Optimal evolutionary tree comparison by sparse dynamic programming

Sparsification of MAST shows that MAST is equivalent to Unary Weighted Bipartite Matching (UWBM) modulo an O(nc/sup /spl radic/(log n/) additive overhead) time algorithm for the special case of bounded degrees.

Abstract

In computational biology one is often interested in finding the concensus between different evolutionary trees for the same set of species. A popular formalizations is the Maximum Agreement Subtree Problem (MAST) defined as follows: given a set A and two rooted trees /spl Tscr//sub 0/ and /spl Tscr//sub 1/ leaf-labeled by the elements of A, find a maximum cardinality subset B of A such that the restrictions of /spl Tscr//sub 0/ and /spl Tscr//sub 1/ to B are topologically isomorphic. Polynomial time solutions exist, but they rely on a dynamic program with /spl Theta/(n/sup 2/) nodes-and /spl Theta/(n/sup 2/) running time. We sparsify this dynamic program and show that MAST is equivalent to Unary Weighted Bipartite Matching (UWBM) modulo an O(nc/sup /spl radic/(log n/) additive overhead. Applying the best bound for UWBM, we get an O(n/sup 1.5/ log n) algorithm for MAST. From our sparsification follows an O(nc/sup /spl radic/(log n/)) time algorithm for the special case of bounded degrees. Also here the best previous bound was /spl Theta/(n/sup 2/).>