Investigation of the robustness of star graph networks

The star interconnection network has been known as an attractive alternative to n-cube for interconnecting a large number of processors. It possesses many nice properties, such as vertex/edge symmetry, recursiveness, sublogarithmic degree and diameter, and maximal fault tolerance, which are all desirable when building an interconnection topology for a parallel and distributed system. Investigation of the robustness of the star network architecture is essential since the star network has the potential of use in critical applications. In this study, three different reliability measures are proposed to investigate the robustness of the star network. First, a constrained two-terminal reliability measure referred to as Distance Reliability (DR) between the source node u and the destination node I with the shortest distance, in an n-dimensional star network, Sn, is introduced to assess the robustness of the star network. A combinatorial analysis on DR especially for u having a single cycle is performed under different failure models (node, link, combined node/link failure). Lower bounds on the special case of the DR: antipode reliability, are derived, compared with n-cube, and shown to be more fault-tolerant than n-cube. The degradation of a container in a Sn having at least one operational optimal path between u and I is also examined to measure the system effectiveness in the presence of failures under different failure models. The values of MTTF to each transition state are calculated and compared with similar size containers in n-cube. Meanwhile, an upper bound under the probability fault model and an approximation under the fixed partitioning approach on the ( n-1)-star reliability are derived, and proved to be similarly accurate and close to the simulations results. Conservative comparisons between similar size star networks and n-cubes show that the star network is more robust than n-cube in terms of ( n-1)-network reliability

Analysis of wormhole routings in cayley graphs of permutation groups.

Over a decade, a new class of switching technology, called wormhole routing, has been investigated in the multicomputer interconnection network field. Several classes of wormhole routing algorithms have been proposed. Most of the algorithms have been centered on the traditional binary hypercube, k-ary n-cube mesh, and torus networks. In the design of a wormhole routing algorithm, deadlock avoidance scheme is the main concern. Recently, new classes of networks called Cayley graphs of permutation groups are considered very promising alternatives. Although proposed Cayley networks have superior topological properties over the traditional network topologies, the design of the deadlock-free wormhole routing algorithm in these networks is not simple. In this dissertation, we investigate deadlock free wormhole routing algorithms in the several classes of Cayley networks, such as complete-transposition and star networks. We evaluate several classes of routing algorithms on these networks, and compare the performance of each algorithm to the simulation study. Also, the performances of these networks are compared to the traditional networks. Through extensive simulation we found that adaptive algorithm outperformed deterministic algorithm in general with more virtual channels. On the network performance comparison, the complete transposition network showed the best performance among the similar sized networks, and the binary hypercube performed better compared to the star graph

Symmetrizing dynamics: from classical to quantum applications

Among the issues regarding networked systems, the “consensus problem” and the related algorithms have received a significant share of attention during the last ten years. In this problem the network agents asymptotically have to attain agreement on the value of some objective variable under local communication constraints. A number of algorithms have been developed to address this problem, among which the celebrated gossip algorithm. The latter relays on switching dynamics and, under rather weak assumptions, exhibits robust convergence under variations in the interaction constraints, i.e. the network topology. In this dissertation we reinterpret the goal of the consensus problem as a symmetrisation problem, and we address it by a switching-type dynamics based on convex combinations of actions of a finite group. In order to study the convergence of our class of algorithms we lift the dynamics to an abstract, group-theoretic level that allow us to derive general conditions for convergence. Such conditions, in fact, are independent of the particular group action, and focus only on the group itself and the way the iterations are selected. Convergence is guaranteed provided that some mild assumptions on the selection rule for the iterations are fulfilled. Furthermore, this class of algorithms retains the robustness features and unsupervised character of the consensus algorithm. Our reformulation allow to devise algorithms for application as diverse as randomized discrete Fourier transform and random state generation. We pose a special emphasis on the extension of the consensus problem to the quantum domain. In this setting we highlight how, due to the richer mathematical structure over which the internal state is encoded, the definition of the consensus goal admits various extensions, each of them exhibiting different features. We also propose a suitable dissipative dynamics enacting the symmetrising gossip interactions and then use our general result on convergence to prove it ensures asymptotic convergence. Beside the technical results, one of the main contributions of our work is a new, generalized view point on consensus, which allows us to extend the robustness of consensus-inspired algorithms to new problems in apparently unrelated fields. This reinforces the role of consensus algorithms as fundamental tools for distributed computing, both in the classical and the quantum setting

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum