Paired compressed cover trees guarantee a near linear parametrized complexity for all $k$-nearest neighbors search in an arbitrary metric space

This paper studies the important problem of finding all $k$-nearest neighbors to points of a query set $Q$ in another reference set $R$ within any metric space. Our previous work defined compressed cover trees and corrected the key arguments in several past papers for challenging datasets. 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 time complexity for $k=1$. The current work fills all previous gaps and improves the nearest 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 $k\geq 1$.