Approximating Gromov-Hausdorff Distance in Euclidean Space
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| _version_ | 1866910458831699968 |
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| author | Majhi, Sushovan Vitter, Jeffrey Wenk, Carola |
| author_facet | Majhi, Sushovan Vitter, Jeffrey Wenk, Carola |
| contents | 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 $\left(1+\frac{1}{4}\right)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_13008 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Approximating Gromov-Hausdorff Distance in Euclidean Space Majhi, Sushovan Vitter, Jeffrey Wenk, Carola Metric Geometry Computational Geometry 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 $\left(1+\frac{1}{4}\right)$. |
| title | Approximating Gromov-Hausdorff Distance in Euclidean Space |
| topic | Metric Geometry Computational Geometry |
| url | https://arxiv.org/abs/1912.13008 |