We present a new efficient combinatorial algorithm for recognizing if a given symmetric matrix is Robinsonian, i.e., if its rows and columns can be simultaneously reordered so that entries are monotone nondecreasing in rows and columns when moving toward the diagonal. As the main ingredient we introduce a new algorithm, named Similarity-First Search (SFS), which extends lexicographic breadth-rst search (Lex-BFS) to weighted graphs and which we use in a multisweep algorithm to recognize Robinsonian matrices. Since Robinsonian binary matrices correspond to unit interval graphs, our algorithm can be seen as a generalization to weighted graphs of the 3-sweep Lex-BFS algorithm of Corneil for recognizing unit interval graphs. This new recognition algorithm is extremely simple and it exploits new insight on the combinatorial structure of Robinsonian matrices. For an nxn nonnegative matrix with m nonzero entries, it terminates in n-1 SFS sweeps, with overall running time O(n2 + nmlog n).

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SIAM Journal on Discrete Mathematics
Networks and Optimization

Laurent, M, & Seminaroti, M. (2017). Similarity-First Search: A new algorithm with application to Robinsonian matrix recognition. SIAM Journal on Discrete Mathematics, 31(3), 1765–1800. doi:https://dx.doi.org/10.1137/16M1056791