Fast range searching with Delaunay Triangulations

This paper studies the idea of answering range searching queries using simple data structures. The only data structure we need is the Delaunay Triangulation of the n input points. We show that when the query polygons are well-shaped the expected query time is O(n1/3 + Q + n ⁗ area(Q)), where Q is the size of the query polygon Q and n ⁗ area(Q) is the expected number of output points, which improves the best known average query time for range searching achieved by the bucketing method. Some empirical results, together with some results on generating simple polygons in a given domain, are also presented.