Finding equitable convex partitions of points in a polygon efficiently

Abstract

Previous work has developed algorithms for finding an equitable convex partition that partitions the plane into n convex pieces each containing an equal number of red and blue points. Motivated by a vehicle routing heuristic, we look at a related problem where each piece must contain one point and an equal fraction of the area of some convex polygon. We first show how algorithms for solving the older problem lead to approximate solutions for this new equitable convex partition problem. Then we demonstrate a new algorithm that finds an exact solution to our problem in O ( N n log N ) time or operations, where n is the number of points, m the number of vertices or edges of the polygon, and N := n + m the sum.