Routing and Broadcasting Algorithms for Generalized-Star Cube

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.

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.

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

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.

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.

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.

An interconnection network based on a star graph of arbitrary order n, is analysed. Star graphs belong to a family of Cayley graphs, in which vertices correspond to elements of a permutation group, and edges correspond to generators of the group. Generators of the star graph of order n are n-1 transpositions, which may be represented with a star transposition tree. The algebraic properties of these graphs are investigated through analysis of the corresponding transposition tree. The algebraic expression for the shortest path between two nodes is found, and it is shown that every shortest path consists of a number of subpaths, which may be combined in an arbitrary order. From the analysis of shortest paths follows the minimum number of virtual channels, into which every physical channel must be split in order to avoid deadlock. Finally, a routing algorithm is developed which repetitively extracts subpaths, by sorting the source permutation to the destination permutation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Some topological properties of star graphs: The surface area and volume

In a hypercube or a star graph, given an unknown number of nodes located at unknown positions each intending to broadcast a message, we propose an efficient routing algorithm to solve this problem using asymptotically optimal or near-optimal transmission time.

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.

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}.

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.

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.