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

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 $2$.