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Approximating Gromov-Hausdorff Distance in Euclidean Space

Sushovan Majhi, Jeffrey Vitter, Carola Wenk

TL;DR

This work establishes a tight relationship between the Gromov-Hausdorff distance $d_{GH}$ and the Hausdorff distance under Euclidean isometries $d_{H,iso}$ for compact subsets of the real line. It proves the upper bound $d_{H,iso}(X,Y) \le \frac{5}{4} d_{GH}(X,Y)$ in 1D and shows the bound is tight, enabling a polynomial-time $O(n\log n)$ algorithm to approximate $d_{GH}$ within a factor of $1.25$ for finite 1D datasets. The authors develop a comprehensive case analysis distinguishing no-double-crossings and double-crossings, using translations and flips of $Y$ to construct near-optimal correspondences and bound distortions. They also discuss Nearest Neighbor correspondences and provide insight into why $d_{GH}$ and $d_{H,iso}$ can differ in higher dimensions, and outline open questions for extending results to higher dimensions. Overall, the paper delivers a concrete, efficient method for 1D GH distance approximation with a tight theoretical bound, informing shape comparison and related computational geometry tasks.

Abstract

The Gromov-Hausdorff distance $(d_{GH})$ proves to be a useful distance measure between shapes. In order to approximate $d_{GH}$ for compact subsets $X,Y\subset\mathbb{R}^d$, we look into its relationship with $d_{H,iso}$, the infimum Hausdorff distance under Euclidean isometries. As already known for dimension $d\geq 2$, the $d_{H,iso}$ cannot be bounded above by a constant factor times $d_{GH}$. For $d=1$, however, we prove that $d_{H,iso}\leq\frac{5}{4}d_{GH}$. We also show that the bound is tight. In effect, this gives rise to an $O(n\log{n})$-time algorithm to approximate $d_{GH}$ with an approximation factor of $\left(1+\frac{1}{4}\right)$.

Approximating Gromov-Hausdorff Distance in Euclidean Space

TL;DR

This work establishes a tight relationship between the Gromov-Hausdorff distance and the Hausdorff distance under Euclidean isometries for compact subsets of the real line. It proves the upper bound in 1D and shows the bound is tight, enabling a polynomial-time algorithm to approximate within a factor of for finite 1D datasets. The authors develop a comprehensive case analysis distinguishing no-double-crossings and double-crossings, using translations and flips of to construct near-optimal correspondences and bound distortions. They also discuss Nearest Neighbor correspondences and provide insight into why and can differ in higher dimensions, and outline open questions for extending results to higher dimensions. Overall, the paper delivers a concrete, efficient method for 1D GH distance approximation with a tight theoretical bound, informing shape comparison and related computational geometry tasks.

Abstract

The Gromov-Hausdorff distance proves to be a useful distance measure between shapes. In order to approximate for compact subsets , we look into its relationship with , the infimum Hausdorff distance under Euclidean isometries. As already known for dimension , the cannot be bounded above by a constant factor times . For , however, we prove that . We also show that the bound is tight. In effect, this gives rise to an -time algorithm to approximate with an approximation factor of .

Paper Structure

This paper contains 39 sections, 20 theorems, 131 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Our Contribution
  4. Gromov-Hausdorff vs Hausdorff Distance under Isometry
  5. Comparing $d_{GH}$ with $d_{H,iso}$
  6. Nearest Neighbor Correspondence
  7. Approximating Gromov-Hausdorff Distance in ${\mathbb R}^1$
  8. No Double Crossings
  9. ($\pmb{{\cal I}_1}$)
  10. ($\pmb{{\cal I}_3}$)
  11. ($\pmb{{\cal I}_6}$)
  12. ($\pmb{{\cal J}_1}$)
  13. ($\pmb{{\cal J}_3}$)
  14. ($\pmb{{\cal J}_5}$)
  15. Double Crossings
  16. ...and 24 more sections

Key Result

Lemma 1.7

For any two compact metric spaces $(X,d_X)$ and $(Y,d_Y)$, the following holds:

Figures (12)

  • Figure 1: The points of $X$ and $Y$ are shown in green and yellow, respectively. The correspondence with minimum distortion is shown by the blue edges and the nearest neighbor correspondence (see Definition \ref{['def:gh-cnn']}) is shown by the red edges.
  • Figure 2: On the left, the (sorted) $X=\{x_1,x_2\}$ and $Y=\{y_1,y_2,y_3\}$ are identified as subsets of the top and the bottom lines respectively. The points of $X$ are shown in green, and the points of $Y$ are shown in yellow. We visualize the correspondence ${\cal C}=\{(x_1,y_1),(x_2,y_2),(x_2,y_3)\}$ by the red edges between the respective points. Also, the edges $(x_1,y_1)$ and $(x_2,y_2)$ are crossing. On the right, the distortion $D$ of a correspondence is attained by the pairs $(x',y')$ and $(x,y)$.
  • Figure 3: The points of $X$ and $Y$ are shown in green and yellow, respectively, on two copies of the real line. The optimal correspondence is shown by the red edges. The distortion for the correspondence is $2\delta$, consequently $d_{GH}(X,Y)=\delta$. We also note that the optimal correspondence is not crossing free.
  • Figure 4: The no double crossing case is shown. The sets $A,A'$ and $B,B'$ are shown as subsets of the thick, blue intervals in the top and bottom.
  • Figure 5: No double crossing Case (1) is shown. The set $A$ is a subset of the thick, blue interval in the top.
  • ...and 7 more figures

Theorems & Definitions (54)

  • Definition 1.1: Directed Hausdorff Distance
  • Definition 1.2: Hausdorff Distance
  • Definition 1.3: Isometry
  • Definition 1.4: Gromov-Hausdorff Distance gromov_metric_2007
  • Definition 1.5: Correspondence
  • Definition 1.6: (Additive) Distortion of Correspondence
  • Lemma 1.7
  • Definition 2.1: Hausdorff under Isometry
  • Remark 2.2
  • Example 2.3: $d_{GH}<d_{H,iso}$ in ${\mathbb R}^2$
  • ...and 44 more