Fast, precise and dynamic distance queries

Yair Bartal, Lee-Ad Gottlieb, Tsvi Kopelowitz, Moshe Lewenstein, Liam Roditty

Abstract: We present an approximate distance oracle for a point set S with n points and doubling dimension Λ. For every ε > 0, the oracle supports (1 + ε)-approximate distance queries in (universal) constant time, occupies space [ε−O(Λ) + 2O(Λ log Λ)]n, and can be constructed in [2O(Λ) log3 n + ε−O(Λ) + 2O(Λ log Λ)]n expected time. This improves upon the best previously known constructions, presented by Har-Peled and Mendel [13]. Furthermore, the oracle can be made fully dynamic with expected O(1) query time and only 2O(Λ) log n + ε−O(Λ) + 2O(Λ log Λ) update time. This is the first fully dynamic (1 + ε)-distance oracle.