Optimal Double-loop Networks Research Articles

Generation and Analysis of a Dataset of Optimal Double-Loop Circulant Networks

Optimal circulant networks are of practical interest as graph models of reliable communication networks for supercomputer systems and networks on a chip. A large dataset of parameters of optimal double-loop circulant networks with a number of nodes of up to 50,000 and 451,000 points has been constructed. The dataset contains, for each number of vertices, all generators corresponding to optimal or suboptimal (in the absence of optimal) descriptions of the graph. This made it possible to find analytical dependencies of the parameters that define families of optimal graphs. Based on the analysis of the constructed dataset, the solution to the problem of determining optimal two-dimensional circulant networks is studied. A process is automated for determining analytical, parametric descriptions of the families of optimal double-loop networks, described polynomials of diameter. Having applied the methods of integrating differential evolution and reduced local search to the analysis of the generated dataset, we rediscovered a number of previously known families and found more than 200 new, different from all those known in literature, families of optimal double-loop networks with generators of linear and quadratic types of polynomials. Our new ap­proach can be applied to studing datasets of other promising classes of graphs and networks.

Procreating tiles of double commutative-step digraphs

Double commutative-step digraph generalizes the double-loop digraph. A double commutative-step digraph can be represented by an L-shaped tile, which periodically tessellates the plane. Given an initial tile L(l, h, x, y), Aguilo et al. define a discrete iteration L(p) = L(l + 2p, h + 2p, x + p, y + p), p = 0, 1, 2, …, over L-shapes (equivalently over double commutative-step digraphs), and obtain an orbit generated by L(l, h, x, y), which is said to be a procreating k-tight tile if L(p)(p = 0, 1, 2, …) are all k-tight tiles. They classify the set of L-shaped tiles by its behavior under the above-mentioned discrete dynamics and obtain some procreating tiles of double commutative-step digraphs. In this work, with an approach proposed by Li and Xu et al., we define some new discrete iteration over L-shapes and classify the set of tiles by the procreating condition. We also propose some approaches to find infinite families of realizable k-tight tiles starting from any realizable k-tight L-shaped tile L(l, h, x, y), 0 ≤ |y − x| ≤ 2k + 2. As an example, we present an infinite family of 3-tight optimal double-loop networks to illustrate our approaches.

The construction of infinite families of any k-tight optimal and singular k-tight optimal directed double loop networks

The double loop network (DLN) is a circulant digraph with n nodes and outdegree 2. It is an important topological structure of computer interconnection networks and has been widely used in the designing of local area networks and distributed systems. Given the number n of nodes, how to construct a DLN which has minimum diameter? This problem has attracted great attention. A related and longtime unsolved problem is: for any given non-negative integer k, is there an infinite family of k-tight optimal DLN? In this paper, two main results are obtained: (1) for any k ⩾ 0, the infinite families of k-tight optimal DLN can be constructed, where the number n(k, e, c) of their nodes is a polynomial of degree 2 in e with integral coefficients containing a parameter c. (2) for any k ⩾ 0, an infinite family of singular k-tight optimal DLN can be constructed.

A new method for constructing infinite families of k-tight optimal double loop networks

The double loop network (DLN) is a circulant digraph with n nodes and outdegree 2. DLN has been widely used in the designing of local area networks and distributed systems. In this paper, a new method for constructing infinite families of k-tight optimal DLN is presented. For k = 0, 1, ..., 40, the infinite families of k-tight optimal DLN can be constructed by the new method, where the number n k (t, a) of their nodes is a polynomial of degree 2 in t and contains a parameter a. And a conjecture is proposed.

An efficient algorithm to find a double-loop network that realizes a given L-shape

Double-loop networks have been widely studied as an architecture for local area networks. It is well known that the minimum distance diagram of a double-loop network yields an L-shape. Given a positive integer N , it is desirable to find a double-loop network with its diameter being the minimum among all double-loop networks with N nodes. Since the diameter of a double-loop network can be easily computed from its L-shape, one method is to start with a desirable L-shape and then find a double-loop network to realize it. This is a problem discussed by many authors [F. Aguiló, M.A. Fiol, An efficient algorithm to find optimal double loop networks, Discrete Math. 138 (1995) 15–29, R.C. Chan, C.Y. Chen, Z.X. Hong, A simple algorithm to find the steps of double-loop networks, Discrete Appl. Math. 121 (2002) 61–72, C.Y. Chen, F.K. Hwang, The minimum distance diagram of double-loop networks, IEEE Trans. Comput. 49 (2000) 977–979, P. Esqué, F. Aguiló, M.A. Fiol, Double commutative-step diagraphs with minimum diameters, Discrete Math. 114 (1993) 147–157] and it has been open for a long time whether this problem can be solved in O ( log N ) time. In this paper, we will provide a simple and efficient O ( log N ) -time algorithm for solving this problem.

Optimal Double-Loop Networks with Non-Unit Steps

A double-loop digraph $G(N;s_1,s_2)=G(V,E)$ is defined by $V={\bf Z}_N$ and $E=\{(i,i+s_1), (i,i+s_2)|\; i\in V\}$, for some fixed steps $1\leq s_1 < s_2 < N$ with $\gcd(N,s_1,s_2)=1$. Let $D(N;s_1,s_2)$ be the diameter of $G$ and let us define $$ D(N)=\min_{\scriptstyle1\leq s_1 < s_2 < N,\atop\scriptstyle\gcd(N,s_1,s_2)=1}D(N;s_1,s_2),\quad D_1(N)=\min_{1 < s < N}D(N;1,s). $$ Some early works about the diameter of these digraphs studied the minimization of $D(N;1,s)$, for a fixed value $N$, with $1 < s < N$. Although the identity $D(N)=D_1(N)$ holds for infinite values of $N$, there are also another infinite set of integers with $D(N) < D_1(N)$. These other integral values of $N$ are called non-unit step integers or nus integers. In this work we give a characterization of nus integers and a method for finding infinite families of nus integers is developed. Also the tight nus integers are classified. As a consequence of these results, some errata and some flaws in the bibliography are corrected.

An infinite family of 4-tight optimal double loop networks

An infinite family of 4-tight optimal double loop networks

Designing of optimal double loop networks

The double loop networkG(N;r,s) hasN vertices and2N directed edges. A natural question is how to chooser ands such thatG(N;r,s) has diameter as short as possible for a givenN. In 1993, Li, Xu and Zhang proposed a method of constructing double loop networks with the minimum diameter for the case ofr=1. The method is developed to construct such networks that none of their minimum diameters can be reached atr=1. As a by-product, a flaw in an assertation by Esque et al. is pointed out.

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 2 ∈ Z n , such that the digraph G( n; s 1, s 2) with set of vertices V = Z n and adjacencies given by i → i + s 1 ( mod n) and i → i + s 2 ( mod n) has minimum diameter d( n). A lower bound of this diameter is known to be be lb( n)=⌈√3 n⌉−2. So, given n, the algorithm has as outputs s 1, s 2 and the minimum integer κ = κ( n) such that d( n; s 1, s 2)= d( n)= lb( n)+ κ The running time complexity of the algorithm is O( κ 3) O( log n )- O( κ) is unknown but it is upper-bounded by O(4√ n). Moreover, in most of the cases the algorithm also gives (as a by-product) an infinite family of digraphs with increasing order and diameter as above, to which the obtained digraph G( n; S 1, S 2) belongs.

The infinite families of optimal double loop networks

Let N, s be integers, 1 < s < N. The double loop network (DLN) G(N; s) is a digraph with N nodes 0, 1,. ..,N 1 and 2N arcs {i +i+I,i+s:i=O,l,..., N-l),where nodes are always represented by residues modulo N. DLN’s have been widely studied lately as practical models in the design of local area networks and parallel processing architectures [ 1,3,5]. Let d(N; s) be the diameter of G(N; s), and let d(N) = min{d(N;s): 1 < s < N}. The network G(N;s) is said to be optimal if d(N; s) = d(N). Wong and Coppersmith initiated the studies of finding optimal G(N; s) for every N 2 4 and they established a good lower bound lb(N) = L fiN A2 for d(N) [S]. A network G(N;s) is said to be tight optimal if d(N;s) = lb(N). Obviously, a tight optimal DLN is certainly optimal but the converse is not true. For example, if N = 3(t + 1)’ then lb(N) = 3t + 1 but d(N) = d(N;3t + 5) = 3t + 2 for every t 2 1. In accordance with this situation, a network G(N; s) is said to be nearly tight optimal if d(N;s) = d(N) = lb(N) + 1. A lot of infinite families of tight optimal and only one infinite family of nearly tight optimal DLN’s, namely {G(3(t + l)*; 3t + 2: t 2 11, have been found by several authors since 1987. However the union of all the known infinite families of optimal DLN’s can not even include a DLN with N nodes for every N I 50 [l, 2, 43. In this note we list explicitly 69 infinite families of tight optimal and 33 infinite families of nearly tight optimal DLN’s such that for every N, 4 I N 5 300, at least one of our families contain an optimal G(N; s) (see Tables 1,2,3).

Performance evaluation of a simulated data-flow computer with low-resolution actors

In the ambition to go beyond a single-processor architecture, to enhance programmability, and to take advantage of the power brought by VLSI devices, data-flow systems and languages were devised. Indeed, due to their functional semantics, these languages offer promise in the area of multiprocessor systems design and will possibly enable the development of computers comprising large numbers of processors with a corresponding increase in performance. Several important design problems have to be surmounted and are described here. We thus present a “variable-resolution” scheme, where the level of primitives can be selected so that the overhead due to the data-flow mode of operation is reduced. A deterministic simulation of a data-flow machine with a variable number of processing elements was undertaken and is described here. The tests were performed using various program structures such as directed acyclic graphs, vector operations, and array handling. The performance results observed confirm the advantage of actors with variable size and indicate the presence of a trade-off between overhead control and the need to control parallelism in the program. We also look at some of the communication issues and examine the effect of several interconnection networks (dual counter-rotating rings, daisy chain, and optimal double loop network) on the performance. It is shown how increasing communication costs induce a performance degradation that can be masked when the size of the basic data-flow actor is increased. The asociative memory cycle time is also changed with similar conclusions. Finally, the lower-resolution scheme is applied to the array handling case; the observations confirm the advantage of a more complex actor at the array level.