Computing the rooted triplet distance between galled trees by counting triangles

We consider a generalization of the rooted triplet distance between two phylogenetic trees to two phylogenetic networks. We show that if each of the two given phylogenetic networks is a so-called galled tree with n   leaves then the rooted triplet distance can be computed in o(n2.687)o(n2.687) time. Our upper bound is obtained by reducing the problem of computing the rooted triplet distance between two galled trees to that of counting monochromatic and almost-monochromatic triangles in an undirected, edge-colored graph. To count different types of colored triangles in a graph efficiently, we extend an existing technique based on matrix multiplication and obtain several new algorithmic results that may be of independent interest: (i) the number of triangles in a connected, undirected, uncolored graph with m   edges can be computed in o(m1.408)o(m1.408) time; (ii) if G is a connected, undirected, edge-colored graph with n vertices and C is a subset of the set of edge colors then the number of monochromatic triangles of G with colors in C   can be computed in o(n2.687)o(n2.687) time; and (iii) if G is a connected, undirected, edge-colored graph with n vertices and R   is a binary relation on the colors that is computable in O(1)O(1) time then the number of R-chromatic triangles in G   can be computed in o(n2.687)o(n2.687) time.