On the least redundancy problem of the queries of order two in combinatorial filing scheme

Abstract

The paper concerns a least redundancy problem of queries of order two in a combinatorial file organization scheme. Every record will be assumed to have m attributes, each of them having n levels, and the queries of order two will be identified with edges of a complete m -partite graph K m ( n,…, n ). S. Yamamoto, S. Tazawa, K. Ushio, and H. Ikeda have proved that if c ⩽ ( m − 1), then the graph, termed “claw with degree c ,” has the least redundancy among all the graphs consisting of c edges over K m ( n,…, n ), and they presented a file organization scheme realizing the least redundancy. S. Tazawa and S. Yamamoto have proved that the claw with degree c has the least redundancy even in the case of c ⩽ n ( m − 1). The purpose of this paper is to introduce some transformations of graphs over K m ( n,…, n ) and to prove that a graph termed “complete normal form” has the least redundancy in any case of c > 0. In mathematical language, the problem here is stated as follows: Let V be an n -dimensional lattice point space {1,…, m } × … × {1,…, m }. For fixed i, j ( i ≠ j ), p, p′ , we define a subset V ( i,j,p,p′ ) = { v } ∈ V ; v i = p}, v j = p′} ⊂ V . For a given possible integer c , how should we select c mutually different V ( i,j, p, p′ ) such that the number of lattice points contained in the union of them is minimum. The solution is Theorem 5, and Theorem 7 gives a formula for finding the minimum number.