Optimal Shortest Path and Minimum-link Path Queries between Two Convex Polygons in the Presence of Obstacles Optimal Shortest Path and Minimum-link Path Queries between Two Convex Polygons in the Presence of Obstacles

We present eecient 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 + 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(log h + logn) (plus O(k) to report the k links), with O(n 3) space and preprocessing time. We also extend our results to the dynamic case, and give a uniied 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 2 n) time, and performs shortest-path and minimum-link-path queries in times O(log h + log 2 n) (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 eeciently 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.