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arXiv · 2602.23122

Reconstructing a giant component of a point set in $\mathbb{R}$

Julien Portier
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Abstract

Let $V \subset \mathbb{R}$ be a finite set with $|V| = n $ and suppose we are given each pairwise distance independently with probability $p$. We show that if $p = (1+\epsilon)/n$, for some fixed $\epsilon >0$, then we can reconstruct a subset of size $\Omega_{\epsilon}(n)$, up to translation and reflection, with high probability. This confirms a conjecture posed by Gir\~ao, Illingworth, Michel, Powierski, and Scott. We also study a deterministic variant proposed by Benjamini and Tzalik. We show that if we are given $m$ distinct pairwise distances of a point set $V \subset \mathbb{R}$ with $|V|=n$, then we can reconstruct a subset of size $\Omega(m/ (n \log n)) $, up to translation and reflection. Moreover, we show that this is optimal, which also disproves a conjecture posed by Benjamini and Tzalik.

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Julien Portier. 2026-02-26. Reconstructing a giant component of a point set in $\mathbb{R}$. https://arxiv.org/abs/2602.23122

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