An Optimal Algorithm To Recognize Robinsonian Dissimilarities

Abstract

A dissimilarity D on a finite set S is said to be Robinsonian if S can be totally ordered in such a way that, for every i < j < k, D(i, j) ≤ D(i, k) and D(j, k) ≤ D(i, k). Intuitively, D is Robinsonian if S can be represented by points on a line. Recognizing Robinsonian dissimilarities has many applications in seriation and classification. In this paper, we present an optimal O(n 2) algorithm to recognize Robinsonian dissimilarities, where n is the cardinal of S. Our result improves the already known algorithms.

A Lex-BFS-based recognition algorithm for Robinsonian matrices

“…We consider the same matrix A as the one used in the example in Section 5 of [30]. However, since [30] handles Robinsonian dissimilarities, we first transform it into a similarity matrix and thus we use instead the matrix a max J − A, where a max denotes the largest entry in the matrix A.…”

“…The same sorting preprocessing was used by Seston [37], who improved the algorithm to O(n 2 log n) by constructing paths in the threshold graphs of A. Very recently, Préa and Fortin [30] presented a more sophisticated O(n 2 ) algorithm, which uses the fact that the maximal cliques of the graph G B are in one-to-one correspondence with the row/column indices of A. Roughly speaking, they use the algorithm from Booth and Leuker [3] to compute a first PQ-tree which they update throughout the algorithm.…”

“…One can also build the intersection graph G B of B, where the balls are the vertices and connecting two vertices if the corresponding balls intersect. Most of the existing algorithms are then based on the fact that a matrix A is Robinsonian if and only if the ball hypergraph H B is an interval hypergraph or, equivalently, if the intersection graph G B is an interval graph (see [28,30]). …”

An Optimization Parameter for Seriation of Noisy Data

“…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]).…”

“…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].…”