Fault Diameter Research Articles

Edge congestion and topological properties of crossed cubes

An n-dimensional crossed cube, CQ/sub n/, is a variation of hypercubes. In this paper, we give a new shortest path routing algorithm based on a new distance measure defined herein. In comparison with Efe's algorithm, which generates one shortest path in O(n/sup 2/) time, our algorithm can generate more shortest paths in O(n) time. Based on a given shortest path routing algorithm, we consider a new performance measure of interconnection networks called edge congestion. Using our shortest path routing algorithm and assuming that message exchange between all pairs of vertices is equally probable, we show that the edge congestion of crossed cubes is the same as that of hypercubes. Using the result of edge congestion, we can show that the bisection width of crossed cubes is 2/sup n-1/. We also prove that wide diameter and fault diameter are [n/2]+2. Furthermore, we study embedding of cycles in cross cubes and construct more types than previous work of cycles of length at least four.

WIDE DIAMETERS OF BUTTERFLY NETWORKS

Reliability and efficiency are important criteria in the design of interconnection networks. Recently, the $w$-wide diameter $d_w(G)$, the $(w-1)$-fault diameter $D_w(G)$, and the $w$-Rabin number $r_w(G)$ have been used to measure network reliability and efficiency. In this paper, we study wide diameters for an important class of parallel networks---butterfly networks. The main result of this paper is to determine their wide diameters.

FAULT DIAMETER FOR SUPERCUBES

Assume that N and s are positive integers with 2s < N ≤ 2s+1. It is claimed by Auletta, Rescigno, and Scarano that the fault diameter of the supercube with N nodes, is exactly s+1 if N∉{2s+1-1, 2s+1 - 2, 2s-1+1}, and s+2 otherwise. In this paper, we will argue that the above claim is not correct. Instead, we will show that the fult diameter of the supercube with N nodes is s+2 if N ∈ {2s+1 - 2i+1|0 ≤ i ≤ s-1}.

Rabin numbers of Butterfly networks

Reliability and efficiency are important criteria in the design of interconnection networks. Recently, the w-wide diameter d w ( G), the ( w - 1)-fault diameter D w ( G), and the w-Rabin number r w ( G) have been used to measure network reliability and efficiency. In this paper, we study these parameters for an important class of parallel networks — Butterfly networks. The main result of this paper is to determine the Rabin number of Butterfly networks.

WIDE DIAMETERS OF BUTTERFLY NETWORKS

Reliability and efficiency are important criteria in the design of interconnection networks. Recently, the $w$-wide diameter $d_w(G)$, the $(w-1)$-fault diameter $D_w(G)$, and the $w$-Rabin number $r_w(G)$ have been used to measure network reliability and efficiency. In this paper, we study wide diameters for an important class of parallel networks---butterfly networks. The main result of this paper is to determine their wide diameters.

Combinatorial properties of two-level hypernet networks

The purpose of the paper is to investigate combinatorial properties of the hypernet network. The hypernet network owns two structural advantages: expansibility and equal degree. In addition, it was shown to be efficient in both communication and computation. Since the number of nodes contained in the hypernet network increases very rapidly with expansion level, we emphasize the hypernet network of two levels (denoted by HN(d, 2)) with a practical view. Recently, combinatorial properties such as container (i.e., node-disjoint paths), wide diameter, and fault diameter have received much attention due to their increasing importance and applications in networks. The following results are obtained for HN(d, 2): (1) best containers with width d-1, (2) containers with (maximum) width d, (3) the (d-1)-wide diameter, (4) the d-wide diameter, (5) the (d-2)-fault diameter, and (6) the (d-1)-fault diameter. More specifically, between every two nodes of HN(d, 2), d (or d-1) packets can be transmitted simultaneously with at most D+2 (or D+1) parallel steps, where D=2d+1 is the diameter of HN(d, 2). Besides, the diameter of HN(d, 2) will increase by at most two (or one), if there are at most d-1 (or d-2) node faults. Our results reveal that HN(d, 2) is not only efficient in parallel transmission, but robust in fault tolerance.

Fault tolerance properties of pyramid networks

In this paper, we study the pyramid network (also called pyramid), one of the important architectures in parallel computing, network computing, and image processing. Some properties of pyramid networks are investigated. We determine the line connectivity and the fault diameters in pyramid networks. We show how to construct a path between two nodes in the faulty pyramid networks in polynomial time. A polynomial-time algorithm is also given for generating the containers in pyramid networks. Our results show that pyramid networks have very good fault tolerance properties.

Topological properties of twisted cube

Twisted cube, TQ n , is derived by changing some connections of hypercube Q n according to specific rules. Recently, many topological properties of this variation cube are studied. In this paper, we prove that its connectivity is n, its wide diameter and fault diameter are [ n/2] + 2. Furthermore, we show that TQ n , is a pancyclic network that is cycles of an arbitrary length at least four.

Product-closed networks

We present a uniform mathematical characterization of interconnection network classes referred to as product-closed networks (PCN). A number of popular network classes fall under this characterization including binary hypercubes, tori, k -ary n -cubes, meshes, and generalized hypercubes. An unlimited number of other networks can be defined using the presented mathematical characterization. An important common feature for all PCN classes is their closure under the Cartesian product of graphs. This provides a tool for generating new PCN classes of interconnection graphs. We evaluate a number of commonly used metrics for all PCN networks including the size, degree, diameter, average distance, connectivity, and fault diameter. We show how all PCN networks share various desirable properties such as simple distributed routing, hierarchical structure, complete sets of node-disjoint paths between arbitrary nodes, attractive embeddings, distributed broadcasting, and fault tolerance properties. The proposed characterization provides a unified model for representing and further analyzing the various known PCN networks, and for building new ones with predetermined properties and characteristics.

Properties and performance of folded cube-connected cycles

This paper analyzes a new multiprocessor interconnection network — the folded cube-connected cycles (FCCC). This topology is obtained by replacing each node of an n -dimensional folded hypercube by an ( n + 1)-cycle. The FCCC n is a regular network of degree 3 and has ( n + 1)2 n nodes. We analyze the topological properties of the FCCC and compare them with those of the cube-connected cycles (CCC). The FCCC has a lower diameter and average distance than that of the CCC. In terms of the most commonly used cost metric (node degree x diameter), the FCCC is better than any of the hypercube-variant networks. It is strongly resilient, and has a fault diameter less than that of the CCC. In terms of bisection width and connectivity also, the FCCC is better than that of the CCC. We propose algorithms for one-to-one communication and broadcasting in FCCC. We also propose an optimal VLSI layout.

Generalized diameters and rabin numbers of networks

Reliability and efficiency are important criteria in the design of interconnection networks. Recently, the w-wide diameter dw(G), the (w − 1)-fault diameter Dw(G), and the w-Rabin number rw(G) have been used to measure network reliability and efficiency. In this paper, we study dw(G), Dw(G) and rw(G) using the strong w-Rabin number rw*(G) for 1 ≤ w ≤ k(G) and G is a circulant network G(dn; {1, d,..., dn−1}), a d-ary cube network C(d, n), a generalized hypercube GH(mn− 1,..., m0), a folded hypercube FH(n) or a WK-recursive network WK(d, t).

On the Rabin number problem

Given positive integers κ and l, let R(κ, l) denote the class of interconnection networks so that for any κ + 1 distinct nodes x, y1, y2, …, yκ there exist κ node-disjoint paths (except at x) of length at most l from x to y1, y2, …, yκ, respectively. The Rabin number of a network W with connectivity κ, denoted by rκ(W), is defined as min{l|W ∈ R(κ, l)}. The Rabin number problem is to compute rκ(W) for a given network W. Computing Rabin numbers for nontrivial networks is not easy, in general. Thus far, only the Rabin number of the hypercube has been computed. In this paper, some new results on the Rabin number problem are derived. First, a theorem relating the Rabin number with both wide diameter and fault diameter is obtained. This theorem can help in the derivation of lower bounds for the Rabin numbers of circulant networks, folded hypercubes, hierarchical folded-hypercube networks, generalized hypercubes, and WK-recursive networks. Moreover, upper bounds are also obtained for WK-recursive networks and a special class of generalized hypercubes. The Rabin numbers of a special class of circulant networks and a special class of generalized hypercubes are determined as a consequence of the derived upper and lower bounds. © 1997 John Wiley & Sons, Inc. Networks 30:219–230, 1997

Fault diameter of k-ary n-cube networks

We obtain the fault diameter of k-ary n-cube interconnection networks (also known as n-dimensional k-torus networks). We start by constructing a complete set of node-disjoint paths (i.e., as many paths as the degree) between any two nodes of a k-ary n-cube. Each of the obtained paths is of length zero, two, or four plus the minimum length except for one path in a special case (when the Hamming distance between the two nodes is one) where the increase over the minimum length may attain eight. These results improve those obtained by B. Bose et al. (1995) where the length of some of the paths has a variable increase (which can be arbitrarily large) over the minimum length. These results are then used to derive the fault diameter of the k-ary n-cube, which is shown to be /spl Delta/+1 where /spl Delta/ is the fault free diameter.

Incrementally extensible hypercube networks and their fault tolerance

The incrementally extensible hypercube or IEH graph, a generalization of the hypercube, was introduced in [1]. It is shown that IEH graphs are incrementally extensible in steps of 1, optimally fault tolerant and its diameter is logarithmic in the number of nodes. Moreover, for any given number of nodes, the difference of the maximum and the minimum degree of a node in the graph is ≤ 1; i.e., the graph is almost regular. We also analyze these graphs in presence of node failures. We develop a fault tolerant routing algorithm which can compute a minimal path between two arbitrary nodes in an IEH graph in the presence of the maximum number of allowable faults. We also determine the fault diameter of the graph. Thus, we show that the IEH graphs, while they can be defined for any integer number of computing nodes, retain all the desirable properties of regular hypercubes including fault tolerance and a tight fault diameter.

Conditional Fault Diameter of Star Graph Networks

It is well known that star graphs are strongly resilient like thencubes in the sense that they are optimally fault tolerant and the fault diameter is increased only by one in the presence of maximum number of allowable faults. We investigate star graphs under the conditions offorbidden faulty sets, where all the neighbors of any node cannot be faulty simultaneously; we show that under these conditions star graphs can tolerate upto (2n− 5) faulty nodes and the fault diameter is increased only by 2 in the worst case in presence of maximum number of faults. Thus, star graphs enjoy the similar property of strong resilience under forbidden faulty sets like then-cubes. We have developed algorithms to trace the vertex disjoint paths under different conditions.

Combinatorial properties of generalized hypercube graphs

This paper investigates combinatorial properties of generalized hypercube graphs including best containers, wide diameter, and fault diameter. These properties have received much attention recently in the study of interconnection networks.

Transposition networks as a class of fault-tolerant robust networks

The paper proposes designs of interconnection networks (graphs) which can tolerate link failures. The networks under study belong to a subclass of Cayley graphs whose generators are subsets of all possible transpositions. We specifically focus on star and bubble sort networks. Our approach is to augment existing dimensions (or generators) with one or more dimensions. If the added dimension is capable of replacing any arbitrary failed dimension, it is called a wildcard dimension. It is shown that, up to isomorphism among digits used in labeling the vertices, the generators of the star graph are unique. The minimum number of extra dimensions needed to acquire i wildcard dimensions is derived for the star and bubble sort networks. Interestingly, the optimally augmented star network coincides with the Transposition network, T/sub n/. Transposition networks are studied rigorously. These networks are shown to be optimally fault tolerant. T/sub n/ is also shown to possess wide containers with short length. Fault diameter of T/sub n/ is shown to be n. While the T can efficiently embed star and bubble sort graphs, it can also lend itself to an efficient embedding of meshes and hypercubes.

Deadlock-free wormhole routing algorithms for star graph topology

For constructing massively parallel multicomputers with over 5000 processing nodes, the star graph topology is known to be better than the hypercube in terms of the average routing distance, the number of links per node, and the fault diameter. The authors present two deadlock-free algorithms for routing in star graph, assuming the wormhole routing model. Both the algorithms use the concept of virtual channels introduced by W. Dally and C.L. Seitz (1987). The first algorithm is non-optimal in terms of the average routing distance, but uses fewer virtual channels on the whole. The second algorithm is optimal in terms of routing performance, but requires a somewhat larger number of virtual channels per node.

Topological Properties of WK-Recursive Networks

The WK-recursive networks, which have been proposed by Vecchia and Sanges, offer high degree of regularity, scalability, and symmetry. They have also the properties of expansibility and equal degree. Many algorithmic results have been obtained for the WK-recursive networks, and a VLSI implementation of a 16-node WK-recursive network has been realized. In this paper, we investigate some topological properties of the WK-recursive networks, including container, wide diameter, and fault diameter.

Fault tolerance of the star graph interconnection network

This paper computes the exact fault diameter of the star graph. The fault diameter of a graph is the maximum diameter achievable, when we delete from the original graph any set of nodes that is smaller than its vertex connectivity. We show that for S 3 and S 4 the exact fault diameter is the original diameter plus two, and for S n , n>4, the fault diameter is the original diameter plus one.