On rectangular partitionings in two dimensions: algorithms, complexity and applications

Abstract. Partitioning a multi-dimensional data set into rectangular partitions subject to certain constraints is an important problem that arises in many database applications, including histogram-based selec-tivity estimation, load-balancing, and construction of index structures. While provably optimal and ecient algorithms exist for partitioning one-dimensional data, the multi-dimensional problem has received less attention, except for a few special cases. As a result, the heuristic parti-tioning techniques that are used in practice are not well understood, and come with no guarantees on the quality of the solution. In this paper, we present algorithmic and complexity-theoretic results for the fundamental problem of partitioning a two-dimensional array into rectangular tiles of arbitrary size in a way that minimizes the number of tiles required to sat-isfy a given constraint. Our main results are approximation algorithms for several partitioning problems that provably approximate the optimal solutions within small constant factors, and that run in linear or close to linear time. We also establish the NP-hardness of several partitioning problems, therefore it is unlikely that there are ecient, i.e., polynomial time, algorithms for solving these problems exactly. We also discuss a few applications in which partitioning problems arise. One of the applications is the problem of constructing multi-dimensional histograms. Our results, for example, give an ecient algorithm to con-struct the V-Optimal histograms which are known to be the most ac-curate histograms in several selectivity estimation problems. Our algo-rithms are the rst to provide guaranteed bounds on the quality of the solution.