A symbolical algorithm on additive basis and double-loop networks

A double-loop digraph G=G(N ; s1; s2), with gcd(N; s1; s2)=1, has the set of vertices V =ZN and the adjacencies are given by u → u + si (mod N ) i = 1; 2. The diameter of G, denoted by D(N ; s1; s2), is known to be lower bounded by lb(N ) with D(N ) = min 16s1¡s2¡N; gcd(N;s1 ; s2)=1 D(N ; s1; s2)¿ ⌈√ 3N ⌉ − 2 = lb(N ): This lower bound is known to be sharp. For a 0xed N ∈N, some algorithms to 0nd D(N ) and steps 16 ¡ ¡N such that D(N ; ; )=D(N ) are known through the bibliography, being of numerical nature all of them. In this paper we propose a symbolic algorithm on the following problem: Given a number of vertices N0 ∈N, 0nd if possible an in0nite family of tight double-loop digraphs G(x) = G(N (x); s1(x); s2(x)) such that N (x0) = N0 and D(x) = D(N (x); s1(x); s2(x)) = lb(N (x)) ∀x¿ x0. This family is parameterized by the integer x with N (x)∈Z and s1(x); s2(x)∈Z=(N (x)). As a direct consequence of such an explicit family of digraphs G(x), we also have an additive basis {s1(x); s2(x)} which covers the elements of Z=(N (x)) with optimal order. The time cost of this algorithm is O( √ N0 logN0), in the worst case. c © 2004 Elsevier B.V. All rights reserved.