A Lex-BFS-Based Recognition Algorithm for Robinsonian Matrices

Abstract

Robinsonian matrices arise in the classical seriation problem and play an important role in many applications where unsorted similarity (or dissimilarity) information must be reordered. We present a new polynomial time algorithm to recognize Robinsonian matrices based on a new characterization of Robinsonian matrices in terms of straight enumerations of unit interval graphs. The algorithm is simple and is based essentially on lexicographic breadth-first search (Lex-BFS), using a divide-and-conquer strategy. When applied to a nonnegative symmetric n × n matrix with m nonzero entries and given as a weighted adjacency list, it runs in O(d(n + m)) time, where d is the depth of the recursion tree, which is at most the number of distinct nonzero entries of A.

A Structural Characterization for Certifying Robinsonian Matrices

“…The former does not hold by (13), while the latter does not hold since (x, y, z) is not Robinson (as observed just before Claim 3.10). Thus (14) holds. Then, by x = ψa y ≺ ψa z, (14) implies z = c. We next claim that y ψ b x.…”

“…Thus (14) holds. Then, by x = ψa y ≺ ψa z, (14) implies z = c. We next claim that y ψ b x. For this assume for contradiction that x ≺ ψ b y.…”

Similarity-First Search: A New Algorithm with Application to Robinsonian Matrix Recognition

“…Relations (19) and (20) hold for j = 1 and we use the shifting property (17) to get the general case. Analogously for relation (21), since it holds for j = 5.…”

Uniform Embeddings for Robinson Similarity Matrices

“…Atkins et al [1] gave a spectral algorithm for the general seriation problem. Laurent and Seminaroti [9,10] give a combinatorial algorithm that generalizes the algorithm from [4]. A general overview of the seriation problem and its applications can be found in [11].…”