Fast Equi-partitioning of Rectangular Domains Using Stripe Decomposition

This paper presents a fast algorithm that provides optimal or near optimal solutions to the minimum perimeter problem on a rectangular grid. The minimum perimeter problem is to partition a grid of size M×N into P equal area regions while minimizing the total perimeter of the regions. The approach taken here is to divide the grid into stripes that can be filled completely with an integer number of regions. This striping method gives rise to a knapsack integer program that can be efficiently solved by existing codes. The solution of the knapsack problem is then used to generate the grid region assignments. An implementation of the algorithm partitioned a 1000×1000 grid into 1000 regions to a provably optimal solution in less than one second. With sufficient memory to hold the M×N grid array, extremely large minimum perimeter problems can be solved easily. Introduction The focus of the algorithm presented here is the Minimum Perimeter Equi-partition problem, MPE(M, N, P). In this problem one is to partition an M×N rectangular grid into P equal area regions while minimizing the total perimeter of the partition. The one restriction of this algorithm is that all regions must have the same area. The area of each region is defined by A MN P = so the restriction is equivalent to P evenly dividing MN. The minimum perimeter problem has several applications in parallel computer systems. In solving partial differential equations numerically, a grid is partitioned among the available processors. Using a five point numerical method, each grid element must communicate with its North, East, South, and West neighbors [DT91]. In assigning processors to the regions of the grid, one wants to minimize the communication between the processors while equalizing the number of grid elements assigned to each processor. This assignment process is analogous to the minimum perimeter problem. Another area of application is in image processing and edge detection in computer vision systems implemented on parallel hardware [Sch89]. Here again the rectangular image needs to partitioned among the processors to minimize inter-processor communication. In order to calculate a lower bound for the minimum perimeter problem, Yackel and Meyer [YM92] have shown that the minimum perimeter of a single region with area A is determined by Π*(A). (1)   Π( ) A A = 2 2 If the entire grid could be tiled with shapes of the optimal perimeter without overlapping then an optimal solution would be found. Because one cannot do any better than this optimal tiling, a lower bound for the objective function of MPE(M, N, P) is given by z. (2) z = P Π*(A) The minimum perimeter problem is a special case of the graph partitioning problem which is NPcomplete. MPE(M, N, P) can be formulated as a quadratic assignment problem with MNP binary variables and MN+P constraints. Details of this formulation are given in Christou and Meyer