Abstract

Abstract. We describe a new technique for dynamically maintaining the trapezoidal decomposition of a connected planar map dX/ [ with n vertices and apply it to the development of a unified dynamic data structure that supports pointlocation, ray-shooting, and shortest-path queries in A4. The space requirement is O(n log n). Point-location queries take time O(log n). Ray-shooting and shortest-path queries take time O(log n) (plus O(k) time if the k edges of the shortest path are reported in addition to its length). Updates consist of insertions and deletions of vertices and edges, and take O(log n) time (amortized for vertex updates). This is the first polylog-time dynamic data structure for shortest-path and ray-shooting queries. It is also the first dynamic point-location data structure for connected planar maps that achieves optimal query time. Key words, point location, ray shooting, shortest path, computational geometry, dynamic algorithm

Citations