Reconstructing a point set from a random subset of its pairwise distances

TL;DR

This paper determines the probability threshold at which a set of points on a line can be fully reconstructed from randomly known pairwise distances, improving previous bounds and analyzing the process's hitting time.

Contribution

It establishes the sharp threshold for reconstructing the entire point set from random pairwise distances and analyzes the process's hitting time for partial reconstruction.

Findings

Sharp threshold at p = (log n)/n for full reconstruction

Weak threshold at 1/n for reconstructing a linear proportion

Improves previous results by Benjamini and Tzalik

Abstract

Let $V$ be a set of $n$ points on the real line. Suppose that each pairwise distance is known independently with probability $p$. How much of $V$ can be reconstructed up to isometry? We prove that $p = (\log n)/n$ is a sharp threshold for reconstructing all of $V$ which improves a result of Benjamini and Tzalik. This follows from a hitting time result for the random process where the pairwise distances are revealed one-by-one uniformly at random. We also show that $1/n$ is a weak threshold for reconstructing a linear proportion of $V$.