Fast Algorithms for Computing Path-Difference Distances

Tree comparison metrics are an important tool for the study of phylogenetic trees. Path-difference distances measure the dissimilarity between two phylogenetic trees (on the same set of taxa) by comparing their path-length vectors. Various norms can be applied to this distance. Three important examples are the $l_{1}\text{-},\;l_{2}\text{-}$l1-,l2-, and $l_{{\infty }}$l∞-norms. The previous best algorithms for computing path-difference distances all have $O(n^{2})$O(n2) running time. In this paper, we show how to compute the $l_{1}$l1-norm path-difference distance in $O(n\;{\log}^{2}\;n)$O(nlog2n) time and how to compute the $l_{2}$l2- and $l_{{\infty }}$l∞-norm path-difference distances in $O(n\;{\log}\;n)$O(nlogn) time. By extending the presented algorithms, we also show that the $l_{p}$lp-norm path-difference distance can be computed in $O(pn\;{\log}^{2}\;n)$O(pnlog2n) time for any positive integer $p$p. In addition, when the integer $p$p is even, we show that the distance can be computed in $O(p^{2}n\;{\log}\;n)$O(p2nlogn) time as well.