Equi-partitioning of Higher-dimensional Hyper-rectangular Grid Graphs

Abstract

A d-dimensional grid graph G is the graph on a finite subset in the integer lattice Z^d in which a vertex x = (x₁, x₂, …, x_n) is joined to another vertex y = (y₁, y₂, …, y_n) if for some i we have |x_i − y_i| = 1 and x_j=y_j for all j ≠ i. G is hyper-rectangular if its set of vertices forms [K₁] ×[K₂] ×…×[K_d], where each K_i is a nonnegative integer, [K_i] = {0, 1, …, K_i−1}. The surface area of G is the number of edges between G and its complement in the integer grid Z^d. We consider the Minimum Surface Area problem, MSA(G,V), of partitioning G into subsets of cardinality V so that the total surface area of the subgraphs corresponding to these subsets is a minimum. We present an equi-partitioning algorithm for higher dimensional hyper-rectangles and establish related asymptotic optimality properties. Our algorithm generalizes the two dimensional algorithm due to Martin []. It runs in linear time in the number of nodes (O(n), n=|G|) when each K_i is O(n^1/d). Utilizing a result due to Bollabas and Leader [], we derive a useful lower bound for the surface area of an equi-partition. Our computational results either achieve this lower bound (i.e., are optimal) or stay within a few percent of the bound.