Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gromov-Hausdorff Distance for Directed Spaces

Lisbeth Fajstrup, Brittany Terese Fasy, Wenwen Li, Lydia Mezrag, Tatum Rask, Francesca Tombari, Živa Urbančič

TL;DR

The paper extends Gromov--Hausdorff theory to directed spaces by introducing a zigzag distance $d_{zz}$ induced by a d-structure and defining a directed Gromov--Hausdorff distance $\,\vec{d}_{GH}$ via d-isometries. It also develops directed analogues based on distortion and codistortion of directed maps (and correspondences), highlighting that these three notions are non-equivalent in the directed setting. The results show that, when using the zigzag metric, the directed GH distance coincides with the classical GH distance on the induced metric spaces, while the directed distortions can yield strictly larger values and capture directional information. The framework enables comparisons of directed metric spaces (e.g., directed graphs, posets, and geometric models with direction) and lays groundwork for stability analyses and applications in networks and directed topology. Future work will compare the three notions, study stability, and explore practical uses in applied topology and data analysis.

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.

Gromov-Hausdorff Distance for Directed Spaces

TL;DR

The paper extends Gromov--Hausdorff theory to directed spaces by introducing a zigzag distance induced by a d-structure and defining a directed Gromov--Hausdorff distance via d-isometries. It also develops directed analogues based on distortion and codistortion of directed maps (and correspondences), highlighting that these three notions are non-equivalent in the directed setting. The results show that, when using the zigzag metric, the directed GH distance coincides with the classical GH distance on the induced metric spaces, while the directed distortions can yield strictly larger values and capture directional information. The framework enables comparisons of directed metric spaces (e.g., directed graphs, posets, and geometric models with direction) and lays groundwork for stability analyses and applications in networks and directed topology. Future work will compare the three notions, study stability, and explore practical uses in applied topology and data analysis.

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.
Paper Structure (14 sections, 15 theorems, 64 equations, 2 figures)

This paper contains 14 sections, 15 theorems, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Directed Spaces
  4. Path Length
  5. Some Operations on Directed Spaces
  6. Cartesian Product
  7. Disjoint Union
  8. Quotient
  9. Gromov--Hausdorff Distance
  10. Directed Gromov--Hausdorff Distance
  11. Zigzag Distance
  12. Directed Gromov--Hausdorff Distance
  13. Distortion and Codistortion of Directed Maps
  14. Discussion

Key Result

Proposition 2.1

Let $(X,d^X)$ and $(Y, d^Y)$ be metric spaces. Then, the following holds:

Figures (2)

  • Figure 1: Let $\vec{I}^2$ be the d-space where $\vec{P}(I^2)$ is given by the product order on $I^2$. The single green arrow (left) is a d-path from $(0,0)$ to $(1,1)$. The multiple green arrows (middle) give a zigzag path between $(1,0)$ and $(0,1)$, whereas the red arrow (right) is not a zigzag path.
  • Figure 2: Balls in the zigzag metric on a directed flat torus.

Theorems & Definitions (58)

  • Remark 2.1: The Unit Interval
  • Remark 2.2: Taking Subpaths
  • Definition 2.1
  • Remark 2.3
  • Definition 2.2
  • Remark 2.4
  • Example 2.1: Partially Ordered Space
  • Example 2.2: Reversed Path Space
  • Example 2.3: Directed Graphs
  • Example 2.4: Directed Hollow Hypercubes
  • ...and 48 more