Improved Implementation of Point Location in General Two-Dimensional Subdivisions

TL;DR

This paper introduces an improved, exact, and general point-location data structure for 2D subdivisions, supporting unbounded and non-linear surfaces, with guaranteed logarithmic query time and efficient preprocessing.

Contribution

It presents a revamped point-location structure with linear size, supporting complex subdivisions and modifications, and guarantees logarithmic query time in a general setting.

Findings

Linear size directed acyclic graph (DAG) for subdivisions.

Logarithmic query time for point location and nearest-neighbor queries.

Efficient preprocessing with expected O(n log n) time.

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 G 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(log n) 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 log n) preprocessing time while providing the above (deterministic) space and query-time guarantees. We describe an efficient preprocessing algorithm, which explicitly verifies the length L of the longest query path in O(n log n) time. 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 Theta(n/log n), we conjecture, based on our experimental results, that this solution achieves expected O(n log n) preprocessing time.