Robinsonian Matrices: Recognition Challenges

Abstract

Ultrametric inequality is involved in different operations on (dis)similarity matrices. Its coupling with a compatible ordering leads to nice interpretations in seriation problems. We accurately review the interval graph recognition problem for its tight connection with recognizing a dense Robinsonian dissimilarity (precisely, in the anti-ultrametric case). Since real life matrices are prone to errors or missing entries, we address the sparse case and make progress towards recognizing sparse Robinsonian dissimilarities with lexicographic breadth first search. The ultrametric inequality is considered from the same graph point of view and the intimate connection between cocomparability graph and dense Robinsonian similarity is established. The current trend in recognizing special graph structures is examined in regard to multiple lexicographic search sweeps. Teaching examples illustrate the issues addressed for both dense and sparse symmetric matrices.