Logarithmic-time link path queries in a simple polygon

We develop a data structure for answering link distance queries between two arbitrary points in a simple polygon. The data structure requires O(n3) time and space for its construction and answers link distance queries in O(log n) time, after which a minimum-link path can be reported in time proportional to the number of links. Here, n denotes the number of vertices of the polygon. Our result extends to link distance queries between pairs of segments or polygons. We also propose a simpler data structure for computing a link distance approximately, where the error is bounded by a small additive constant. Finally, we also present a scheme for approximating the link and the shortest path distance simultaneously.