kth Nearest Neighbor Sampling in the Plane

Let B be a square region in the plane. We give an efficient algorithm that takes a set P of n points from B, and produces a set M ⊂ B with the property that the distance to the second nearest point in M approximates the distance to the kth nearest point of P . That is, there are constants α, β ∈ R such that for all x ∈ B, we have αdP,k(x) ≤ dM,2(x) ≤ βdP,k(x), where dM,2 and dP,k denote the second nearest and kth nearest neighbor distance functions to M and P respectively. The algorithm is based on Delaunay refinement. The output set M also has the property that its Delaunay triangulation has a lower bound on the size of the smallest angle. The primary application is in statistical density estimation and robust geometric inference. 1 Robust Sizing Functions Since the pioneering work of Chew [3] and Ruppert [9], Delaunay refinement has remained an important approach to mesh generation (see for example the book [2]). The algorithm: Starting from the Delaunay triangulation of the input points P (restricted to a bounding box B), repeatedly add the circumcenter of any triangle whose circumradius is more than a constant times larger than the length if its shortest edge. Such a triangulation is said to have bounded radius-edge ratio and will be referred to as a quality mesh and will necessarily also have a lower bound on the size of the smallest angle. Ruppert showed that not only does this remarkably simple algorithm terminate, it produces a point set that is asymptotically optimal in size [9]. The key to Ruppert’s analysis is the so-called feature size function, which for a point set P is the distance to the second nearest point of P , denoted dP,2 : R → R. There is a constant γ such that output set M has the property that γdP,2 ≤ dM,2 ≤ dP,2. The optimality of the approach comes from proof that if M is the vertex set of a quality mesh containing P , ∗This work was partially supported under grants CCF-1464379 and CCF-1525978. 1Ruppert’s analysis also works for more general inputs including line segments as well. then |M | = Θ (∫