Well-Separated Pair Decompositions for High-Dimensional Datasets

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...

Pruning spanners and constructing well-separated pair decompositions in the presence of memory hierarchies

Given a geometric graph G = (S,E) in R with constant dilation t, and a positive constant ε, we show how to construct a (1 + ε)-spanner of G with O(|S|) edges using O(sort(|E|)) memory transfers in the cacheoblivious model of computation. The main building block of our algorithm, and of independent interest in itself, is a new cache-oblivious algorithm for constructing a well-separated pair deco...

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...