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 $\left(1+\frac{1}{4}\right)$.
翻译：Gromov-Hausdorff 距离 $( d ⁇ GH}) 证明是测量形状之间距离的有用尺度。 为了对紧凑子集 $X,Y\subset\mathb{R ⁇ d$ 约 $d ⁇ GH} 美元,我们调查它与美元H,iso} 美元下Hausdorf 距离(Euclidean isometries) 下 的 imimimimum Hausdorf 距离 $( d\geq 2 美元) 。 已经知道, $d ⁇ H, iso} 美元无法被一个恒定系数乘以$d ⁇ GH 。 但是, $d=1美元, 我们证明它与美元, i ⁇ leq\\ frac{5 ⁇ 4d} 美元的关系。 我们还证明这个界限很紧。 实际上,这会产生一个$(n\log{n) $-time lograg to ach to $( $d) $dq_GHHHH) $(1\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
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