The paper presents a description of the results of computational experiments aimed at constructing heuristic approximations of the spectra of the number of intercalates in the Latin squares and diagonal Latin squares of orders 10–28. Starting from orders 18–20, the calculations become computationally complex and require the using of distributed computing systems. The heuristic approximations of the spectra obtained as a result of the computational experiments made it possible to impose the strongest upper and lower bounds known to date on the corresponding terms of the numerical series A307163, A307164, and A345760 (for diagonal Latin squares) and A092237 and A368182 (for Latin squares) in the OEIS.

We enumerate all extended self-orthogonal diagonal Latin squares of order up to 10. Our method reduces the problem of enumerating extended self-orthogonal diagonal Latin squares to a satisfiability (SAT) problem, and we find all solutions of the SAT problem using a SAT solver. We additionally show that there is no triple of mutually orthogonal diagonal Latin squares of order 10 containing an extended self-orthogonal diagonal Latin square.

Article aimed to the description of equivalent transformations that are allow to get at least one diagonal Latin square (DLS) from all main classes of DLS included in main class of given Latin square (LS) if they are exist. Detailed description of corresponding algorithms for LS of odd and even orders is given. Estimations for time and memory complexities are presented, algorithms are provided with detailed examples. Description of the results of diagonalization and canonization is shown. They allow to get collections of orthogonal diagonal Latin squares in a more efficient way comparing with direct usage of Euler-Paker method using volunteer distributed computing projects Gerasim@Home and RakeSearch on BOINC platform. The possibility of obtaining stronger upper and lower bounds for some numerical series in OEIS connected with DLS using suggested transformations is shown. Prospects for further application of these transformations using distributed software implementation of corresponding algorithms are outlined.

Volunteer computing is a cheap yet efficient type of distributed computing, where desktops of private persons are united into projects. Some of these projects are aimed at finding new mathematical objects based on orthogonal systems of Latin squares. In 2021, new systems of orthogonal diagonal Latin squares of order 10 were found in a volunteer computing project. This was done using cells mapping schemes related to extended self-orthogonal diagonal Latin squares. In the present study, a classification of such schemes is proposed. The classification is built upon a structure of a multiset of cycle lengths, when a scheme is considered a permutation. For orders 1–9, the classification is constructed completely on a computer, while for order 10 only some classes are determined via volunteer computing. Finally, for order 10 new orthogonal systems were investigated using the cells mapping schemes and a SAT solver on a computer. It is described how the latter results can lead to finding the remaining classes for order 10 via volunteer computing.

Purpose of research. The purpose of this study is to solve the problem of restoring the external load on the rack– and-beam structural system and to assess the impact on the accuracy of solving the problem of the error of noisy deflections – constructive input data of the computing system.Methods. The main scientific methods used in this study are methods of modeling and identification of boundary conditions, the grid method of regularization of solving inverse incorrect problems. Measurement reduction and approximation methods, methods for evaluating the quality of input data processing, regularization and approximation algorithms using the Lebesgue grid function, and numerical methods are also used.Results. The main result of this work is two theorems about the external load on a rack-and-beam structural system. The existence and uniqueness of the solution is proved. Also, the results are the formulas of Lagrange multipliers in linear Lagrangian approximation and the optimal plan of coordinates of the nodes of the approximation grid for the equation of deflections of the beam of the fourth and fifth degree with Chebyshev alternance. An assessment of the quality of the approximation of the external load on the rack-and-beam structure by the values of the target parameters was carried out.Conclusion. This article proposes a method for restoring the external load on a rack-and-beam structure using the results of solving the inverse Cauchy problem for the equation of deflections of a beam with minimizing the influence of the error of noisy input data.

The concept of X-based diagonal fillings in diagonal Latin squares (DLS) is introduced, their enumeration is carried out them, isomorphism classes are constructed, and the interconnection of these problems with problems based on card decks is shown. On their basis, a method for generating strongly normalized canonical forms of the DLS using a limited set of ESODLS CMS schemes is proposed. It makes possible to reduce the necessary computational costs by 3-4 orders of magnitude compared to a software implementation based on a Brute Force based enumeration of the DLS. Using the developed software implementations and the Gerasim@ Home volunteer distributed computing project on the BOINC platform, the number of main classes of DLS of order 9 has been calculated. This number was previously unknown and amounted to 3 292 326 155 394.

The main goal of this work is to create a parallel application that performs computationsusing a multithreaded execution model, optimized to make the best utilization of all availablehardware resources. One of the main implementation requirements is to optimize application performanceon different computer architectures, and to enable parallel execution of the applicationon various computing devices that are part of a heterogeneous computing system. The possibilityof applying various methods of software and algorithmic optimization on multiprocessor architecturesof different generations was investigated as well as the effectiveness of their use for highlyloaded multithreaded applications was estimated. The problem of quasi-optimal dynamic distributionof computational tasks among all currently available computing devices of a heterogeneouscomputing system was also solved. Currently, not only multiprocessor computing systems are usedto solve large computational problems, but also various types of distributed systems. Distributedcomputing systems have a number of features: possible failures of nodes and communicationchannels, unstable operating time of nodes, possible errors in calculations, heterogeneity of computingnodes. By heterogeneity of computing nodes, we will understand not only the different computingcapacity and different architectures of central processors, but also the presence of otherdevices on the node capable of performing calculations. Such devices include video cards andmathematical coprocessors. A node of a distributed computing system will be called heterogeneousif, in addition to one or more central processing units, it contains additional computing devices.When solving a computational problem on a distributed system, it is necessary to maximize theutilization of all available computing resources. To do this, it is necessary not only to distributecomputing subtasks to nodes in accordance with their computing capacity, but also to take intoaccount the features of additional computing devices. This work is devoted to the study of methodsfor maximizing the resources utilization of heterogeneous nodes.

This study focuses on searching for pairs of orthogonal diagonal Latin squares of order 10. Consider a cells mapping in accordance to which one diagonal Latin square is mapped to another one. Given a certain cells mapping schema, the problem is to find a pair of orthogonal diagonal Latin squares of order 10 such that they match the schema (or to prove that such a pair does not exist). The problem is reduced to the Boolean satisfiability problem (SAT). Three mapping schemes are considered, and for each of them a SAT instance is constructed. If a satisfying assignment is found for an instance, the corresponding pair of orthogonal Latin squares can be easily extracted from it. The Cube-and-Conquer approach is used to solve the instances. The cubing phase is performed on a sequential look-ahead SAT solver, while on the conquer phase an experiment in a BOINC-based volunteer computing project is launched. In the experiment, for two out of three schemes orthogonal pairs are found.

The main goal of the work was aimed to create a parallel application using a multithreaded execution model, which will allow the most complete and efficient use of all available computing resources. At the same time, the main attention was paid to the issues of maximizing the performance of the multithreaded computing part of the application and more efficient use of available hardware. During the development process, the effectiveness of various methods of software and algorithmic optimization was evaluated, taking into account the features of the functioning of a highly loaded multithreaded application, designed to run on systems with a large number of parallel computing threads. The problem of loading all available computing resources at the moment was solved, including the dynamic distribution of the involved CPU cores/threads and the computing accelerators, installed in the system.

A large number of personal mobile devices and the presence of powerful computing processors on them allows one to use them as computing nodes. The idle computing power of mobile devices can be used in distributed computing systems. This paper discusses the use of personal mobile devices in the project of voluntary distributed computing. The features of computing on mobile devices are discussed. The results of computational experiments on a desktop grid system are presented. Setting up a BOINC-based desktop grid system for efficient use of mobile devices as computing nodes is discussed.

Рассмотрены вопросы синтеза систем логического управления в базисе логических мультиконтроллеров. Приведен ряд дискретных комбинаторных оптимизационных задач, возникающих в процессе синтеза, наряду с методами и алгоритмами их решения в рамках синтеза разбиений граф-схем параллельных алгоритмов логического управления, оценки их асимптотических сложностей, результаты сравнительного анализа и их влияние на аппаратную сложность синтезируемых систем. Предназначена для научных работников и инженеров, занимающихся вопросами проектирования систем логического управления, а также аспирантов, магистрантов и студентов старших курсов, обучающихся по направлениям подготовки 09.03.01 Информатика и вычислительная техника, 09.04.01 Информатика и вычислительная техника.

The article describes a number of simple transformations (rotation of intercalates, loops, Latin subrectangles, replacement of transversals) in the problem of constructing a collection of orthogonal diagonal Latin squares (ODLS) of order 10 aiming to try to find a triple of pairwise orthogonal diagonal Latin squares of order 10. The averaged time spent on obtaining one canonical form of ODLS using efficient software implementations of square generators based on nested loops and bit arithmetic, on the one hand, and the Euler-Parker method for checking the DLS for the presence of ODLS (together with the composition of the canonizer, if necessary), on the other hand, is 8.3 h using single threaded CPU implementation. It is shown that the application of the indicated transformations in some cases is capable of providing new canonical forms (CFs) of ODLS with significantly lower computational time. The article presents the results of comparing the effectiveness of simple transformations, as well as estimates of the minimum and maximum number of certain structural elements of the DLS (intercalates, loops, Latin subrectangles) depending on its order N (sequences A307163, A307164, A307166, A307167, A307170, A307171, A307841, A307842, A307839, A307840, A287645, A287644 in OEIS). The postprocessor of the found CF of ODLS, working within the framework of the Gerasim@Home volunteer distributed computing project based on this subset of effective simple transformations, allows increasing the output of new CF of ODLS by 10–15

Purpose of research. We have discovered a wide range of problems that are important in practice and which can be reduced in polynomial time to discrete combinatorial optimization problems, many of which can be solved using graph theory. One of these tasks is finding the chromatic number of a graph and its corresponding coloring. Taking into account the fact that the combinatorial problem of finding the chromatic number of a graph belongs to the complexity class and does not allow obtaining an optimal solution in a rational time for problems of practically important dimension, the search for a suitable heuristic method that allows obtaining high-quality solutions with low costs required for computation is demanded and relevant. The aim of the study is to analyze the results of using the bee colony method in the task at hand. The tasks of this research are: description of algorithmic techniques in a formalized form, which make it possible to apply the bee colony method in the problem to be solved, making modifications to the bee colony method that increase the efficiency of the method, namely the quality of the resulting final colorings, as well as the determination of factors affecting the quality and the time spent in finding solutions. Methods. To conduct research in the selected area, computational experiments were organized based on the use of heuristic methods in the problem under consideration. Meta-optimization of the tuning parameters of the methods and determination of their convergence rate was carried out, as well as a comparison of the quality and time of obtaining solutions. Results. As a result of the study, the convergence rate of the method was found to be higher than that of the random walk method; the dependence of the quality of the resulting final colorings on the graph size N and density d was found. It was found that the chosen method is faster than the method of weighted random enumeration with the variation of vertices according to the minimum of admissible colors on »67% , which currently generates solutions with the lowest chromatic number, while losing quality to it on »7% . A higher rate of convergence was noticed when compared with the method of random walks, the principle of which is the same as that of foraging bees. Conclusion. It was found that the bee colony method finds colorings with the same average chromatic number in fewer iterations than the random walk method, i.e. it has a higher convergence rate, while remaining significantly fast relative to the method of random search with a variation of vertices to reduce the allowed colors.

The article deals with the problem of the analysis of effectiveness of the heuristic methods based on the modification of earlier found decisions in the test problem for getting the shortest path in graph.The article briefly describes the selected group of methods used to solve the problem.The methodology considers the experimental comparison for estimating the quality of solutions based on the performance of computational experiments with the samples of pseudo-randomly structured graphs that uses the BOINC platform.It also presents the description of obtained experimental results which allow to identify the areas of the preferable usage of selected subset of methods depending on the size of the problem and power of constraints.It is shown that the particle swarm optimization, random walks, simulated annealing and bee colony methods are ineffective in the selected problem and significantly inferior to the quality of solutions that are provided by ant colony optimization method and genetic algorithms.

The paper discusses the reduction of problems based on Latin squares to the exact cover problem aiming at its subsequent solution using the dancing links algorithm. The former problems include generation of Latin squares and diagonal Latin squares of a general form/with a given normalization, generation of orthogonal Latin and diagonal Latin squares directly/through the set of transversals, obtaining a set of transversals for a given square, forming a subset of disjoint transversals. For each subproblem, we describe in detail the process of forming the corresponding binary coverage matrices. We show that the use of the proposed approach in comparison with the classical one, i.e. the formation of sets of transversals and their coverages using exhaustive enumeration, allows one to increase the eective processing pace of diagonal Latin squares by 2.5{5.6 times. The developed software implementations of the algorithms are used in computational experiments as part of the Gerasim@Home volunteer distributed computing project on the BOINC platform

We propose an algorithm for enumerating diagonal Latin squares. It relies on specific properties of diagonal Latin squares to employ symmetry breaking techniques. Furthermore, the algorithm employs several heuristic optimizations and bit arithmetic techniques. We use the algorithm to enumerate diagonal Latin squares of order at most 9.