Multicomputer interconnection network based on a star graph

Distance formula and shortest paths for the [formula omitted]-star graphs

In this paper a fault tolerant routing algorithm on star graph is proposed. Each node in an n-star is associated with a fault-vector of d=\(\lceil \frac{3(n-1)}{2} \rceil\) bits, which is an approximate measure of the number and distribution of faults in the neighborhood. The routing algorithm based on the fault-vector finds the shortest path between any source-destination pair in presence of large number of faulty nodes or links. Simulation results show that the % of cases where the algorithm fails to find a shortest path even when a shortest path exists, is as low as 2.7 with 140 node or link faults in a 7-star.

A new class of interconnection networks based on a family of graphs, called cubic graphs are introduced. These latter graphs arise as Cayley graphs of certain subgroups of the symmetric group. It turns out that these Cayley graphs are a hybrid between the binary hypercube and the star graph, and hence are called cubic star graphs, and are denoted by CS(m, n), m≥1 and n≥1. CS(m, n) inherits several of the properties of the hypercube and the star graph. In this paper, we present an analysis of the symmetric and topological properties. In particular, it is shown that CS(m, n) is edge transitive and hence maximally fault tolerant. We give an algorithm for finding the shortest path and provide an enumeration of the node disjoint paths. Optimal algorithms for single source and all-source broadcasting (also called gossiping) are derived. It is shown that CS(m, n) is Hamiltonian and interesting embeddings of several cycles, grids, and binary trees are derived. The paper concludes with a comparison of CS(m, n) with the binary hypercube and the star graph.

The adaptive routing approach has been expected as a promising way to improve network performance by utilizing available network bandwidth. Previous adaptive routing strategies in wormhole-routed multicomputer networks restrict the routing of messages by the routing algorithm to prevent deadlock. This results in low degree of adaptivity and low utilization of physical or virtual channels. In this paper, we examine the possibility of performing restriction-free adaptive routing in wormhole-routed networks as an approach to further improving the performance of these networks. A new flow control policy, called message cutting-in, is proposed, and two adaptive routing strategies are presented. Freedom of communication deadlock is achieved by the proposed flow control policy. The proposed adaptive routing strategies do not restrict routing and maximally utilize the physical and virtual channels. Simulation results show that the restriction-free adaptive routing approach is promising from the fact that it has the lowest latency and highest throughput depending on the number of virtual channels per physical channel and patterns of message traffic.

A new deadlock prevention scheme for nonminimal adaptive wormhole routing

Constructing vertex–disjoint paths in ( n, k)-star graphs

In this paper, another version of the star cube called the generalized-star cube, GSC(n, k, m), is presented as a three level interconnection topology. GSC(n, k, m) is a product graph of the (n, k)-star graph and the m-dimensional hypercube (m-cube). It can be constructed in one of two ways: to replace each node in an m-cube with an (n, k)-star graph, or to replace each node in an (n, k)-star graph with an m-cube. Because there are three parameters m, n, and k, the network size of GSC(n, k, m) can be changed more flexibly than the star graph, star-cube, and (n, k)-star graph. We first investigate the topological properties of the GSC(n, k, m), such as the node degree, diameter, average distance, and cost. Also, the regularity and node symmetry of the GSC(n, k, m) are derived. Next, we present a formal shortest-path routing algorithm. Then, we give broadcasting algorithms for both of the single-port and all-port models. To develop these algorithms, we use the spanning binomial tree, the neighborhood broadcasting algorithm, and the minimum dominating set. The complexities of the routing and broadcasting algorithms are also examined.

In this paper, another version of the star cube called the generalized-star cube, GSC(n, k, m), is presented as a three level interconnection topology. GSC(n, k, m) is a product graph of the (n, k)-star graph and the m-dimensional hypercube (m-cube). It can be constructed in one of two ways: to replace each node in an m-cube with an (n, k)-star graph, or to replace each node in an (n, k)-star graph with an m-cube. Because there are three parametersm, n, and k, the network size of GSC(n, k, m) can be changed more flexibly than the star graph, star-cube, and (n, k)-star graph. This paper describes the topology of the GSC(n, k, m), gives a formal shortest-path routing algorithm, and examines the topological properties of the GSC(n, k, m), such as the node degree, diameter, average distance, and cost. Also, the regularity and node symmetry of the GSC(n, k, m) are derived.

A star graph Sn [1], of order n, is defined to be a symmetric graph G=(V, E) where V is the set of n! vertices, each representing a distinct permutation of n elements and E is the set of symmetric edges such that two permutations (nodes) are connected by an edge iff one can be reached from the other by interchanging its first symbol with any other symbol. The star graph Sn is a (n−1)-regular graph with n! nodes and n!(n−1)/2 edges. Recursive hierarchical structure is one of the most attractive and well known properties of star graphs. A dimension n star graph can be divided into n substars of dimension n−1 by grouping the nodes with the same symbol at the ith position together, 2≤i≤n [see Figure 1]. In this paper, we propose a new recursive hierarchical structure of star graphs. The objective is to redesign shortest routing in star graphs in the light of this new structure and design new efficient algorithms for shortest path multicast algorithms [2] adaptibe to bandwidth and latency requirements.

A bipartite graph is Hamiltonian laceable if any two of its vertices in different partite sets are connected by a Hamiltonian path. A Hamiltonian laceable graph $G$ is called strongly Hamiltonian laceable if any two of its vertices in the same partite set are connected by a path of length $|V(G)|-2$. A Hamiltonian laceable graph $G$ (with two partite sets $V_0, V_1$) is called hyper-Hamiltonian laceable, if for any vertex $v \in V_{i}$ for $i \in \{0,1\}$, there is a Hamiltonian path of $G-\{v\}$ between any two vertices in $V_{1-i}$. In this paper, we focus on the edge-fault-tolerant strongly Hamiltonian laceability and hyper-Hamiltonian laceability on the class of Cayley graphs generated by transposition trees, which are a generalization of star graph and bubble-sort graph. For every $n$-dimensional Cayley graph generated by a transposition tree $\Gamma _n$, we show that $\Gamma _{n}-F$ is strongly Hamiltonian laceable for any $F \subseteq E(\Gamma _{n})$ with $|F|\leq n-3$, which generalizes results in [ 1, 11], and show that $\Gamma _{n}-F$ is hyper-Hamiltonian laceable for any $F \subseteq E(\Gamma _{n})$ with $|F|\leq n-4$.

We present a method to enhance wormhole routing algorithms for deadlock-free fault-tolerant routing in tori. We consider arbitrarily-located faulty blocks and assume only local knowledge of faults. Messages are routed via shortest paths when there are no faults, and this constraint is only slightly relaxed to facilitate routing in the presence of faults. The key concept we use is that, for each fault region, a fault ring consisting of fault free nodes and physical channels can be formed around it. These fault rings can be used to route messages around fault regions. We prove that at most four additional virtual channels are sufficient to make any fully-adaptive algorithm tolerant to multiple faulty blocks in torus networks. As an example of this technique, we present simulation results for a fully-adaptive algorithm and show that good performance can be obtained with as many as 10% links faulty.

Recently, the merits of cooperative communication in the physical layer have been explored. However, the impact of cooperative communication on the design of the higher layers has not been well-understood yet. Cooperative routing in wireless networks has gained much interest due to its ability to exploit the broadcast nature of the wireless medium in designing power efficient routing algorithms. Most of the existing cooperation based routing algorithms are implemented by finding a shortest path route first and then improving the route using cooperative communication. As such, these routing algorithms do not fully exploit the merits of cooperative communications, since the optimal cooperative route might not be similar to the shortest path route. In this paper, we propose a cooperation-based routing algorithm, namely, the minimum power cooperative routing (MPCR) algorithm, which makes full use of the cooperative communications while constructing the minimum-power route. The MPCR algorithm constructs the minimum-power route, which guarantees certain throughput, as a cascade of the minimum-power single-relay building blocks from the source to the destination. Thus, any distributed shortest path algorithm can be utilized to find the optimal cooperative route with polynomial complexity. Using analysis, we show that the MPCR algorithm can achieve power saving of 65.61% in regular linear networks and 29.8% in regular grid networks compared to the existing cooperation-based routing algorithms, where the cooperative routes are constructed based on the shortest-path routes. From simulation results, MPCR algorithm can have 37.64% power saving in random networks compared to those cooperation-based routing algorithms.

We study end-to-end delay performance of different routing algorithms in networks with random failures. Specifically, we compare delay performances of differential backlog (DB) and shortest path (SP) routing algorithms and show that DB routing outperforms SP routing in terms of throughput when the network is heavily loaded and/or the failure rate is high while SP routing achieves better delay performance in the low load regime. Then, we investigate delay performance of a hybrid routing algorithm that combines principles of both SP and DB routing algorithms and show that it outperforms both of these routing algorithms. Finally, we demonstrate improvements in delay performance of DB routing through the use of a digital fountain approach which was originally proposed for multicast applications. In addition, our results show that there exists an optimal coding rate where digital fountain based DB routing achieves minimum end-to-end delay. To the best of our knowledge, this is the first work which investigates delay performance of DB routing and its enhanced versions for networks with link failures.

Gossiping and routing in second-kind Frobenius graphs

One of the topics in graph theory that is interesting and developed continuously is metric dimension. It has some new variation concepts, such as star metric dimension. The order set of Z={z1,z2,…,zn }⊆V(G) called star resolving set of connected graph G if Z is Star Graph and for every vertex in G has different representation to the set Z. The representation is expressed as the distance d(u,z), it is the shortest path from vertex u to z for every u,z∈V(G). Star basis of a graph is the smallest cardinality of star resolving set. The number of vertex in star basis is called star metric dimension of G which denoted by Sdim(G). The purpose of this article is to determine the characteristic of star metric dimension and the value of star metric dimension of some classes of graphs. The method which is used in this study is library research. Some of the results of this research are complete graph has Sdim(Kn )=n-1, for n≥3, bipartite graph K(2,n) has Sdim (K(2,n) )=n, for n≥3. Besides complete bipartite graph hasn’t star metric dimension or for m,n≥3 or it can said that Sdim (K(m,n) )=0. Another graph, that is Fan graph has Sdim (Fn )=2 for 2≤n≤5 and for n≥6 Sdim (Fn )=(2n+3)/5.