# Min-Max-Boundary Domain Decomposition

## Abstract

Domain decomposition is one of the most effective 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) |*G* _{i}| = *Θ*(|*G*|/*k*) for all *i* ∈ 1, . . . , *k*; 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, finds a partition of *G* into *k* subgraphs *G* _{1}, . . . , *G* _{k}, such that for all *i*, *G* _{i} has *Θ*(|*G*|/*k*) vertices and the number of edges connecting *G* _{i} with the other subgraphs is *O*((|*G*|/*k*)^{1−1/d }). Our algorithm can find such a partition in *O*(|*G*| log *k*) time. 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 sacrificing the communication overhead.

## Keywords

Planar Graph Domain Decomposition Decomposition Tree Separator Theorem Degree Graph## Preview

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