An Optimal Algorithm for On-line Bipartite Matching. University of California at Berkeley & International Computer Science Institute

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1 A Optimal Algorithm for O-li Bipartit Matchig Richard M. Karp Uivrsity of Califoria at Brkly & Itratioal Computr Scic Istitut Umsh V. Vazirai Uivrsity of Califoria at Brkly Vijay V. Vazirai Corll Uivrsity 1. Itroductio Thr has b a grat dal of itrst rctly i th rlativ powr of o-li ad off-li algorithms. A o-li algorithm rcivs a squc of rqusts ad must rspod to ach rqust as soo as it is rcivd. A off-li algorithm may wait util all rqusts hav b rcivd bfor dtrmiig its rsposs. O approach to valuatig a o-li algorithm is to compar its prformac with that of th bst possibl off-li algorithm for th sam problm. Thus, giv a masur of "profit", th prformac of a o-li algorithm ca b masurd by th worst-cas ratio of its profit to that of th optimal off-li algorithm. This gral approach has b applid i a umbr of cotxts, icludig data structurs [SITa], bi packig [CoGaJo], graph colorig [GyL] ad th k-srvr problm [MaMcSI]. Hr w apply it to bipartit matchig ad show that a simpl radomizd o-li algorithm achivs th bst possibl prformac. 2. Problm Statmt Lt G (U,V,E) b a bipartit graph o 2 vrtics such that G cotais a prfct matchig. Lt B b a x matrix rprstig th structur of G (U,V,E). Th rows of B corrspod to vrtics i U (th boys) ad th colums to vrtics i V (th girls); ach dg is rprstd by a 1 i th appropriat positio. W cosidr th problm of costructig a larg matchig i G (U,V,E) o-li. Assum that th girl vrtics arriv i a prslctd ordr, ad that th dgs icidt to a vrtx ar rvald to us oly wh th vrtx arrivs. Th task is to dcid, as ach girl vrtx arrivs, which boy vrtx to match hr to, so that th siz of th matchig obtaid is maximizd. Altrativly, w ca viw th matchig as big costructd whil th matrix is rvald colum-by-colum. As a covtio w will assum that colums ar rvald i th ordr,-1, Th prformac of a radomizd algorithm A for this task is dotd by p (A) ad is dfid to b: MIN MIN E [siz of matchig achivd by A ] G ordr of girl vrtic~ whr th xpctatio is tak ovr th itral coi flips of A. Rmark: A grdy algorithm which always matchs a girl if possibl (to a arbitrarily chos boy amog th ligibl os), achivs a maximal matchig - ad thr for a matchig of siz at last ~-. O th othr had a advrsary ca limit ay dtrmiistic algorithm to a matchig of siz ~: for xampl, by lttig th first ~2 colums cotai all os ad th last ~ colums cotai os oly i thos rows which ar matchd by th dtr miistic algorithm i th first ~- stps. Prmissio to copy without f all or part of this matrial is gratd providd that th copis ar ot mad or distributd for dirct commrcial advatag, th ACM copyright otic ad th titl of th publicatio ad its dat appar, ad otic is giv that copyig is by prmissio of th Associatio for Computig Machiry. To copy othrwis, or to rpublish, rquirs a f ad/or spcific prmissio. May of th rsults i th litratur of o-li algorithms cocr th prformac of radomizd o-li algorithms agaist a adaptiv o-li advrsary [BBoKa- TaWi]. I th cotxt of th prst problm, adaptiv ACM /90/0005/0352 $

2 ss mas that th advrsary is prmittd to spcify th matrix colum-by-colum, ad to tak ito accout, i spcifyig ay giv colum, th dcisios that th radomizd algorithm has mad i rspos to th arrivals of arlir colums. Th fact that th advrsary is o-li mas that th advrsary must costruct his ow prfct matchig colum-by-colum, choosig th row to b matchd i ach colum at th sam tim as h spcifis th colum. A adaptiv o-li advrsary ca limit ay radomizd o-li algorithm to a matchig of xpctd siz /2+0 (log) by choosig th matrix, ad his ow prfct matchig, as follows: for i--0 to /2, thr is a 1 i positio j,-i if ad oly if row j dos ot li i th matchig costructd so far by th algorithm, ad also dos ot li i th matchig costructd so far by th advrsary; for his prfct matchig, th advrsary chooss a 1 i colum -i at radom. For i=/2+1 to, thr is a 1 i positio j,-i if ad oly if row j dos ot li i th matchig costructd so far by th advrsary; i this cas also, th advrsary chooss for his prfct matchig a radom 1 i colum -i. To show that o radomizd algorithm ca achiv mor tha /2+O (log) o th avrag agaist this advrsary, w argu as follows. First, ay o-grdy radomizd algorithm ca b rplacd by a grdy o that prforms at last as wll o th avrag. Scodly, for ay grdy algorithm A, lt T(A) b th st of rows that ar matchd i colums, /2+l by both A ad th advrsary. Th th xpctd cardiality of T(A) is O(log), ad th siz of th matchig producd by algorithm A dos ot xcd /2+ I T(A) I. Th Rakig Algorithm: W shall aalyz th prformac of th followig radomizd o-li matchig algorithm, which w shall rfr to as th RANKING algorithm: Iitializatio: Pick a radom prmutatio of th boy vrtics - thrby assigig to ach boy a radom priority or rakig. Matchig Phas: As ach girl arrivs, match hr to th ligibl boy (if ay) of highst rak. Rmark: At first sight it might appar mor atural to aalyz th algorithm RANDOM, which picks a boy at radom from amog th ligibl boys ach tim a girl arrivs. Howvr, RANDOM prforms arly as poorly as a dtrmiistic grdy algorithm; it achivs a matchig of xpctd siz oly +o (log) o th followig matrix: Bii=l if i=j or if ~<j< ad l~</2, ad 0 othrwis. RANDOM prforms poorly i this xampl bcaus it coctrats too much ffort o th ds uppr half of th matrix for th first -~ movs, thrby missig out o th crucial dgs i th spars lowr half of th matrix. RANKING has a implicit slfcorrctig mchaism that tds to favor thos currdy ligibl boys who hav b ligibl last oft i th past. It is this fatur of RANKING that allows it to prform wll v o graphs whr local dsity cosidratios ar misladig. 2. Aalysis of th Rakig Algorithm Th Duality Pricipl: Aftr th iitializatio phas of RANKING, thr is a ordrig o both th boy ad girl vrtics (th prslctd ordrig o girls ad th radomly chos ordrig o th boys). At this poit thr is a symmtry btw th boy vrtics ad th girl vrtics: th prformac of RANKING rmais uchagd if w itrchag th rols of th boys ad girls by lttig th boys arriv accordig to thir rakig ad pickig th highst rakd ligibl girl. Lmma 1: For ay fixd ordrigs of th boy ad girl vrtics, th matchig pickd durig th matchig phas of RANKING rmais uchagd if th rols of th boy ad girl vrtics ar itrchagd. Proof: Th proof is by iductio o th umbr of boys ad girls. Lt b b th highst rakd boy, ad g, th highst rakd girl that b has a dg to. Now, if th matchig is foud from th boys' sid, b will b matchd to g i th first stp. Also, if th matchig is foud from th girls' sid, th first tim that b is ligibl to b matchd is wh g arrivs; clarly, thy ar matchd at that tim. Th lmma follows by rmovig b ad g from th graph, ad applyig th iductio hypothsis to th rmaiig graph. Hcforth w shall rgard th colums as ordrd from 1 to with colum havig highst rak ad colum 1 lowst, ad th rows as arrivig i radom ordr. As ach row arrivs it is matchd to th highst rakig availabl ligibl colum. Viwig rows as 353

3 arrivig i radom ordr givs us a w otio of tim which is crucial to our aalysis of th algorithm. Lt 6(1).. or() b a ordrig of th rows. By tim t w ma th istat of th t th row arrival, i.. 6(t). W xt giv a tchical lmma that will b usful at svral poits. Cosidr a variat of RANKING which, as ach boy arrivs, ithr matchs him to th highst rakig ligibl girl, or ls rfuss to match him at all, v though o or mor ligibl girls may b availabl. Th rul that dtrmis whthr this algorithm rfuss may b quit arbitrary. Lmma 2: For ay fixd ordrig of th boys ad rakig of th girls, th st of girls matchd by RANKING is a suprst of th st matchd by ay rfusal algorithm. Proof: By iductio o t. By th iductio hypothsis, th st of girls ligibl to b matchd at tim t+l by th rfusal algorithm forms a suprst of thos ligibl to RANKING. Now, sic both algorithms us th sam rakig o th girls, if th rfusal algorithm chooss to match a girl who is also ligibl for RANKING, th RANKING must match hr too. Thus, i all cass, th st of girls matchd by RANKING rmais a suprst of th st matchd by th rfusal algorithm. Nxt, w prov that w ca assum w.l.o.g, that th adjaccy matrix B of th graph is uppr-triagular. Lmma 3: Th xpctd siz of th matchig producd by RANKING is miimum for som uppr-triagular matrix. Proof: Lt B b ay matrix. Rumbr th rows of B so that a prfct matchig sits o th mai diagoal (i.. Bil = 1 for l #_/<). This rumbrig has o ffct o th prformac of RANKING, sic th rows arriv i a radom ordr. Lt B' b th matrix obtaid wh all tris of B blow th mai diagoal ar rplacd by 0 (i.. B'ij = Bij if i <j ad 0 if i >j). Now RANKING o B" may b viwd as a rfusal algorithm o B. Thus, by lmma 2, th xpctd siz of matchig obtaid by rakig o B" is at most as larg as o B. [] Rmark: W cojctur that i fact th xpctd siz of matchig achivd by RANKING is miimizd by th complt uppr-triagular matrix. Howvr, w do ot kow how to prov this dirctly. W shall show a prformac guarat for RANKING that is matchd to withi low ordr trms by its prformac o th complt uppr-triagular matrix, thus provig idirctly that this is th worst cas (to withi lowr ordr trms). Provig th cojctur will yild th strogr rsult that RANKING has th bst prformac guarat. Hcforth w will assum that B is upprtriagular, with diagoal tris 1, corrspodig to th uiqu prfct matchig i th graph. Cosidr th symmtric diffrc of this prfct matchig with th maximal matchig M producd by RANKING. If I M I = /2, ach coctd compot of th symmtric diffrc is a augmtig path is of lgth 3, ad o diagoal tris ar pickd. I this cas, for ach i, ithr row i or colum i is matchd, but ot both. O th othr had, whvr may diagoal lmts ar chos or may log augmtig paths occur, thr will corrspodigly b a larg umbr of idics i such that row i ad colum i ar both matchd. Th ida bhid our proof is that, udr a radom ordrig of th rows, RANKING is likly to yild a larg umbr of such idics, ad hc a larg matchig. This last implicatio is mad prcis i th followig lmma. Lmma 4: Lt B b a x uppr triagular matrix with diagoal tris 1. Lt M b ay matchig i th associatd graph such that for ach i.ithr row i or colum i is matchd, ad lt D ={i: row i ad colum i ar both matchd i M}. Th IMI= + ID I 2 Proof: For ach i, ithr row i or colum i is matchd, i.. covrd by som dg i M. I D I= umbr of i such that both row i ad colum i ar covrd. Now, th umbr of vrtics covrd by M is + I DI ad th +ldi umbr of dgs i M is 2 Corollary: E[IMI] =/2+ 1/2E[IDI]. W will lowr-boud E [ IM I ] by lowr-boudig E[IDI] = ~; Pr [colum i ad row i both gt matchd], i=1 whr th probability is ovr radom row arrivals. For th purpos of aalyzig th prformac of RANK- ING, it is usful to cosidr a modificatio - th algorithm EARLY - which rfuss to match row i if it arrivs aftr colum i has alrady b matchd. Notic that o th complt uppr-triagular matrix algorithm EARLY is idtical to RANKING. Lmma 5: For vry ordrig of th rows, RANKING producs at last as larg a matchig as that producd by algorithm EARLY. 354

4 Proof: This follows from Lmma 2, sic EARLY is a rfusal algorithm. W will lowr-boud E[IDI] for algorithm EARLY. Algorithm EARLY has th proprty that row i gts matchd if ad oly if colum i is ot alrady matchd wh row i arrivs. I particular, if i som ordrig colum i gts matchd at tim t ad row i arrivd at tim <t th row i must also gt matchd (bcaus, i particular, colum i was availabl for row i). Idx i trs th st D i prcisly this way. Dfiitio: Lt cr b a prmutatio of th rows, ad lt c~ ~) b th squc obtaid by dltig i from its origial positio i ff ad movig it to th last positio. If EARLY dos ot match colum i udr a, th dfi W(c,i) = 0. Othrwis, dfi W(~,/) to b th tim at which colum i gts matchd udr th prmutatio ff i); if colum i rmais umatchd udr c~ i) th dfi W (or,i) =. Now, dfi w[ = Pr [W(~,i) = t] for O~_t<. whr ff is a radom prmutatio of th rows. Clarly, Pr [colum i gts matchd] = Z w~. Th xt lmma shows what fractio of this probability corrspods to th favorabl vt that colum i ad row i both gt matchd. Lmma 6: Lt W(c,i) = t ad t<. Obtai prmutatio o" from c~ i) by movig row i ito th jth positio. Th, udr o", EARLY will match row i as wll as colum i by tim t+l, ifj<_t, ad will ot match row i at all ifj>t. Proof." Ifj>t th colum i will gt matchd udr o" at tim t, bfor row i arrivs, so row i will ot gt matchd. If j<t th th i th colum is ligibl wh row i arrivs; thrfor EARLY matchs row i. Ruig EARLY o ~(z) for t stps ca b rgardd as a rfusal algorithm o o J ru for t+l stps. So by lmma 2, th colums matchd udr o by tim t+l form a supr-st of th colums matchd udr a i) by tim t; hc colum i gts matchd udr o" by tim t+l. Lmma 7: Pr[row i ad colum i both gt matchd] = --' w;.!l Proof: Firstly, otic that if W~r,/) =, th row i must hav arrivd at or bfor th ~ wh colum i got matchd i a, ad hc row~'must also hav gott matchd. Cosidr ay tim t, l<t<, ad cosidr th ordrigs such that W(c~,/) = t. Say that two such ordrigs ff ad rc ar quivalt if c~ I) = ~ i). Clarly, ach quivalc class has ordrigs, ad row i falls i o of th first t positios i t of ths. Th proof follows by Lmma 6. [] - ~:t Ltwt= Zlw ~., ByLmma6, E[IDI]= --W,. W will ow lowr-boud E[ID I] by lowr-boudig th right had sid. W first prst a asy boud stablishig that th xpctd siz of th matchig is at last (2-~r-2). Lmma 8: If colum i gts matchd at tim t udr c~ th udr O "(i) colum i ithr rmais umatchd, or gts matchd at som tim >t-1. Proof: Th algorithm udr a (i) for th first t-1 stps ca b rgardd as a rfusal algorithm for our algorithm ru o c~ for t stps. Now th lmma follows by applyig lmma 2. Dfiitio: Lt mt = Pr [som colum is matchd at tim t ]. Corollary: E ws < E ms. s~t s~t+l Lmma 9: EARLY producs a matchig of siz at last (2--/--2) o a >< uppr-triagular matrix. Proof: Lt c~ b th siz of matchig producd. Th, by Lmmas 3 ad 6, I ~ Iwt. ff. > ~ + -~,=1 Sic mt < 1 ad ~: wt = c~, w s by th corollary to lmma 7 that Z twt is miimizd by sttig ml=m2 = -'' =ma= 1~, Wl=ml+m 2 ad wt =mr+l, t>l. Substitutig th rsultig boud ito th abov iquality yilds ~ 2-~--2. [] I Lmma 9, w hav mad th pssimistic assumptio that mt = 1 for l<t<_ct, which would ma that th first x rows to arriv all gt matchd. This is, of cours, ot th cas, sic, v arly i th procss, a row may arriv aftr its colum is alrady matchd. Thus th mt 's, ad hc also th wt 's, ar sprad out i tim. Lmma 10 maks this obsrvatio mor prcis. Lmma 10: For all t, mt= 1 - s~<'t " Proof: Lt m[ =Pr[row i occurs at tim t ad gts 355

5 matchd]. Th clarly mt = Z m~. Now, Pr [row i occurs at i--1 tim t ad dos ot gt matchd] = 1 II E w,/. i.. pick a prmutatio o such that W(c,i) <t, ad mov row i ito th t ~ plac. Thrfor, m[ = 1 1 y. w~. i Th lmma follows by summig ovr i. [] II tl s <t Lt om b th xpctd siz of matchig producd pl by EARLY. W d to lowr-boud Y. twt subjct to: (i). ~ wt=a (ii). mt= 1-1 I; ws, ad s<t (iii). Zws< Z ms. s~ s~t+l Th solutio is much simplr if coditio (iii) is rplacd by coditio (iii)' blow: (iii)' Y. ws -< 2; ms s.~t sst Also, w will drop coditio (ii) for t= (this dos ot affct th validity of our boud). Lmma 11 stablishs that rplacig (iii) by coditio (iii)' dos ot chag th dsird lowr-boud by much. Lmma 12 assrts that, subjct to (i), (ii) ad (iii)', ~ twt is miimizd by pick ig th wi's grdily, i.. by makig ach w i,i=1,2, i tur as larg as possibl. Lmma 11: Lt ff = (w 1,w2, w, ) b ay solutio to coditios (i), (ii), ad (iii). Th thr is a solutio x = (xl,x2, "" x~ ) to coditios (i), (ii) ad (iii)' such that th L 1 orm of (ff - ~) is at most 2. s<t Proof: By coditios (ii) ad (iii) w hav (iv). Y. ws < t+l- 1 y. (t+l-s)ws. a~ a~t ~" is obtaid from ~ by movig o uit from th lowst k possibl idics to w, Pick k such that ~ wi --- I ad k+l i=i 5". wi > 1. St xl = 0 for 1~ < k, i=1 k xk+l = wk+l -- (1-- ie1 wi ) ad x, = w~+l. Th rmaiig idics of ar th sam as thos of ft. Clarly if satisfis (iv), th ~ satisfis coditio (v) statd blow, ad th L 1 orm of (ff - x--) is at most 2. Lmma 12: Subjct to coditios (i), (ii) ad (iii)' ~; twt is miimizd by pickig wi "s grdily. Proof: By coditios (ii) ad (iii)' w hav: (v) y. w, _< t - ± z (t-s) w, s.~t I'1 s <t Suppos for cotradictio that th wi "s that miimiz Ft ~: twt ar ot pickd grdily accordig to coditios (i) ad (v). Lt t b th last tim such that wt is ot as larg as possibl. Lt th dficicy i wt b. Icras wt by, dcras wt+ 1 by (l+l/), ad icras w, by /. Th w wi's satisfy (i) ad (v), ad hav a smallr Y. twt. Cotradictio. [] Rmark: Th grdy solutio rsultig from coditio (v) is w, = (1-1 ),-1 Thorm 1: Th prformac of algorithm EARLY is (1-1)+o () Proof: By Lmma I0 ad I 1, it is sufficit to pick wi "s grdily subjct to coditios (i), (ii) ad (iii)'. This yilds w, =(1-1) t-i,for,2... T T whr T is such that E wt = o~. Substitutig for wt ad solvig for T yilds T< - I(l-c0. Lt (1-1) = 0. Th, 0 r = 1-a. Now, T T Z t w, = Y~ to'-1= (1--(0r)-- TOT (1--0)) > (1-0) 2 2 (Ot+(1-01(1 -a)) Substitutig this ito our lowr-boud of /2 + E [ID I] o th siz of th matchig yilds: 1 T ~. >- "-~ + -~ t~=l twt > -~ + ~ (~ + (1-~) I (1-~)) This givs ( t-1) > (1-o01(/- 0 Thus c~2 (1-1 ). [] Rmark: A simpl cosquc of our proof is that if RANKING is applid to a matrix B for which th siz of th maximum matchig is m <, th th xpctd siz of th matchig producd by RANKING is at last (l-lira + o(m). 356

6 3. Boudig th Prformac of Ay O-Li Algorithm I this sctio w will show that RANKING is optimal, up to lowr ordr trms. Thorm 2: Th prformac of ay o-li bipartit matchig algorithm is < (1- ) + o (). Lt T b th x complt uppr-triagular matrix. As bfor, w assum that th colums of T arriv i th ordr, By th k *h colum arrival w ma th arrival of colum umbr -k+l. Cosidr th algorithm RANDOM, which matchs ach colum to a radomly chos ligibl row. Dfiitio: Lt T b th complt uppr-triagular matrix. With vry prmutatio o (1, -.. } associat a problm istac (T,), whr th adjaccy matrix is obtaid by prmutig th rows of T udr, ad th colums arriv i th ordr,-i, 1. Lt P dot th uiform probability distributio ovr ths! istacs. Lmma 13: Lt A b a dtrmiistic o-li algorithm that is 'grdy' i th ss that it vr lavs a colum umatchd if thr is a ligibl row. Th, th xpctd siz of matchig producd by A wh giv a istac (T,~) from P is th sam as th xpctd siz of matchig producd by RANDOM o T. Proof: Th lmma follows from th two claims listd blow, which may b provd by a straightforward iductio o tim: 1. For algorithm A o (T,r0, as wll as for RANDOM o T, if thr ar k ligibl rows at tim t, th thy ar qually likly to b ay st of k rows from amog th first -t+l rows of T. 2. For ach k, th probability that thr ar k ligibl rows at tim t is th sam for RANDOM ru o T as it is for A ru o (T,). [] Lmma 14: Th prformac of ay o-li matchig algorithm is uppr boudd by th xpctd siz of matchig producd by th algorithm RANDOM o th complt uppr-triagular matrix. Proof: Lt E [R (T,~)] dot th xpctd siz of matchig producd by th giv radomizd o-li algorithm, ad lt E[A(P)] dot th xpctd siz of matchig producd by a dtrmiistic algorithm A wh giv a iput from distributio P. By Yao's lmma [Ya], mi{e [R (T,r0]} <_max{e [A(P)]}. A whr th maximum is ovr all dtrmiistic algorithms. W.l.o.g. th bst dtrmiistic algorithm is grdy (by simulatig A, ad matchig th currt colum to th row matchd by A, if th row is availabl, ad to a arbitrary ligibl row othrwis). Th proof follows from Lmma 13. [] Lmma 16: Th xpctd siz of matchig producd by algorithm RANDOM o T is (1-1) + o(). Proof: Th proof rsts o th followig crucial obsrvatio mad i Lmma 13: giv that thr ar l rows still ligibl at th k *h arrival colum, thy ar qually likly to b ay st of I rows from amog th first -k+l rows of T. Lt x (t) ad y (t) b radom variabls rprstig th umbr of colums rmaiig ad th umbr of rows still ligibl at tim t. Lt Ax =x(t+l)-x(t) ad Ay =y(t+l)-y(t). Th Ax =-1 ad Ay is -2 if th diagoal try i th t+l a colum was ligibl but was ot matchd, ad -1 othrwis. Usig th fact that th st of ligibl rows is radomly chos from amog th first -t: E[Ay] =-1- y(t). y(t)--1 =-1- y(t)-i x(t) y(t) x(t) Thrfor E[Ay] = 1+ y(t)-i E[Ax] x(t) " Kurtz's thorm [Ku] says that with probability tdig to 1 as tds to ifiity, th solutios of th abov stochastic diffrc quatio ar closly approximatd by th solutio of th diffrtial quatio: dy = 1+Y-1 dx x Solvig this diffrtial quatio with th iitial coditio x--y =, w gt y = 1 + x (-1 - I x) So, wh oly o row is ligibl, th umbr of colums rmaiig is + o (). Thrfor, th xpctd siz of, i matchig producd is (1--~ ) + o (). Rmark: 1) Thr is a itrstig altrativ dscriptio of th bhavior of algorithm RANDOM o T. I this dscriptio, th algorithm bgis by spcifyig a radom prmutatio r=(cs(1), s(2)... cs()) of {1, 2... }. Th, as ach colum -i arrivs, RANDOM matchs that colum with row x, whr x is th first lmt of cs which has ot prviously b matchd ad is lss tha or 357

7 qual to -i. It is asy to s that this is a faithful dscriptio of RANDOM, ad as a cosquc, th followig two radom variabls hav th sam distributio: (i) th siz of th matchig producd by RANDOM o T; (ii) th lgth of th logst subsquc of a radom prmutatio such that, for all k, th k th lmt of th subsquc is gratr tha or qual to k. Thus, as a byproduct of Lmma 16 w obtai th itrstig combiatorial rsult that th xpctatio of this lattr radom, i variabl is (1---I)+o (). 2). It is asy to show that th xpctd siz of matchig producd by RANDOM ad RANKING is th sam o T. So, provig th cojctur that T is th worst matrix for RANKING togthr with Lmmas 13 ad 14 will show that RANKING is th bst possibl o-li bipartit matchig algorithm. [Ku] T. G. Kurtz, "Solutios of Ordiary Diffrtial Equatios as Limits of Pur Jump Markov Procsss', Joural of Applid Probability, vol. 7, 1970, pp [MaMcSI] M. Maass, L.A. McGoch, D. Slator, "Comptitiv Algorithms for Oli Problms', STOC 1988, pp [S1,Ta] D. Slator, R.E. Tarja, "Amortizd Efficicy of List Updat ad Pagig Ruls', Comm. ACM, vol. 28, 1985, pp [Ya] A.C. Yao, "Probabilistic Computatios: Towards a Uifid Masur of Complxity', FOCS 1977, pp Op Qustios: 1. Is th complt uppr-triagular matrix th worst-cas iput for RANKING? 2. Is RANKING a optimal o-li matchig algorithm i th o-bipartit cas? Ackowldgmts: W would lik to ackowldg hlpful discussios with Rajv Motwai. Rfrcs: [BBoKaTaWi] S. B-David, A. Borodi, R. Karp, G. Tardos, A. Wigdrso, "O th Powr of Radomizatio i O-Li Algorithms', STOC [CoGaJo] E. G. Coffma, M. R. Gary, D. S. Johso, 'Dyamic Bi Packig', SIAM J. comput., vol 12, 1983, pp [Gy,L] A. Gyarfas, J. Lhl, 'Oli ad First Fit Colorigs of Graphs', J. Graph thory, Vol. 12, No. 2, pp ,