Subtree Distances, Tight Spans and Diversities

TL;DR

This paper characterizes when a set of distances can be represented by a subtree structure in a real tree, extending previous finite case results to general distance spaces and diversities, with implications for metric embedding theory.

Contribution

It extends Hirai's finite case characterization of subtree representations to infinite and general distance spaces, introducing new tight span results for diversities.

Findings

Characterization of subtree representations for general distance spaces.

Proof that the tight span of a distance space is hyperconvex.

First characterization of tree-like tight spans for diversities.

Abstract

Metric embeddings are central to metric theory and its applications. Here we consider embeddings of a different sort: maps from a set to subsets of a metric space so that distances between points are approximated by minimal distances between subsets. Our main result is a characterization of when a set of distances $d(x,y)$ between elements in a set $X$ have a subtree representation, a real tree $T$ and a collection $\{S_x\}_{x \in X}$ of subtrees of~$T$ such that $d(x,y)$ equals the length of the shortest path in~$T$ from a point in $S_x$ to a point in $S_y$ for all $x,y \in X$. The characterization was first established for {\em finite} $X$ by Hirai (2006) using a tight span construction defined for distance spaces, metric spaces without the triangle inequality. To extend Hirai's result beyond finite $X$ we establish fundamental results of tight span theory for general distance spaces, including the surprising observation that the tight span of a distance space is hyperconvex. We apply the results to obtain the first characterization of when a diversity -- a generalization of a metric space which assigns values to all finite subsets of $X$, not just to pairs -- has a tight span which is tree-like.