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, denoted ALSH (Approximate Labelled Subtree Homeomorphism) is to compute the homeomorphic subtree of T which also maximizes the overall node-to-node resemblance. We describe an O(mn/ log m + mn 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(mn) algorithm for rooted ordered trees.