W e present an optimal parallel algorithm t o construct the well-separated pair decomposition of a point set P in Zd. W e show how this leads to a deterministic opt imal O(1ogn) t ime parallel algorithm for finding the k nearest neighbors of each point in P, where k is a constant. W e discuss several additional applications of the well-separated pair decomposition for which we can derive fas t ...

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...

The fast multipole method (FMM) performs fast approximate kernel summation to a specified tolerance ǫ by using a hierarchical division of the domain, which groups source and receiver points into regions that satisfy local separation and the well-separated pair decomposition properties. While square tilings and quadtrees are commonly used in 2D, we investigate alternative tilings and associated ...

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...

Aimed at an audience of researchers and graduate students in computational geometry and algorithm design, this book uses the Geometric Spanner Network Problem to showcase a number of useful algorithmic techniques, data structure strategies, and geometric analysis techniques with many applications, practical and theoretical. The authors present rigorous descriptions of the main algorithms and th...

We construct Belyi maps having specified behavior at finitely many points. Specifically, for any curve C defined over Q, and any disjoint finite subsets S, T ⊂ C(Q), we construct a finite morphism φ : C → P such that φ ramifies at each point in S, the branch locus of φ is {0, 1,∞}, and φ(T ) ∩ {0, 1,∞} = ∅. This refines a result of Mochizuki’s. We also prove an analogous result over fields of p...