INDEXING METHODS FOR FORMING COMBINATORIAL CONFIGURATIONS OF THE CLASS OF SYSTEMS OF DISTINCT REPRESENTATIVES

Abstract

Abstract. Objectives In this paper, methods for solving of a class of combinatorial tasks, known as systems of distinct representatives (SDR), are considered. The objective is to develop methods and algorithms for the formation of a combinatorial SDR configuration that includes as rows, columns or row- and column subsets, which are composed of elements of the original family of nxn – sets occupying different positions in the initial sets, as well to determine the possible number of proposed configurations. Method Index ordering methods are used for the arrangement of elements in the formed systems of distinct representatives, the essence of which is to formulate requirements for the process of configuration having specified properties through the regularity of indexing elements within these configurations. Results The general formulation of the issue of constructing an SDR is considered in terms of a problem of formation from the elements of sets and subsets, which include one element from each initial set, with each of these elements being located at different positions in the original sets. The task was reformulated in reference to the requirements for indexing the elements of these subsets. Each element in set systems has a two-index designation, with the first element in the index indicating membership of a specific initial set and the second – to its location. In order to fulfil the requirements formulated in the task, it is necessary for indices of the SDR elements to have values from 1 to n. Conclusion Two methods for solving the problem are proposed: cyclic shifts of rows and columns of the matrix configuration formed by the original sets, and by a given law for indexing the elements of the environment. The number of possible options for the formation of representative systems is determined. The reasons for the propagation of the proposed methods for solving the problem are established only for initial sets of odd dimensions.