A Lex-BFS-based recognition algorithm for Robinsonian matrices

TL;DR

This paper introduces a new polynomial-time recognition algorithm for Robinsonian matrices using Lex-BFS and a novel characterization involving unit interval graphs, improving efficiency in seriation applications.

Contribution

The paper presents a simple, divide-and-conquer Lex-BFS-based algorithm for recognizing Robinsonian matrices, based on a new characterization via straight enumerations of unit interval graphs.

Findings

Algorithm runs in O(d(n+m)) time, where d is the recursion depth.

Efficient recognition of Robinsonian matrices from sparse data.

Applicable to matrices with nonnegative symmetric entries.

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 $n\times n$ 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 $A$.