An efficient algorithm for link distance problems

Abstract

The link distance between two points inside a simple polygon P is defined to be the minimum number of edges required to form a polygonal path inside P that connects the points. A link furthest neighbor of a point p € P is a point of P whose link distance is the maximum from p. The Iink center of P is the collection of points whose link distances to their link furthest neighbors are minimized. We present an O(n log n) time and O(n) space algorithm for computing the link center of a. simple polygon P, where n is the number of vertices of P. This improves the previous O(n2) time and space algorithm. Our algorithm essentially sweeps a chord through the polygon and spends O(log n) time at each step. We demonstrate that the output of the algorithm, a sequence of sets of chords, is a powerful tool for solving several other link distance problems.