Improved Dynamic Geodesic Nearest Neighbor Searching in a Simple Polygon

We present an efficient dynamic data structure that supports geodesic nearest neighbor queries for a set S of point sites in a static simple polygon P . Our data structure allows us to insert a new site in S, delete a site from S, and ask for the site in S closest to an arbitrary query point q ∈ P . All distances are measured using the geodesic distance, that is, the length of the shortest path that is completely contained in P . Our data structure achieves polylogarithmic update and query times, and uses O(n log3 n logm + m) space, where n is the number of sites in S and m is the number of vertices in P . The crucial ingredient in our data structure is an implicit representation of a vertical shallow cutting of the geodesic distance functions. We show that such an implicit representation exists, and that we can compute it efficiently. 2012 ACM Subject Classification Computational Geometry