A Lex-BFS-based recognition algorithm for Robinsonian matrices

Laurent, Monique; Seminaroti, Matteo

doi:10.1016/j.dam.2017.01.027

Cited by 12 publications

(3 citation statements)

References 25 publications

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“…Interval graphs are boxicity 1 graphs since they can be represented as overlapping intervals in one dimension. Deciding if a network is an interval graph can be done in linear time 34,35 . Practically, however, a method based on the fact that the clique-membership matrix of an interval graph has the C1P 36 may be preferred.…”

Section: Political Niches and Overlap Networkmentioning

confidence: 99%

Legislators’ roll-call voting behavior increasingly corresponds to intervals in the political spectrum

Schoch

Brandes

2020

Sci Rep

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Scaling techniques such as the well known NOMINATE position political actors in a low dimensional space to represent the similarity or dissimilarity of their political orientation based on roll-call voting patterns. Starting from the same kind of data we propose an alternative, discrete, representation that replaces positions (points and distances) with niches (boxes and overlap). In the one-dimensional case, this corresponds to replacing the left-to-right ordering of points on the real line with an interval order. As it turns out, this seemingly simplistic one-dimensional model is sufficient to represent the similarity of roll-call votes by U.S. senators in recent years. In a historic context, however, low dimensionality represents the exception which stands in contrast to what is suggested by scaling techniques.

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Section: Political Niches and Overlap Networkmentioning

confidence: 99%

Legislators’ roll-call voting behavior increasingly corresponds to intervals in the political spectrum

Schoch

Brandes

2020

Sci Rep

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“…See [19] for the first polynomial time algorithm for this problem, and [24,20,15,14] for more recent efficient algorithms. Most of these algorithms are based on a similar principle; namely the connection between Robinsonian similarity matrices and unit interval graphs ( [15,14]) or interval (hyper) graphs ( [19,24,20]). A spectral algorithm based on reordering the matrix according to the components of the second eigenvector of the Laplacian, or the Fiedler vector, was given in [1], and was then applied to the ranking problem in [11].…”

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confidence: 99%

“…The problem of recognizing Robinsonian matrices, and finding their Robinson orderings, can be solved in polynomial time. See [19] for the first polynomial time algorithm for this problem, and [24,20,15,14] for more recent efficient algorithms. Most of these algorithms are based on a similar principle; namely the connection between Robinsonian similarity matrices and unit interval graphs ( [15,14]) or interval (hyper) graphs ( [19,24,20]).…”

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confidence: 99%

An Optimization Parameter for Seriation of Noisy Data

Ghandehari¹,

Janssen²

2019

SIAM J. Discrete Math.

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A square symmetric matrix is a Robinson similarity matrix if entries in its rows and columns are non-decreasing when moving towards the diagonal. A Robinson similarity matrix can be viewed as the affinity matrix between objects arranged in linear order, where objects closer together have higher affinity. We define a new parameter, Γ 1 , which measures how badly a given matrix fails to be Robinson similarity. Namely, a matrix is Robinson similarity precisely when its Γ 1 attains zero, and a matrix with small Γ 1 is close (in the normalized ℓ 1 -norm) to a Robinson similarity matrix. Moreover, both Γ 1 and the Robinson similarity approximation can be computed in polynomial time. Thus, our parameter recognizes Robinson similarity matrices which are perturbed by noise, and can therefore be a useful tool in the problem of seriation of noisy data.

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Uniform Embeddings for Robinson Similarity Matrices

Janssen

Zhang

2021

Lecture Notes in Computer Science

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A Robinson similarity matrix is a symmetric matrix where all entries in all rows and columns are increasing towards the diagonal. A Robinson matrix can be decomposed into the weighted sum of k adjacency matrices of a nested family of unit interval graphs. We study the problem of finding an embedding which gives a simultaneous unit interval embedding for all graphs in the family. This is called a uniform embedding. We give a necessary and sufficient condition for the existence of a uniform embedding, derived from paths in an associated graph. We also give an efficient combinatorial algorithm to find a uniform embedding or give proof that it does not exist, for the case where k = 2.

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A Lex-BFS-based recognition algorithm for Robinsonian matrices

Cited by 12 publications

References 25 publications

Legislators’ roll-call voting behavior increasingly corresponds to intervals in the political spectrum

Legislators’ roll-call voting behavior increasingly corresponds to intervals in the political spectrum

An Optimization Parameter for Seriation of Noisy Data

Uniform Embeddings for Robinson Similarity Matrices

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