Fast, precise and dynamic distance queries

TL;DR

This paper introduces a new approximate distance oracle for point sets with doubling dimension, offering constant-time queries, efficient space and construction, and the first fully dynamic version with minimal update time.

Contribution

It presents the first fully dynamic (1+ε)-approximate distance oracle with constant query time and efficient update, improving previous static and dynamic solutions.

Findings

Supports (1+ε)-approximate distance queries in constant time.

Constructed with space complexity depending on ε and λ.

First fully dynamic (1+ε)-distance oracle with low update time.

Abstract

We present an approximate distance oracle for a point set S with n points and doubling dimension {\lambda}. For every {\epsilon}>0, the oracle supports (1+{\epsilon})-approximate distance queries in (universal) constant time, occupies space [{\epsilon}^{-O({\lambda})} + 2^{O({\lambda} log {\lambda})}]n, and can be constructed in [2^{O({\lambda})} log3 n + {\epsilon}^{-O({\lambda})} + 2^{O({\lambda} log {\lambda})}]n expected time. This improves upon the best previously known constructions, presented by Har-Peled and Mendel. Furthermore, the oracle can be made fully dynamic with expected O(1) query time and only 2^{O({\lambda})} log n + {\epsilon}^{-O({\lambda})} + 2^{O({\lambda} log {\lambda})} update time. This is the first fully dynamic (1+{\epsilon})-distance oracle.