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, � � � , xn) is joined to another vertex y = (y1, y2, � � � , yn) if for some i we have |xi − yi| = 1 and xj = yj for all j 6 i. G is hyper-rectangular if its set of vertices forms [K1] × [K2] × � � � × [Kd], where each Ki is a nonnegative integer, [Ki] = {0,1, � � � , Ki −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 [8]. It runs in linear time in the number of nodes (O(n), n = |G|) when each Ki is O(n 1/d ). Utilizing a result due to Bollabas and Leader [3], 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.