TIGHT COMPLEXITY BOUNDS FOR PARALLEL COMPARISON SORTING. Yossi Azar Department of Computer Science School of Mathematical Sciences Tel Aviv University

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1 TIGHT COMPLEXITY BOUNDS FOR PARALLEL COMPARISON SORTING Noga Alo Dpartmt of Mathmatics Tl Aviv Uivrsity ad Bll Commuicatios Rsarch Yossi Azar Dpartmt of Computr Scic School of Mathmatical Scics Tl Aviv Uivrsity Uzi Vishki Dpartmt of Computr Scic Courat Istitut of Mathmatical Scics Nw York Uivrsity ad Tl Aviv Uivrsity ABSTRACT Th tim complxity of sortig lmts usig p ~ procssors o Valiat's paralll compariso tr modl is cosidrd. Th followig rsults ar obtaid. 1. W show that this tim complxity is (Iog/log(1 +p/». This complmts th AKS sortig twork i sttlig th widr problm of compariso sort of lmts by p procssors, whr th problm for p ~ was rsolvd. To prov th lowr boud, w show that to achiv tim k ~ log, w d o (k l + l/k ) comparisos. Haggkvist ad Hll provd a similar rsult oly for fixd k. 2. For vry fixd tim k, w show that: (a) O( l +l/ k 10g l/k ) comparisos ar rquird, (0 ( 1+11k log) ar kow to b sufficit i this cas), ad (b) thr xists a radomizd algorithm for compariso sort i tim k with a xpctd umbr of O( l +l/ k ) comparisos. This implis that for vry fixd k, ay dtrmiistic compariso sort algorithm must b asymptotically wors tha this radomizd algorithm. Th lowr boud improvs o Haggkvist Hll's lowr boud. 3. W show that "approximat sortig" i tim 1 rquirs asymptotically mor tha log procssors. This sttls a problm raisd by M. Rabi. I. INTRODUCTION Appartly, thr is o problm i Computr Scic which rcivd mor atttio tha sortig. K-73], for istac, foud that xistig computrs dvot approximatly a quartr of thir tim to sortig. Th advt of paralll computrs stimulatd itsiv rsarch of sortig with rspct to various modls. of paralll computatio. Extsiv lists of rfrcs which rcordd this activity ar giv i Ak-85], BH-86] ad Th-83]. Most of th fastst srial ad paralll sortig algorithms ar basd o biary comparisos. I ths algorithms th umbr of comparisos is typically th primary masur of tim complxity. Ay lowr boud o th umbr of comparisos rquird for a problm, clarly implis a tim lowr boud for such algorithms. I th prst papr, w rstrict our atttio to a paralll compariso modl, itroducd by Valiat Va 75], whr oly comparisos ar coutd. I masurig tim complxity withi this modl, w do ot cout stps i which commuicatio amog th procssors, movmt of data ad mmory addrssig ar prformd. W also avoid coutig stps i which cosqucs ar dducd from comparisos that wr prformd. Not that our lowr bouds apply to all algorithms, basd o comparisos, i ay paralll accss machi (PRAM) icludig PRAMs which allow simultaous accss to th sam commo mmory locatio for rad ad writ purposs. S BHo-82] for a discussio o hirarchy of modls that implis this. I a srial dcisio tr modl, w wish to miimiz th umbr of comparisos. Th goal of a algorithm i a paralll compariso modl is to miimiz th umbr of compariso rouds as wll as th total umbr of comparisos prformd. Lt k stad for th umbr of compariso rouds (tim) of a algorithm i th paralll compariso modl. Lt c (k, ) dot th miimum total umbr of comparisos rquird to sort ay lmts i k rouds (ovr all possibl algorithms). Th kow 0 ( log) comparisos lowr boud for sortig i a srial dcisio tr modl implis that, for ay k, c(k, ) = O(log). This lowr boud ca b matchd by uppr bouds as follows: For k = c log, th sortig twork of AKS-83] implis c(k, ) = O(log), whr c > 0 is a costat which is implid by th twork. For' k > clog, th rsult c (k, ) = 0 ( log) also holds. To s this, simply simulat th AKS twork by slowig it dow to work i k rouds. For k = 1, c(l, ) = lh( 2 -). This is sic ay sortig algorithm which works i o roud must prform all comparisos. Othrwis, suppos that a dispsd compariso is btw two succssiv lmts i th sortd ordr; th algorithm will clarly fail to distiguish thir ordr. O th othr had, obsrv that prformig all comparisos simultaously yilds a o /86/0000/0502$ IEEE 502 Authorizd licsd us limitd to: TEL AVIV UNIVERSITY. Dowloadd o April 11,2010 at 10:09:47 UTC from IEEE Xplor. Rstrictios apply.

2 roud algorithm i th paralll compariso modl that matchs xactly this lowr boud. So, it rmais to cosidr th situatio for 1 < k ~ c log. RESULTS W start with th mai rsult of Sctio 2: Rsult 1. c(k, ) > k 1+.1 k 1 : t - for ay k, whr is th bas of th atural logarithm. Corollaris ofrsult 1: Proof p > --- implis log thrfor k > 1 + log (1 + l!...) k = log. logo + ;) 1 ~ 1, Suppos w hav p procssors with th itrprtatio that ach procssor ca prform at most o compariso at ach roud. Obsrv that kp ~ c (k, ) or p ~ c (k, ) / k. Thrfor, Corollary 1. Ay k -roud (k ~ 1) raralll algorithm 1+ k for sortig lmts ds p > -- - procssors This yilds p = 0 ( k) for k ~ c log whr c is ay costat such that 0 < c < 1. Corollary 2. Th umbr of rouds rquird to sort lmts usig p ~ procssors is k=o log 10g(1 + l!...) 1 + l!... > k ad Hc, for p ~, Corollary 3. If p = logfj for (j > 0 th th umbr of rouds rquird to sort lmts is k =,8I~;~g)' This is a immdiat corollary of Corollary 2. A paralll algorithm is said to achiv optimal spd up 'f..... I Sq () h 1 Its ruig tim IS proportioa to,wr p Sq () is a lowr boud o th srial ruig tim, is th siz of th problm big cosidrd ad p is th umbr of procssors usd. Corollary 4. If th umbr of procssors is largr tha by a ordr of magitud th it is impossibl to dsig a optimal spd up compariso sortig algorithm. Mor formally, suppos that th umbr of procssors p is ot 0 () (i.., = 0 (p» th thr is o (compariso) sortig algorithm which rus i tim Rsult 2. Sctio 2 prsts also uppr bouds which match th lowr bouds of Rsult 1. Spcifically, w dscrib a xplicit paralll compariso algorithm that sorts Imts. I 0 log rouds usig 10g(1 + l!...) p ~ procssors. By xplicit w ma that w actually dscrib such a algorithm, ad ot mrly prov its xistc usig coutig argumts. To udrstad bttr th sigificac of ths lowr ad uppr bouds (rsults 1 ad 2) w will us o mor quivalt formulatio of th rsults. Corollary 5. Suppos w ar giv p ~ procssors to sort lmts. Th total umbr of comparisos prformd by th fastst paralll sortig algorithm is log (t~pi) IOg). Th factor log rprsts th srial lowr ad uppr bouds for sortig usig comparisos. Th othr factor rprsts th dviatio from optimal spd up. All th rmaiig rsults, apparig i Sctio 3, apply to a fixd umbr of rouds k. Our mai rsult i this part is that for vry fixd k, thr is a xplicit radomizd algorithm for sortig lmts i k rouds whos xpctd umbr of comparisos is smallr tha ay possibl dtrmiistic algorithm. This is a immdiat corollary of rsults 3 ad 4 blow. Rsult 3. W prst a radomizd algorithm whos xpctd umbr of comparisos is 0 ( 1+11k). Rsult 4. For vry dtrmiistic paralll sortig 1+.1 algorithms c (k, ) = 0 ( k (log) 11k). This improvs o Haggkvist ad Hll who showd that for vry fixd k, c(k, ) = ( 1 + 1Ik ). Notic that th oly diffrc btw our improvd lowr boud ad th prviously kow o, is a xtra factor of (log ) 11k. Nvrthlss, this is prcisly th factor that sparats th asymptotic bhavior of th bst radomizd algorithm from that of th bst dtrmiistic o. Suppos w hav to sort lmts ad lt A b a st of pairs of ths lmts. Dot p = IA I. Th st A is a approximat sortig i o roud if kowig th rlativ ordr of ach fair i A, provids th rlativ ordr of 0-0 (I) (~ out of th (~ ) pairs without ay furthr comparisos of pairs of lmts. Rsult 5. W show that hr p must b asymptotically biggr tha log (i.., log = 0 (p», thus sttlig a problm posd by Rabi (cf. BH-8S]). 503 Authorizd licsd us limitd to: TEL AVIV UNIVERSITY. Dowloadd o April 11,2010 at 10:09:47 UTC from IEEE Xplor. Rstrictios apply.

3 Usig a similar tchiqu w ca show that for vry fixd k, ( /(2 k -O. Oog)2/(2 k -O) comparisos ar dd to fid th mdia of lmts i k rouds. (A uppr boud of 0( /(2 k -O. Oog)2-2/(2 k -O) was provd by Pippgr Pi-86]') This improvs by a factor of Oog )2/{2 k -I> Haggkvist-Hll's lowr boud HH-801 ad sparats th asymptotic bhavior of th bst algorithm for slctig th maximum (which is ( f(2 k -O), s HH-80]) from that for slctig th mdia. Th dtaild proof of this last rsult will appar somwhr ls. Mor o th sigificac of th rsults. I studyig th limit of paralll algorithms it is itrstig to idtify asymptotically th miimal tim k that ca b achivd by a optimal spd up algorithm. W call this miimal tim th paralllism brak poit of th problm big cosidrd. Va-75] provd that Ooglog) is th brak poit for fidig th maximum amog lmts. BHo-82] gav a lowr boud ad Kr-83] a uppr boud to prov that Ooglog) is th brak poit for mrgig two sortd lists, whr is th lgth of ach list. Th abov two lowr bouds wr also obtaid i a paralll compariso Iodl (which is thrfor oft rfrrd to as Valiat's modl). Th prst papr abls us to add sortig to th list of problms for which th brak poit was idtifid. Spcifically, Corollary 4 complmts th sortig twork of AKS 83] i provig that Oog) is th brak poit for sortig lmts. It is itrstig to compar th "pattr" i which th brak poit occurs i ths thr problms. Th lgat lowr boud proofs of Valiat ad Borodi Hopcroft show that Ooglog) rouds ar rquird if procssors ar usd for th problms of fidig th maximum ad mrgig, rspctivly. Th algorithms of Valiat ad Kruskal ru i 0 Ooglog) rouds usig I I procssors for ach of ths problms, og og rspctivly. This isolats distictly th brak poits for ths two problms sic th asymptotic tim boud ca ot b improvd by icrasig th umbr of procssors from I I to. O th othr had, such dgrat og og isolatio dos ot occur i th sortig problm. Spcifically, Corollary 5 implis that icrasig th umbr of procssors asymptotically always yilds asymptotic dcras i th umbr of compariso rouds. Mor o xtat work. Lt us rviw works o sortig lmts i a paralll compariso modl. Rcall that Ha'ggkvist ad Hll HH-81] provd that if k, th umbr of rouds, is costat, th ( 1+lIk) procssors ar rquird to sort lmts. Usig radom graphs, Bollobas ad Thomaso BT-83] provd that thr is a algorithm that uss p =0 ( 3f210g) procssors ad sorts lmts i two rouds. Bollobas ad Hll BH-85] (s also Pi-86D showd that lmts ca b sortd i a costat umbr of rouds k usig O( 1 + 1/k log) comparisos. This almost matchs th Haggkvist-Hll lowr boud. Rmark. Covrsly, ths rsults imply that for p = 0 ( 1+E) procssors, it is impossibl to sort i lss tha k = lie rouds, but w ca sort i k = lie + I rouds. So ths uppr ad lowr bouds ar at most o roud apart wh k is costat. Howvr, a closr look at this lowr boud of Haggkvist ad Hll rvals th followig. Thy actually provd that if k, th umbr of rouds, is a variabl, l+l/k th p > 2 k + l k - 2k procssors ar rquird to sort lmts. For costat k, Rsult 4 provids a asymptotically bttr boud. Nxt, w compar Haggkvist-Hll's rsult with Corollary J. Obsrv, that thir proof implis that p = ( I +11k) oly wh k is costat ad thrfor for o-costat k Corollary 1 is strogr. Morovr, thir rsult bcoms trivial for k ~.Jlog. This is sic for this rag thir rsult implis a asymptotic boud which is 0 () for th umbr of procssors p as ca b radily vrifid. O th othr had, Corollary 1 stats that p > l + 1 / k l - for vry k. As was idicatd abov, this implis that p = ( I +11k), for ay k < clog, whr 0 < c < 1 is a costat. W ot a fw additioal paprs whos titls ar rlatd to th titl of th prst papr. L-84] proposd a adaptatio of AKS twork to boudd dgr -od tworks. MW-85] gav a.jlog lowr boud for paralll sortig by procssors i som variat of PRAM (s also B-86] for a strogr rsult). Thir modl is ot comparabl to th paralll compariso modl cosidrd hr. Th trivial log lowr boud for paralll sortig by procssors i th paralll compariso modl dos ot allow o compariso algorithms lik buckt sort. O th othr had, rakig a lmt amog othr lmts ca b do i o roud of comparisos usig procssors i th paralll compariso modl, whil thir PRAM sms to rquir o costat tim usig procssors. Rsults 3 ad 4 sparat dtrmiistic ad radomizd complxity for sortig i a fixd ulbr of rouds. A rsult of a similar flavor for th problm of slctig th l-th out of lmts is kow. Spcifically, Rischuk R-81] gav a radomizd compariso paralll algorithm for slctio whos xpctd ruig tim is boudd by a costat, usig procssors. Togthr with th lowr boud of Va-75] for fidig th maximum amog lmts, w coclud that thr xists a radomizd algorithm for slctio that prforms bttr tha ay of its dtrmiistic coutrparts. 2. TIGHT LOWER AND UPPER BOUNDS FOR NOT NECESSARILY CONSTANT NUMBER OF ROUNDS 504 Authorizd licsd us limitd to: TEL AVIV UNIVERSITY. Dowloadd o April 11,2010 at 10:09:47 UTC from IEEE Xplor. Rstrictios apply.

4 2.1 Th paralll computatio modl Lt V b a st of lmts tak from a totally ordrd domai. Th paralll compariso modl of computatio allows algorithms that work as follows. Th algorithm cosists of tim stps calld rouds. I ach roud biary comparisos ar prformd simultaously. Th iput for ach compariso ar two lmts of V. Th output of ach compariso is o of th followig two: < or >. Not that w do ot allow quality btw two lmts of V. This ca b do without loss of grality, sic w dfi th ordr bt\\' two qual iput lmts to b th ordr of thir idics. Each itm may tak part i svral comparisos durig th sam roud. Rmark. Our discussio uss th followig corrspodc btw ach roud ad a graph. Th lmts ar th vrtics. Each compariso to b prformd is a udirctd dg which cocts its iput lmts. Each computatio rsults i oritig this dg from th largst lmt to th smallst. Thus i ach roud w gt a acyclic oritatio of th corrspodig graph, ad th trasitiv closur of th uio of th r oritd graphs obtaid util roud r rprsts th st of all pairs of lmts whos rlativ ordr is kow at th d of roud r. Suppos w prformd r rouds whr r > 0 is som itgr. Cosidr ay fuctio of V that ca b computd usig th comparisos prformd i ths r rouds without ay furthr comparisos of lmts i V. Our modl dfis such a fuctio to b computabl followig roud r. Not that this dfiitio supprsss all computatioal stps that do ot ivolv comparisos of lmts i V. Which comparisos to prform at roud r + 1 ad th iput for ach such compariso should b fuctios which ar computabl followig roud r. War itrstd i sortig th lmts i V from th smallst to th largst i k rouds, whr th itgr k ca b ithr costat or a fuctio of. Rcall that c (k, ) dots th miimum total umbr of comparisos rquird to sort ay lmts i k rouds (ovr all possibl algorithms). 2.2 Th lowr boud Lt us rstat th mai thorm of this sctio. Th Lowr Boud Thorm: k c(k, ) > k(-- - ) for ay k, ~ 1, whr is th bas of th atural logarithm. Proof By iductio o k ad. Th bas of th iductio. For k = 1 ad vry 2-2 ~ 1, clarly c(l, ) = --- > - -. For 2 = 1, 2 ad vry k ~ 1 c (k, 1) ~ 0 > k (l - 1), / c (k, k 2) > 0 > k (- - 2) ~ k(-- - 2). Th iductiv assumptio: Giv k,, if k' ~ k ad ' <, or k' < k ad ' ~, th 1+1..,, k c(k', ') > k'(-- - '). Tak ay k-roud algorithm for sortig a st V of lmts. Th first roud of th algorithm cosists of som st E of comparisos. Rcall that w look at thm as dgs i th graph G = (V, E). A idpdt st i G is a subst of vrtics from V such that o two vrtics ar adjact by a dg i E. A idpdt st is maximal if it is ot a propr subst of aothr idpdt st. Cosidr th graph of th first roud of comparisos. Lt S b a maximal idpdt st i this graph ~d dot x = IS I. Each of th - x lmts of S must shar a dg with a lmt of S, or othrwis S is ot maximal. For our lowr boud proof, w rstrict our atttio to liar ordrs o V, i 'Y.hich ach lmt of S is gratr tha ach lmt of S. For ay of ths ordrs it is impossibl to obtai ay iformatio rgardig th~rlatio btw two lmts of S or two lmts of S usig com.parisos btw a lmt of S ad a lmt of S. Thrfor, asid from ths - x comparisos, thr must b at last c (k - 1, x) comparisos to sort S ad at last c (k, - x) comparisos to sort S. This implis th followig rcursio, c (k, ) ~ c (k, - x) + - x + c (k - 1, x), by th iductiv assumptio ( - x) 1+Ilk ] > k - ( - x) + ( - x) xl+l/(k-i) ] + (k - 1) - x. By opig parthss ad prmutig trms w gt!. ( - x)i+i/k +!-..=..l x 1+1/(k-I) + - k =:I+l/k l-~ ] 1+I/k 1 ] x /(k-i) 1 ] k 1+1/k + k 1/k - k. Rcall th Gomtric-Arithmtic Ma Iquality: aa +{3b ~ a cx b{3, whr a+{3= 1 a,{3,a, b ~ O. By takig 1 1 x 1+1/ (k - I) a = 1 - k' {3 = k' a = 1+IIk ' b = 1/k ' w gt that th last xprssio is 505 Authorizd licsd us limitd to: TEL AVIV UNIVERSITY. Dowloadd o April 11,2010 at 10:09:47 UTC from IEEE Xplor. Rstrictios apply.

5 '" ~ l + l / k ~ Rcall that th icrasig squc 1 + ~ )k covrgs to ad thrfor, Ilk > 1 + ~. This implis ~ ; I+I/k 1_ ~ ]I+llk + ~ 1 + ~]] - k. Rcall Broulli's Iquality: (1 - a)t ~ 1 - at for t ~ 1, a ~ 1. This implis, ~ ; I+ llk 1 - ~ 1 + ~] + ~ 1 + ~]] - k k l + llk ) = -; I+ llk - k = k This complts th proof of th Lowr Boud Thorm. 2.3 Th uppr boud Thorm. ] 1+l/k 11k ] 1 - ~ + l:llk2 ;llk2 - k. Giv lmts from a totally ordrd domai, thr is a xplicit algorithm i a paralll compariso modl for sortig ths lmts i o IOg(;o~~/) ] rouds usig p ~ procssors. Proof First rcall th AKS compariso twork. It sorts lmts i 0 Oog) rouds usig p = 12 procssors (i.., l2 comparisos i ach roud). W giv a algorithm i a paralll compariso modl. Each roud of th w algorithm is calld suprroud. Th algorithm is drivd from AKS twork by simply shrikig 0 = O.510g(1 +pl) rouds of this twork ito o suprroud. Th costructio of th algorithm is basd o th followig ida. W aim that th followig Assrtio will hold. Assrtio. Aftr suprroud r, th followig thigs ar availabl: (1) Th pair of iput lmts for ach compariso prformd i th first or rouds of AKS twork. (2) Th rsult of ach such compariso. This Assrtio implis that aftr 0 (I (IOg /» og 1+p suprrouds th rsults of all comparisos of AKS twork ar availabl ad th sortig is compltd (sic it is computabl). W show how to satisfy th Assrtio for ay suprroud r. For r = 0 th Assrtio triviaiiy holds. W show how to satisfy th Assrtio for suprroud r assumig that it is satisfid for ay suprroud < r. Th fact that w rlat to a compariso twork implis that ach lmt, which is compard i roud o(r - 1) + i, whr 1 ~ i ~ 0, is o of at most 2 i - l lmts which ar outputs of comparisos of th first o(r - 1) rouds (or iput lmts). By th iductiv assumptio, ach of ths outputs is availabl followig suprroud r - 1. Thrfor, ach ~om~ariso i roud o(r - 1) + i, is actually o of (2'-1) possibl pairs. All w do is prform all ths possibl comparisos simultaously (for 1 ~ i ~ 0). Ths comparisos clarly iclud th actual comparisos prformd by AKS twork i ths rouds. But 0 = 0.510g (1 +..), ad thrfor, this umbr of log(l+.) comparisos is ot mor tha - 2 ~- 2 ~ ;] = ; p ~ p. So, thr ar ough procssors to prform all ths comparisos. 3. SORTING IN A FIXED NUMBER OF ROUNDS 3.1 Radomizd algorithms It rmais to show that this costructio also yilds th pairs of iput lmts to ach compariso which was actually prformd i ach of ths rouds. For this w show by simpl iductio that th actual pair of ach compariso, as wll as its rsult ar availabl, for all rouds ~ 0(, -:- 1) + i, 0 ~ i ~ o. For i = 0, this follows from th iductiv assumptio of th Assrtio for, - 1. Suppos that for all rouds < 0(' - 1) + i, th actual pairs compard, as wll as thir rsult ar availabl. Each lmt participatig i roud 0(, - 1) + i is a outcom of th actual comparisos of prcdig rouds ad thir rsults. Thy ar kow by th iductiv assumptio. Thrfor, th iput pair for ach such compariso is kow. W alrady argud that th rsult of ach such compariso was foud by our algorithm. This complts th proof of th iductio for i. Takig i = 0, w complt th iductiv proof of th Assrtio. Th umbr of comparisos that th algorithm has to prform i ach suprroud is: o 4 0!!:.. ~ (2i- l )2 <!!:.. _ <!! i-i Thorm 3.1 For ay k ~ 1, thr is a xplicit radomizd algorithm for sortig lmts i k rouds, whos xpctd umbr of comparisos E (, k) is at most c(k). l +l/ k, whr c(k) is som costat dpdig o k oly. Proof By iductio o k. For k = 1 th rsult is trivial. Assumig it holds for k - 1 ad vry, w prov it for k. Put t = r 11k 1. I th first roud our algorithm chooss radomly a st T of t - 1 lmts from th st V of lmts w hav to sort ad compars a~h of thm to vry v E V (icludig th othr lmts of T). Aftr this roud, th st V - T will b brok ito t blocks AI' A 2,..., At, such that for ach i < j ad Qi E Ai' Qj E A j Qi is smallr tha Qj' 506 Authorizd licsd us limitd to: TEL AVIV UNIVERSITY. Dowloadd o April 11,2010 at 10:09:47 UTC from IEEE Xplor. Rstrictios apply.

6 W ow apply, rcursivly, our radomizd algorithm for sortig i k - 1 rouds, to ach A;, i paralll. W claim that Bollobas ad Thomaso BT-83] improvd it ad showd that E (, k) ~ (1 - -l+2 1). + 1 ~ ;-1. ]1-2 1+_1 ~ c (k - 1). (i - 1) k-l - 1 Idd, thr ar (/ ::. 1) ways to choos th st T, ad for ach fixd j, 1 ~ j ~ 1, th umbr of ths choics with IA j I = i-i (for 1 ~ i ~ ) is prcisly th umbr of ways to writ ~ (j-i) -l <; ~j.!!::l ]1-2 1+_1 ~. (i - 1) k-l - 1 l-l 1+_ 1 _ ~ (1 + 0 (1» k. -(j-i). j k-l.. 1+_1_ Sic th sum ~ j k-l. - j covrgs, this implis l+l that E (, k) ~ c (k). k for a proprly dfid costat c (k). This complts th proof. Rmark 3.2 j~1 W ca show that Thorm 3.1 is sharp for k = 2 i th ss that for vry radomizd algorithm for sortig lmts i two rouds thr is a iput for which th xpctd umbr of comparisos of th algorithm is ( 3/2). W do ot kow if th thorm is sharp for largr valus of k. 3.2 Lowr ad uppr dtrmiistic bouds E:v th first otrivial cas, that of sortig lmts i two rouds, rcivd cosidrabl atttio. Haggkvist ad Hll HH-81] showd that l.. 3 / 2 - l.. ~ c (2 ) = 0 ( 5 / 3 10g). 8 2 ~, i + 2 as a ordrd sum of 1-1 o-gativ itgrs (rprstig th cardialitis of th blocks As bsids A j ), which is (7 -=-1) To stimat th right had sid of th last iquality w brak th sum ito coscutiv blocks, ach of siz l-l -- ( - 1) /(1-2) ::::: k, ad otic that for ay cl <.JjJj, if > (Cl). Explicit algorithms for sortig i two rouds with o ( 2 ) comparisos ar giv i Pi-85], AI-85] ad Pi-861 Hr w slightly improv both bouds ad show Thorm 3.3 ( 3 / 2.Jlog) ~ c (2, ) ~ 0 3 / 2 log ]..Jloglog W also prov: Thorm 3.4 For vry fixd k ~ 2 A uppr boud of O( l + l / k log) for c(k, ) is kow, as idicatd i th itroductio. Our mthods also abl us to improv th kow bouds for approximat sortig i o roud. A algorithm that approximatly sorts lmts i o roud with p comparisos is a st of p pairs of lmts (a;, b;)f-l from th st V of lmts w hav to sort, such that for ach possibl st of aswrs for th p qustios "is a; < b;" th rlativ ordr of all but 0 ( 2 ) of th pairs of lmts of V will b kow. Lt a () dot th miimum p such that a approximat sortig algorithm i o roud with p comparisos xists. Bollobas ad Rosfld studid ths algorithms i BR-81] (also s BH-85]) ad thir rsults imply that for vry fixd E > 0 a () = 0 ( 1+E). M. Rabi (cf. BH-85]) askd whthr a () = 0 ( log ). Th xt propositio shows that this is fals. Propositio 3.5 c (k, ) = ( / k Oog) Ilk). (i) lim a ()/ log ~ 00. Mor prcisly; for -oo vry E > 0, ay two rouds sortig algorithm that uss at most E 2 comparisos i th scod roud must us (1- log ) comparisos i th first roud. E (io For ay fuctio w() ~ 00, a () ~. log. loglog. w(). Th uppr bouds i Thorm 3.3 ad i Propositio 3.5 ar provd by combiig crtai probabilistic argumts with som of th idas of BT-83] ad Pi 861 Th dtails will appar somwhr ls. Hr w 507 Authorizd licsd us limitd to: TEL AVIV UNIVERSITY. Dowloadd o April 11,2010 at 10:09:47 UTC from IEEE Xplor. Rstrictios apply.

7 prst th proofs of th lowr bouds i Thorms 3.3, 3.4 ad Propositio 3.5. A crucial lmma hr is th followig rsult. Lmma 3.6 Evry graph with vrtics ad at most d dgs, cotais a iducd subgraph with I/4J vrtics ad maximum dgr at most 4d which has a 4d propr vrtx colorig with color classs VI V 2,..., V 4d such that for ach 1 ~ i < i + j ~ 4d a~d ach v E Vj, v has at most 2 j + I ighbors i V i + j Proof Lt G = (V, E) b a graph with vrtics ad at most d dgs. Sic th sum of th dgrs of all vrtics of G is at most 2d, ot mor tha half of th vrtics hav dgrs ~ 4d, ad thus G cotais a iducd subgraph K o a st U of at last /2 vrtics with maximum dgr smallr tha 4d. By a stadard rsult from xtrmal graph thory, K has a propr 4d vrtx colorig. Lt U l' U2,..., U4d b th color classs. For vry vrtx u of K, lt N (u) dot th st of all its ighbors i K. For a prmutatio 7r of 1, 2,..., 4d ad ay vrtx u of K, dfi th 7r-dgr d (7r, u) of u as follows; lt i satisfy u E U1r(;) th 4d-i d(7r, u) = ~ IN(u) U 1r (i+j) I/2 j. j-o W claim that th xpctd valu of d(7r, u) ovr all prmutatios 7r of {I,..., 4d} is at most 1. Idd, for a radom prmutatio 7r th probability that a fixd ighbor v of u will cotribut 1/2 r to d (7r, u) is at most 1/4d for all r > O. Hc ach ighbor cotributs to this xpctd valu at most.l...d ~ 1/2 r = 1/4d, ad th dsird rsult follows sic 4 r>o IN(u) I ~ 4d. Cosidr ow th sum ~ d(7r, u). Th xpctd u E U valu of this sum (ovr all 7r's) is at most IU I, by th prcdig paragraph. Hc thr is a fixd prmutatio (1 such that ~ d «(1, u) ~ IU I. It follows u E U that d «(1, u) ~ 2 for at last IU I/2 ~ /4 vrtics u of K. Lt W b a st of l/4j of ths vrtics, lt H b th iducd subgraph of G o Wad dfi Vi = U(1(;) W (1 ~ i ~ 4d). Clarly, for vry 1 ~ i < 4d ad vry v E Vi ~ IN(u) V i + j I/2 j ~ 2 j>o ad thus v has at most 2 j + I ighbors i V i + j. This complts th proof. Lmma 3.7 Evry graph G = (V, E) with vrtics ad at most. d dgs, whr d = 0 () ad d = 0 (log), has a acyclic oritatio whos trasitiv closur has at most (~) - ~ log ; ) dgs. Proof By Lmma 3.6 thr is a subst W of cardiality I/4J of V ad a propr 4d vrtx colorig of th iducd subgraph of G o W with color classs VI'...' V 4d satisfyig th coclusios of th lmma. Put V o = V - (VI U... U V 4d ) ad orit ach dg (u, v) of G with u E Vi' v E V j ad 0 ~ i < j ~ 4d from u to v. Th othr dgs of V (that joi two mmbrs of Yo) will b oritd i a arbitrary acyclic ordr. Lt T b th trasitiv closur of this oritd graph. For v E V, lt NT (v) dot th st of ighbors of v i T. Suppos v E Yj, 1 ~ i < i + j ~ 4d. W claim that th umbr of dirctd paths i our oritd G that start j at v ad d at som mmbr of U V i + r 3" is at most 2 J. r-l Idd, ach such path must b of th form v, vi I vi 2 Vi,' whr i < i l < i 2 <... < i r ~ i + j, vii EVil,..., vi, E Vi,. Thr ar 2 j possibilitis for choosig ii' i 2,., i r, ad sic ach vrtx of th path is a ighbor of th. h 2il- i + I h C p~~lous o, t r ar at most c olcs lor Vi I' 2' 2- ' 1+ 1 h C C olcs lor Vi 2, tc. Hc th total umbr of paths is smallr tha 2 3j ad thus if v E Vi th Put r = l ~ INT(v) (0 V i + r ) I < 2 3j. r-i log2 4 d ) J, ad partitio th st of blocks VI' V 2,.., V 4d ito s = r4d1 blocks WI,..., W s of r coscutiv Vi-s' ach cotaiig at most r blocks. By th prcdig paragraph, th total umbr of dgs of T i ach block Jfj is ot biggr tha IWj I 2 3r ~ IWi I.J"fi Thus thr ar at last ± I~il) _!!:.. ~)1/2 i-i 4 4d pairs of lmts that ar ot adjact i T. covxity of th fuctio g (x) = ~] s IWi I) ~ 2 log (;) ~ 2 ~s 2 =0 i-i 2 log ; =0 d ad thus T dos ot cotai at last d By th 508 Authorizd licsd us limitd to: TEL AVIV UNIVERSITY. Dowloadd o April 11,2010 at 10:09:47 UTC from IEEE Xplor. Rstrictios apply.

8 2 1/2 2 ] ] ] d log d - 4" 4d = d log d dgs. This complts th proof. W ca ow prov th lowr bouds i Thorms 3.3, 3.4 ad i Propositio 3.5. To prov th lowr boud i Thorm 3.3, cosidr ay two rouds algorithm that sorts a st V of lmts. Th first roud of th algorithm cosists of som st E of comparisos. Dfi d by IE ~ =. d. Clarly w may assum that d = 0 ( 13) ad d = O( I/3 ). By Lmma 3.7 th graph G = (V, E) has a acyclic oritatio whos trasitiv closur misss ; log ] dgs. If th aswrs i th first roud corrspod to this oritatio th clarly i th scod roud th algorithm has to compar all ths ; log 1 pairs. Thus, by th trivial iquality a +b ~ 2~ c(2, ) ~ d + ; log ] ~ ( 3/2 Oog) 1/2), as dd. rsults of th first roud, ad such a iformatio ca b obtaid oly from comparisos btw lmts of V;. Thus, i th xt k rouds, all th sts VI'.'.' V s hav to b sortd. By th iductio hypothsis th umbr of comparisos for this task is at last s ~ ck IV j /k Oog IV j 1)l/k ;-1 Th total umbr of comparisos is thus at last d + (1+l/k(Iog ;)I/k /d 1 / k ). O ca asily chck that this umbr is ~ ( l+l/(k+i). Oog) I/(k+I). (Idd, at last o of th two summads must b that big). This complts th iductio ad Thorm 3.4 follows. Ackowldgmts. W would lik to thak A. Borodi, M. Dubir ad M. Patrso for hlpful commts. Th proof of Propositio 3.5 part 0) is aalogous. If a two rouds sortig algorithm uss C log comparisos i th first roud, th by Lmma 3.7, it must us 2 ] comparisos i th scod roud. c Thorm 3.4 is drivd from th lowr boud of Thorm 3.3 provd abov by iductio o k, startig with k == 2. For k = 2, th rsult is just th statmt of Thorm 3.3. Suppos, by iductio, that 1+.1 c(k, ) ~ ck k Oog) 11k, whr ck > 0 is a costat, dpdig oly o k. Cosidr a algorithm for sortig a st V of lmts i k + 1 rouds. Lt E b th st of comparisos btw pairs of lmts of V mad i th first roud. As bfor, E corrspods to a st of dgs of a graph G == (V, E). Dfi d by IE I == d.. By a stadard rsult from xtrmal graph thory (that follows,.g., from th trivial part of Lmma 3.6), ay graph with m vrtics ad avrag dgr j, cotais a idpdt st of siz 0 (m /j). By a rpatd applicatio of this, w coclud that G cotais o (d) idpdt sts, ach of siz (/d). Dot ths sts by VI,..., V s (s == (d» ad dfi s Vo - V -.U Vi' Rstrict our atttio ow oly to ;-1 liar ordrs o V for which ach v; E V; is smallr tha ach Vj E Vj' for all 0 ~ i < j ~ s. Clarly, if o < i ~ s, ad u, v E Vi w do ot hav ay iformatio about th rlativ ordr of u ad v from th REFERENCES Ak-85] S. Akl, "Paralll Sortig Algorithms", Acadmic Prss, AI-85l N. Alo, Expadrs, sortig i rouds ad suprcoctrators of limitd dpth, Proc. 17th ACM Symposium o Thory of Computig (1985), AKS-83] M. Ajtai, J. Komlos ad E. Szmrdi, A O(log) sortig twork, Proc. 15th ACM Symposium o Thory of Computig (1983), 1-9. Also, M. Ajtai, J. Komlos ad E. Szmrdi, Sortig i c log paralll stps, Combiatorica 3(1983), AV-861 Y. Azar ad U. Vishki, Tight compariso bouds o th complxity of paralll sortig, SIAM J. Comput., to appar. B-86l P. Bam, Limits o th powr of cocurrt-writ paralll machis, Proc. 18th ACM Symposium o Thory of Computig (1986), BR-82l B. Bollobas ad M. Rosfld, Sortig i o roud, Isral J. Math. 38(1981) BT-83] B. Bollobas ad A. Thomaso, Paralll sortig, Discrt Applid Math. 6(1983) I ll. 509 Authorizd licsd us limitd to: TEL AVIV UNIVERSITY. Dowloadd o April 11,2010 at 10:09:47 UTC from IEEE Xplor. Rstrictios apply.

9 BH-85l B. Bollobas ad P. Hll, Sortig ad Graphs, i: Graphs ad ordrs, I. Rival d., D. Ridl (1985), BO-86l B. Bollobas, Radom Graphs, Acadmic Prss (1986), Chaptr 15 (Sortig algorithms). BHo-82l A. Borodi ad J.E. Hopcroft, Routig, mrgig ad sortig o paralll modls of computatio, Proc. 14th ACM Symposium o Thory of Computig (1982), HH-80l R. Haggkvist ad P. Hll, Graphs ad paralll compariso algorithms, Cogr. Num. 29 (1980) HH-81 l R. Haggkvist ad P. Hll, Paralll sortig with costat tim for comparisos, SIAM J. Comput. 10(1981) HH-82l R. Haggkvist ad P. Hll, Sortig ad mrgig i rouds, SIAM J. Alg. ad Disc. Math. 3(1982) K-73l D.E. Kuth, "Th Art of Computr Programmig, Vol. 3: Sortig ad Sarchig", Addiso Wsly Kr-83l C.P. Kruskal, Sarchig, mrgig ad sortig i paralll computatio, IEEE Tras. Computrs c-32(1983) L-84l F.T. Lighto, Tight bouds o th complxity of paralll sortig, Proc. 16th ACM Symposium o Thory of Computig (1984) Pi-85l N. Pippgr, Explicit costructio of highly xpadig graphs, prprit (1985). Pi-86l N. Pippgr, Sortig ad slctig i rouds, prprit (1986). Th-83l C. Thompso, Th VLSI complxity of sortig, IEEE Tras. Computrs C-32, 12(1983). R-81 l R. Rischuk, A fast probabilistic sortig algorithm, Proc. 22d IEEE Symp. o Foudatios of Computr Scic (1981), RV-83l J. Rif ad L.G. Valiat, A logarithmic tim sort for liar siz twork, Proc. 15th ACM Symposium o Thory of Computig (1983) lsv-81] Y. Shiloach ad U. Vishki, Fidig th maximum, mrgig ad sortig i a paralll modl of computatio, J. Algorithms 2,1 (1981) Va-75l L.G. Valiat, Paralllism i compariso problms, SIAM J. Compo 4(1975) Authorizd licsd us limitd to: TEL AVIV UNIVERSITY. Dowloadd o April 11,2010 at 10:09:47 UTC from IEEE Xplor. Rstrictios apply.

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