Optimal Equi - Partition of Rectangular Domains for ParallelComputationIoannis

We present an eecient method for the partitioning of rectangular domains into equi-area sub-domains of minimum total perimeter. For a variety of applications in parallel computation, this corresponds to a load-balanced distribution of tasks that minimize interprocessor communication. Our method is based on utilizing, to the maximum extent possible, a set of optimal shapes for sub-domains. We prove that for a large class of these problems, we can construct solutions whose relative distance from a computable lower bound converges to zero as the problem size tends to innnity. PERIX-GA, a genetic algorithm employing this approach, has successfully solved to optimality million-variable instances of the perimeter-minimization problem and for a one-billion-variable problem has generated a solution within 0.32% of the lower bound. We report on the results of an implementation on a CM-5 supercomputer and make comparisons with other existing codes. 1 The Minimum Perimeter Problem We consider the Minimum Perimeter Equi-partition problem MPE(M; N; P), a geometric problem with intrinsic beauty that nds numerous applications in parallel computing. It is essentially a graph partitioning problem that, when restricted to rectangular grids (the main focus of this paper), can be stated as follows: given a rectangular grid of dimensions MN and a number of processors 1 P, where P divides MN, nd the partition of the grid that minimizes the total perimeter induced subject to the constraint that each processor is assigned the same number of grid cells. Geometrically, the problem may be thought of as partitioning the grid into P equi-area regions (each of area A := MN=P) of minimum total perimeter. Since graph partitioning is itself a special case of a more general problem, the so-called Quadratic Assignment Problem (QAP), it follows that MPE can be formulated as a QAP ((PRW93]). In terms of binary variables in an integer programming formulation ((NW85]) the problem may be described