Optimal evolutionary tree comparison by sparse dynamic programming

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/).<
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