Let G be a 3-connected ordered graph with n vertices and m edges. Let \(\textbf{p}\) 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 \(\textbf{p}\). In this paper, we show that, with high probability over \(\textbf{p}\), we can efficiently reconstruct both G and \(\textbf{p}\) 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.

令 G 为具有 n 个顶点和 m 个边的 3 连通有序图。令 \(\textbf{p}\) 为这 n 个顶点到整数范围 \(\{1, 2,3, \ldots , 2^b\}\) 的随机选择映射，其中 \(b\ge m ^2\)。令 \(\ell \) 为 \(\textbf{p}\) 下 G 边的 m 欧几里得长度的向量。在本文中，我们证明，在 \(\textbf{p}\) 上的高概率下，我们可以从 \(\ell \) 有效地重建 G 和 \(\textbf{p}\)。即使给出了 G 和 \(\ell \)，这个重建问题在最坏的情况下也是 NP-HARD。我们还表明，我们的结果在 \(\ell \) 中存在少量误差，并且在实际设置中具有足够准确的长度测量。我们的方法将之前用于解决随机子集和问题的格约化与 Seymour 算法结合起来，该算法可以在给定拟阵独立预言的情况下有效地重建有序图。