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
-; l
₂
-, and l
_∞
-norms. The previous best algorithms for computing path-difference distances all have O(n
²
) running time. In this paper, we show how to compute the l
₁
-norm path-difference distance in O(n log
²
n) time and how to compute the l
₂
- and l
_∞
-norm path-difference distances in O(n log n) time. By extending the presented algorithms, we also show that the l
_p
-norm path-difference distance can be computed in O(pn log2 n) time for any positive integer p. In addition, when the integer p is even, we show that the distance can be computed in O(p
²
n log n) time as well.