Determining a Points Configuration on the Line from a Subset of the Pairwise Distances

TL;DR

This paper studies how to uniquely determine point configurations on the line from a subset of pairwise distances, establishing extremal bounds and analyzing reconstruction from random distances with high probability.

Contribution

It provides new extremal bounds for point determination from distances and analyzes the probabilistic reconstruction of points from random distance sets.

Findings

Large subsets of points can be uniquely reconstructed from rac{|\u211d|}{n} of the distances when rac{|\u211d|}{n^{3/2}} distances are known.

Reconstruction of points is highly probable when each pair's distance is known independently with probability rac{C \, ext{log}(n)}{n}.

The paper introduces a linear-time randomized algorithm for embedding points on the line from partial distance information.

Abstract

We investigate rigidity-type problems on the real line and the circle in the non-generic setting. Specifically, we consider the problem of uniquely determining the positions of $n$ distinct points $V = {v_1, \ldots, v_n}$ given a set of mutual distances $\mathcal{P} \subseteq {V \choose 2}$. We establish an extremal result: if $|\mathcal{P}| = \Omega(n^{3/2})$, then the positions of a large subset $V' \subseteq V$, where large means $|V'| = \Omega(\frac{|\mathcal{P}|}{n})$, can be uniquely determined up to isometry. As a main ingredient in the proof, which may be of independent interest, we show that dense graphs $G=(V,E)$ for which every two non-adjacent vertices have only a few common neighbours must have large cliques. Furthermore, we examine the problem of reconstructing $V$ from a random distance set $\mathcal{P}$. We establish that if the distance between each pair of points is known independently with probability $p = \frac{C \ln(n)}{n}$ for some universal constant $C > 0$, then $V$ can be reconstructed from the distances with high probability. We provide a randomized algorithm with linear expected running time that returns the correct embedding of $V$ to the line with high probability. Since we posted a preliminary version of the paper on arxiv, follow-up works have improved upon our results in the random setting. Gir\~ao, Illingworth, Michel, Powierski, and Scott proved a hitting time result for the first moment at which an time at which one can reconstruct $V$ when $\mathcal{P}$ is revealed using the Erd\"os--R\'enyi evolution, our extremal result lies in the heart of their argument. Montgomery, Nenadov and Szab\'o resolved a conjecture we posed in a previous version and proved that w.h.p a graph sampled from the Erd\"os--R\'enyi evolution becomes globally rigid in $\mathbb{R}$ at the moment it's minimum degree is $2$.