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This paper studies the important problem of finding all $k$-nearest neighborsto points of a query set $Q$ in another reference set $R$ within any metricspace. Our previous work defined compressed cover trees and corrected the keyarguments in several past papers for challenging datasets. In 2009 Ram, Lee,March, and Gray attempted to improve the time complexity by using pairs ofcover trees on the query and reference sets. In 2015 Curtin with the aboveco-authors used extra parameters to finally prove a time complexity for $k=1$.The current work fills all previous gaps and improves the nearest neighborsearch based on pairs of new compressed cover trees. The novel imbalanceparameter of paired trees allowed us to prove a better time complexity for anynumber of neighbors $k\geq 1$.

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