Computing Asymmetric Median Tree of Two Trees via Better Bipartite Matching Algorithm

Abstract

Maximum bipartite matching is a fundamental problem in computer science with many applications. The HopcroftKarp algorithm can find a maximum bipartite matching of a bipartite graph G in $O(\sqrt{n} m)$ time where n and m are the number of nodes and edges, respectively, in the bipartite graph G. However, when G is dense (i.e., $m=O(n^2)$), the Hopcroft–Karp algorithm runs in $O(n^{2.5})$ time.

In this paper, we consider a special case where the bipartite graph G is formed as a union of $\ell$ complete bipartite graphs. In such case, even when G has $O(n^2)$ edges, we show that a maximum bipartite graph can be found in $O(\sqrt{n} (n + \ell ) \log n)$ time.

We also describe how to apply our solution to compute the asymmetric median tree of two phylogenetic trees. We improve the running time from $O(n^{2.5})$ to $O(n^{1.5} \log ^3 n)$.