Improved implementation of point location in general two-dimensional subdivisions

Abstract

We present a major revamp of the point-location data structure for general two-dimensional subdivisions via randomized incremental construction, implemented in Cgal, the Computational Geometry Algorithms Library. We can now guarantee that the constructed directed acyclic graph is of linear size and provides logarithmic query time. Via the construction of the Voronoi diagram for a given point set S of size n, this also enables nearest-neighbor queries in guaranteed O(logn) time. Another major innovation is the support of general unbounded subdivisions as well as subdivisions of two-dimensional parametric surfaces such as spheres, tori, cylinders. The implementation is exact, complete, and general, i.e., it can also handle non-linear subdivisions. Like the previous version, the data structure supports modifications of the subdivision, such as insertions and deletions of edges, after the initial preprocessing. A major challenge is to retain the expected O(n logn) preprocessing time while providing the above (deterministic) space and query-time guarantees. We describe efficient preprocessing algorithms, which explicitly verify the length L of the longest query path. However, instead of using L, our implementation is based on the depth D of G. Although we prove that the worst case ratio of D and L is Θ(n/logn), we conjecture, based on our experimental results, that this solution achieves expected O(n logn) preprocessing time. © 2012 Springer-Verlag.