Fast, precise and dynamic distance queries

We present an approximate distance oracle for a point set <i>S</i> with <i>n</i> points and doubling dimension λ. For every ε > 0, the oracle supports (1 + ε)-approximate distance queries in (universal) constant time, occupies space [ε<sup>−<i>O</i>(λ)</sup> + 2<sup><i>O</i>(λ log λ)</sup>]<i>n</i>, and can be constructed in [2<sup><i>O</i>(λ)</sup> log<sup>3</sup> <i>n</i> + ε<sup>−<i>O</i>(λ)</sup> +2<sup><i>O</i>(λ log λ)</sup>]<i>n</i> 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 <i>O</i>(1) query time and only 2<sup><i>O</i>(λ)</sup> log <i>n</i> + ε<sup>−<i>O</i>(λ)</sup> +2<sup><i>O</i>(λ log λ)</sup> update time. This is the first fully dynamic (1 + ε)-distance oracle.

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