ON RECTANGULAR COVERING PROBLEMS

Abstract

Many applications like image processing, data compression or pattern recognition require a covering of a set of n points most often located in the (discrete) plane by rectangles due to specific cost constraints. In this paper we provide exact dynamic programming algorithms for covering point sets by regular rectangles, that have to obey certain boundary conditions, namely all rectangles must have lengths of at least k, which is a prescribed problem parameter. The concrete objective function underlying our optimization problem is the sum of area, circumference and a constant over all rectangles forming a covering. This objective function can be motivated by requirements of numerically solving PDE's by discretization over (adaptive multi-)grids. More precisely, we propose exact deterministic algorithms for such problems based on a (set theoretic) dynamic programming approach yielding a time bound of O(n23n). In a second step this bound is (asymptotically) decreased to O(n62n) by exploiting some structural features. Finally, a generalization of the problem and its solution methods is discussed for the case of arbitrary (finite) space dimension.