  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 $\left(1+\frac{1}{4}\right)$.
