Optimal Shortest Path and Minimum-Link Path Queries between Two Convex Polygons inside a Simple Polygonal Obstacle

Abstract

We present e cient algorithms for shortest-path and minimum-link-path queries between two c o n vex 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 numberof vertices of the query polygons. We s h o w 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) ( p l u s O(k) to report the k links), with O(n 3 ) space and preprocessing t i m e . 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 n umber of vertices in S. The data structure uses O(n) space, supports updates in O(log 2 n) time, and performs shortest-path and minimumlink-path queries in times O(log h + l o g 2 n) ( p l u s O(k) to report the k edges of the path) and O(log h + k log 2 n), respectively. P erforming 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-linkpath queries in a dynamic environment, even for two-point queries.

Single-Point Visibility Constraint Minimum Link Paths in Simple Polygons

“…This window partition is essentially a shortest path map, because it divides the simple polygon into faces of equal link distance from a fixed source point. By contrast, the work of Arkin et al [3] supports O(log n) time queries between any two points inside P after building n shortest path maps for all vertices of P , i.e., the total time complexity for this construction is O(n 2 ) (an optimal algorithm for this case was presented by Chiang et al in [4]). On the other hand, when there are holes in the polygon, Mitchell et al [10] proposed an incremental algorithm with O(Eα(n) log 2 n)) time bound, where n is the total number of edges of the obstacles, E is the size of the visibility graph, and α(n) denotes the extremely slowly growing inverse of Ackermann's function.…”

Single-Point Visibility Constraint Minimum Link Paths in Simple Polygons

“…This window partition is essentially a shortest path map, because it divides the simple polygon into faces of equal link distance from a fixed source point. By contrast, the work of Arkin et al [3] supports O(log n) time queries between any two points inside P after building n shortest path maps for all vertices of P , i.e., the total time complexity for this construction is O(n 2 ) (an optimal algorithm for this case was presented by Chiang et al in [4]). On the other hand, when there are holes in the polygon, Mitchell et al proposed an incremental algorithm with O(Eα(n) log 2 n)) time bound, where E is the size of the visibility graph, α(n) is a slow increasing function, which is the inverse of the Ackermann function and n is the number of vertices of obstacles [10].…”