Coarse Grained Parallel Maximum Matching In Convex Bipartite Graphs

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1 Coarse Grained Parallel Maximum Matchin In Convex Bipartite Graphs P. Bose, A. Chan, F. Dehne, and M. Latzel School o Computer Science Carleton University Ottawa, Canada K1S 5B6 jit,achan,dehne,mlatzel@scs.carleton.ca Abstract We present a coarse rained parallel alorithm or computin a maximum matchin in a convex bipartite raph G = (A; B; E). For p processors with N=p memory per processor, N = jaj + jbj, N=p p, the alorithm requires O(lo p) communication rounds and O(T sequ ( n p ; m p ) + n p lo p) local computation, where n = jaj, m = jbj and T sequ (n; m) is the sequential time complexity or the problem. For the BSP model, this implies O(lo p) supersteps with O(N + n p lo p) communication cost and 1 Introduction 1.1 The Problem We study the problem o computin the maximum matchin in convex bipartite raphs deined as ollows. Deinition 1 A Bipartite raph G =(A; B; E) is an undirected raph with vertex set A [ B where A and B are disjoint and E (A B). For the remainder let n = jaj, m = jbj and N = n + m. Deinition 2 A Convex bipartite raph is a bipartite raph G =(A; B; E) with an orderin B =(b 1 ;b 2 ; b n ) such that or any a 2 A, i(a; b i ) 2 E and (a; b j ) 2 E (i j) then (a; b k ) 2 E or all i k j. Deinition 3 AmatchinM in a raph G = (V;E) is a subset o E such that no two edes in M are incident to the same vertex. Deinition 4 A maximum matchin in a raph G =(V;E) is a matchin o G o which the size (number o edes) is maximum. The problem o indin a maximum matchin in a bipartite raph or a convex bipartite raph is a classical and well studied problem [7, 12, 14, 15]. The case o convex bipartite raphs has several interestin applications as outlined in [7, 15]. A particularly interestin industrial application or matchin parts with products was described in [15]. Sequential solutions or maximum matchin in bipartite raphs and convex bipartite raphs with time complexities O(n 5=2 ) and O(n + (m)) were described in [14] and [15], respectively, where (m) is a very slowly rowin unction related to the inverse Ackermann unction. A PRAM alorithm or maximum matchin in convex bipartite raphs requirein O(lo 2 n) time and n=2 processors was presented in [7]. In this paper, we consider the maximum matchin problem or the coarse rained parallel and BSP models o computation described in the next subsection. These new parallel models have recently received much attention because they allow the desin o parallel alorithms that have much improved practical perormance in actual implementations on commercial multiprocessors compared to previous models like the PRAM or ine rained network models (e.. mesh or hypercube). Beore proceedin to the description o the coarse rained parallel and BSP models o computation, we note that indin a maximum matchin in a convex bipartite raph can be transormed into the ollowin problem: Deinition 5 Given a set o intervals I = I 1 ; ;I n, each o which represents an inteer rane, i.e. I i =(l i ;r i ). A maximum interval assinment consists o assinin, or a maximum number o intervals, one inteer label to each interval such that the inteer or each interval is within the interval s rane and no two intervals are assined the same inteer. For a iven convex bipartite raph G = (A; B; E) let I(G) be the set o jaj intervals containin or each a 2 A an interval I i =(l i ;r i ) where l i and r i are the smallest and larest ranks, in B, o all element b 2 B with (a; b) 2 E.

2 Observation 1 The maximum matchin problem or a convex bipartite raph G can be reduced to indin a maximum interval assinment in I(G). 1.2 The Model The Coarse Grained Multicomputer (CGM) A CGM (Coarse Grained Multicomputer) is a special case o a BSP (Bulk Synchronous Parallel) multiprocessor [18] where all communication within a superstep is reduced to one h-relation. A CGM consists o a collection o p processors with N=p local memory each, connected by a router that can deliver messaes in a point to point ashion. A CGM alorithm consists o an alternatin sequence o computation rounds and communication rounds separated by barrier synchronization. A computation round is equivalent to a computation superstep in the BSP model, and the total computation cost T comp is deined analoously. A communication roundconsists o a sinle h-relation with h N=p. The total communication cost,t comm, is measured by the number o communication rounds. Consult [3, 4, 5] or more details. In a recent overview o dierent BSP and related models, Goodrich [13] reerred to the CGM as the weak-crew BSP. The CGM model aims at desinin simple and practical, yet theoretically optimal or eicient, parallel alorithms or coarse rained parallel systems (N=p >> 1). Alorithms do usually require a lower bound on N=p, e..n=p p or N=p p. The CGM model tarets in particular the case where the overall computation speed is considerably larer than the overall communication speed, which is usually the case. Since the messae size is maximal, the model also minimizes the messae overhead associated with sendin a messae, which is very important in practice. Relationship Between BSP And CGM In the remainder o this paper, we will present our alorithms in the CGM model. The relationship to the BSP model is iven by the ollowin Observation 2 A CGM alorithm with rounds and computation cost T comp corresponds to a BSP alorithm with supersteps, communication cost O( N p ) and the same computation cost T comp. 1.3 The Result In this paper, we present a coarse rained parallel alorithm or computin a maximum matchin o a convex bipartite raph G = (A; B; E). For a p processor CGM with N=p memory per processor, N=p p, the alorithm requires O(lo p) communication rounds and O(T sequ ( n p ; m p ) + n p lo p) local computation where T sequ (n; m) is the sequential time complexity or the problem. For the BSP model, this implies O(lo p) supersteps with O(N + n p lo p) communication cost and 2 Alorithm Description In the ollowin Section 2.1, we irst present an alorithm that solves a special case: all intervals start at the same let end point. This alorithm will be used in our solution or the eneral case which is presented in Section Special Case: All Intervals Start At The Same Point Alorithm 1 All intervals have the same let end point l i = l; i =1n. Input: Asetonintervals I = I 1 ;:::;I n with their associated let and riht end points I i =(l i ;r i );i =1:::n, distributed over a p processor CGM with n=p intervals per processor. Note that, we assume n p p. (1) All intervals are sorted by their riht end points usin CGM parallel inteer sort [6, 13]. (2) Each processor P i ;i = 1 :::p, 1, determines the larest riht end point, e i, received in Step 1 and sends it to the next processor P i+1. (3) Each processor P i ;i = 2 :::p,setss i = e i,1 +1. The irst processor sets s 1 = l. We call (s i ;e i ) the controlled rane o P i. This is the rane o inteers that the processor can use to label intervals. (4) Each processor P i ;i = 1 :::p, temporarily chanes the let end points o its intervals to s i and then solves the modiied problem locally usin a sequential alorithm (essentially sequential inteer sort). (5) Each processor P i ;i = 1 :::p, calculates how many labels in the controlled rane are let unused (a i )and how many intervals did not yet receive an inteer label (b i ). These two numbers per processor are broadcast to all processors (in one h-relation). This is possible since n p p. (6) Based on the data received in the previous step, each processor can now calculate rom where it should request labels and to where it should send unused labels. See Alorithm 2 or more details. (7) Each processor selects unused labels and sends them to the processors that need them. (8) Upon receivin the needed labels, each processor can now assin them to the respective intervals. End o Alorithm Alorithm 2 Sequential computation or Step 6 in Alorithm 1, executed by processor P k, 1 k p.

3 Input: An array a = (a 1 ;:::;a p ) representin the numbers o ree (unused) labels in the p processors. An array b = (b 1 ;:::;b p ) representin the numbers o intervals so ar without labels in the p processors. Output: An array et = (et 1 ;:::;et p ) representin numbers o labels that are to be sent to processor P k by the other processors. An array ive =(ive i ;:::;ive p ) representin numbers o labels that are expected rom processor P k by the other processors. or (i=1; ip; i++) et i =0;ive i =0; i=1; j=2; while ((i<p) and (jp)) i (i==j) j++; while ((a i 6=0) and (jp)) i++; i (a i b j ) i (P k =i) et j =b j ; i (P k =j) receive i =b j ; a i =a i -b j ; b j =0; j++; else End o Alorithm i (P k =i) et j =a i ; i (P k =j) ive i =a i ; b j =b j -a i ; a i =0; Lemma 1 The sequential time complexity o Alorithm 2 is O(p). Proo. The variable i counts rom 1 to p-1, and j counts rom 2 to p. Since both variables are incremented independently, the total time complexity o the alorithm is O(p). 2 Lemma 2 The inteer label assinment produced by Alorithm 1 is maximum. Proo. It is easy to see that the label assinment based on the order o the riht end points is maximum. Let A o be such an assinment. Let A 1 be the label assinment produced by Alorithm 1. Since A o is maximum, ja o j ja 1 j.we compare assinments A 1 and A o. We irst remove the intervals that are not assined labels in both assinments. I the remainin intervals all have labels assined in both assinments, then we have ja o j = ja 1 j and the lemma ollows. I the kth interval (lk 0 ;r0 k ) (with respect to the riht end points) is assined a label in A o but not in A 1, then the irst k, 1 intervals must be assined labels l; l +1;:::;l+k, 2 in A o and the kth interval must be assined label l + k, 1. This also means rk 0 l + k, 1. Now consider A 1.Itheirstk, 1 intervals are also assined labels l; l +1;:::;l+ k, 2, thenl + k, 1 will be available or the kth intervals. I some o the irst k,1 intervals are not assined labels in the rane l; l+1;:::;l+k,2, then there will be a hole in this rane that can be used or the kth interval. Hence, or every interval with a label assined in A o, there is a label assined to that interval in A 1. Thereore, ja 1 jja o j. Thus, the label assinment obtained rom Alorithm 1 is maximum. 2 Theorem 1 Alorithm 1 inds a maximum label assinment or n intervals with the same let endpoints stored on a p processor CGM with n=p local memory per processor, n p p, in O(1) communication rounds with O( n p ) local computation. Proo. The correctness o the alorithm is shown in Lemma 2. The communication rounds required in each step o Alorithm 1 is O(1). Hence the total number o communication rounds needed is O(1). In each round, the total messae size per processor is O( n p ). Step 1 uses CGM inteer sort [6, 13] which requires O(1) communication rounds and O( n p ) local computation. The local computation o Step 6 is O(p) (rom Lemma 1). For all other steps, it is either O(1) or O( n p ). Thereore, the total local computation per processor is O( n p )+O(p) =O( n p ) since n p p The General Case In order to solve the maximum interval assinment problem or arbitrary intervals, we now combine Alorithm 1 with a mere/unmere scheme presented in [7]. Alorithm 3 General case where the intervals can have dierent let end points. Input: Aseton intervals I = I 1 ;:::;I n with their associated let and riht end points I i =(l i ;r i );i =1n, distributed over a p processor CGM with n=p intervals per processor. Note that, we assume that n p p. (1) All intervals are sorted by their let end points usin CGM parallel inteer sort [6, 13]. (In case o a tie, intervals are compared by their riht end points.)

4 (2) Intervals with the same let end points are combined into roups. All roups that are stored completely within a processor are mered into a sinle roup. Let be the number o roups. Note that is at most 2p +1. (3) Each roup is assined a controlled rane (l i ;r i );i = 1 ::: where l i is equal to the smallest let end point o that roup and r i = l i+1, 1;i =1:::, 1. Let r be the larest riht end point o the intervals. (4) Usin a sequential alorithm, each processor solves the problem or interval roups that are completely within the processor. Remove the label o all those intervals that received a label outside their roup s controlled rane. Classiy the intervals usin one o the ollowin three types. Matchable (M): all labeled intervals. To-be-determined (T ): all non labeled intervals that extend beyond the rihtmost label iven by that processor. Unmatchable (U): all remainin intervals. Remove all unmatchable intervals. (5) Usin Alorithm 1, solve the problem or interval roups that cross a processor boundary (or each such roup in isolation). Remove the label o all intervals that received a label outside their roup s controlled rane. Classiy the intervals usin one o the ollowin three types. Matchable (M): all labeled intervals. To-be-determined (T ): all nonlabeled intervals that extend beyond the rihtmost label iven by that processor. Unmatchable (U): all remainin intervals. Remove all unmatchable intervals. (6) Mere the intervals in adjacent roups. Let M L, T L be the intervals rom the let roup, let M R, T R be the intervals rom the riht roup, and let (l L ;r L ) and (l R ;r R ) be the controlled ranes o the let and riht roups, respectively. Usin Alorithm 1, solve the problem or T L [ M R over the rane (l R ;r R ). Let the resultin sets o machable and to-be-determined intervals be M n,andt n. For the combined roup, set M = M L [ M n and T = T n [ T R. The controlled rane o the combined roup is (l L ;r R ). (7) Repeat Step 6 O(lo ) times until a sinle roup is obtained. (8) Usin a reverse process o the previous steps, usin O(lo ) iterations, redistribute the intervals back into the roups obtained at the end o Step 3. In each splittin phase, let M be the roup we are splittin and let M L and M R be those two components. Let V be the set o intervals that have a let end point smaller than the startin value o the controlled rane o M. Note that, V can be empty. Let W be the union o V and M L. W should be distributed to M L and the rest to M R. However, the controlled rane o M L may not allow the entire W to be assined to M L. Hence, we assin only the irst min(jw j;lr,lj) intervals to M L and the remainder to M R. (9) Step 8 is repeated until roups are obtained. (10) Usin a sequential alorithm, each processor solves the problem or its interval roups over each roup s controlled rane. For roups that span over more than one processor, adjust the let end points to the startin values o the controlled rane and then apply Alorithm 1. End o Alorithm Theorem 2 Alorithm 3 solves the label assinment problem in O(lo p) communication rounds and Proo. The correctness o the split/mere scheme ollows rom [7]. Step 6 is executed O(lo ) times. In Step 6, we invoke Alorithm 1 which requires O(1) communication rounds. Since 2p +1, the total number o communication rounds is O(lo p). The local computation time is dominated by the O(1) executions o the sequential alorithm on each processor and the linear local time in each o the O(lo ) iterations. 2 As described in Section 1.1, this theorem implies an alorithm or computin the maximum matchin o a convex bipartite raph with the same number o communication rounds and local computation. 3 Conclusion In this paper, we have presented a coarse rained parallel alorithm or computin a maximum matchin o a convex bipartite raph G =(A; B; E). Forp processors with N=p memory per processor, N=p p, the alorithm requires O(lo p) communication rounds and O(T sequ ( n p ; m p ) + n p For the BSP model, this implies O(lo p) supersteps with O(N + n p lo p) communication cost and From a practical point o view, i.e. or implementin this method on commercial multiprocessors, it is very important that the problem size is not a parameter or the number o communication rounds. The result obtained, i.e. O(lo p) communication rounds, is a unction o p only. Since p is usually ixed or rows on very slowly in practice, and lo p is a slowly rowin unction, the number o communication rounds is essentially a ixed constant or most practical arranements. This is important because empirical studies show that the number o communication rounds is the most important parameter inluencin the observed runnin time. From a theoretical point o view it is an important open problem to study whether it is possible to ind an alorithm with even ewer communication rounds (is O(1) possible?) and/or determine lower bounds or the number o communication rounds.

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