Optimal Shortest Path and Minimum-Link Path Queries Between Two Convex Polygons Inside a Simple Polygonal Obstacle

We present e cient 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 logh logn 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 logh logn plus O k to report the k links with O n space and preprocessing time We also extend our results to the dynamic case and give a uni ed data structure that supports both queries for convex polygons in the same region of a connected planar subdivision S The update operations consist of insertions and deletions of edges and vertices Let n be the current number of vertices in S The data structure uses O n space supports updates in O log n time and performs shortest path and minimum link path queries in times O logh log n plus O k to report the k edges of the path and O logh k log n respectively Performing shortest path queries is a variation of the well studied separation problem which has not been e ciently 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