Constructing a Maximal Independent Set in Parallel. Mark Goldberg ( ), Thomas Spencer Department of Computer Science Rensselaer Polytechnic Institute

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1 Constructng a Maxma Independent Set n Parae Mark Godberg ( ), Thomas Spencer Department of Computer Scence Rensseaer Poytechnc Insttute ABSTRACT The probem of constructng n parae a maxma ndependent set of a gven graph s consdered. A new determnstc NC -agorthm mpemented n the EREW PRAM mode s presented. On graphs wth n ver- tces and m edges, t uses O ((n +m )/ogn ) processors and runs n O (og3n ) tme. Ths reduces by a factor of ogn both the runnng tme and the pro- cessor count of the prevousy fastest determnstc agorthm whch soves the probem usng a near number of processors. Key words: parae computaton, NC, graph, maxma ndependent set, determnstc. 1. Introducton The probem of constructng n parae a maxma ndependent set of a gven graph, MIS, has been nvestgated n severa recent papers. Karp and Wgderson proved n [KW] that the probem s n NC. Ther agorthm fnds a maxma ndependent set of an n - vertex graph n O (og4n ) tme and uses O (n 3/og3n ) processors. In successve papers, the authors proposed agorthms whch ether are faster, or use a smaer number of proces- sors. Luby n [L1] and Aon et a n [ABI] presented probabstc agorthms runnng n O (og2n ) tme on a EREW PRAM wth a near number of processors. Luby aso descrbed a technque for convertng probabstc agorthms nto determnstc ones; the technque preserves the runnng tme but requres an ncrease n the number of processors used to O (n 2m ), where m s the number of edges n the graph. The frst determnstc NC -agorthm on a near number of processors (EREW mode) was constructed n [GS]; ts runnng tme s O (og4n ). Recenty, Luby [L2] proposed a genera method for convertng randomzed parae agorthms nto determnstc ones, whch does not requre an ncrease n the number of processors. In the case of MIS, the method yeds a new ago- rthm runnng on a near number of processors n poyogarthmc tme; however, t runs ( ) supported n part by the Natona Scence Foundaton under grant DCR

2 - 2 - sower than than that n [GS]. A NC -agorthms for MIS mentoned above use the foowng top-eve desgn proposed n [KW]: Start wth an empty ndependent set I. Fnd an ndependent set I, add t to I, and deete I and the vertces adjacent to a vertex n I from the graph. Repeat the pre- vous step unt the graph s empty. Ca FINDSET the procedure whch constructs I. One can easy prove that an agorthm wth such a structure beongs to NC f FINDSET s desgned so that, on any n -vertex graph (n >0), t runs n poyogarthmc tme and devers an ndependent set C such that C N (C ) =Ω(n /ogs n ) for some fxed s 0; (N (C ) s the set of neghbors of C ). In ths paper, we present a new determnstc agorthm for MIS whch mproves the runnng tme of the agorthm n [GS] by a factor of ogn and smutaneousy reduces, aso by a factor of ogn, the number of processors used. The agorthm s mpemented n the EREW mode of computaton. Thus, the processor-tme product of the agorthm n [ GS] s mproved by og2n. These gans are due to the new verson of the FINDSET procedure. The new procedure fnds n O (og2n ) tme an ndependent set I such that removto be caed ony O (ogn ) tmes. ng I N (I ) reduces the sze of the graph by haf. Thus, the procedure FINDSET needs To reduce the number of processors used, we modfy the defnton of the sze of the graph so that t takes nto account the number of edges deeted. The resut of ths modfcaton s that the processor-reducton technque of Mer and Ref [MR] can be apped to reduce the number of processors necessary by a factor of ogn. Both the agorthm of [GS] and the one of ths paper use the dea of a parta coorng. A parta coorng s an assgnment of coors to some, not necessary, a of the ver- tces such that no two adjacent vertces have the same coor. Exampes of parta coorcoorng where no vertex s coored. If a coorng assgns coors to a of the vertces, ngs ncude the trva coorng where every vertex has a dfferent coor, and the empty then t s caed compete. The set of vertces wth the same coor s caed a coor cass. FINDSET constructs a sequence of parta coorngs startng wth the trva coorng. The objectve s to fnd a "bg" coor cass C (the defnton of "bg" s gven ater). If such a coor cass C s found, FINDSET deetes C N (C ) from the graph; f no coor cass s "bg", some of the coor casses are merged. A sde-effect of ths s that some vertces become uncoored. FINDSET hats when a vertces are ether deeted or uncoored. It returns the unon of a the "bg" coor casses t found. The success of such an approach depends on the defnton of a "bg" coor cass and on the effcency of mergng. When an agorthm decdes to merge, t shoud do t "fast" and so that "not-too-many" vertces are decoored. The technca means by whch ths task s accompshed n our mpementaton s the comparson of two representatons of the edge set E of a coored graph. The frst one vews E as the unon of the sets L (C ) of edges wth an endpont n the coor cass C, where C ranges over the set of a coor casses. Roughy speakng, L (C ) measures the sze of the part of the graph that woud be deeted f C s added to I. The second representaton of E parttons the edges nto casses so that the membershp of an edge s determned by the coors of both ts endwoud be decoored f mergng were performed accordng to ths partcuar cass. The ponts. Every cass of the second representaton measures the sze of the graph whch Propostons 1 and 2 estabsh that f there are no bg casses, then there s effcent

3 - 3 - mergng. We expect that the consderaton of these representatons can be hepfu n the desgn of parae agorthms for other graph probems. 2. Termnoogy The defntons of cass NC and modes of computatons can be found n [P], [V], [KR]. The graph-theoretc termnoogy used n ths paper s standard [BM]. The degree of a vertex v n a subgraph H s denoted deg H (v ). Gven a set K V (G ) of vertces, σ H (K ) = v K (1+deg H (v )) s caed the weght of K n H. If the graph s understood, the ndces n degh and σ H (K ) are omtted. We use weght as the defnton of sze when we say that FINDSET fnds an ndependent set I such that the deeton of I N (I ) reduces the sze of H by haf. Let C =[C 0,...,C r 1] be the coecton of coor casses of a parta coorng φ. Then, L denotes the set of edges whch have one endpont n C, and N denotes the set of ver- tces whch have neghbors n C. The consderaton of L s yeds the frst method of cassfyng the edges. To understand the second method, we need to defne the functons p χ(p ) = p 1 f p s odd, f p s even. rev (,q ;p )=(q )mod χ(p ). ndex (,j ;p ) = ( +j )mod χ(p ) 2 mod χ(p ) 2j mod χ(p ) f 0,j χ(p ) 1, f j =χ(p ), f =χ(p ). One can ready prove that for every q and (0,q χ(p ) 1), j =rev (,q ;p ) s the ony nteger n the nterva [0,χ(p ) 1] wth ndex (,j ;p )=q. Let φ be a vertex coorng and C =[C 0,...,C r 1] be the set of ts coor casses. For k =0,...,χ 1 et π k = {(C,C j ): ndex (,j ;r )=k } be a partton of C nto r /2 pars and poscan ready check, that for a j, π and πj do not share a common par. Thus, the co- sby one sngeton. These χ parttons of C are caed the reguar parttons of C. One ecton {π 0,..., π χ 1} presents a mnma edge coorng of the compete graph whose vertces are C 0,...,C r 1. Fx k and et π=π. For each =0,..., r /2, defne B =(C N (C )) (C N (C )) an r /2 1 = 0 k j j B = B. The weght σ(π) of π s then gven by σ(π) = σ(b ). 3. An Outne of the Agorthm d In ths secton, we present a descrpton of FINDSET and prove that every appcaton of FINDSET reduces the weght of the graph by haf.

4 - 4 - Let G be a graph wth n vertces and m edges. FINDSET starts by defnng an empty ndependent set I and a trva vertex coorng on the vertces of G. Then, t proceeds n phases. At every phase, one of the foowng actons s performed: <1> A coor cass C* s found for whch σ(n (C* )) (n +m )/ogn. A vertces of C* are added to I and a vertces from C* N (C* ) are deeted. <2> The coor casses are parttoned nto pars (C,C ), wth possby one coor cass eft over. The two coor casses of each par are merged. To make the merged coor cass ndependent, the weghts of the sets C N (C ) and C N (C ) are compared and the vertces of the set wth the smaer weght are decoored. Acton <1> s done whenever possbe; acton <2> s done ony when there s no sutabe coor cass C*. The actons are executed unt at most one coor cass s eft. If t s ndeed one coor cass, then ndependent of ts weght the coor cass s added to I. Acton <2> s done by the procedure HALF. It constructs a reguar parttonng π of the mnma weght, and merges every par of coor casses matched by π. The foowng propostons show that acton <2> does not decoor too many vertces. Proposton 1. Let C=[C 0,..., C r 1] be the set of coor casses of a coorng φ and {π,...,π } be the set of reguar parttonngs of C. Then 0 χ 1 χ 1 j =0 σ(π ) σ(n (C )). (* ) j r 1 =0 Proof. The set of parttonngs {π j } (j =0,...,χ 1) contans every par of coor casses exacty once. Therefore, for every coored vertex v, ts contrbuton to the eft part of the nequaty s equa to σ(v ) (the number of coor casses that contan vertces adjacent to v ). Obvousy, the contrbuton of v to the rght part s as bg as that. Proposton 2. Let φ be a coorng and C=[C 0,..., C r 1] be the set of ts coor casses. If for every 0, σ(n (C )) (n +m )/ogn, then there s a reguar parttonng π of C and a set D of vertces such that (1) for every par C,C of coor casses matched by π, C C D s an ndependent set; ( 2) σ(d ) n +m 2ogn r. χ(r ) Proof. If for every 0, σ(n (C )) (n +m )/ogn, then t foows from (*) that χ 1 j =0 σ(π j ) (n +m ) r, ogn whch n turn mpes the exstence of a reguar parttonng πk of C wth σ(π k ) n +m r. ogn χ(r ) Let {(C,C )} be the set of pars of π. For every =0,..., r /2 1, compute j k

5 - 5 - σ(c N (C j )) and σ(cj N (C )) and denote D the set havng the smaest vaue of σ. Then, the unon D = D satsfes condtons (1) and (2). Theorem. FINDSET executes at most O (ogn ) actons. If n and m, respectvey, are the number of vertces and edges of the graph G nduced on the set of decoored vertces, then n +m ( 2 1 +o (1))(n +m ). Proof. If acton <1> s apped, then the tota number of edges and vertces deeted s at east (n +m )/ogn ; thus these actons are executed at most ogn tmes. Obvousy, the number of tmes actons of the second type are apped s aso at most ogn. Let D be the set of vertces that are decoored by the th appcaton of an acton of type 2 and et A = D. Then, and σ(a ) ogn = 0 n +m σ(a ) σ(d ), σ(d ) n +m 2ogn n +m 2ogn r, χ(r ) r n +m χ(r ) 2ogn ogn = 0 r, χ(r ) where r s the number of coor casses just before the th appcaton of the type 2 acton. ogn r To evauate, note that χ <r for even r s ony, and for such r s, every =0 χ(r ) appcaton of an acton of type 2 reduces the number of coor casses by haf; f r 3 s odd, r (r /2)+1. Thus, +1 whch mpes the theorem. ogn =0 r ogn χ(r ) =1 2 ogn +2, 2 1 Coroary. Any appcaton of FINDSET yeds an ndependent set I, such that I N (I ) (1/2 o (1))(n +m ). In the next secton, we w show that FINDSET can be mpemented to run n O (og2n ) tme. Ths w mpy that the runnng tme of the agorthm s O (og3n ).

6 Impementaton of the agorthm We w frst see how to mpement FINDSET to run n O (og2n ) tme on n +m processors. Obvousy, for every appcaton of FINDSET, acton of ether type s executed at most O (ogn ) tmes. Thus, we shoud show that the executon of each acton requres ony O (ogn ) tme. A graph G s represented by a st L =L (G ) of ts vertces and edges. Each edge s n ths st twce, once n each drecton. Thus, the ength of L s n +2m. For each entry of L, there s a processor attached to t. The processors are numbered by 1,...,n +2m ; the frst n of them represent the vertces. In addton to ts endponts, each edge knows the coors (f any) of the endponts and the ocaton of tsef gven n the other drecton. Amost a of the operatons are done by sortng ths st based on some key. >From [ AKS, C] we know that t s possbe to sort m records n O (ogm ) tme on m processors. To deete a coor cass C, each edge checks the coors of ts endponts to see f one has the same coor as C. If one does, the edge deetes tsef. Then FINDSET sorts the st of edges to remove the deeted edges. To decde whether to do acton <1> or acton <2>, FINDSET needs to cacuate σ(c ), for each C n φ. It can cacuate the degree of each vertex by sortng the st of edges by ther frst vertex. It can then sort the st of vertces by coor, and add up the degrees of the vertces. When HALF performs acton <2>, t fnds the reguar partton π that mnmzes σ(π). For ths purpose, t needs to know the degrees of each vertex. It can cacuate them tsef, or t can nhert them from the prevous step. To cacuate σ(π), HALF cacuates, for each edge wth two coored endponts, ndex (,j ;r ), where and j are the coors of the endponts. It then sorts the edge st by ndex, breakng tes by frst vertex. The dfferent frst vertces that occur wth a gven ndex q are the vertces n D q for the part- ton π q. HALF then sums the degrees of these vertces to cacuate the weghts σ(π t ) (t =0,..,χ(r ) 1) and seects the partton wth the mnmum weght. Let q be the ndex of the seected partton π. At ths stage, HALF s consderng the st F of edges whose ndex s q. For every par (C,C j ) of π, the vertces of one of the coor casses w be ether decoored or recoored when C and C j are merged; we ca ths coor cass egbe. To fnd egbe casses, HALF computes σ =σ(c N (C )) for j every =0,..., r /2 1, and sets cass C egbe ff σ j σ j. Once HALF has dentfed the egbe coor casses, t ooks at each edge and decoors those vertces v beongng to the egbe coor cass Ct that are connected to a vertex wth coor rev (t,q ;r ). Fnay, the pars of coor casses determned by q are merged nto snge casses by recoorng, for every par, a the vertces n the coor cass wth the arger coor,.e. each vertex v wth orgna coor c (v ) changes ts coor to rev (c (v ),q ;r ) ff rev (c (v ),q ;r ) c (v ). In genera, the new coor casses do not necessary have consecutve numbers. To fx ths, the vertces are sorted by coor, so that the set of coors n use can be determned. Next, HALF renumbers the coors and gves each vertex ts new coor. To update the coors of the frst vertex stored wth the edges, the st of edges s sorted by frst vertex. Then, the ponters to the other representaton of the edges are foowed to update the coor of the second vertex. Each executon of the man oop of FINDSET requres between one and three sorts

7 - 7 - and O (ogn ) tme spent dong other msceaneous work. Thus, FINDSET requres ony O (og2n ) tme n a. The number of tmes FINDSET s caed s O (ogn ) mpyng that the runnng tme of the agorthm s O (og3 n ). So far, we have assumed that m +n processors are avaabe. However, usng the processor reducton technque of Mer and Ref [MR], the agorthm can be executed on ( m +n )/ogn processors wthout ncreasng the runnng tme by more than a constant. For an arbtrary k >1, f there are ony (m +n )/k processors, then each rea processor can smu- ate k vrtua processors n the agorthm. Snce a ca to FINDSET haves the vaue n +m of the graph, the number of vrtua processors that every rea processor smuates decreases by a factor of 2 after each ca to FINDSET. Thus, there s a constant C >0, such that for every =1,2,..., the th ca of FINDSET s executed n C 2 og3n tme. Ths ncreases the runnng tme of the agorthm by a factor 2. There s no probem aocatng vrtua processors to rea processors. Each vrtua proony necessary to reaocate vrtua processors after each ca to FINDSET. The new cessor s responsbe for a snge tem n st L of vertces and edges n the graph. It s representaton of G (I N (I )) can easy be cacuated by sortng. The reaocaton can be even done after each executon of the oop n FINDSET. Whe ths w speed up the agorthm for a typca graph, the worst-case graphs do not get smaer fast enough to mprove the asymptotc performance of the agorthm. Note that the same technque can be apped to reduce by a factor of ogn the number of processors used by the probabstc agorthms from [L1] and [ABI]. 5. Concuson An mportant property of a parae agorthm s the tota work W t does. Obvts runnng tme. Thus, our determnstc agorthm for MIS does O ((m +n )og2n ) work ousy, W P T, where P s the number of the processors used by the agorthm and T s whch s a factor of og2n more than the work done by the obvous sequenta agorthm. It woud be nce to fnd an agorthm that does ess work. One approach woud be to mprove the sortng agorthm that our agorthm uses. Ths mght be possbe snce the keys of a the sorts are sma ntegers. In fact, Ref has an agorthm that sorts n sma ntegers whe dong ony O (n ) work on a randomzed concurrent-read, concurrent-wrte, PRAM [R]. However, f randomzaton s ntroduced n the mode, then the agorthms from [L1] and [ABI] are preferabe. Thus, rea mprovement woud be acheved f better determnstc sortng agorthms were used. Obvousy, they woud be of nterest for other reasons as we. Another approach s to fnd a way to sort ess often. Both approaches appear to be very dffcut. A more promsng way to mprove the agorthm woud be to reduce ts runnng tme wthout ncreasng the work that t does very much (f at a). It may be possbe to reduce the runnng tme of the agorthm by executng dfferent cas to FINDSET n parae. There are severa ways to do that; the dffcuty seems to be n deveopng a better anaytca technque for estmatng the runnng tme. The dua representaton of the edge set may aso be usefu for other probems, edge- coorng and vertex-coorng beng among the frst canddates. Another potentay frutfu appcaton of ths representaton s the probem of constructng an ndependent set of a guaranteed sze. In [G], an agorthm was descrbed whch runs n O (og3n ) tme on

8 - 8 - O (n +m ) processors and constructs an ndependent set wth n 2/32m (m n /2) vertces. Concevaby, an approprate change n the defnton of the weght of a set w convert our agorthm nto one whch buds an ndependent set contanng n 2/(2m +n ) vertces. Ths s guaranteed by Túran s theorem [T] whch aso states that ths bound s best possbe n terms of n and m. Acknowedgment. We gratefuy thank the three referees for ther hepfu comments on the frst verson of the paper. References [AKS] Ajta, M., J. Komos, E. Szemered, An O (n ogn ) sortng network, Combnatorca, 3 (1983), pp [ ABI]Aon, N., L. Baba, A. Ita, A fast and smpe randomzed parae agorthm for the maxma ndependent set probem, J. of Agorthms, 7(1986), pp [ C] Coe, R., Parae merge sort, Proc. 27th Annua Symp. on Foundaton of Computer Scence, (1986), pp [ BM]Bondy, J.A., U.S.R. Murty, Graph Theory wth Appcatons, North Hoand, New York, [ G] Godberg, M., Parae agorthms for three graph probems, CONGRESSUS NUMERANTIUM, vo. 54, (1986), pp [ GS] Godberg, M., T. Spencer, A New Parae Agorthm for the Maxma Independent Set Probem, SIAM J. Comp., 18 (1989), pp [ KR]Karp, R.M., V. Ramachandran, Handbook of Theoretca Computer Scence, Preprnt, Unversty of Caforna, Berkeey, (1987). [ KW]Karp, R.M., A. Wgderson, A Fast parae agorthm for the maxma ndependent set probem, Proc. 16th ACM Symp. on Theory of Computng, 1984, pp [ L1] Luby, M., A Smpe parae agorthm for the maxma ndependent set probem, SIAM J. Comp., 15 (1986), pp [ L2] Luby, M., Removng Randomness n Parae Computaton Wthout a Processor Penaty, Proc. 29th Annua Symp. on Foundaton of Computer Scence, (1988), pp [MR]Mer, G.L., J.H. Ref, Parae Tree Contracton and ts Appcatons, Proc. 26th Annua Symp. on Foundaton of Computer Scence, (1985), pp [ P] Pppenger N., On Smutaneous Resource Bounds, Proc. 20th Annua Symp. on Foundaton of Computer Scence (1979), pp

9 - 9 - [R] Ref, John H., An Optma Parae Agorthm for Integer Sortng, Proc. 26th Annua Symp. on Foundaton of Computer Scence, (1985), pp [T] Túran, P., On the theory of graphs, Cooq. Math. 3 (1954), pp [ V] Vshkn, U., Synchronous parae computaton - a survey, Preprnt, Courant Insttute, New York Unversty (1983).

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