The infinite families of optimal double loop networks

Optimal routing in double loop networks

In this paper, we study the problem of finding the shortest path in circulant graphs with an arbitrary number of jumps. We provide algorithms specifically tailored for weighted undirected and directed circulant graphs with two jumps which compute the shortest path. Our method only requires O(log N ) arithmetic operations and the total bit complexity is O(log2 N log log N log log log N ), where ...

Some Optimal Designs of Undirected Double-loop Networks

An efficient algorithm to find optimal double loop networks

The problem of finding optimal diameter double loop networks with a fixed number of vertices has been widely studied. In this work, we give an algorithmic solution of the problem by using a geometrical approach. Given a fixed number of vertices n, the general problem is to find "steps" s 1 , s z e ~',, such that the digraph G(n; sl,s2) with set of vertices V = 7 / a n d adjacencies given by i ~...

Optimal double-loop networks with non-unit steps∗

A double-loop digraph G(N ; s1, s2) = G(V,E) is defined by V = ZN and E = {(i, i + s1), (i, i + s2)| i ∈ V }, for some fixed steps 1 ≤ s1 < s2 < N with gcd(N, s1, s2) = 1. Let D(N ; s1, s2) be the diameter of G and let us define D(N) = min 1≤s1<s2<N, gcd(N,s1,s2)=1 D(N ; s1, s2), D1(N) = min 1<s<N D(N ; 1, s). Some early works about the diameter of these digraphs studied the minimization of D(N...

Equivalent Double - Loop Networks

Hwang and Xu defined equivalent double-loop networks and gave one such result showing that the L-shapes of the two equivalent networks are recombinations of three rectangles. Recently, Rödseth gave an elegant algebraic theorem for equivalent multi-loop networks. We show that its double-loop version yields equivalent networks of the 3rectangle version. We also show that other seemingly different...