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We consider rigidity type problems on $\mathbb{R}, S^1$ in a non-genericsetting. Given $n$ distinct points $V = \{ v_1,...,v_n \}$ and a set of mutualdistances $\mathcal{P} \subseteq { V \choose 2}$, does $\mathcal{P}$ definesthe position of $V$ uniquely up to isometry?  We show that if $|\mathcal{P}| =\Omega(n^{3/2})$, one can determine theposition of a subset $V' \subseteq V$, $|V'| =\Omega(\frac{|\mathcal{P}|}{n})$. The main ingredient in the proof is showingthat dense graphs $G=(V,E)$ for which every two non-adjacent vertices have onlya few common neighbours must have large cliques.  We also consider random distance set $\mathcal{P}$. If the distance of eachpair of points is known with probability $p = \frac{C \ln(n)}{n}$ for largeenough $C$ then it is possible to reconstruct $V$ from the distances w.h.p. andprovide a randomized algorithm with linear expected running time that returnsan embedding to the line w.h.p.

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