A new compressed cover tree guarantees a near linear parameterized complexity for all $k$-nearest neighbors search in metric spaces

This paper studies the classical problem of finding all $k$ nearest neighbors to points of a query set $Q$ in another reference set $R$ within any metric space. The well-known work by Beygelzimer, Kakade, and Langford in 2006 introduced cover trees and claimed to guarantee a near linear time complexity in the size $|R|$ of the reference set for $k=1$. Our previous work defined compressed cover trees and corrected the key arguments for $k\geq 1$ and previously unknown challenging data cases. In 2009 Ram, Lee, March, and Gray attempted to improve the time complexity by using pairs of cover trees on the query and reference sets. In 2015 Curtin with the above co-authors used extra parameters to finally prove a similar complexity for $k = 1$. Our work fills all previous gaps and substantially improves the neighbor search based on pairs of new compressed cover trees. The novel imbalance parameter of paired trees allowed us to prove a better time complexity for any number of neighbors $k\geq 1$.