Algorithms for Dynamic Closest Pair and n-Body Potential Fields

We present a general technique for dynamizing certain problems posed on point sets in Euclidean space for any xed dimension d. This technique applies to a large class of structurally similar algorithms, presented previously by the authors , that make use of the well-separated pair decomposition. We prove eecient worst-case complexity for maintaining such computations under point insertions and deletions, and apply the technique to several problems posed on a set P containing n points. In particular, we show how to answer a query for any point x that returns a constant-size set of points, a subset of which consists of all points in P that have x as a nearest neighbor. We then show how to use such queries to maintain the closest pair of points in P. We also show how to dynamize the fast multipole method, a technique for approximating the potential eld of a set of point charges. All our algorithms use the algebraic model that is standard in computational geometry, and have worst-case deterministic O(log 2 n) complexity for updates and queries.