Frechet-Like Distances between Two Merge Trees

TL;DR

This paper introduces a Frechet-Like distance measure for rooted trees and merge trees, proves its computational complexity, and explores its relation to interleaving distance, extending the concept of curve similarity to tree structures.

Contribution

It defines a novel Frechet-Like distance for trees, proves its SNP-hardness, and relates it to the interleaving distance for merge trees.

Findings

Frechet-Like distance between trees is SNP-hard to compute.

Modified Frechet-Like distance can measure similarity between merge trees.

The paper establishes a relation between the Frechet-Like distance and interleaving distance.

Abstract

The purpose of this paper is to extend the definition of Frechet distance which measures the distance between two curves to a distance (Frechet-Like distance) which measures the similarity between two rooted trees. The definition of Frechet-Like distance is as follows: Tow men start from the roots of two trees. When they reach to a node with the degree of more than $2$, they construct $k-1$ men which $k$ is the outgoing degree of the node and each man monitor a man in another tree (there is a rope between them). The distance is the minimum length of the ropes between the men and the men whom are monitored and they all go forward (the geodesic distance between them to the root of the tree increases) and reach to the leaves of the trees. Here, I prove that the Frechet-Like distance between two trees is SNP-hard to compute. I modify the definition of Frechet-Like distance to measure the distance between tow merge trees, and I prove the relation between the interleaving distance and the modified Frechet-Like distance.