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

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 ∈ 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(nlog n) time while the nearest neighbor search can be done in O(nlog 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.