Finding Simple Paths on Given Points in a Polygonal Region

Abstract

Given a set X of points inside a polygonal region P, two distinguished points s, t ∈ X, we study the problem of finding the simple polygonal paths that turn only at the points of X and avoid the boundary of P, from s to t. We present an O((n ² + m) logm) time, O(n ² + m) space algorithm for computing a simple path or reporting no such path exists, where n is the number of points of X and m is the number of vertices of P. This gives a significant improvement upon the previously known O(m ² n ²) time and space algorithm, and O(n ³ logm + m n) time, O(n ³ + m) space algorithm.

An important result of this paper, termed the Shortest-path Dependence Theorem, is to characterize the simple paths of the minimum link distance, in terms of the shortest paths between the points of X inside P. It finally turns out that the visibility graph of X, together with an implicit representation of the shortest paths between all pairs of the points of X, is sufficient to compute a simple path from s to t or report no simple paths exist. The Shortest-path Dependence Theorem is of interest in its own right, and might be used to solve other problems concerning simple paths or polygons.

This work was partially supported by the Grant-in-Aid (MEXT/JSPS KAKENHI 23500024) for Scientific Research from Japan Society for the Promotion of Science and the National Natural Science Foundation of China under grant 61173034.