Min-Max-Boundary Domain Decomposition

Domain decomposition is one of the most eeective and popular parallel computing techniques for solving large scale numerical systems. In the special case when the amount of computation in a subdomain is proportional to the volume of the subdomain, domain decomposition amounts to minimizing the surface area of each subdomain while dividing the volume evenly. Motivated by this fact, we study the following min{max boundary multi{way partitioning problem: Given a graph G and an integer k > 1, we would like to divide G into k subgraphs G 1 ; : : : ; G k (by removing edges) such that (i) jG i j = (jGj=k) for all i 2 f1; : : : ; kg; and (ii) the maximum boundary size of any subgraph (the set of edges connecting it with other subgraphs) is minimized. We provide an algorithm that given G, a well{shaped mesh in d dimensions, nds a partition of G into k subgraphs G 1 ; : : : ; G k , such that for all i, G i has (jGj=k) vertices and the number of edges connecting G i with the other subgraphs is O((jGj=k) 1?1=d). Our algorithm can nd such a partition in O(jGj log k) time. We show that extensions of our main algorithm, when d = 2, can nd similar partitions in planar graphs and graphs with bounded genus and forbidden minors. Finally, we extend our results to vertex{weighted and vertex{based graph decomposition. Our results can be used to simultaneously balance the computational and memory requirement on a distributed{memory parallel computer without sacriicing the communication overhead.