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.
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