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

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 BreadthFirst 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 × n nonnegative matrix with m nonzero entries, it terminates in n − 1 SFS sweeps, with overall running time O(n 2 + nm log n).

Perfect elimination orderings for symmetric matrices

“…Corneil [5] gives an algorithm for recognizing unit interval graphs based on three sweeps of Lex-BFS. In [18] a weighted generalization of Lex-BFS, called Similarity First Search (SFS), is introduced, which applies to symmetric matrices. It is shown in [18] that n sweeps of SFS can recognize Robinsonian matrices of size n by returning a Robinson ordering.…”

“…In [18] a weighted generalization of Lex-BFS, called Similarity First Search (SFS), is introduced, which applies to symmetric matrices. It is shown in [18] that n sweeps of SFS can recognize Robinsonian matrices of size n by returning a Robinson ordering. It is natural to ask whether SFS can also be used to find perfect elimination orderings.…”

A Structural Characterization for Certifying Robinsonian Matrices

“…Different algorithms were recently introduced in [14,15], based on a link between Robinsonian matrices and unit interval graphs (pointed out in [23]) and exploiting the fact that unit interval graphs can be recognized efficiently using a simple graph search algorithm, namely Lexicographic Breadth-First Search (Lex-BFS) (see [3,4]). The algorithm of [14] is based on expressing Robinsonian matrices as conic combinations of (adjacency matrices of) unit interval graphs and iteratively using Lex-BFS to check whether these are unit interval graphs; its overall running time is O(L(m + n)), where L is the number of distinct values in the matrix and m is its number of nonzero entries.…”

“…The algorithm of [14] is based on expressing Robinsonian matrices as conic combinations of (adjacency matrices of) unit interval graphs and iteratively using Lex-BFS to check whether these are unit interval graphs; its overall running time is O(L(m + n)), where L is the number of distinct values in the matrix and m is its number of nonzero entries. The algorithm of [15] relies on a new search algorithm, Similarity-First Search (SFS), which can be seen as a generalization of Lexicographic Breadth-First Search (Lex-BFS) to the setting of weighted graphs. The SFS algorithm runs in O(n + m log n) time and the recognition algorithm for Robinsonian matrices terminates after at most n iterations of SFS, thus with overall running time O(n 2 + nm log n) [15].…”

An Optimization Parameter for Seriation of Noisy Data

“…The problem of recognizing Robinsonian matrices, and finding their Robinson orderings, can be solved in polynomial time. See [19] for the first polynomial time algorithm for this problem, and [24,20,15,14] for more recent efficient algorithms. Most of these algorithms are based on a similar principle; namely the connection between Robinsonian similarity matrices and unit interval graphs ( [15,14]) or interval (hyper) graphs ( [19,24,20]).…”