A Dual of the Rectangle-Segmentation Problem for Binary Matrices

We consider the problem to decompose a binary matrix into a small number of binary matrices whose 1-entries form a rectangle. We show that the linear relaxation of this problem has an optimal integral solution corresponding to a well known geometric result on the decomposition of rectilinear polygons.

Download Full-text

A Divide-and-Conquer Approach for Solving Fuzzy Max-Archimedeant-Norm Relational Equations

Abstract and Applied Analysis ◽

10.1155/2014/315290 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10

Author(s):

Jun-Lin Lin ◽

Hung-Chjh Chuan ◽

Laksamee Khomnotai

Keyword(s):

Original Problem ◽

Relevant Literature ◽

The Other ◽

Divide And Conquer ◽

Fuzzy Relational Equations ◽

Original Matrix ◽

Binary Matrix ◽

The Matrix ◽

Binary Matrices ◽

Test Matrices

A system of fuzzy relational equations with the max-Archimedeant-norm composition was considered. The relevant literature indicated that this problem can be reduced to the problem of finding all the irredundant coverings of a binary matrix. A divide-and-conquer approach is proposed to solve this problem and, subsequently, to solve the original problem. This approach was used to analyze the binary matrix and then decompose the matrix into several submatrices such that the irredundant coverings of the original matrix could be constructed using the irredundant coverings of each of these submatrices. This step was performed recursively for each of these submatrices to obtain the irredundant coverings. Finally, once all the irredundant coverings of the original matrix were found, they were easily converted into the minimal solutions of the fuzzy relational equations. Experiments on binary matrices, with the number of irredundant coverings ranging from 24 to 9680, were also performed. The results indicated that, for test matrices that could initially be partitioned into more than one submatrix, this approach reduced the execution time by more than three orders of magnitude. For the other test matrices, this approach was still useful because certain submatrices could be partitioned into more than one submatrix.

Download Full-text

RELATIONAL TOPOLOGICAL MAP

International Journal of Computational Intelligence and Applications ◽

10.1142/s146902681000294x ◽

2010 ◽

Vol 09 (04) ◽

pp. 353-370

Author(s):

LAZHAR LABIOD ◽

NISTOR GROZAVU ◽

YOUNÈS BENNANI

Keyword(s):

Qualitative Data ◽

A Priori ◽

Large Data ◽

Large Data Sets ◽

Data Sets ◽

Topological Order ◽

Topological Map ◽

Binary Matrix ◽

Relational Analysis ◽

Binary Matrices

This paper introduces a relational topological map model, dedicated to multidimensional categorial data (or qualitative data) arising in the form of a binary matrix or a sum of binary matrices. This approach is based on the principle of Kohonen's model (conservation of topological order) and uses the Relational Analysis formalism by maximizing a modified Condorcet criterion. This proposed method is developed from the classical Relational Analysis approach by adding a neighborhood constraint to the Condorcet criterion. We propose a hybrid algorithm, which deals linearly with large data sets, provides a natural clusters identification and allows a visualization of the clustering result on a two-dimensional grid while preserving the a priori topological order of this data. The proposed approach called Relational Topological Map (RTM) was validated on several databases and the experimental results showed very promising performances.

Download Full-text

Partition of a Binary Matrix intok(k ≥ 3) Exclusive Row and Column Submatrices Is Difficult

Mathematical Problems in Engineering ◽

10.1155/2014/934630 ◽

2014 ◽

Vol 2014 ◽

pp. 1-13

Author(s):

Peiqiang Liu ◽

Daming Zhu ◽

Jinjie Xiao ◽

Qingsong Xie ◽

Yanyan Mao

Keyword(s):

Positive Integer ◽

Bipartite Graphs ◽

Np Hard ◽

Binary Matrix ◽

Attribute Vector ◽

The Matrix ◽

Binary Matrices

A biclustering problem consists of objects and an attribute vector for each object. Biclustering aims at finding a bicluster—a subset of objects that exhibit similar behavior across a subset of attributes, or vice versa. Biclustering in matrices with binary entries (“0”/“1”) can be simplified into the problem of finding submatrices with entries of “1.” In this paper, we consider a variant of the biclustering problem: thek-submatrix partition of binary matrices problem. The input of the problem contains ann×mmatrix with entries (“0”/“1”) and a constant positive integerk. Thek-submatrix partition of binary matrices problem is to find exactlyksubmatrices with entries of “1” such that theseksubmatrices are pairwise row and column exclusive and each row (column) in the matrix occurs in exactly one of theksubmatrices. We discuss the complexity of thek-submatrix partition of binary matrices problem and show that the problem is NP-hard for anyk≥3by reduction from a biclustering problem in bipartite graphs.

Download Full-text

Mathematical Modeling in the Textile Industry

Bulletin of Mathematical Sciences and Applications ◽

10.18052/www.scipress.com/bmsa.1.16 ◽

2012 ◽

Vol 1 ◽

pp. 16-28 ◽

Cited By ~ 2

Author(s):

Krasimir Yordzhev ◽

Hristina Kostadinova

Keyword(s):

Textile Industry ◽

Cardinal Number ◽

Object Oriented ◽

Combinatorial Problems ◽

Binary Matrix ◽

C Programming Language ◽

C Programming ◽

Binary Matrices ◽

Factor Set ◽

Made In

A mathematical model, describing some different weaving structures, is made in this article. The termsself-mirror and rotation-stable weaving structure are initiated here. There are used the properties and operationsin the set of the binary matrices and an equivalence relation in this set. Some combinatorial problems aboutfinding the cardinal number and the elements of the factor set according to this relation is discussed. We proposean algorithm, which solves these problems. The presentation of an arbitrary binary matrix using sequence ofnonnegative integers is discussed. It is shown that the presentation of binary matrices using ordered n-tuples ofnatural numbers makes the algorithms faster and saves a lot of memory. Implementing these ideas a computerprogram, which receives all of the examined objects, is created. In the paper we use object-oriented programmingusing the syntax and the semantic of C++ programming language. Some advantages in the use of bitwiseoperations are shown. The results we have received are used to describe the topology of the different weavingstructures.

Download Full-text

The effective algorithm of ensuring the integrity of the transmitted information

Proceedings of the National Academy of Sciences of Belarus Physics and Mathematics Series ◽

10.29235/1561-2430-2021-57-4-506-512 ◽

2021 ◽

Vol 57 (4) ◽

pp. 506-512

Author(s):

I. L. Kuznetsova ◽

A. S. Poljakov

Keyword(s):

Error Detection ◽

Communication Systems ◽

High Speed ◽

Correction Method ◽

Detection Algorithm ◽

Binary Matrix ◽

Transmitted Information ◽

Information And Communication ◽

Binary Matrices ◽

Transfer Of Information

The problem of ensuring the integrity of the transmitted information in modern information and communication systems is considered in this paper. An optimized algorithm for detecting and correcting errors in the information transmitted over communication lines is proposed. It was developed on the basis of the results of previous studies of the error correction method based on the parity values of the coordinates of a binary matrix. An easy-to-implement, high-speed and efficient error detection algorithm is proposed which is focused on the use of small binary matrices, for example, (4 × 8) or (7 × 8) bits. In such matrices, the possible number of errors that appear in them during the transfer of information is relatively small and easily detected.

Download Full-text

Efficiency of the error correction method by the parity values of binary matrix coordinates

Proceedings of the National Academy of Sciences of Belarus Physics and Mathematics Series ◽

10.29235/1561-2430-2019-55-3-375-382 ◽

2019 ◽

Vol 55 (3) ◽

pp. 375-382

Author(s):

A. S. Poljakov ◽

I. L. Kuznetsova

Keyword(s):

Error Correction ◽

Bit Error Rate ◽

Correction Method ◽

Optimal Choice ◽

Binary Matrix ◽

Error Correction Method ◽

Binary Matrices ◽

Information Errors ◽

Communication Lines ◽

Intensity Density

The results of study of the characteristics of the proposed method [1] for correction of errors arising during information transmission via communication lines are presented. The estimates of the efficiency of search for errors and the performance of an algorithm developed to realize the proposed method using the parity values of binary matrix coordinates are obtained; among these errors are rows, columns, main and auxiliary diagonals, are obtained. We have determined the dependence of algorithm characteristics on the intensity (density) of bit errors in the message obtained after transmission via communication lines and on the size of matrices, into which a transmitted message is divided.The time spent for calculating the parity values of matrix coordinates and for the algorithm used to find transmitted information errors are given. Recommendations on an optimal choice of sizes of binary matrices are presented. It is shown that, when the bit error rate is 10–2 and less, the algorithm detects all the available errors.

Download Full-text