We present efficient algorithms for shortest-path and minimum-link-path queries between two convex polygons inside a simple polygon P, which acts as an obstacle to be avoided. Let n be the number of vertices of P, and h the total number of vertices of the query polygons. We show that shortest-path queries can be performed optimally in time O( log h + log n) (plus O(k) time for reporting the k edges of the path) using a data structure with O(n) space and preprocessing time, and that minimum-link-path queries can be performed in optimal time O( log h + log n) (plus O(k) to report the k links), with O(n3) space and preprocessing time. We also extend our results to the dynamic case, and give a unified data structure that supports both queries for convex polygons in the same region of a connected planar subdivision [Formula: see text]. The update operations consist of insertions and deletions of edges and vertices. Let n be the current number of vertices in [Formula: see text]. The data structure uses O(n) space, supports updates in O( log 2 n) time, and performs shortest-path and minimum-link-path queries in times O( log h+ log 2n) (plus O(k) to report the k edges of the path) and O( log h + k log 2 n), respectively. Performing shortest-path queries is a variation of the well-studied separation problem, which has not been efficiently solved before in the presence of obstacles. Also, it was not previously known how to perform minimum-link-path queries in a dynamic environment, even for two-point queries.
Optimal Shortest Path and Minimum-Link Path Queries between Two Convex Polygons inside a Simple Polygonal Obstacle