Simple and Efficient Distribution-Sensitive Point Location, in Triangulations

We analyze, implement, and evaluate a distributionsensitive point location algorithm based on the classical Jump & Walk, called Keep, Jump, & Walk. For a batch of query points, the main idea is to use previous queries to improve the current one. In practice, Keep, Jump, & Walk is actually a very competitive method to locate points in a triangulation. Regarding point location in a Delaunay triangulation, we show how the Delaunay hierarchy can be used to answer, under some hypotheses, a query q with a O(log #(pq)) randomized expected complexity, where p is a previously located query and #(s) indicates the number of simplices crossed by the line segment s. The Delaunay hierarchy has O(n logn) time complexity and O(n) memory complexity in the plane, and under certain realistic hypotheses these complexities generalize to any finite dimension. Finally, we combine the good distribution-sensitive behavior of Keep, Jump, & Walk, and the good complexity of the Delaunay hierarchy, into a novel point location algorithm called Keep, Jump, & Climb. To the best of our knowledge, Keep, Jump, & Climb is the first practical distributionsensitive algorithm that works both in theory and in practice for Delaunay triangulations.