A new near-linear time algorithm for k-nearest neighbor search using a compressed cover tree
Authors: Yury Elkin, Vitaliy Kurlin

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 of Q in the set R in a near-linear time. In the paper at ICML 2006, Beygelzimer, Kakade, and Langford introduced a cover tree and attempted to prove that this tree can be built in O(n log n) time while the nearest neighbor search can be done O(n log m) time with a hidden dimensionality factor. In 2015, section 5.3 of Curtin’s PhD pointed out that the proof of the latter claim can have a serious gap in time complexity estimation. A paper at TopoInVis 2022 reported explicit counterexamples for a key step in the proofs of both claims. The past obstacles will be overcome by 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 sets R,Q but not on their sizes.