A Parallel Distance-2 Graph Coloring Algorithm

>A Parallel Distance-2 Graph Coloring Algorithm for Distributed Memory Computers star Doruk Bozda˘g 1 , Umit Catalyurek 1 , Assefaw H. Gebremedhin 2 , Fredrik Manne 3 , Erik G. Boman 4 ,andF¨usun ¨ Ozg¨uner 1 1 Ohio State University, USA {bozdagd, ozguner}@ece.osu.edu, umit@bmi.osu.edu 2 Old Dominion University, USA assefaw@cs.odu.edu 3 University of Bergen, Norway Fredrik.Manne@ii.uib.no 4 Sandia National Laboratories, USA egboman@sandia.gov Abstract. The distance-2 graph coloring problem aims at partitioning the vertex set of a graph into the

fewest sets consisting of vertices pairwise at distance greater than two from each other. Application examples include numerical opti- mization and channel assignment. We present the first distributed-memory heuris- tic algorithm for this NP-hard problem. Parallel speedup is achieved through graph partitioning, speculative (iterative) coloring, and a BSP-like organization of computation. Experimental results show that the algorithm is scalable, and com- pares favorably with an alternative approach—solving the problem on a graph G by first constructing the square graph G 2 and then applying a parallel distance-1 coloring algorithm on G 2 . 1 Introduction An archetypal problem in the efficient computation of sparse Jacobian and Hessian matrices is the distance-2 (D2) vertex coloring problem in an appropriate graph [1]. D2 coloring also finds applications in channel assignment [2] and facility location problems [3]. It is closely related to a strong coloring of a hypergraph which in turn models problems that arise in the design of multifiber WDM networks [4]. The D2 coloring problem is known to be NP-hard [5]. In many parallel applications where a graph coloring is required, the graph is al- ready distributed among processors. Under such circumstances, gathering the graph on one processor to perform the...

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