Optimal shortest path and minimum-link path queries between two convex polygons inside a simple polygonal obstacle

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(logh+log n) (plus O(κ) time for reporting the κ edges of the path) using a data structure with O(κ) space and preprocessing time, and that minimum-link-path queries can be performed in optimal time O(logh+f log n) (plus O(κ) to report the κ 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 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(log² n) time, and performs shortest-path and minimum-link-path queries in times O(log h+log²n) (plus O (κ) to report the κ edges of the path) and O(log h + κlog² 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.