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A Lex-BFS-based recognition algorithm for Robinsonian matrices

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 $A$.