The k-Delaunay tree

Abstract

We have shown that the k-Delaunay Tree of n points can be constructed in O(nlogn+k ³ n) (resp. O(k ^[d+1/2]+1 n ^[d+1/2]) randomized expected time in the plane (resp. in d space). Its randomized expected size is O(k ² n) (resp. O(k ^[d+1/2] n ^[d+1/2]). The k-Delaunay Tree allows to compute the order≤k Voronoi diagrams of n points within the same bounds. Any order l≤k Voronoi diagram can be deduced from the k-Delaunay Tree in time proportional to its size, which is O(ln) in two dimensions. Moreover, the k-Delaunay Tree can be used to find the l nearest neighbors of a given point.

The structure could be extended to deal with k-levels of general arrangements. But in this case, it is not sufficient to store the Delaunay Tree augmented with additional neighborhood relations, since the hyperplanes are not supposed to have one face on the lowest level. We must store parent pointers between old and new triangles of all widths.

An important point is that these results hold whatever the point distribution may be.

The algorithm is simple and, moreover, the numerical computations involved are also quite simple : they consist mostly of comparisons of (squared) distances in order to check if a point lies inside or outside a ball. Experimental results, for uniform as well as degenerate distributions of points, have provided strong evidence that this algorithm is very effective in practice, for small values of k. For large values of k, a similar structure, based on the order k furthest neighbors Voronoi diagrams, could be derived. It would provide results similar to the ones above to construct all order≥n-k Voronoi diagrams and to find l furthest neighbors for l≤k.