An Optimal Reconciliation Algorithm for Gene Trees with Polytomies

Reconciliation is a method widely used to infer the evolutionary relationship between the members of a gene family. It consists of comparing a gene tree with a species tree, and interpreting the incongruence between the two trees as evidence of duplication and loss. In the case of binary rooted trees, linear-time algorithms have been developed for the duplication, loss, and mutation (duplication + loss) costs. However, a strict prerequisite to reconciliation is to have a gene tree free from error, as few misplaced edges may lead to a completely different result in terms of the number and position of inferred duplications and losses. How should the weak edges be handled? One reasonable answer is to transform the binary gene tree into a non-binary tree by removing each weak edge and collapsing its two incident vertices into one. The created polytomies are “apparent” as they do not reflect a true simultaneous divergence of many copies from a common ancestor, but rather a lack of resolution. In this paper, we consider the problem of reconciling a non-binary rooted gene tree

G

with a binary rooted species tree

S

, were polytomies of

G

are assumed to be apparent. We give a linear-time algorithm that infers a reconciliation of minimum mutation cost between a binary refinement of a polytomy and

S

, improving on the best known result, which is cubic. This implies a straightforward generalization to a gene tree

G

with nodes of arbitrary degree, that runs in time

O

(|

S

||

G

|), which is shown to be an optimal algorithm.