Equi-depth multidimensional histograms
SIGMOD '88 Proceedings of the 1988 ACM SIGMOD international conference on Management of data
Histogram-based estimation techniques in database systems
Histogram-based estimation techniques in database systems
On approximating rectangle tiling and packing
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Rectangular tiling in multi-dimensional arrays
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
On Rectangular Partitionings in Two Dimensions: Algorithms, Complexity, and Applications
ICDT '99 Proceedings of the 7th International Conference on Database Theory
ICALP '97 Proceedings of the 24th International Colloquium on Automata, Languages and Programming
Approximation algorithms for min-max generalization problems
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Relations between two common types of rectangular tilings
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
A new approximation algorithm for multidimensional rectangle tiling
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
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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.