A new near-linear time algorithm for k-nearest neighbor search using a compressed cover tree

TL;DR

This paper introduces a simplified compressed cover tree and a near-linear time algorithm for efficiently finding k-nearest neighbors in metric spaces, improving upon previous methods with better theoretical guarantees.

Contribution

It presents a new compressed cover tree construction algorithm and a k-nearest neighbor search algorithm with proven near-linear time complexity, filling gaps in prior proofs.

Findings

Constructed a compressed cover tree in O(n log n) time.

Developed a k-NN search algorithm with O(m(k+log n) log k) complexity.

Achieved near-linear time performance with theoretical guarantees.

Abstract

Given a reference set $R$ of $n$ points and a query set $Q$ of $m$ points in a metric space, this paper studies an important problem of finding $k$-nearest neighbors of every point $q \in Q$ in the set $R$ in a near-linear time. In the paper at ICML 2006, Beygelzimer, Kakade, and Langford introduced a cover tree on $R$ and attempted to prove that this tree can be built in $O(n\log n)$ time while the nearest neighbor search can be done in $O(n\log m)$ time with a hidden dimensionality factor. This paper fills a substantial gap in the past proofs of time complexity by defining a simpler compressed cover tree on the reference set $R$. The first new algorithm constructs a compressed cover tree in $O(n \log n)$ time. The second new algorithm finds all $k$-nearest neighbors of all points from $Q$ using a compressed cover tree in time $O(m(k+\log n)\log k)$ with a hidden dimensionality factor depending on point distributions of the given sets $R,Q$ but not on their sizes.