Minimum K-Adjacent Rectangles of Orthogonal Polygons and its Application

Abstract

This paper presents the problem of partitioning a rectangle $\Re$ which contains several non-overlapping orthogonal polygons, into a minimum number of rectangles. By introducing maximally horizontal line segments of largest total length, the number of rectangles intersecting with any vertical scan line over the interior surface of $\Re$ is less than or equal to k, a positive integer. Our methods are based on a construction of the directed acyclic graph G = (V, E) corresponding to the structures of the orthogonal polygons contained in $\Re$. According to this, it is easy to verify whether a horizontal segment can be introduced in the partitioning process. It is demonstrated that an optimal partition exists if only if all path lengths from the source to the sink in G are less than or equal to k + 1. Using this technique, we propose two integer program formulations with a linear number of constraints to find an optimal partition. Our goal is motivated by a problem involving utilization of memory descriptors applying to a memory protection mechanism for the embedded systems. We discuss our motivation in greater details.