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 Km(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 Km(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 Km(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; vi = p}, vj = 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