4142 39 4 v 3045 4210 a(4)3093 4160 y(\010)3148 4168 y Fq(i)3175 4160 y Fr(\()p Fu(D)3268 4168 y Fq(i)3295 4160 y Fr(\))3325 4069 y Fj(o)3401 4160 y Fu(>)3492 4111 y Fr(1)p 3492 4142 V 3492 4210 a(4)3559 4160 y Fu(:)2049 4351 y Fi(Pr)m(oof:)74 b Fv(W)-6 b(e)17 b(\002rst)g(use)h(the)f(T)-6 b(op-Hea)o(vy)19 b(Deques)f(Lemma)f(to)h (sho)n(w)g(that)f(if)2049 4434 y(a)g(thro)n(w)g(tar)o(gets)g(a)f (process)i(with)f(a)g(non-empty)h(deque)g(as)f(its)g(victim,)f(then) 2049 4517 y(the)26 b(potential)h(decreases)g(by)f(at)g(least)g Fr(1)p Fu(=)p Fr(2)h Fv(of)f(the)g(potential)h(associated)2049 4600 y(with)g(that)g(victim)f(process.)49 b(W)-6 b(e)26 b(then)i(consider)g(the)f Fu(P)38 b Fv(thro)n(ws)27 b(as)g(ball)2049 4683 y(tosses,)i(and)e(we)f(use)h(the)g(Balls)f(and)h(W)-6 b(eighted)27 b(Bins)g(Lemma)g(to)f(sho)n(w)2049 4766 y(that)17 b(with)g(more)g(than)h Fr(1)p Fu(=)p Fr(4)g Fv(probability)-5 b(,)18 b(the)g(total)f(potential)g(decreases)h(by) 2049 4849 y Fr(1)p Fu(=)p Fr(4)h Fv(of)f(the)h(potential)f(associated)h (with)f(all)g(processes)h(with)f(a)g(non-empty)2049 4932 y(deque.)2126 5015 y(Consider)25 b(an)o(y)f(process)h Fu(q)i Fv(in)d Fu(D)3004 5023 y Fq(i)3031 5015 y Fv(,)h(and)g(let)e Fu(u)h Fv(denote)h(the)g(node)g(at)f(the)2049 5098 y(top)g(of)f Fu(q)s Fv(')l(s)g(deque)h(at)f(round)h Fu(i)p Fv(.)37 b(From)22 b(the)i(T)-6 b(op-Hea)o(vy)24 b(Deques)g(Lemma)2049 5181 y(\(Lemma)29 b(6\),)j(we)d(ha)o(v)o(e)h Fu(\036)2774 5189 y Fq(i)2800 5181 y Fr(\()p Fu(u)p Fr(\))41 b Fm(\025)g Fr(\(3)p Fu(=)p Fr(4\)\010)3275 5189 y Fq(i)3302 5181 y Fr(\()p Fu(q)s Fr(\))p Fv(.)54 b(No)n(w)-5 b(,)32 b(consider)e(an)o (y)2049 5264 y(thro)n(w)e(that)g(occurs)g(at)g(a)f(round)i Fu(k)41 b Fm(\025)d Fu(i)p Fv(,)29 b(and)g(suppose)g(this)f(thro)n(w)g (tar)o(-)2049 5347 y(gets)36 b(process)g Fu(q)i Fv(as)e(the)g(victim.) 72 b(W)-6 b(e)36 b(consider)g(tw)o(o)g(cases.)73 b(In)36 b(the)2049 5430 y(\002rst)27 b(case,)j(the)e(thro)n(w)g(is)f (successful)i(with)e Fk(popTop)i Fv(returning)f(a)g(node.)1929 5679 y Fw(7)p eop %%Page: 8 8 8 7 bop -150 -75 a Fv(If)31 b(the)h(returned)h(node)g(is)e(node)i Fu(u)p Fv(,)i(then)d(after)g(round)h Fu(k)r Fv(,)i(node)e Fu(u)e Fv(has)-150 8 y(been)f(assigned)h(and)f(possibly)h(already)f(e)o (x)o(ecuted.)57 b(At)29 b(the)g(v)o(ery)h(least,)-150 91 y(node)f Fu(u)g Fv(has)f(been)h(assigned,)j(and)d(the)f(potential)h (has)g(decreased)g(by)g(at)-150 174 y(least)23 b Fr(\(2)p Fu(=)p Fr(3\))p Fu(\036)230 182 y Fq(i)257 174 y Fr(\()p Fu(u)p Fr(\))p Fv(.)35 b(If)23 b(the)g(returned)g(node)h(is)f(not)g (node)i Fu(u)p Fv(,)e(then)h(node)g Fu(u)-150 257 y Fv(has)i(already)h (been)f(assigned)h(and)g(possibly)g(already)f(e)o(x)o(ecuted.)45 b(Again,)-150 340 y(the)29 b(potential)g(has)h(decreased)g(by)f(at)g (least)g Fr(\(2)p Fu(=)p Fr(3\))p Fu(\036)1284 348 y Fq(i)1311 340 y Fr(\()p Fu(u)p Fr(\))p Fv(.)52 b(In)29 b(the)g(other)-150 423 y(case,)22 b(the)f(thro)n(w)g(is)g(unsuccessful) i(with)e Fk(popTop)h Fv(returning)g(either)f Fk(NULL)-150 506 y Fv(or)28 b Fk(ABORT)p Fv(.)h(If)e Fk(popTop)i Fv(returns)g Fk(NULL)p Fv(,)f(then)g Fu(q)s Fv(')l(s)g(deque)h(w)o(as)f(empty)h(at) -150 589 y(some)f(time)g(during)h(round)g Fu(k)r Fv(,)g(so)f(node)h Fu(u)f Fv(has)g(already)h(been)g(assigned)-150 672 y(and)e(possibly)h (e)o(x)o(ecuted.)47 b(Y)-7 b(et)26 b(again,)j(the)d(potential)h(has)g (decreased)h(by)-150 755 y(at)23 b(least)h Fr(\(2)p Fu(=)p Fr(3\))p Fu(\036)308 763 y Fq(i)335 755 y Fr(\()p Fu(u)p Fr(\))p Fv(.)36 b(Finally)-5 b(,)24 b(if)g Fk(popTop)g Fv(returns)g Fk(ABORT)p Fv(,)g(then)g(some)-150 838 y(other)33 b Fk(popTop)h Fv(or)e Fk(popBottom)j Fv(returned)e(a)g(top-most)g(node) h(at)e(some)-150 921 y(time)22 b(during)i(round)f Fu(k)r Fv(,)h(and)f(after)f(round)i Fu(k)r Fv(,)f(node)h Fu(u)e Fv(has)h(been)h(assigned)-150 1004 y(and)j(possibly)g(already)g(e)o(x)o (ecuted.)46 b(In)26 b(all)g(cases,)i(the)e(potential)h(has)g(de-)-150 1087 y(creased)f(by)h(at)e(least)g Fr(\(2)p Fu(=)p Fr(3\))p Fu(\036)661 1095 y Fq(i)688 1087 y Fr(\()p Fu(u)p Fr(\))p Fv(.)43 b(Thus,)27 b(if)e(a)h(thief)f(tar)o(gets)h(process)g Fu(q)-150 1170 y Fv(as)c(the)g(victim)f(at)h(a)g(round)g Fu(k)30 b Fm(\025)c Fu(i)p Fv(,)c(then)h(the)e(potential)i(drops)f(by)h (at)e(least)-150 1253 y Fr(\(2)p Fu(=)p Fr(3\))p Fu(\036)70 1261 y Fq(i)97 1253 y Fr(\()p Fu(u)p Fr(\))g Fm(\025)g Fr(\(2)p Fu(=)p Fr(3\)\(3)p Fu(=)p Fr(4\)\010)706 1261 y Fq(i)735 1253 y Fr(\()p Fu(q)s Fr(\))g(=)g(\(1)p Fu(=)p Fr(2\)\010)1163 1261 y Fq(i)1191 1253 y Fr(\()p Fu(q)s Fr(\))p Fv(.)-73 1336 y(W)-6 b(e)29 b(no)n(w)h(consider)h(all)e Fu(P)40 b Fv(processes)31 b(and)g Fu(P)40 b Fv(thro)n(ws)30 b(that)g(occur)g(at)-150 1419 y(or)39 b(after)f(round)h Fu(i)p Fv(.)82 b(F)o(or)38 b(each)h(process)h Fu(q)h Fv(in)d Fu(D)1278 1427 y Fq(i)1305 1419 y Fv(,)43 b(if)38 b(one)h(or)f(more)-150 1502 y(of)c(the)g Fu(P)45 b Fv(thro)n(ws)35 b(tar)o(gets)e Fu(q)k Fv(as)e(the)f(victim,)j(then)e(the)f(potential)h (de-)-150 1585 y(creases)27 b(by)g Fr(\(1)p Fu(=)p Fr(2\)\010)422 1593 y Fq(i)450 1585 y Fr(\()p Fu(q)s Fr(\))p Fv(.)45 b(If)26 b(we)g(think)h(of)f(each)h(thro)n(w)g(as)f(a)h(ball)f(toss,) -150 1668 y(then)f(we)f(ha)o(v)o(e)h(an)g(instance)g(of)f(the)h(Balls)f (and)h(W)-6 b(eighted)24 b(Bins)h(Lemma)-150 1751 y(\(Lemma)34 b(7\).)69 b(F)o(or)34 b(each)h(process)g Fu(q)i Fv(in)d Fu(D)1079 1759 y Fq(i)1106 1751 y Fv(,)k(we)c(assign)h(it)e(a)i(weight) -150 1834 y Fu(W)-77 1842 y Fq(q)-2 1834 y Fr(=)42 b(\(1)p Fu(=)p Fr(2\)\010)329 1842 y Fq(i)356 1834 y Fr(\()p Fu(q)s Fr(\))p Fv(,)32 b(and)e(for)f(each)h(other)g(process)g Fu(q)j Fv(in)c Fu(A)1571 1842 y Fq(i)1597 1834 y Fv(,)j(we)d(as-)-150 1917 y(sign)20 b(it)f(a)h(weight)g Fu(W)404 1925 y Fq(q)461 1917 y Fr(=)j(0)p Fv(.)j(The)20 b(weights)g(sum)g(to)g Fu(W)33 b Fr(=)23 b(\(1)p Fu(=)p Fr(2\)\010)1655 1925 y Fq(i)1683 1917 y Fr(\()p Fu(D)1776 1925 y Fq(i)1802 1917 y Fr(\))p Fv(.)-150 2000 y(Using)k Fu(\014)41 b Fr(=)36 b(1)p Fu(=)p Fr(2)28 b Fv(in)e(Lemma)h(7,)i(we)e(conclude)h (that)f(the)f(potential)i(de-)-150 2083 y(creases)f(by)g(at)f(least)g Fu(\014)t(W)46 b Fr(=)36 b(\(1)p Fu(=)p Fr(4\)\010)927 2091 y Fq(i)955 2083 y Fr(\()p Fu(D)1048 2091 y Fq(i)1074 2083 y Fr(\))26 b Fv(with)g(probability)i(greater)-150 2166 y(than)19 b Fr(1)f Fm(\000)f Fr(1)p Fu(=)p Fr(\(\(1)h Fm(\000)e Fu(\014)t Fr(\))p Fu(e)p Fr(\))21 b Fu(>)g Fr(1)p Fu(=)p Fr(4)p Fv(.)p 1809 2166 42 42 v -150 2343 a Fw(4.3)74 b(Anal)o(ysis)18 b(f)o(or)h(dedicated)f(en)m(vir)o(onments) -150 2453 y Fv(In)32 b(this)f(section)h(we)g(analyze)g(the)g (performance)h(of)f(the)g(non-blocking)-150 2536 y(w)o(ork)38 b(stealer)g(in)g(dedicated)g(en)m(vironments.)82 b(In)38 b(a)f(dedicated)i(\(non-)-150 2619 y(multiprogrammed\))26 b(en)m(vironment,)i(all)c Fu(P)35 b Fv(processes)26 b(are)f(scheduled)i (in)-150 2702 y(each)20 b(round,)f(so)g(we)g(ha)o(v)o(e)g Fu(P)614 2710 y Fq(A)686 2702 y Fr(=)i Fu(P)11 b Fv(.)-150 2862 y Fw(Theorem)19 b(9)41 b Fi(Consider)28 b(any)f(multithr)m(eaded)h (computation)g(with)e(work)-150 2945 y Fu(T)-105 2953 y Ft(1)-53 2945 y Fi(and)19 b(critical-path)f(length)g Fu(T)717 2953 y Fs(1)800 2945 y Fi(being)h(e)o(xecuted)f(by)g(the)g (non-bloc)o(king)-150 3028 y(work)30 b(stealer)f(with)g Fu(P)40 b Fi(pr)m(ocesses)31 b(in)e(a)h(dedicated)g(en)m(vir)m(onment.) 56 b(The)-150 3111 y(e)o(xpected)19 b(e)o(xecution)g(time)e(is)h Fu(O)r Fr(\()p Fu(T)780 3119 y Ft(1)815 3111 y Fu(=P)26 b Fr(+)14 b Fu(T)1047 3119 y Fs(1)1112 3111 y Fr(\))p Fi(.)22 b(Mor)m(eo)o(ver)-8 b(,)20 b(for)e(any)h Fu(")i(>)-150 3194 y Fr(0)p Fi(,)d(the)g(e)o(xecution)g(time)f(is)g Fu(O)r Fr(\()p Fu(T)684 3202 y Ft(1)719 3194 y Fu(=P)23 b Fr(+)12 b Fu(T)946 3202 y Fs(1)1024 3194 y Fr(+)g(lg)r(\(1)p Fu(=")p Fr(\)\))17 b Fi(with)g(pr)m(obability)-150 3277 y(at)i(least)g Fr(1)e Fm(\000)g Fu(")p Fi(.)-150 3436 y(Pr)m(oof:)74 b Fv(Lemma)21 b(5)g(bounds)i(the)d(e)o(x)o(ecution)i (time)e(in)h(terms)g(of)f(the)h(num-)-150 3519 y(ber)c(of)g(thro)n(ws.) 23 b(W)-6 b(e)16 b(shall)h(pro)o(v)o(e)h(that)f(the)g(e)o(xpected)h (number)g(of)f(thro)n(ws)g(is)-150 3602 y Fu(O)r Fr(\()p Fu(T)-15 3610 y Fs(1)50 3602 y Fu(P)11 b Fr(\))p Fv(,)20 b(and)h(that)f(the)g(number)i(of)e(thro)n(ws)h(is)f Fu(O)r Fr(\(\()p Fu(T)1338 3610 y Fs(1)1421 3602 y Fr(+)e(lg)q(\(1)p Fu(=")p Fr(\)\))p Fu(P)11 b Fr(\))-150 3685 y Fv(with)19 b(probability)g(at)g(least)f Fr(1)g Fm(\000)f Fu(")p Fv(.)-73 3768 y(W)-6 b(e)18 b(analyze)h(the)g(number)g(of)g(thro)n(ws)g (by)g(breaking)h(the)e(e)o(x)o(ecution)i(into)-150 3851 y Fl(phases)h Fv(of)f Fr(\002\()p Fu(P)11 b Fr(\))19 b Fv(thro)n(ws.)28 b(W)-6 b(e)20 b(sho)n(w)h(that)f(with)g(constant)i (probability)-5 b(,)21 b(a)-150 3935 y(phase)c(causes)g(the)g (potential)f(to)g(drop)h(by)g(a)f(constant)i(f)o(actor)m(,)e(and)h (since)g(we)-150 4018 y(kno)n(w)h(that)f(the)g(potential)h(starts)e(at) h Fr(\010)847 4026 y Ft(0)903 4018 y Fr(=)22 b(3)1023 3986 y Ft(2)p Fq(T)1089 3994 y Fb(1)1150 3986 y Fs(\000)p Ft(1)1250 4018 y Fv(and)c(ends)g(at)e(zero,)i(we)-150 4101 y(can)24 b(use)f(this)g(f)o(act)h(to)f(analyze)h(the)g(number)g (of)f(phases.)37 b(The)24 b(\002rst)e(phase)-150 4184 y(be)o(gins)30 b(at)e(round)j Fu(t)387 4192 y Ft(1)461 4184 y Fr(=)41 b(1)29 b Fv(and)g(ends)h(at)f(the)g(\002rst)f(round)i Fu(t)1510 4152 y Fs(0)1510 4197 y Ft(1)1573 4184 y Fv(such)g(that)-150 4267 y(at)23 b(least)g Fu(P)34 b Fv(thro)n(ws)24 b(occur)g(during)g (the)g(interv)n(al)f(of)h(rounds)g Fr([)p Fu(t)1527 4275 y Ft(1)1562 4267 y Fu(;)13 b(t)1624 4235 y Fs(0)1624 4280 y Ft(1)1658 4267 y Fr(])p Fv(.)37 b(The)-150 4350 y(second)20 b(phase)g(be)o(gins)g(at)e(round)i Fu(t)771 4358 y Ft(2)827 4350 y Fr(=)h Fu(t)936 4318 y Fs(0)936 4363 y Ft(1)987 4350 y Fr(+)c(1)p Fv(,)i(and)h(so)f(on.)-73 4433 y(Consider)28 b(a)f(phase)h(be)o(ginning)h(at)d(round)j Fu(i)p Fv(,)g(and)f(let)e Fu(j)32 b Fv(be)27 b(the)h(round)-150 4516 y(at)37 b(which)i(the)f(ne)o(xt)g(phase)g(be)o(gins.)81 b(W)-6 b(e)37 b(will)g(sho)n(w)h(that)g(we)g(ha)o(v)o(e)-150 4599 y Fr(Pr)13 b Fm(f)p Fr(\010)38 4607 y Fq(j)93 4599 y Fm(\024)21 b Fr(\(3)p Fu(=)p Fr(4\)\010)403 4607 y Fq(i)431 4599 y Fm(g)j Fu(>)g Fr(1)p Fu(=)p Fr(4)p Fv(.)k(Recall)20 b(that)h(the)f(potential)h(can)f(be)h(parti-)-150 4682 y(tioned)d(as)f Fr(\010)188 4690 y Fq(i)236 4682 y Fr(=)22 b(\010)373 4690 y Fq(i)399 4682 y Fr(\()p Fu(A)487 4690 y Fq(i)513 4682 y Fr(\))11 b(+)g(\010)680 4690 y Fq(i)707 4682 y Fr(\()p Fu(D)800 4690 y Fq(i)827 4682 y Fr(\))p Fv(.)22 b(Since)17 b(the)g(phase)i(contains)f(at)f(least)-150 4765 y Fu(P)27 b Fv(thro)n(ws,)17 b(Lemma)g(8)f(implies)g(that)h Fr(Pr)c Fm(f)p Fr(\010)1014 4773 y Fq(i)1058 4765 y Fm(\000)k Fr(\010)1190 4773 y Fq(j)1244 4765 y Fm(\025)k Fr(\(1)p Fu(=)p Fr(4\)\010)1554 4773 y Fq(i)1582 4765 y Fr(\()p Fu(D)1675 4773 y Fq(i)1702 4765 y Fr(\))p Fm(g)g Fu(>)-150 4848 y Fr(1)p Fu(=)p Fr(4)p Fv(.)41 b(W)-6 b(e)24 b(need)h(to)f(sho)n (w)h(that)g(the)f(potential)h(also)f(drops)i(by)f(a)f(constant)-150 4931 y(fraction)d(of)f Fr(\010)240 4939 y Fq(i)267 4931 y Fr(\()p Fu(A)355 4939 y Fq(i)381 4931 y Fr(\))p Fv(.)28 b(Consider)21 b(a)f(process)i Fu(q)h Fv(in)d Fu(A)1239 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Fv(.)h(Thus,)-150 5346 y(no)i(matter)f(ho)n(w)h Fr(\010)367 5354 y Fq(i)417 5346 y Fv(is)f(partitioned)h(between)h Fr(\010)1165 5354 y Fq(i)1191 5346 y Fr(\()p Fu(A)1279 5354 y Fq(i)1305 5346 y Fr(\))e Fv(and)i Fr(\010)1545 5354 y Fq(i)1571 5346 y Fr(\()p Fu(D)1664 5354 y Fq(i)1691 5346 y Fr(\))p Fv(,)f(we)-150 5429 y(ha)o(v)o(e)19 b Fr(Pr)13 b Fm(f)q Fr(\010)196 5437 y Fq(i)240 5429 y Fm(\000)j Fr(\010)371 5437 y Fq(j)426 5429 y Fm(\025)21 b Fr(\(1)p Fu(=)p Fr(4\)\010)736 5437 y Fq(i)763 5429 y Fm(g)h Fu(>)f Fr(1)p Fu(=)p Fr(4)p Fv(.)2126 -75 y(W)-6 b(e)16 b(shall)g(say)h(that)g(a)f(phase)h(is)g Fl(successful)e Fv(if)h(it)g(causes)h(the)g(potential)g(to)2049 8 y(drop)h(by)g(at)f(least)g(a)g Fr(1)p Fu(=)p Fr(4)i Fv(fraction.)k(A)17 b(phase)h(is)f(successful)h(with)f(probabil-)2049 101 y(ity)f(at)g(least)f Fr(1)p Fu(=)p Fr(4)p Fv(.)24 b(Since)15 b(the)i(potential)f(starts)g(at)f Fr(\010)3387 109 y Ft(0)3444 101 y Fr(=)21 b(3)3563 69 y Ft(2)p Fq(T)3629 77 y Fb(1)3691 69 y Fs(\000)p Ft(1)3789 101 y Fv(and)c(ends)2049 184 y(at)f Fr(0)g Fv(\(and)h(is)f(al)o(w)o(ays)h(an)f(inte)o(ger\),)h (the)f(number)h(of)f(successful)h(phases)g(is)f(at)2049 267 y(most)j Fr(\(2)p Fu(T)2326 275 y Fs(1)2408 267 y Fm(\000)d Fr(1\))e(log)2665 285 y Ft(4)p Fq(=)p Ft(3)2774 267 y Fr(3)22 b Fu(<)f Fr(8)p Fu(T)2998 275 y Fs(1)3063 267 y Fv(.)i(The)c(e)o(xpected)h(number)g(of)f(phases)2049 350 y(needed)27 b(to)e(obtain)h Fr(8)p Fu(T)2664 358 y Fs(1)2754 350 y Fv(successful)g(phases)g(is)f(at)g(most)g Fr(32)p Fu(T)3755 358 y Fs(1)3821 350 y Fv(.)42 b(Thus,)2049 433 y(the)21 b(e)o(xpected)h(number)g(of)e(phases)i(is)e Fu(O)r Fr(\()p Fu(T)3202 441 y Fs(1)3268 433 y Fr(\))p Fv(,)g(and)h(because)h(each)g(phase)2049 516 y(contains)17 b Fu(O)r Fr(\()p Fu(P)11 b Fr(\))16 b Fv(thro)n(ws,)h(the)g(e)o (xpected)g(number)h(of)e(thro)n(ws)h(is)e Fu(O)r Fr(\()p Fu(T)3876 524 y Fs(1)3942 516 y Fu(P)c Fr(\))p Fv(.)2049 599 y(W)-6 b(e)18 b(no)n(w)i(turn)f(to)g(the)g(high)g(probability)h (bound.)2126 682 y(Suppose)d(the)g(e)o(x)o(ecution)h(tak)o(es)f Fu(n)22 b Fr(=)f(32)p Fu(T)3247 690 y Fs(1)3321 682 y Fr(+)9 b Fu(m)16 b Fv(phases.)23 b(Each)17 b(phase)2049 765 y(succeeds)i(with)f(probability)g(at)g(least)f Fu(p)k Fr(=)g(1)p Fu(=)p Fr(4)p Fv(,)e(so)f(the)g(e)o(xpected)h(number)2049 848 y(of)k(successes)i(is)e(at)g(least)g Fu(np)30 b Fr(=)g(8)p Fu(T)3042 856 y Fs(1)3127 848 y Fr(+)21 b Fu(m=)p Fr(4)p Fv(.)36 b(W)-6 b(e)23 b(no)n(w)h(compute)h(the)2049 931 y(probability)f(that)f(the)h(number)g Fu(X)30 b Fv(of)23 b(successes)i(is)e(less)g(than)h Fr(8)p Fu(T)3832 939 y Fs(1)3897 931 y Fv(.)36 b(W)-6 b(e)2049 1014 y(use)19 b(the)g(Chernof)n(f)h(bound)g([2)q(,)e(Theorem)h(A.13],)2595 1188 y Fr(Pr)13 b Fm(f)q Fu(X)27 b(<)21 b(np)c Fm(\000)g Fu(a)p Fm(g)k Fu(<)h(e)3297 1151 y Fs(\000)3370 1131 y Fc(a)3402 1114 y Fa(2)p 3354 1140 96 4 v 3354 1172 a(2)p Fc(np)3482 1188 y Fu(;)2049 1325 y Fv(with)d Fu(a)k Fr(=)f Fu(m=)p Fr(4)p Fv(.)k(Thus)19 b(if)g(we)h(choose)h Fu(m)h Fr(=)h(32)p Fu(T)3397 1333 y Fs(1)3480 1325 y Fr(+)17 b(16)d(ln\(1)p Fu(=")p Fr(\))p Fv(,)19 b(then)2049 1408 y(we)g(ha)o(v)o(e)2485 1585 y Fr(Pr)13 b Fm(f)p Fu(X)28 b(<)21 b Fr(8)p Fu(T)2873 1593 y Fs(1)2938 1585 y Fm(g)84 b Fu(<)e(e)3238 1543 y Fs(\000)3354 1515 y Fa(\()p Fc(m=)p Fa(4\))3508 1498 y(2)p 3296 1532 305 4 v 3296 1566 a(16)p Fc(T)3387 1574 y Fb(1)3448 1566 y Fa(+)p Fc(m=)p Fa(2)3060 1750 y Fm(\024)g Fu(e)3238 1708 y Fs(\000)3334 1680 y Fa(\()p Fc(m=)p Fa(4\))3488 1663 y(2)p 3296 1697 263 4 v 3296 1732 a Fc(m=)p Fa(2+)p Fc(m=)p Fa(2)3060 1868 y Fr(=)g Fu(e)3238 1832 y Fs(\000)3299 1812 y Fc(m)p 3296 1821 57 4 v 3296 1853 a Fa(16)3060 1996 y Fm(\024)g Fu(e)3238 1960 y Fs(\000)3296 1932 y Fa(16)11 b(ln\(1)p Fc(=")p Fa(\))p 3296 1949 247 4 v 3391 1981 a(16)3060 2104 y Fr(=)82 b Fu(")19 b(:)2049 2241 y Fv(Thus,)k(the)g(probability)g(that)g(the)f(e)o(x)o(ecution)i (tak)o(es)f Fr(64)p Fu(T)3559 2249 y Fs(1)3645 2241 y Fr(+)c(16)14 b(ln\(1)p Fu(=")p Fr(\))2049 2324 y Fv(phases)31 b(or)e(more)h(is)g(less)f(than)h Fu(")p Fv(.)55 b(W)-6 b(e)29 b(conclude)j(that)d(the)h(number)h(of)2049 2407 y(thro)n(ws)18 b(is)f Fu(O)r Fr(\(\()p Fu(T)2500 2415 y Fs(1)2577 2407 y Fr(+)12 b(lg)r(\(1)p Fu(=")p Fr(\)\))p Fu(P)f Fr(\))17 b Fv(with)g(probability)h(at)g(least)f Fr(1)12 b Fm(\000)g Fu(")p Fv(.)p 4008 2407 42 42 v 2049 2580 a Fw(4.4)74 b(Anal)o(ysis)18 b(f)o(or)h(m)o(ultipr)o(ogrammed)f (en)m(vir)o(onments)2049 2690 y Fv(W)-6 b(e)27 b(no)n(w)h(generalize)g (the)g(analysis)f(of)h(the)f(pre)n(vious)i(section)f(to)f(bound)2049 2773 y(the)h(e)o(x)o(ecution)h(time)e(of)h(the)g(non-blocking)i(w)o (ork)e(stealer)g(in)g(multipro-)2049 2856 y(grammed)g(en)m(vironments.) 51 b(Recall)27 b(that)g(in)g(a)h(multiprogrammed)g(en)m(vi-)2049 2939 y(ronment,)c(the)f(k)o(ernel)h(is)e(an)h(adv)o(ersary)h(that)f (may)g(choose)h(not)f(to)g(sched-)2049 3022 y(ule)j(some)g(of)f(the)h (processes)g(at)g(some)g(or)f(all)g(rounds.)45 b(In)25 b(particular)m(,)i(at)2049 3105 y(each)19 b(round)h Fu(i)p Fv(,)e(the)h(k)o(ernel)g(schedules)h Fu(p)3122 3113 y Fq(i)3166 3105 y Fv(processes)g(of)f(its)f(choosing.)24 b(W)-6 b(e)2049 3188 y(consider)24 b(three)g(dif)n(ferent)f(classes)h (of)f(adv)o(ersaries,)i(with)e(each)h(class)g(be-)2049 3271 y(ing)c(more)g(po)n(werful)h(than)f(the)g(pre)n(vious,)h(and)f(we) g(consider)h(increasingly)2049 3354 y(po)n(werful)j(forms)g(of)f(the)h (yield)f(system)h(call.)36 b(In)24 b(all)e(cases,)j(we)e(\002nd)h(that) 2049 3437 y(the)19 b(e)o(xpected)h(e)o(x)o(ecution)g(time)f(is)f Fu(O)r Fr(\()p Fu(T)3104 3445 y Ft(1)3139 3437 y Fu(=P)3226 3445 y Fq(A)3294 3437 y Fr(+)e Fu(T)3415 3445 y Fs(1)3480 3437 y Fu(P)6 b(=P)3622 3445 y Fq(A)3673 3437 y Fr(\))p Fv(.)2126 3520 y(W)-6 b(e)20 b(pro)o(v)o(e)h(our)g(upper)h(bounds)g (for)e(multiprogrammed)i(en)m(vironments)2049 3604 y(using)j(the)f (results)g(of)h(Section)f(4.2)g(and)h(the)f(same)h(general)g(approach)h (as)2049 3687 y(is)i(used)h(to)g(pro)o(v)o(e)g(Theorem)g(9.)51 b(The)29 b(only)g(place)g(in)f(which)h(the)f(proof)2049 3770 y(of)d(Theorem)h(9)f(depends)i(on)e(the)g(assumption)i(of)e(a)g (dedicated)h(en)m(viron-)2049 3853 y(ment)d(is)g(in)f(the)h(analysis)h (of)f(progress)h(being)f(made)h(by)f(those)h(processes)2049 3936 y(in)g(the)g(set)f Fu(A)2410 3944 y Fq(i)2436 3936 y Fv(.)37 b(In)24 b(particular)m(,)h(in)f(pro)o(ving)h(Theorem)f(9,)h (we)e(considered)2049 4019 y(a)f(round)h Fu(i)e Fv(and)i(an)o(y)f (process)h Fu(q)h Fv(in)e Fu(A)3047 4027 y Fq(i)3073 4019 y Fv(,)g(and)g(we)g(sho)n(wed)h(that)f(at)f(round)i Fu(i)p Fv(,)2049 4102 y(the)i(potential)h(decreases)g(by)g(at)f(least)g Fr(\(5)p Fu(=)p Fr(9\)\010)3333 4110 y Fq(i)3360 4102 y Fr(\()p Fu(q)s Fr(\))p Fv(,)h(because)h(process)f Fu(q)2049 4185 y Fv(e)o(x)o(ecutes)21 b(its)f(assigned)h(node,)g(if)f(an)o(y)-5 b(.)27 b(This)20 b(conclusion)i(is)e(not)g(v)n(alid)h(in)f(a)2049 4268 y(multiprogrammed)f(en)m(vironment,)g(because)g(the)f(k)o(ernel)g (may)h(choose)g(not)2049 4351 y(to)g(schedule)i(process)f Fu(q)i Fv(at)d(round)i Fu(i)p Fv(.)j(F)o(or)19 b(this)g(reason,)h(we)f (need)h(the)g(yield)2049 4434 y(system)f(calls.)2126 4517 y(The)25 b(use)i(of)e(yield)h(system)g(calls)g(ne)n(v)o(er)g (constrains)h(the)e(k)o(ernel)i(in)e(its)2049 4600 y(choice)g(of)g(the) f(number)h Fu(p)2761 4608 y Fq(i)2812 4600 y Fv(of)f(processes)h(that)g (it)e(schedules)j(at)e(a)h(step)f Fu(i)p Fv(.)2049 4683 y(Y)l(ield)e(calls)g(constrain)h(the)g(k)o(ernel)g(only)g(in)f(its)g (choice)h(of)f Fi(whic)o(h)h Fu(p)3878 4691 y Fq(i)3926 4683 y Fv(pro-)2049 4766 y(cesses)e(it)e(schedules.)29 b(W)-6 b(e)20 b(wish)g(to)g(a)o(v)o(oid)g(constraining)i(the)e(k)o (ernel)h(in)f(its)2049 4849 y(choice)i(of)e(the)h(number)h(of)f (processes)h(that)e(it)h(schedules,)h(because)g(doing)2049 4932 y(so)27 b(w)o(ould)h(admit)e(tri)n(vial)g(solutions.)48 b(F)o(or)26 b(e)o(xample,)j(if)d(we)h(could)h(force)2049 5015 y(the)c(k)o(ernel)h(to)e(schedule)j(only)e(one)h(process,)g(then)g 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Fi(.)37 b(Mor)m(eo)o(ver)-8 b(,)26 b(for)d(any)h Fu(")30 b(>)g Fr(0)p Fi(,)25 b(the)e(e)o(xecution)-150 1012 y(time)15 b(is)h Fu(O)r Fr(\()p Fu(T)195 1020 y Ft(1)229 1012 y Fu(=P)316 1020 y Fq(A)372 1012 y Fr(+)5 b(\()p Fu(T)512 1020 y Fs(1)582 1012 y Fr(+)g(lg)q(\(1)p Fu(=")p Fr(\)\))p Fu(P)h(=P)1051 1020 y Fq(A)1102 1012 y Fr(\))16 b Fi(with)f(pr)m (obability)h(at)g(least)-150 1095 y Fr(1)h Fm(\000)g Fu(")p Fi(.)-150 1243 y(Pr)m(oof:)74 b Fv(As)27 b(in)g(the)g(proof)h (of)f(Theorem)g(9,)i(we)e(bound)i(the)e(number)h(of)-150 1326 y(thro)n(ws)f(by)g(sho)n(wing)g(that)f(in)h(each)g(phase,)i(the)d (potential)h(decreases)g(by)-150 1409 y(a)g(constant)i(f)o(actor)f (with)f(constant)h(probability)-5 b(.)50 b(W)-6 b(e)27 b(consider)h(a)g(phase)-150 1492 y(that)c(be)o(gins)i(at)e(round)i Fu(i)p Fv(.)40 b(The)24 b(potential)h(is)f Fr(\010)1122 1500 y Fq(i)1181 1492 y Fr(=)32 b(\010)1328 1500 y Fq(i)1355 1492 y Fr(\()p Fu(A)1443 1500 y Fq(i)1469 1492 y Fr(\))21 b(+)g(\010)1656 1500 y Fq(i)1683 1492 y Fr(\()p Fu(D)1776 1500 y Fq(i)1802 1492 y Fr(\))p Fv(.)-150 1575 y(From)26 b(Lemma)g(8,)i(we)e(kno)n(w)h(that)f(the)g(potential)g(decreases)i(by)e (at)g(least)-150 1658 y Fr(\(1)p Fu(=)p Fr(4\)\010)79 1666 y Fq(i)107 1658 y Fr(\()p Fu(D)200 1666 y Fq(i)226 1658 y Fr(\))c Fv(with)g(probability)h(more)g(than)f Fr(1)p Fu(=)p Fr(4)p Fv(.)35 b(It)21 b(remains)i(to)f(pro)o(v)o(e)-150 1741 y(that)c(with)g(constant)h(probability)f(the)h(potential)f(also)g (decreases)i(by)e(a)g(con-)-150 1824 y(stant)h(fraction)g(of)g Fr(\010)397 1832 y Fq(i)424 1824 y Fr(\()p Fu(A)512 1832 y Fq(i)537 1824 y Fr(\))p Fv(.)-73 1907 y(Consider)h(a)f(process)h Fu(q)j Fv(in)c Fu(A)703 1915 y Fq(i)729 1907 y Fv(.)24 b(If)19 b Fu(q)j Fv(is)d(scheduled)i(at)f(some)f(round)i(dur)o(-)-150 1990 y(ing)j(the)g(phase,)i(then)e(the)h(potential)f(decreases)h(by)f (at)g(least)g Fr(\(5)p Fu(=)p Fr(9\)\010)1727 1998 y Fq(i)1754 1990 y Fr(\()p Fu(q)s Fr(\))-150 2073 y Fv(as)f(in)g(Theorem) g(9.)35 b(During)24 b(the)e(phase,)j(at)e(least)f Fu(P)34 b Fv(thro)n(ws)23 b(occur)m(,)h(so)f(at)-150 2156 y(least)e Fu(P)31 b Fv(processes)22 b(are)f(scheduled,)i(with)e(some)h(processes) g(possibly)g(be-)-150 2239 y(ing)h(scheduled)h(multiple)e(times.)34 b(These)22 b(scheduled)i(processes)g(are)e(cho-)-150 2322 y(sen)27 b(at)g(random,)j(so)d(we)g(can)g(treat)g(them)g(lik)o(e)g (random)h(ball)f(tosses)g(and)-150 2405 y(appeal)20 b(to)g(the)g(Balls) f(and)h(W)-6 b(eighted)20 b(Bins)f(Lemma)h(\(Lemma)f(7\).)25 b(In)20 b(f)o(act,)-150 2488 y(this)g(selection)g(of)g(processes)g(at)g (random)h(is)e(not)h(the)g(same)g(as)g(ball)g(tosses,)-150 2571 y(because)25 b(the)o(y)f(are)g(not)h(all)e(independent:)35 b(A)24 b(process)h(cannot)g(be)f(sched-)-150 2654 y(uled)i(more)g(than) h(once)f(in)g(a)g(gi)n(v)o(en)h(round.)45 b(But)25 b(this)h(dependence) i(only)-150 2737 y(increases)18 b(the)g(probability)h(that)e(a)h(bin)g (recei)n(v)o(es)g(a)g(ball.)k(F)o(or)c(each)g(process)-150 2820 y Fu(q)k Fv(in)e Fu(A)42 2828 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Fv(cannot)f(be)e(scheduled)i(at)f(round)g Fu(j)k Fv(as)15 b(the)h(sched-)-150 5347 y(ule)24 b(calls)f(for)m(,)i(we)f (schedule)h(process)f Fu(r)i Fv(instead.)38 b(That)24 b(is,)g(we)g(schedule)-150 5430 y(process)i Fu(r)h Fv(in)e(place)g(of)h Fu(q)s Fv(.)41 b(Observ)o(e)26 b(that)f(this)g(change)h(in)f(the)h (schedule)2049 -75 y(does)18 b(not)g(change)h(the)f(number)h(of)f (processes)h(scheduled)g(at)e(an)o(y)h(round;)h(it)2049 8 y(only)h(changes)g(which)f(processes)h(are)f(scheduled.)2126 91 y(The)h(non-blocking)i(w)o(ork)f(stealer)e(uses)i Fk(yieldTo)g Fv(to)f(yield)g(to)g(the)g(last)2049 174 y(chosen)27 b(victim)f(process)g(before)g(the)g(ne)o(xt)g(steal)g (attempt.)43 b(Speci\002cally)-5 b(,)2049 257 y(Line)19 b(8)g(of)g(the)g(scheduling)h(loop)g(\(Figure)e(1\))h(is)g Fk(yieldTo\(victim\))p Fv(.)2049 416 y Fw(Theorem)g(11)41 b Fi(Consider)19 b(any)g(multithr)m(eaded)g(computation)h(with)e(work) 2049 499 y Fu(T)2094 507 y Ft(1)2146 499 y Fi(and)h(critical-path)f (length)g Fu(T)2916 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4818 y Fq(i)2873 4810 y Fv(with)g(probability)h(greater)g(than)g Fr(1)p Fu(=)p Fr(16)p Fv(.)24 b(The)2049 4893 y(remainder)g(of)f(the)g(proof)h(is)f (the)g(same)g(as)g(that)g(of)h(Theorem)f(9,)h(b)o(ut)f(with)2049 4976 y(dif)n(ferent)c(constants.)p 4008 4976 V 2049 5153 a Fw(4.4.3)73 b(Adaptive)18 b(ad)o(ver)o(sar)q(y)2049 5263 y Fv(An)h Fl(adaptiv)o(e)i Fv(adv)o(ersary)f(selects)g(both)f(the) h(number)g Fu(p)3510 5271 y Fq(i)3555 5263 y Fv(of)g(processes)g(and) 2049 5346 y(which)f(of)g(the)g Fu(p)2480 5354 y Fq(i)2524 5346 y Fv(processes)h(e)o(x)o(ecute)f(at)f(each)i(round)g Fu(i)p Fv(,)e(and)h(it)f(may)h(do)h(so)1929 5679 y Fw(9)p eop %%Page: 10 10 10 9 bop -150 -75 a Fv(in)25 b(an)g(on-line)h(f)o(ashion.)42 b(The)25 b(adapti)n(v)o(e)h(adv)o(ersary)g(is)f(constrained)h(only)-150 8 y(by)19 b(the)g(requirement)h(to)f(obe)o(y)h(yield)f(system)g(calls.) -73 91 y(T)-6 b(o)20 b(deal)h(with)g(an)g(adapti)n(v)o(e)g(adv)o (ersary)-5 b(,)22 b(we)f(emplo)o(y)h(a)e(po)n(werful)i(yield)-150 174 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b(oung.)37 b(Mach:)31 b(A)21 b(ne)n(w)h(k)o(ernel)h(foundation)h(for)d(UNIX)g(de)n(v)o(elop-)2201 5347 y(ment.)e(In)14 b Fn(Pr)m(oceedings)i(of)f(the)g(Summer)g(1986)g (USENIX)f(Confer)n(ence)p Fo(,)j(pages)2201 5422 y(93\226112,)h(July)f (1986.)1908 5679 y Fw(10)p eop %%Page: 11 11 11 10 bop -117 -75 a Fo([2])42 b(Noga)27 b(Alon)f(and)h(Joel)g(H.)f (Spencer)l(.)56 b Fn(The)26 b(Pr)m(obabilistic)j(Method)p Fo(.)56 b(John)2 -1 y(W)m(ile)o(y)18 b(&)f(Sons,)f(1992.)-117 107 y([3])42 b(K)n(en)23 b(Arnold)g(and)g(James)f(Gosling.)42 b Fn(The)23 b(J)n(ava)g(Pr)m(o)o(gr)o(amming)g(Langua)o(g)o(e)p Fo(.)2 182 y(Addison-W)-5 b(esle)o(y)l(,)18 b(1996.)-117 290 y([4])42 b(Guy)17 b(E.)f(Blelloch,)j(Phillip)g(B.)d(Gibbons,)h(and) h(Y)-7 b(ossi)16 b(Matias.)25 b(Pro)o(v)n(ably)19 b(ef)n(\002-)2 365 y(cient)i(scheduling)h(for)d(languages)j(with)e(\002ne-grained)i (parallelism.)34 b(In)19 b Fn(Pr)m(o-)2 439 y(ceedings)24 b(of)e(the)h(7th)f(Annual)g(A)n(CM)g(Symposium)h(on)f(P)-5 b(ar)o(allel)23 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b(1984.)2049 664 y([22])42 b(M.)26 b(Herlihy)i(and)f(J.)f(W)m(ing.)55 b(Axioms)27 b(for)g(concurrent)i (objects.)57 b(In)26 b Fn(Pr)m(o-)2201 738 y(ceedings)21 b(of)e(the)h(14th)f(A)n(CM)g(Symposium)h(on)e(Principles)j(of)e(Pr)m(o) o(gr)o(amming)2201 813 y(Langua)o(g)o(es)p Fo(,)g(pages)e(13\22626,)h (January)g(1987.)2049 921 y([23])42 b(Maurice)18 b(Herlihy)l(.)k(A)16 b(methodology)h(for)f(implementing)j(highly)e(concurrent)2201 996 y(data)h(structures.)24 b(In)17 b Fn(Pr)m(oceedings)h(of)f(the)g (Second)h(A)n(CM)e(SIGPLAN)g(Sympo-)2201 1070 y(sium)23 b(on)g(Principles)h(and)f(Pr)o(actice)h(of)f(P)-5 b(ar)o(allel)24 b(Pr)m(o)o(gr)o(amming)f(\(PP)-5 b(oPP\))p Fo(,)2201 1145 y(pages)18 b(197\226206,)g(Seattle,)h(W)-5 b(ashington,)18 b(March)g(1990.)2049 1253 y([24])42 b(Maurice)25 b(Herlihy)l(.)44 b(W)-5 b(ait-free)24 b(synchronization.)47 b Fn(A)n(CM)23 b(T)l(r)o(ansactions)i(on)2201 1328 y(Pr)m(o)o(gr)o(amming)17 b(Langua)o(g)o(es)i(and)d(Systems)p Fo(,)h(11\(1\):124\226149,)i 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b(Demand-based)c(cosched-)2201 3727 y(uling)23 b(of)f(parallel)j(jobs)d(on)g(multiprogrammed)i (multiprocessors.)41 b(In)22 b Fn(Pr)m(o-)2201 3802 y(ceedings)h(of)d (the)i(IPPS)d('95)i(W)-6 b(orkshop)21 b(on)g(J)n(ob)g(Sc)o(heduling)i (Str)o(ate)m(gies)g(for)2201 3877 y(P)-5 b(ar)o(allel)18 b(Pr)m(ocessing)p Fo(,)g(April)f(1995.)2049 3984 y([32])42 b(Andre)n(w)26 b(T)m(uck)o(er)g(and)g(Anoop)f(Gupta.)50 b(Process)26 b(control)h(and)e(scheduling)2201 4059 y(issues)17 b(for)g(multiprogrammed)i(shared-memory)f(multiprocessors.)26 b(In)16 b Fn(Pr)m(o-)2201 4134 y(ceedings)k(of)f(the)g(T)-5 b(welfth)20 b(A)n(CM)e(Symposium)h(on)f(Oper)o(ating)i(Systems)f(Prin-) 2201 4209 y(ciples)27 b(\(SOSP\))p Fo(,)d(pages)i(159\226166,)i (Litch\002eld)f(P)o(ark,)g(Arizona,)h(December)2201 4283 y(1989.)2049 4391 y([33])42 b(Jef)n(fre)o(y)27 b(D.)e(Ullman.)52 b(NP-complete)27 b(scheduling)h(problems.)52 b Fn(J)n(ournal)27 b(of)2201 4466 y(Computer)18 b(and)g(System)g(Sciences)p Fo(,)h(10:384\226393,)f(1975.)2049 4574 y([34])42 b(Mark)21 b(T)-5 b(.)19 b(V)-7 b(ande)n(v)o(oorde)22 b(and)e(Eric)h(S.)e (Roberts.)35 b(W)-5 b(orkCre)n(ws:)28 b(An)20 b(abstrac-)2201 4649 y(tion)d(for)e(controlling)k(parallelism.)24 b Fn(International)19 b(J)n(ournal)e(of)f(P)-5 b(ar)o(allel)16 b(Pr)m(o-)2201 4723 y(gr)o(amming)p Fo(,)h(17\(4\):347\226366,)i(August)f(1988.)1908 5679 y Fw(11)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF