19 research outputs found
Combinatorial algorithms for the seriation problem
In this thesis we study the seriation problem, a combinatorial problem arising in data analysis, which asks to sequence a set of objects in such a way that similar objects are ordered close to each other. We focus on the combinatorial structure and properties of Robinsonian matrices, a special class of structured matrices which best achieve the seriation goal. Our contribution is both theoretical and practical, with a particular emphasis on algorithms. In Chapter 2 we introduce basic concepts about graphs, permutations and proximity matrices used throughout the thesis. In Chapter 3 we present Robinsonian matrices, discussing their characterizations and recognition algorithms existing in the literature. In Chapter 4 we discuss Lexicographic Breadth-First search (Lex-BFS), a special graph traversal algorithm used in multisweep algorithms for the recognition of several classes of graphs. In Chapter 5 we introduce a new Lex-BFS based algorithm to recognize Robinsonian matrices, which is derived from a new characterization of Robinsonian matrices in terms of straight enumerations of unit interval graphs. In Chapter 6 we introduce the novel Similarity-First Search algorithm (SFS), a weighted version of Lex-BFS which we use in a multisweep algorithm for the recognition of Robinsonian matrices. In Chapter 7 we model the seriation problem as an instance of Quadratic Assignment Problem (QAP) and we show that if the data has a Robinsonian structure, then one can find an optimal solution for QAP using a Robinsonian recognition algorithm. In Chapter 8 we discuss how to solve the seriation problem when the data does not have a Robinsonian structure, by finding a Robinsonian approximation of the original data. Finally, in Chapter 9 we discuss some experiments which we have carried out in order to compare the performance of the algorithms introduced in the thesis
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 re-
ordered. 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 non-
negative symmetric n × 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
Perfect Elimination Orderings for Symmetric Matrices
We introduce a new class of structured symmetric matrices by extending the
notion of perfect elimination ordering from graphs to weighted graphs or
matrices. This offers a common framework capturing common vertex elimination
orderings of monotone families of chordal graphs, Robinsonian matrices and
ultrametrics. We give a structural characterization for matrices that admit
perfect elimination orderings in terms of forbidden substructures generalizing
chordless cycles in graphs.Comment: 16 pages, 3 figure
A structural characterization for certifying Robinsonian matrices
A symmetric matrix is Robinsonian if its rows and columns can be simultaneously reordered in such a way that entries are monotone nondecreasing in rows and columns when moving toward the diagonal. The adjacency matrix of a graph is Robinsonian precisely when the graph is a unit interval graph, so that Robinsonian matrices form a matrix analogue of the class of unit interval graphs. Here we provide a structural characterization for Robinsonian matrices in terms of forbidden substructures, extending the notion of asteroidal triples to weighted graphs. This implies the known characterization of unit interval graphs and leads to an efficient algorithm for certifying that a matrix is not Robinsonian
A structural characterization for certifying robinsonian matrices
A symmetric matrix is Robinsonian if its rows and columns can be
simultaneously reordered in such a way that entries are monotone nondecreasing
in rows and columns when moving toward the diagonal. The adjacency matrix of a
graph is Robinsonian precisely when the graph is a unit interval graph, so that
Robinsonian matrices form a matrix analogue of the class of unit interval
graphs. Here we provide a structural characterization for Robinsonian matrices
in terms of forbidden substructures, extending the notion of asteroidal triples
to weighted graphs. This implies the known characterization of unit interval
graphs and leads to an efficient algorithm for certifying that a matrix is not
Robinsonian.Comment: 21 pages, 1 figur