Dynamic Data Structures for $k$-Nearest Neighbor Queries

TL;DR

This paper introduces dynamic data structures for efficient $k$-nearest neighbor queries in the plane, supporting insertions and queries with improved worst-case time bounds, including geodesic and algebraic distances.

Contribution

It presents a general query algorithm for $k$-NN that reduces query time and adapts static structures for dynamic use, including geodesic and algebraic distances.

Findings

Achieves $O(rac{ ext{log}^2 n}{ ext{log} ext{log} n} + k)$ worst-case query time for Euclidean $k$-NN.

Supports fully dynamic updates with polylogarithmic time complexity.

Extends to geodesic and algebraic distance functions in planar settings.

Abstract

Our aim is to develop dynamic data structures that support $k$-nearest neighbors ($k$-NN) queries for a set of $n$ point sites in the plane in $O(f(n) + k)$ time, where $f(n)$ is some polylogarithmic function of $n$. The key component is a general query algorithm that allows us to find the $k$-NN spread over $t$ substructures simultaneously, thus reducing an $O(tk)$ term in the query time to $O(k)$. Combining this technique with the logarithmic method allows us to turn any static $k$-NN data structure into a data structure supporting both efficient insertions and queries. For the fully dynamic case, this technique allows us to recover the deterministic, worst-case, $O(\log^2n/\log\log n +k)$ query time for the Euclidean distance claimed before, while preserving the polylogarithmic update times. We adapt this data structure to also support fully dynamic \emph{geodesic} $k$-NN queries among a set of sites in a simple polygon. For this purpose, we design a shallow cutting based, deletion-only $k$-NN data structure. More generally, we obtain a dynamic planar $k$-NN data structure for any type of distance functions for which we can build vertical shallow cuttings. We apply all of our methods in the plane for the Euclidean distance, the geodesic distance, and general, constant-complexity, algebraic distance functions.