Approximating Minimization Diagrams and Generalized Proximity Search

TL;DR

This paper introduces a general framework for efficiently approximating minimization diagrams and generalized proximity search in high-dimensional spaces, enabling fast queries for complex distance functions including non-metric ones.

Contribution

The authors develop a versatile data-structure for approximating minimization diagrams applicable to a wide range of functions, including non-metric distances, with near-linear size and logarithmic query time.

Findings

Efficient approximation of minimization diagrams in ^d.

Applicability to weighted Voronoi diagrams and convex body metrics.

Works for non-triangle inequality functions, previously not well-understood.

Abstract

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.