Reconstruction in One Dimension from Unlabeled Euclidean Lengths

IF 1 2区 数学 Q1 MATHEMATICS
Robert Connelly, Steven J. Gortler, Louis Theran
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引用次数: 0

Abstract

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.

Abstract Image

从无标注的欧氏长度重建一维图像
让 G 是一个 3 连的有序图,有 n 个顶点和 m 条边。让 \(\textbf{p}\) 是随机选择的这 n 个顶点到整数范围 \(\{1, 2,3, \ldots , 2^b\}\) 的映射,为 \(b\ge m^2\)。让 \(\ell \)成为 G 的边在\(textbf{p}\)下的 m 欧氏长度向量。在本文中,我们证明了在\(\textbf{p}\)上,我们可以以很高的概率从\(\ell \)有效地重建 G 和\(\textbf{p}\)。即使 G 和 (textbf{p})都是给定的,这个重构问题在最坏的情况下也是 NP-HARD。我们还证明了在\(\ell \)中存在少量误差的情况下,以及在真实环境中,在长度测量足够精确的情况下,我们的结果都是成立的。我们的方法结合了之前用于解决随机子集和问题的晶格还原法和西摩算法,后者可以在给定矩阵的独立性神谕的情况下高效地重建有序图。
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来源期刊
Combinatorica
Combinatorica 数学-数学
CiteScore
1.90
自引率
0.00%
发文量
45
审稿时长
>12 weeks
期刊介绍: COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are - Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups). - Combinatorial optimization. - Combinatorial aspects of geometry and number theory. - Algorithms in combinatorics and related fields. - Computational complexity theory. - Randomization and explicit construction in combinatorics and algorithms.
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