Computing the agreement of trees with bounded degrees

Abstract

The Maximum Agreement Subtree (MAST) is a well-studied measure of similarity of leaf-labelled trees. There are several variants, depending on the number of trees, their degrees, and whether or not they are rooted. It turns out that the different variants display very different computational behavior. We address the common situation in biology, where the involved trees are rooted and of bounded degree, most typically simply being binary.

• We give an algorithm which computes the MAST of k trees on n species where some tree has maximum degree d in time O(kn ³+n ^d). This improves the Amir and Keselman FOCS '94 O(kn ^d+1+n ^2d) bound.

• We give an algorithm which computes the MAST of 2 trees with degree bound d in time O(n√d log³ n). This should be contrasted with the Farach and Thorup FOCS '94 $O(nc^{\sqrt {log n} } + n\sqrt d \log n)$ bound. Thus, for d a constant, we get an O(n log³ n) bound, replacing the previous $O(nc^{\sqrt {log n} } )$bound.

Both of our algorithms are quite simple, relying on the combinatorial structure of the problem, rather than on advanced data structures.

Part of this work was done when this author was visiting DIMACS and University of Warwick.