Reconstruction in one dimension from unlabeled Euclidean lengths

TL;DR

This paper demonstrates that, for a certain class of ordered graphs, it is highly probable to efficiently reconstruct the graph and vertex positions from unlabeled Euclidean edge lengths, even with some measurement errors.

Contribution

It introduces a probabilistic method combining lattice reduction and matroid reconstruction to solve the graph and vertex position recovery problem from Euclidean lengths.

Findings

High-probability successful reconstruction of graphs and vertex mappings.

Robustness of the method to small measurement errors.

Application of lattice reduction and matroid algorithms to geometric reconstruction.

Abstract

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.