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-dimensionalarra y 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-dimensionalarra ys it is shown in [KMP98] that the problem is NP-hard and an approximation algorithm is given. This paper offers a new (d2+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.