Approximating Gromov-Hausdorff distance in Euclidean space

Abstract

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