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

TL;DR

This paper proves that under certain probabilistic conditions, a large subset of a one-dimensional point set can be reconstructed up to symmetry, confirming a prior conjecture and analyzing a deterministic variant with optimal bounds.

Contribution

It confirms a conjecture on probabilistic reconstruction of large subsets in 1D point sets and establishes optimal bounds for a deterministic reconstruction method.

Findings

Reconstruction of a large subset with high probability when p=(1+ε)/n.

Deterministic reconstruction bounds are proven to be optimal.

Disproves a previous conjecture on the deterministic variant.

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.