Gromov-Hausdorff Distance for Directed Spaces

TL;DR

This paper extends the Gromov-Hausdorff distance concept to directed metric spaces, introducing a new measure based on zigzag paths and exploring its properties and alternative formulations.

Contribution

It proposes a novel directed Gromov-Hausdorff distance for directed metric spaces and analyzes its differences from the classical case.

Findings

Directed Gromov-Hausdorff distance is not equivalent to classical versions.

Introduces new formulations based on d-maps and d-correspondences.

Provides a framework for comparing directed metric spaces.

Abstract

The Gromov-Hausdorff distance measures the similarity between two metric spaces by isometrically embedding them into an ambient metric space. We introduce an analogue of this distance for metric spaces endowed with directed structures. The directed Gromov-Hausdorff distance measures the distance between two extended metric spaces, where the new metric, defined on the same underlying space, is induced by the length of zigzag paths. This distance is then computed by isometrically embedding the directed metric spaces into an ambient directed space equipped with the zigzag distance. Analogously to the classical Gromov-Hausdorff distance, we also propose alternative formulations based on the distortion of d-maps and d-correspondences. However, unlike the classical case, these directed distances are not equivalent.