Reconstructing almost all of a point set in $\mathbb{R}^d$ from randomly revealed pairwise distances
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917745416732672 |
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| author | Barnes, Douglas Petr, Jan Portier, Julien Shaw, Benedict Randall Sergeev, Alan |
| author_facet | Barnes, Douglas Petr, Jan Portier, Julien Shaw, Benedict Randall Sergeev, Alan |
| contents | Let $V$ be a set of $n$ points in $\mathbb{R}^d$, and suppose that the distance between each pair of points is revealed independently with probability $p$. We study when this information is sufficient to reconstruct large subsets of $V$, up to isometry.
Strong results for $d=1$ have been obtained by Girão, Illingworth, Michel, Powierski, and Scott. In this paper, we investigate higher dimensions, and show that if $p>n^{-2/(d+4)}$, then we can reconstruct almost all of $V$ up to isometry, with high probability. We do this by relating it to a polluted graph bootstrap percolation result, for which we adapt the methods of Balogh, Bollobás, and Morris. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01882 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reconstructing almost all of a point set in $\mathbb{R}^d$ from randomly revealed pairwise distances Barnes, Douglas Petr, Jan Portier, Julien Shaw, Benedict Randall Sergeev, Alan Combinatorics Metric Geometry Probability Let $V$ be a set of $n$ points in $\mathbb{R}^d$, and suppose that the distance between each pair of points is revealed independently with probability $p$. We study when this information is sufficient to reconstruct large subsets of $V$, up to isometry. Strong results for $d=1$ have been obtained by Girão, Illingworth, Michel, Powierski, and Scott. In this paper, we investigate higher dimensions, and show that if $p>n^{-2/(d+4)}$, then we can reconstruct almost all of $V$ up to isometry, with high probability. We do this by relating it to a polluted graph bootstrap percolation result, for which we adapt the methods of Balogh, Bollobás, and Morris. |
| title | Reconstructing almost all of a point set in $\mathbb{R}^d$ from randomly revealed pairwise distances |
| topic | Combinatorics Metric Geometry Probability |
| url | https://arxiv.org/abs/2401.01882 |