A note on consecutive ones in a binary matrix

Education and Information Technologies

The study proves the existence of an algorithm to receive all elements of a class of binary matrices without obtaining redundant elements, e. g. without obtaining binary matrices that do not belong to the class. This makes it possible to avoid checking whether each of the objects received possesses the necessary properties. This significantly improves the efficiency of the algorithm in terms of the criterion of time. Certain useful educational effects related to the analysis of such problems in programming classes are also pointed out.

2001

Matrices with the consecutive ones property and interval graphs are important notations in the field of applied mathematics. We give a theoretical picture of them in first part. We present the earliest work in interval graphs and matrices with the consecutive ones property pointing out the close relation between them. We pay attention to Tucker's structure theorem on matrices with the consecutive ones property as an essential step that requires a deep considerations. Later on we concentrate on some recent work characterizing the matrices with the consecutive ones property and matrices related to them in the terms of interval digraphs as the latest and most interesting outlook on our topic. Within this framework we introduce a classiffcation of matrices with consecutive ones property and matrices related to them. We describe the applications of matrices with the consecutive ones property and interval graphs in different fields. We make sure to give a general view of application a...

2016

An algorithm for obtaining all n × n binary matrices having exactly 2 units in every row and every column is described in the paper. After analysing the work of the algorithm a formula for calculating the number of these matrices has been obtained. This formula is known and has been obtained using other methods, which by their nature are purely analytical and not constructive. Thus a new, constructive proof of this known formula has been obtained.

Springer Proceedings in Mathematics & Statistics, 2015

A 0-1 matrix where in each row the 1s occur consecutively is said to have the consecutive 1s property. Since this property is scarcely fulfilled in real problems and since it is non-deterministic polynomial time (NP)-hard to find the nearest arrangement to the property, we give a quadratic assignment formulation for optimizing the distance to the property. The formulation carries over the sign case with 0, +1, −1 matrix entries. We discuss and compare this exact approach, for both signed and unsigned cases, with spectral approaches based on bisection instead. Keywords 0-1 matrices • Consecutive 1s property • Consecutive sign property •

Zeitschrift für Operations Research, 1986

This note deals with the problem of permuting elements within columns of a real matrix so as to minimize a real-valued function of row sums. The special case dealing with minimization of maximum row sum has been studied by several authors 6, recently. Here we are concerned primarily with the case in which the matrix has two columns only and the function is Schur-convex.

Electronic Journal of Linear Algebra, 2005

Generalizing the Bruhat order for permutations (so for permutation matrices), a Bruhat order is defined for the class of m by n (0, 1)-matrices with a given row and column sum vector. An algorithm is given for constructing a minimal matrix (with respect to the Bruhat order) in such a class. This algorithm simplifies in the case that the row and column sums are all equal to a constant k. When k = 2 or k = 3, all minimal matrices are determined. Examples are presented that suggest such a determination might be very difficult for k ≥ 4.

2016

This work examines the concept of S-permutation matrices, namely n^2 × n^2 permutation matrices containing a single 1 in each canonical n × n subsquare (block). The article suggests a formula for counting mutually disjoint pairs of n^2 × n^2 S-permutation matrices in the general case by restricting this task to the problem of finding some numerical characteristics of the elements of specially defined for this purpose factor-set of the set of n × n binary matrices. The paper describe an algorithm that solves the main problem. To do that, every n× n binary matrix is represented uniquely as a n-tuple of integers.

We discuss an equivalence relation on the set of square binary matrices with the same number of 1's in each row and each column. Each binary matrix is represented using ordered n-tuples of natural numbers. We give a few starting values of integer sequences related to the discussed problem. The obtained sequences are new and they are not described in the On-Line Encyclopedia of Integer Sequences (OEIS). We show a relationship between some particular values of the parameters and the Fibonacci sequence.

International Journal of Modern Education and Computer Science, 2013

Some techniques for the use of bitwise operations are described in the article. As an example, an open problem of isomorphism-free generations of combinatorial objects is discussed. An equivalence relation on the set of square binary matrices having the same number of units in each row and each column is defined. Each binary matrix is represented using ordered n-tuples of natural numbers. It is shown how by using the bitwise operations can be implemented an algorithm that gets canonical representatives which are extremal elements of equivalence classes relative to a double order on the set of considered objects.

Journal of mathematical sciences and applications, 2013

An algorithm for obtaining all n n × binary matrices having exactly 2 units in every row and every column is described in the paper. After analysing the work of the algorithm a formula for calculating the number of these matrices has been obtained. This formula is known and has been obtained using other methods, which by their nature are purely analytical and not constructive. Thus a new, constructive proof of this known formula has been obtained.

Journal of Computer and System Sciences, 2010

We develop an algorithmically useful refinement of a forbidden submatrix characterization of 0/1-matrices fulfilling the Consecutive Ones Property (C1P). This characterization finds applications in new polynomial-time approximation algorithms and fixed-parameter tractability results for the NP-hard problem to delete a minimum number of rows or columns from a 0/1-matrix such that the remaining submatrix has the C1P. algorithms), NI 369/4. 2 The C1P can be defined symmetrically for columns; we focus on rows here. 3 The certifying algorithm of McConnell [32] decides whether a given 0/1-matrix has the C1P or not. If it does not have the C1P, then the algorithm generates a "certificate", that is, a small (compared to the size of the input matrix) proof that can be verified by a "fast and uncomplicated" polynomial-time algorithm (for more details about such certificates see ).

An algorithm for obtaining all n\times n binary matrices having exactly 2 units in every row and every column is described in the paper. After analysing the work of the algorithm a formula for calculating the number of these matrices has been obtained. This formula is known and has been obtained using other methods, which by their nature are purely analytical and not constructive. Thus a new, constructive proof of this known formula has been obtained.

Annals of the Institute of Statistical Mathematics, 2009

A k × n array with entries from a q-letter alphabet is called a t-covering array if each t × n submatrix contains amongst its columns each one of the q t different words of length t that can be produced by the q letters. In the present article we use a probabilistic approach based on an appropriate Markov chain embedding technique, to study a t-covering problem where, instead of looking at all possible t × n submatrices, we consider only submatrices of dimension t × n with its rows being consecutive rows of the original k × n array. Moreover, an exact formula is established for the probability distribution function of the random variable, which enumerates the number of deficient submatrices (i.e., submatrices with at least one missing word, amongst their columns), in the case of a k × n binary matrix (q = 2) obtained by realizing kn Bernoulli variables.

Discrete Applied Mathematics, 2012

Motivated by problems of comparative genomics and paleogenomics, in [6] the authors introduced the Gapped Consecutive-Ones Property Problem (k, δ)-C1P: given a binary matrix M and two integers k and δ, can the columns of M be permuted such that each row contains at most k blocks of ones and no two consecutive blocks of ones are separated by a gap of more than δ zeros. The classical C1P problem, which is known to be polynomial is equivalent to the (1, 0)-C1P problem. They showed that the (2, δ)-C1P Problem is NP-complete for all δ ≥ 2 and that the (3, 1)-C1P problem is NP-complete. They also conjectured that the (k, δ)-C1P Problem is NP-complete for k ≥ 2, δ ≥ 1 and (k, δ) = (2, 1). Here, we prove that this conjecture is true. The only remaining case is the (2, 1)-C1P Problem, which could be polynomial-time solvable.

ArXiv, 2012

In this article we discuss the presentation of a random binary matrix using sequence of whole nonnegative numbers. We examine some advantages and disadvantages of this presentation as an alternative of the standard presentation using two-dimensional array. It is shown that the presentation of binary matrices using ordered n-tuples of natural numbers makes the algorithms faster and saves a lot of memory. In this work we use object-oriented programming using the syntax and the semantic of C++ programming language.