Fully Retroactive Approximate Range and Nearest Neighbor Searching

Abstract

We describe fully retroactive dynamic data structures for approximate range reporting and approximate nearest neighbor reporting. We show how to maintain, for any positive constant $d$, a set of $n$ points in ${\mathbb{R}}^d$ indexed by time such that we can perform insertions or deletions at any point in the timeline in $O(\log n)$ amortized time. We support, for any small constant $\epsilon >0$, $(1+\epsilon )$-approximate range reporting queries at any point in the timeline in $O(\log n+k)$ time, where $k$ is the output size. We also show how to answer $(1+\epsilon )$-approximate nearest neighbor queries for any point in the past or present in $O(\log n)$ time.

24 pages, 4 figures. To appear at the 22nd International Symposium on Algorithms and Computation (ISAAC 2011)