  Let $G$ be a $3$-connected ordered graph with $n$ vertices and $m$ edges. Let
$\mathbf{p}$ be a randomly chosen mapping of these $n$ vertices to the integer
range $\{1, 2,3, \ldots, 2^b\}$ for $b\ge m^2$. Let $\ell$ be the vector of $m$
Euclidean lengths of $G$'s edges under $\mathbf{p}$. In this paper, we show
that, with high probability over $\mathbf{p}$, we can efficiently reconstruct
both $G$ and $\mathbf{p}$ from $\ell$. This reconstruction problem is NP-HARD
in the worst case, even if both $G$ and $\ell$ are given. We also show that our
results stand in the presence of small amounts of error in $\ell$, and in the
real setting, with sufficiently accurate length measurements.
  Our method combines lattice reduction, which has previously been used to
solve random subset sum problems, with an algorithm of Seymour that can
efficiently reconstruct an ordered graph given an independence oracle for its
matroid.
