  The phylogenetic Mean Pairwise Distance (MPD) is one of the most popular
measures for computing the phylogenetic distance between a given group of
species. More specifically, for a phylogenetic tree T and for a set of species
R represented by a subset of the leaf nodes of T, the MPD of R is equal to the
average cost of all possible simple paths in T that connect pairs of nodes in
R.
  Among other phylogenetic measures, the MPD is used as a tool for deciding if
the species of a given group R are closely related. To do this, it is important
to compute not only the value of the MPD for this group but also the
expectation, the variance, and the skewness of this metric. Although efficient
algorithms have been developed for computing the expectation and the variance
the MPD, there has been no approach so far for computing the skewness of this
measure.
  In the present work we describe how to compute the skewness of the MPD on a
tree T optimally, in Theta(n) time; here n is the size of the tree T. So far
this is the first result that leads to an exact, let alone efficient,
computation of the skewness for any popular phylogenetic distance measure.
Moreover, we show how we can compute in Theta(n) time several interesting
quantities in T that can be possibly used as building blocks for computing
efficiently the skewness of other phylogenetic measures.

Quantitative Methods;Computational Engineering, Finance, and Science;Data Structures and Algorithms