A Cubic-Time 2-Approximation Algorithm for rSPR Distance

Due to hybridization events in evolution, studying two different genes of a set of species may yield two related but different phylogenetic trees for the set of species. In this case, we want to measure the dissimilarity of the two trees. The rooted subtree prune and regraft (rSPR) distance of the two trees has been used for this purpose. The problem of computing the rSPR distance of two given trees has many applications but is NP-hard. The previously best approximation algorithm for rSPR distance achieves a ratio of 2 in polynomial time and its analysis is based on the duality theory of linear programming. In this paper, we present a cubic-time approximation algorithm for rSPR distance that achieves a ratio of 2. Our algorithm is based on the notion of key and several structural lemmas; its analysis is purely combinatorial and explicitly uses a search tree for computing rSPR distance exactly.