An Algorithm for Finding a Representation of a Subtree Distance

Abstract

Generalizing the concept of tree metric, Hirai (2006) introduced the concept of subtree distance. A mapping $d:X\times X \rightarrow \mathbb {R}_+$ is called a subtree distance if there exist a weighted tree T and a family $\{T_x\,|\,x \in X\}$ of subtrees of T indexed by the elements in X such that $d(x,y)=d_T(T_x,T_y)$, where $d_T(T_x,T_y)$ is the distance between $T_x$ and $T_y$ in T. Hirai (2006) gave a characterization of subtree distances which corresponds to Buneman’s four-point condition (1974) for the tree metrics. Using this characterization, we can decide whether or not a given matrix is a subtree distance in O$(n^4)$ time. However, the existence of a polynomial time algorithm for finding a tree and subtrees representing a subtree distance has been an open question. In this paper, we show an O$(n^3)$ time algorithm that finds a tree and subtrees representing a given subtree distance.

K. Ando—This work was supported by JSPS KAKENHI Grant Number 15K00033.