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 $\R^d$ 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2011
 DOI:
 10.48550/arXiv.1109.0312
 arXiv:
 arXiv:1109.0312
 Bibcode:
 2011arXiv1109.0312G
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Data Structures and Algorithms;
 E.0
 EPrint:
 24 pages, 4 figures. To appear at the 22nd International Symposium on Algorithms and Computation (ISAAC 2011)