Minimum k-adjacent rectangles of orthogonal polygons and its application
Abstract
This paper presents the problem of partitioning a rectangle R 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 R 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 R. 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.
Keywords
Computation theory
Computational geometry
Directed graphs
Embedded systems
Graph theory
Integer programming
Linear programming
Directed acyclic graph (DAG)
Horizontal segments
Integer program formulations
Interior surfaces
Memory protection
Optimal partitions
Polygon partitioning
Positive integers
Geometry