Approximating Minimization Diagrams and Generalized Proximity Search

We investigate the classes of functions whose minimization diagrams can be approximated efficiently in \Re^d. We present a general framework and a data-structure that can be used to approximate the minimization diagram of such functions. The resulting data-structure has near linear size and can answer queries in logarithmic time. Applications include approximating the Voronoi diagram of (additively or multiplicatively) weighted points. Our technique also works for more general distance functions, such as metrics induced by convex bodies, and the nearest furthest-neighbor distance to a set of point sets. Interestingly, our framework works also for distance functions that do not comply with the triangle inequality. For many of these functions no near-linear size approximation was known before

Robust Proximity Search for Balls using Sublinear Space

Given a set of n disjoint balls b1, . . ., bn in IRd, we provide a data structure, of near linear size, that can answer (1 \pm \epsilon)-approximate kth-nearest neighbor queries in O(log n + 1/\epsilon^d) time, where k and \epsilon are provided at query time. If k and \epsilon are provided in advance, we provide a data structure to answer such queries, that requires (roughly) O(n/k) space; that is, the data structure has sublinear space requirement if k is sufficiently large

Approximating minimization diagrams and generalized proximity search

We investigate the classes of functions whose minimization diagrams can be approximated efficiently in ℝd. We present a general framework and a data-structure that can be used to approximate the minimization diagram of such functions. The resulting data-structure has near linear size and can answer queries in logarithmic time. Applications include approximating the Voronoi diagram of multiplicatively weighted points, but the new technique also works for more general distance functions. For example, we get such data-structures for metrics induced by convex bodies, and the nearest furthest-neighbor distance to a set of point sets. Interestingly, our framework also works for distance functions that do not obey the triangle inequality. For many of these functions no near linear size approximation was known before