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arXiv · 2401.01882

Reconstructing almost all of a point set in $\mathbb{R}^d$ from randomly revealed pairwise distances

Douglas Barnes·Jan Petr·Julien Portier·Benedict Randall Shaw·Alan Sergeev
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Abstract

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 Gir\~ao, 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, Bollob\'as, and Morris.

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Douglas Barnes, Jan Petr, Julien Portier, Benedict Randall Shaw, Alan Sergeev. 2024-01-03. Reconstructing almost all of a point set in $\mathbb{R}^d$ from randomly revealed pairwise distances. https://arxiv.org/abs/2401.01882

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