Similarity-First Search: a new algorithm with application to Robinsonian matrix recognition

TL;DR

This paper introduces a new combinatorial algorithm, Similarity-First Search (SFS), for recognizing Robinsonian matrices, extending Lex-BFS to weighted graphs, with efficient multisweep performance and applications to unit interval graphs.

Contribution

The paper presents the SFS algorithm, a novel extension of Lex-BFS for weighted graphs, enabling efficient recognition of Robinsonian matrices and generalizing existing recognition algorithms.

Findings

Algorithm terminates in n-1 sweeps for n×n matrices.

Overall running time is O(n^2 + nm log n).

Extends recognition of unit interval graphs to weighted graphs.

Abstract

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 main ingredient we introduce a new algorithm, named Similarity-First-Search (SFS), which extends Lexicographic Breadth-First 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 $n\times n$ nonnegative matrix with $m$ nonzero entries, it terminates in $n-1$ SFS sweeps, with overall running time $O(n^2 +nm\log n)$.