Approximate labelled subtree homeomorphism

Given two undirected trees T and P, the Subtree Homeomorphism Problem is to find whether T has a subtree t that can be transformed into P by removing entire subtrees, as well as repeatedly removing a degree-2 node and adding the edge joining its two neighbors. In this paper we extend the Subtree Homeomorphism Problem to a new optimization problem by enriching the subtree-comparison with node-to-node similarity scores. The new problem, called Approximate Labelled Subtree Homeomorphism (ALSH), is to compute the homeomorphic subtree of T which also maximizes the overall node-to-node resemblance. We describe an O (m² n / log m + m n log n) algorithm for solving ALSH on unordered, unrooted trees, where m and n are the number of vertices in P and T, respectively. We also give an O (m n) algorithm for rooted ordered trees and O (m n log m) and O (m n) algorithms for unrooted cyclically ordered and unrooted linearly ordered trees, respectively.