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Let $G$ be a $3$-connected graph with $n$ vertices and $m$ edges. Let $mathbf{p}$ be a randomly chosen mapping of these $n$ vertices to the integer range $[1..2^b]$ for $bge m^2$. Let $mathbf{l}$ be the vector of $m$ Euclidean lengths of $G$s edges under $mathbf{p}$. In this paper, we show that, WHP over $mathbf{p}$, we can efficiently reconstruct both $G$ and $mathbf{p}$ from $mathbf{l}$. In contrast to this average case complexity, this reconstruction problem is NP-HARD in the worst case. In fact, even the labeled version of this problem (reconstructing $mathbf{p}$ given both $G$ and $mathbf{l}$) is NP-HARD. We also show that our results stand in the presence of small amounts of error in $mathbf{l}$, and in the real setting with approximate length measurements. Our method is based on older ideas that apply lattice reduction to solve certain SUBSET-SUM problems, WHP. We also rely on an algorithm of Seymour that can efficiently reconstruct a graph given an independence oracle for its matroid.
Let $mathbf{p}$ be a configuration of $n$ points in $mathbb{R}^d$ for some $n$ and some $d ge 2$. Each pair of points defines an edge, which has a Euclidean length in the configuration. A path is an ordered sequence of the points, and a loop is a pat
Let $mathbf{p}$ be a configuration of $n$ points in $mathbb{R}^d$ for some $n$ and some $d ge 2$. Each pair of points has a Euclidean length in the configuration. Given some graph $G$ on $n$ vertices, we measure the point-pair lengths corresponding t
A subset $X$ in the $d$-dimensional Euclidean space is called a $k$-distance set if there are exactly $k$ distances between two distinct points in $X$. Einhorn and Schoenberg conjectured that the vertices of the regular icosahedron is the only 12-poi
A bar-joint framework $(G,p)$ in a (non-Euclidean) real normed plane $X$ is the combination of a finite, simple graph $G$ and a placement $p$ of the vertices in $X$. A framework $(G,p)$ is globally rigid in $X$ if every other framework $(G,q)$ in $X$
In this note, we give a short solution of the kissing number problem in dimension three.