Generalized-Star Cube: A New Class of Interconnection Topology for Massively Parallel Systems

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.

We propose a new class of interconnection networks, called macro-star networks, which belong to the class of Cayley graphs and use the star graph as a basic building module. A macro-star network can have node degree that is considerably smaller than that of a star graph of the same size, and diameter that is sublogarithmic and asymptotically within a factor of 1.25 from a universal lower bound (given its node degree). We show that algorithms developed for star graphs can be emulated on suitably constructed macro-stars with asymptotically optimal slowdown. This enables us to obtain through emulation a variety of efficient algorithms for the macro-star network, thus proving its versatility. Basic communication tasks, such as the multimode broadcast and the total exchange, can be executed in macro-star networks in asymptotically optimal time under both the single-port and the all-port communication models. Moreover, no interconnection network with similar node degree can perform these communication tasks in time that is better by more than a constant factor than that required in a macro-star network. We show that macro-star networks can embed trees, meshes, hypercubes, as well as star, bubble-sort, and complete transposition graphs with constant dilation. We introduce several variants of the macro-star network that provide more flexibility in scaling up the number of nodes. We also discuss implementation issues and compare the new topology with the star graph and other popular topologies.

The k-ary n-cube, or k-ary n-dimensional torus, is widely used as the interconnection network for constructing commercial supercomputers. Compared to the hypercube, torus has the advantage of ease of implementation but its diameter increases rapidly as the network scales up. This paper investigates the configurations of the k-ary n-cube by adjusting the parameters k and n according to the network size for achieving good cost performance tradeoffs. The node degree and diameter strongly affect the cost performance of the network. An interconnection network consists of routers and links which connect the ports of routers by following up an interconnect topology. The router is commonly implemented with a buffered crossbar switch whose cost increases non-linearly as the number of ports increases. The number of ports in a router is usually larger than the node degree because at least one port is needed to connect the compute node and this port is not counted as part of the node degree. The diameter determines the maximum communication delay between nodes which is a reciprocal proportion to the performance. This paper gives an analytical model for evaluating the relative cost performance (RCP) to hypercube. The model is parameterized with the node degree, the diameter, the router complexity, the number of compute nodes in a node, and the system size. The RCP function for a given n always has a minimum value at a certain system size. This is helpful for us to determine the values of k and n when we design a k-ary n-cube network. We also investigate the k-ary n-dimensional mesh and the unidirectional torus with different parameters. The unidirectional torus has a similar behavior to but better cost performance than the bidirectional torus.

Spontaneous explosive is an abrupt transition to collective behavior taking place in heterogeneous networks when the frequencies of the nodes are positively correlated with the node degree. This explosive transition was conjectured to be discontinuous. Indeed, numerical investigations reveal a hysteresis behavior associated with the transition. Here, we analyze explosive synchronization in star graphs. We show that in the thermodynamic limit the transition to (and out of) collective behavior is indeed discontinuous. The discontinuous nature of the transition is related to the nonlinear behavior of the order parameter, which in the thermodynamic limit exhibits multiple fixed points. Moreover, we unravel the hysteresis behavior in terms of the graph parameters. Our numerical results show that finite-size graphs are well described by our predictions.

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.

A star coloring of a graph $G$ is a proper vertex coloring in which every path on four vertices in $G$ is not bicolored. The star chromatic number $\chi_{s}\left(G\right)$ of $G$ is the least number of colors needed to star color $G$. In this paper, we find the exact values of the star chromatic number of modular product of complete graph with complete graph $K_m \diamond K_n$, path with complete graph $P_m \diamond K_n$ and star graph with complete graph $K_{1,m}\diamond K_n$. \par All graphs in this paper are finite, simple, connected and undirected graph and we follow \cite{bm, cla, f} for terminology and notation that are not defined here. We denote the vertex set and the edge set of $G$ by $V(G)$ and $E(G)$, respectively. Branko Gr\"{u}nbaum introduced the concept of star chromatic number in 1973. A star coloring \cite{alberton, fertin, bg} of a graph $G$ is a proper vertex coloring in which every path on four vertices uses at least three distinct colors. The star chromatic number $\chi_{s}\left(G\right)$ of $G$ is the least number of colors needed to star color $G$. \par During the years star coloring of graphs has been studied extensively by several authors, for instance see \cite{alberton, col, fertin}.

In this paper, we consider the problem of searching a network for intruders. We propose a strategy for capturing the intruder in the popular interconnection topology, the star network. According to the proposed strategy, a team of collaborative software agents are responsible for capturing a hostile intruder (e.g. a virus). These agents asynchronously move along the network links and the intruder has the capability of escaping arbitrarily fast.

Contraction theory provides an elegant way to analyze the behavior of certain nonlinear dynamical systems. In this paper, we discuss the application of contraction to synchronization of diffusively interconnected components described by nonlinear differential equations. We provide estimates of convergence of the difference in states between components, in the cases of line, complete, and star graphs, and Cartesian products of such graphs. We base our approach on contraction theory, using matrix measures derived from norms that are not induced by inner products. Such norms are the most appropriate in many applications, but proofs cannot rely upon Lyapunov-like linear matrix inequalities, and different techniques, such as the use of the Perron-Frobenious Theorem in the cases of $L^1$ or $L^{\infty }$ norms, must be introduced.

A star coloring of a graph G is a proper vertex coloring in which every path on four vertices in G is not bicolored. The star chromatic number χs (G) of G is the least number of colors needed to star color G. Let G = (V,E) be a graph with V = S1 [ S2 [ S3 [ . . . [ St [ T where each Si is a set of all vertices of the same degree with at least two elements and T =V (G) − St i=1 Si. The degree splitting graph DS (G) is obtained by adding vertices w1,w2, . . .wt and joining wi to each vertex of Si for 1 i t. The comb product between two graphs G and H, denoted by G ⊲ H, is a graph obtained by taking one copy of G and |V (G)| copies of H and grafting the ith copy of H at the vertex o to the ith vertex of G. In this paper, we give the exact value of star chromatic number of degree splitting of comb product of complete graph with complete graph, complete graph with path, complete graph with cycle, complete graph with star graph, cycle with complete graph, path with complete graph and cycle with path graph.

This paper shows the power of randomization in designing efficient parallel algorithms for the problems of routing and PRAM emulation. We show that with randomization techniques optimal routing can be obtained for a large class of processor interconnection networks called leveled networks. This class includes well-known networks (e.g., mesh, hypercube and de Bruijn) as well as new ones (e.g., star graph, for which the network diameter is sub-logarithmic in the network size). Using the obtained routing algorithms, optimal emulations of the CROW PRAM on the leveled network and mesh are also given.

Let R be a ring with identity and M be a unitary left R-module. The intersection graph of an R-moduleM, denoted by G(M), is defined to be the undirected simple graph whose vertices are in one to one correspondence with all non-trivial submodules of M and two distinct vertices are adjacent if and only if the corresponding submodules of M have nonzero intersection. We investigate the interplay between the module-theoretic properties of M and the graph-theoretic properties of G(M). We characterize all modules for which the intersection graph of submodules is connected. Also the diameter and the girth of G(M) are determined. We study the clique number and the chromatic number of G(M). Among other results, it is shown that if G(M) is a bipartite graph, then G(M) is a star graph.

Development of a computationally-tractable design method for combined multi-physics optimization of packing and routing problems, at a relevant scale, within compact packaging volumes, will offer benefits across several engineering domains. But for performing multi-physics packing and routing optimization, the generation of spatially feasible initial layouts is essential. Three new and computationally efficient methods are demonstrated in this article to produce automatically interference-free 2D geometric layouts. First, a novel 2D force-directed layout method (FDLM) is proposed that implicitly ensures noninterference between components and/or the interconnect network by utilizing spring force theory without using explicit geometric constraints. Second, the A* algorithm, a well-established 2D shortest path algorithm (SPA), has been modified significantly to perform efficient routing of complex interconnect systems. Third, a new geometric topology (GT) enumeration algorithm is presented that produces all unique interconnect routing configurations for given multi-component system architecture. These layout generation methods are then compared with respect to average computational efficiencies and average success rates in attaining feasible layouts for a restricted class of topologies, including evaluation of how the methods scale to problems with an increased number of components. Limitations and future work items for each method are discussed. These methods are presented as an important step toward solution strategies that are compatible with the currently unmet challenges of real-world 2D and 3D combined packing and routing problems, including efficient navigation of the space of discrete options for interconnect geometric topology, as well as scaling to more complex problems.

On a class of recursive network topologies

The advancements in technology in this contemporary time require special computing devices to able to manage the increase of the computational power. However, the limitation of the sequential computing systems stands as an obstacle in fulfilling these requirements. Thus, the research community looking for a viable solution able to solve the grand computing problems. A parallel computing system is a collection of processing elements that cooperate and communicate to provide a fast solution for a problem. A new symmetric network named Shifted Peterson Network (SPN) has been proposed in this paper. The basic module (BM) of the suggested network is composed of ten nodes, and the BMs of SPN are connected through external nodes to construct an advanced level of this network. Furthermore, the connectivity of SPN is based on a shifting mechanism of the binary digits of each node. The architecture of SPN discussed in this paper, in addition, the static network performance parameters of this network has been evaluated and compared to popular conventional interconnection networks. The networks evaluated in terms of node degree, diameter, average distance, arc connectivity, bisection width, cost, wiring complexity, packing density, and message traffic density. SPN showed good results in almost all aspects compared to these networks. Therefore, SPN is a suitable choice for the massively parallel computer (MPC) systems.

For integer k, r > 0, (k, r) -coloring of graph G is a proper coloring on the vertices of G by k-colors such that every vertex v of degree d(v) is adjacent to vertices with at least min{d(v), r} different color. By a proper k -coloring of graph G, we mean a map c : V (G) → S, where |S| = k, such that any two adjacent vertices are different color. An r -dynamic k -coloring is a proper k -coloring c of G such that |c(N (v))| ≥ min{r, d(v)} for each vertex v in V (G), where N (v) is the neighborhood of v and c(S) = {c(v) : v ∈ S} for a vertex subset S . The r-dynamic chromatic number, written as χr (G), is the minimum k such that G has an r-dynamic k-coloring. Note the 1-dynamic chromatic number of graph is equal to its chromatic number, denoted by χ(G), and the 2-dynamic chromatic number of graph denoted by χd (G). By simple observation with a greedy coloring algorithm, it is easy to see that χr (G) ≤ χr+1(G), however χr+1(G) − χr (G) does not always have the same difference. Thus finding an exact values of χr (G) is significantly useful. In this paper, we investigate the some exact value of χr (G) when G is for an operation product of cycle, star, complete, and path graphs.