Bounds on the length of convex partitions of polygons

Abstract

A heuristic for partitioning rectilinear polygons into rectangles, and polygons into convex parts by drawing lines of minimum total length is proposed. For the input polygon with n vertices, k concave vertices and the perimeter of length p, the heuristic draws partitioning lines of total length O(plogk) and runs in time O(nlogn). To demonstrate that the heuristic comes close to optimal in the worst case, a uniform family of rectilinear polygons Q _k with k concave vertices, k=1, 2, ... and a uniform family of polygons P _k with k concave vertices, k=1, 2, ... are constructed such that any rectangular partition of Q _k has (total line) length Ω(plogk), and any convex partition of P _k has length Ω(plogk/loglogk). Finally, a generalization of the heuristic for minimum length of convex partition of simple polygons to include polygons with polygonal holes is given.