Fully Retroactive Approximate Range and Nearest Neighbor Searching

TL;DR

This paper introduces fully retroactive data structures for approximate range and nearest neighbor searches in high-dimensional spaces, supporting efficient updates and queries at any point in the timeline.

Contribution

It presents the first fully retroactive data structures for approximate range and nearest neighbor searching with polylogarithmic update and query times.

Findings

Supports insertions and deletions in $O(\log n)$ amortized time.

Enables $(1+\epsilon)$-approximate range reporting with $O(\log n + k)$ query time.

Provides $(1+\epsilon)$-approximate nearest neighbor queries in $O(\log n)$ time.

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.