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Reconstruction in one dimension from unlabeled Euclidean lengths

Robert Connelly, Steven J. Gortler, Louis Theran

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.

Reconstruction in one dimension from unlabeled Euclidean lengths

Abstract

Let be a -connected ordered graph with vertices and edges. Let be a randomly chosen mapping of these vertices to the integer range for . Let be the vector of Euclidean lengths of 's edges under . In this paper, we show that, with high probability over , we can efficiently reconstruct both and from . This reconstruction problem is NP-HARD in the worst case, even if both and are given. We also show that our results stand in the presence of small amounts of error in , 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.

Paper Structure

This paper contains 23 sections, 18 theorems, 49 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Connectivity assumption
  3. Integer setting
  4. Errors in input
  5. Real setting
  6. Basic ideas
  7. Related LLL Work
  8. Graph theoretic preliminaries
  9. Graphs, orientations, and measurements
  10. The algorithm
  11. The sampling model
  12. Reconstruct $S^{\sigma_{\bf p}}$
  13. More Error/Fewer Bits
  14. Proof of Proposition \ref{['prop:prob']}
  15. Reconstruct $G$
  16. ...and 8 more sections

Key Result

theorem \oldthetheorem

Let $G$ be a $3$-connected (resp. $2$-connected) graph with $m$ edges. Let $b \ge m^2$. Let each $p_i$ be chosen independently and uniformly at random from $[2^b]$. Then there is a polynomial time algorithm that succeeds, WHP, on the unlabeled (resp. labeled) reconstruction problems.

Figures (2)

  • Figure 1: The graphs, (a) and (d) are related by a $2$-swap. Note that the edge lengths in one dimension are unchanged under a 2-swap.
  • Figure 2: Number of bits required for the LLL step to recover the cycle space with probability $0.9$ vs. number of vertices. Log-Log on right.

Theorems & Definitions (43)

  • theorem \oldthetheorem
  • theorem \oldthetheorem
  • theorem \oldthetheorem
  • definition \oldthetheorem
  • definition \oldthetheorem
  • lemma \oldthetheorem
  • proof
  • definition \oldthetheorem
  • definition \oldthetheorem
  • Proposition \oldthetheorem: See e.g, oxley
  • ...and 33 more