Tiling multi-dimensional arrays

Abstract

We continue the study of the tiling problems introduced in [KMP98]. The first problem we consider is: given a d-dimensional array of non-negative numbers and a tile limit p, partition the array into at most p rectangular, non-overlapping subarrays, referred to as tiles, in such a way as to minimise the weight of the heaviest tile, where the weight of a tile is the sum of the elements that fall within it. For one-dimensional arrays the problem can be solved optimally in polynomial time, whereas for two-dimensional arrays it is shown in [KMP98] that the problem is NP-hard and an approximation algorithm is given. This paper offers a new (d ² +2d−1)/(2d−1) approximation algorithm for the d-dimensional problem (d ≥ 2), which improves the (d+3)/2 approximation algorithm given in [SS99]. In particular, for two-dimensional arrays, our approximation ratio is 7/3 improving on the ratio of 5/2 in [KMP98] and [SS99]. We briefly consider the dual tiling problem where, rather than having a limit on the number of tiles allowed, we must ensure that all tiles produced have weight at most W and do so with a minimaln umber of tiles. The algorithm for the first problem can be modified to give a 2d approximation for this problem improving upon the 2d+1 approximation given in [SS99]. These problems arise naturally in many applications including databases and load balancing.

Article

This work was supported by the Engineering and PhysicalSciences Research Council and in part by the ESPRIT LTR Project no. 20244 — ALCOM-IT.