Cavity Matchings, Label Compressions, and Unrooted Evolutionary Trees

TL;DR

This paper introduces an efficient algorithm for finding maximum agreement subtrees in unrooted evolutionary trees, accommodating mixed tree structures and employing label compression for improved performance.

Contribution

It presents a novel recursive algorithm that achieves optimal time complexity for unrooted trees with mixed structures, extending previous rooted tree methods.

Findings

Achieves O(n^{1.5} log n) time complexity for unrooted trees.

Handles trees with directed and undirected edges simultaneously.

Uses label compression to efficiently compute maximum weight matchings.

Abstract

We present an algorithm for computing a maximum agreement subtree of two unrooted evolutionary trees. It takes O(n^{1.5} log n) time for trees with unbounded degrees, matching the best known time complexity for the rooted case. Our algorithm allows the input trees to be mixed trees, i.e., trees that may contain directed and undirected edges at the same time. Our algorithm adopts a recursive strategy exploiting a technique called label compression. The backbone of this technique is an algorithm that computes the maximum weight matchings over many subgraphs of a bipartite graph as fast as it takes to compute a single matching.