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

A d-dimensional grid graph G is the graph on a finite subset in the integer lattice Z 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. 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 ). 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. Article Type Communicated by Submitted Revised Regular paper X. He April 2005 January 2007 The research of Robert R. Meyer was supported in part by NSF grant DMI-0100220 and DMI-0400294. Gunawardena et al., Partitioning of Grid Graphs, JGAA, 11(1) 83–98 (2007)84 V V T R P V V T R P L L J H F L L J H F D D B B A M M J H F L M J H F W W T R P V W T R P X X U S Q W W T R P N N K I G M M J H F D E B C A D D B B A X X U S Q X Y U S Q N N K I G N O K I G E E C C A Y Y U S Q Y Y U S Q O O K I G O O K I G E E C C A Figure 1: An equi-partition for MSA([5], 5) with bound gap 0.04%. Only component A does not have minimum surface area.