Abstract

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

Highlights

  • One of the known types of combinatorial objects are the Latin Squares (LS) [1, 2], which are square tables of size N × N cells, where N is the order of the square, filled with elements of some alphabet U (for definiteness in this paper, by integers from 0 to N − 1), so that in each row and each column the elements of the alphabet are not repeated

  • One of the known types of combinatorial objects are the Latin Squares (LS) [1, 2], which are square tables of size N × N cells, where N is the order of the square, filled with elements of some alphabet U, so that in each row and each column the elements of the alphabet are not repeated

  • The developed software implementations of the algorithms for constructing cover matrices, DLX, and reconstructing solutions from the found covers are currently actively used in the Gerasim@Home volunteer computing project within the BOINC platform [16], and the speed characteristics of the search for ODLS of order 10 are limited by this combination of software implementations of these algorithms

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Summary

Introduction

One of the known types of combinatorial objects are the Latin Squares (LS) [1, 2], which are square tables of size N × N cells, where N is the order of the square, filled with elements of some alphabet U (for definiteness in this paper, by integers from 0 to N − 1), so that in each row and each column the elements of the alphabet are not repeated. The computations within the RakeSearch project are based on the existence of such ODLS pairs, for some orders [6], in which an orthogonal mate is obtained from the original one by rearranging its rows (a special case of orthogonality of the ESOLS type (Extended SelfOrthogonal Latin Squares) [7], which allows one to raise the processing rate of squares to a value of about 70,000–80,000 DLS/s.

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