Separated-pair approximation and separated-pair pair-density functional theory

The well-separated pair decomposition for balls

Given a real number t > 1, a geometric t-spanner is a geometric graph for a point set in R with straight lines between vertices such that the ratio of the shortest-path distance between every pair of vertices in the graph (with Euclidean edge lengths) to their actual Euclidean distance is at most t. An imprecise point set is modeled by a set R of regions in R. If one chooses a point in each reg...

I/O-Efficient Well-Separated Pair Decomposition

We present an external-memory algorithm to compute a well-separated pair decomposition (WSPD) of a given point set S in Rd in O(sort(N)) I/Os, where N is the number of points in S and sort(N) denotes the I/O-complexity of sorting N items. (Throughout this paper, we assume that the dimension d is fixed). As applications of the WSPD, we show how to compute a linear-size t-spanner for S within the...

Dynamic Well-Separated Pair Decomposition Made Easy

Global Illumination Using Well-Separated Pair Decomposition

Instant radiosity methods rely on using a large number of virtual point lights (VPLs) to approximate global illumination. Efficiency considerations require grouping the VPLs into a small number of clusters that are treated as individual lights with respect to each point to be shaded. Two examples of clustering algorithms are Lightcuts [WFA∗05] and LightSlice [OP11]. In this work we use the noti...

Lecture 19 : Well Separated Pair Decomposition

Let P be a set of n points in R d. For any parameter s ≥ 1, A, B ⊆ P are said to be s-separated or well separated if A and B can be enclosed by congruent d-spheres S A and S B of radius r such that the minimum distance between S A and S B is at least sr. There is a stronger notion of well-separated point sets. For any parameter s ≥ 1, A, B ⊆ P are said to be s-separated if the bounding boxes of...