Path Costs in Evolutionary Tree Reconstruction

This paper describes a dynamic programming algorithm to solve a family of problems in the reconstruction of evolutionary trees from protein sequence data, that of constructing "minimal" colorings. This dynamic programming formulation can be modified to efficiently enumerate the number of minimal colorings and thereby be used to calculate the average cost of any given edge, where the average is taken over the entire set of minimal colorings. An extension of our dynamic programming formulation allows for the calculation of average path costs amongst all minimal colorings. our results resolve questions raised in (Hendy and penny, 1987); in particular, we develop polynomial time procedures to find the minimum, maximum, and average (expected) cost of an edge, and more generally of a path, for a minimal coloring. Our algorithm is distinguished in its flexibility to address further distribution and statistical questions relating to the minimal colorings. Furthermore, the more general concept of calculating statistics describing the set of optimal solutions may be of interest in other combinatorial problems.