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WEIGHTED MULTI-CONNECTED
LOOP NETWORKS
¨OYSTEIN J. R¨ODSETH
University of Bergen, Department of Mathematics,
All´ egt. 55, N-5007 Bergen, Norway
Abstract
Given relatively prime integers N,a1,...,ak, a multi-connected
loop network is defined as the directed graph with vertex set Z/NZ =
{0,1,...,N −1}, and directed edges i → r ≡ i+aj (mod N). If each
edge i → i+ajis given a positive real weight wjfor j = 1,...,k , then
we have a weighted multi-connected loop network. The weight of a
path is the sum of weights on its edges. The distance from a vertex to
another is the minimum weight of all paths from the first vertex to the
second. The diameter of the network is the maximum distance, and
the average diameter is the average distance in the network. In this
paper we study the diameter and the average diameter of a weighted
multi-connected loop network. We give a unified and generalized pre-
sentation of several results in the literature, and also some new results
are obtained.
1 Introduction
Let N ≥ 1,a1,...,ak be integers and w1,...,wkpositive real numbers. A
multi-connected loop network is defined as the directed graph with vertex set
Z/NZ = {0,1,...,N − 1} and directed edges i → r ≡ i + aj (mod N). We
assume that the network is strongly connected, which is equivalent to the
assertion
gcd(N,a1,...,ak) = 1. (1)
For each j, we give each of the edges i → i + aj (mod N) the weight wj.
The resulting network is called a weighted multi-connected loop network.
The weight of a directed path is the sum of weights on its edges. The
distance d(i,j) from a vertex i to a vertex j is the minimum weight of all
paths from i to j. The diameter D = DN(a1,...,ak;w1,...,wk) is given as
D = maxd(i,j), and the average diameter A = AN(a1,...,ak;w1,...,wk) is
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the average of all the distances d(i,j) in the network. Since the network is
vertex symmetric, we have
D = max
0≤j<Nd(0,j),A =1
N
N−1
?
j=0
d(0,j). (2)
For many applications it is a natural requirement of a network that the
diameter is small. Often it is, however, more important that the average
diameter is small. Of course, a network is particularly nice if it satisfies both
these requirements.
In the literature on loop networks partial answers to two problems are
given. We will designate these two problems as the local problem and the
global problem. The local problem consists in designing an algorithm pro-
ducing D and/or A, given the input N,a1,...,ak;w1,...,wk. The global
problem is, given N and w1,...,wk, to find a sequence a1,...,akwhich min-
imizes D and/or A.
Suppose that all the ai are positive, and put wi = ai, i = 1,...,k.
By Lemma 3 of Brauer and Shockley [1], we then have that D − N is the
“Frobenius number” of the sequence N,a1,...,ak, which is the largest integer
which cannot be represented by the linear form Nx0+ a1x1+ ··· + akxkin
non-negative integers xi. By the theorem of Selmer [15], we also have in this
case that A−(N −1)/2 is the number of non-negative integers which cannot
be represented in this way. For results on the corresponding local problem,
see [5], [11], [12], [16]. In [14] we gave some results on the corresponding
global problem. Also an upper bound problem has been studied in this case,
see [3], [4], [13].
Next, for arbitrary integers ai, put all the wi = 1. Then we are back
to the problems studied by Wong and Coppersmith [19], which arouse from
Stone’s description [18] of a “particular organization of a multimodule mem-
ory, designed to facilitate parallell block transfer in the high-speed memory
of a computer system”.
For k = 2 and arbitrary weights, a solution to the local problem is given
by Cheng and Hwang [2]. This solution is a rather simple lifting to general
weights w1, w2of (a variant of) the algorithm of Greenberg [5] for the case
w1 = a1, w2 = a2(the Frobenius case). As Cheng and Hwang remark, a
corresponding lifting can easily be done with our results [11]. We shall do
this at the beginning of Section 3. This we do because it gives us a good
starting point for the study of the global problem for k = 2. However, a
coarse worst case analysis indicates that Cheng and Hwang’s algorithm is
much faster than the one given in Section 3. For the local problem for k = 2,
we therefore recommend the use of Cheng and Hwang’s algorithm.
Wong and Coppersmith [19] showed that
√3N − 2,
DN(1,s;1,1) ≥
AN(1,s;1,1) ≥5
9
√3N − 1. (3)
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On the other hand, Hwang and Xu [6] constructed an s (dependent on N)
satisfying
√3N + 2(3N)1/4+ 5, N ≥ 6348.
However, they did not consider AN(1,s;1,1).
In Section 2 of this paper we give some reduction formulas for A and D, in
Section 3 we give the results corresponding to (3) for DN(a1,a2;w1,w2) and
AN(a1,a2;w1,w2), in Section 4 we use a simplified version of Hwang and Xu’s
construction [6] to give an s such that a1= 1, a2= s is close to minimizing
DN(a1,a2;1,1) and AN(a1,a2;1,1) simultaneously, and in Section 5 we give
some general bounds for D and A.
DN(1,s;1,1) <(4)
2 Reduction formulas
We now put tj= tj(N;a1,...,ak;w1,...,wk) = d(0,j). Then
tj= min{
k
?
i=1
wixi|
k
?
i=1
aixi≡ j (mod N)}, (5)
where all the xiare non-negative integers, and the formulas (2) become
D = max
0≤j<Ntj,A =1
N
N−1
?
j=0
tj.(6)
Clearly, we have
DN(a1,...,ak;w1,...,wk) = w1DN(a1,...,ak;1,w2
w1,...,wk
w1,...,wk
w1),(7)
AN(a1,...,ak;w1,...,wk) = w1AN(a1,...,ak;1,w2
w1). (8)
Suppose that c is an integer prime to N, and let b be the multiplicative
inverse of c mod N. Then
tbj(N;a1,...,ak;w1,...,wk) = tj(N;a1c,...,akc;w1,...,wk). (9)
As j runs through a complete residue system mod N, so does bj. Hence, by
(6),
DN(a1,...,ak;w1,...,wk) = DN(a1c,...,akc;w1,...,wk),(10)
AN(a1,...,ak;w1,...,wk) = AN(a1c,...,akc;w1,...,wk). (11)
In particular, if 1 ≤ d|ai, i = 1,2,...,k, by (1), we then have gcd(d,N) =
1, and by (10) and (11),
DN(a1,...,ak;w1,...,wk) = DN(a1
d,...,ak
d;w1,...,wk),
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