Constructing Two Edge-Disjoint Hamiltonian Cycles in Locally Twisted Cubes

The $n$-dimensional hypercube network $Q_n$ is one of the most popular interconnection networks since it has simple structure and is easy to implement. The $n$-dimensional locally twisted cube, denoted by $LTQ_n$, an important variation of the hypercube, has the same number of nodes and the same number of connections per node as $Q_n$. One advantage of $LTQ_n$ is that the diameter is only about half of the diameter of $Q_n$. Recently, some interesting properties of $LTQ_n$ were investigated. In this paper, we construct two edge-disjoint Hamiltonian cycles in the locally twisted cube $LTQ_n$, for any integer $n\geqslant 4$. The presence of two edge-disjoint Hamiltonian cycles provides an advantage when implementing algorithms that require a ring structure by allowing message traffic to be spread evenly across the locally twisted cube.Comment: 7 pages, 4 figure

Fractional strong matching preclusion for two variants of hypercubes

Let F be a subset of edges and vertices of a graph G. If G-F has no fractional perfect matching, then F is a fractional strong matching preclusion set of G. The fractional strong matching preclusion number is the cardinality of a minimum fractional strong matching preclusion set. In this paper, we mainly study the fractional strong matching preclusion problem for two variants of hypercubes, the multiply twisted cube and the locally twisted cube, which are two of the most popular interconnection networks. In addition, we classify all the optimal fractional strong matching preclusion set of each

Automorphisms generating disjoint Hamilton cycles in star graphs

In the first part of the thesis we define an automorphism φn for each star graph Stn of degree n − 1, which yields permutations of labels for the edges of Stn taken from the set of integers {1, . . . , bn/2c}. By decomposing these permutations into permutation cycles, we are able to identify edge-disjoint Hamilton cycles that are automorphic images of a known two-labelled Hamilton cycle H1 2(n) in Stn. Our main result is an improvement from the existing lower bound of bϕ(n)/10c to b2ϕ(n)/9c, where ϕ is Euler’s totient function, for the known number of edge-disjoint Hamilton cycles in Stn for all odd integers n. For prime n, the improvement is from bn/8c to bn/5c. We extend this result to the cases when n is the power of a prime other than 3 and 7. The second part of the thesis studies ‘symmetric’ collections of edge-disjoint Hamilton cycles in Stn, i.e. collections that comprise images of H1 2(n) under general label-mapping automorphisms. We show that, for all even n, there exists a symmetric collection of bϕ(n)/2c edge-disjoint Hamilton cycles, and Stn cannot have symmetric collections of greater than bϕ(n)/2c such cycles for any n. Thus, Stn is not symmetrically Hamilton decomposable if n is not prime. We also give cases of even n, in terms of Carmichael’s reduced totient function λ, for which ‘strongly’ symmetric collections of edge-disjoint Hamilton cycles, which are generated from H1 2(n) by a single automorphism, can and cannot attain the optimum bound bϕ(n)/2c for symmetric collections. In particular, we show that if n is a power of 2, then Stn has a spanning subgraph with more than half of the edges of Stn, which is strongly symmetrically Hamilton decomposable. For odd n, it remains an open problem as to whether the bϕ(n)/2c can be achieved for symmetric collections, but we are able to show that, for certain odd n, a ϕ(n)/4 bound is achievable and optimal for strongly symmetric collections. The search for edge-disjoint Hamilton cycles in star graphs is important for the design of interconnection network topologies in computer science. All our results improve on the known bounds for numbers of any kind of edge-disjoint Hamilton cycles in star graphs

Estratégias eficientes para identificação de falhas utilizando o diagnóstico baseado em comparações

Orientador: Prof. Dr. Elias Procópio Duarte Jr.Tese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Curso de Pós-Graduaçao em Informática. Defesa: Curitiba, 12/04/2013Bibliografia: fls. 126-148Resumo: O diagnóstico baseado em comparações e uma forma realista para detectar falhas em hardware, software, redes e sistemas distribuídos. O diagnostico se baseia na comparaçao de resultados de tarefas produzidos por pares de unidades para determinar quais sao as unidades falhas e sem-falha do sistema. Qualquer diferenca no resultado da comparacao indica que uma ou ambas as unidades estao falhas. O diagnostico completo do sistema e baseado no resultado de todas as comparações. Este trabalho apresenta um novo algoritmo de diagnostico para identificar falhas em sistemas de topologia arbitraria com base no modelo MM*. A complexidade do algoritmo proposto e O(t2AN) no pior caso para sistemas de N unidades, onde t denota o numero maximo permitido de unidades falhas e A e o grau da unidade de maior grau no sistema. Esta complexidade e significativamente menor que a dos outros algoritmos previamente publicados. Alem da especificacao do algoritmo e das provas de correcão, resultados obtidos atraves da execucao exaustiva de experimentos sao apresentados, mostrando o desempenho me dio do algoritmo para diferentes sistemas. Al em do novo algoritmo para sistemas de topologia arbitraria, este trabalho tambem apresenta duas outras solucoes para deteccão e combate a poluicao de conteudo, ou alteracoes nao autorizadas, em transmissões de mídia contínua ao vivo em redes P2P - a primeira e uma solucão centralizada e que realiza o diagnostico da poluicao na rede, e a segunda e uma solucao completamente distribuída e descentralizada que tem o objetivo de combater a propagacao da poluicao na rede. Ambas as solucoes utilizam o diagnostico baseado em comparacoes para detectar alterações no conteudo dos dados transmitidos. As soluções foram implementadas no Fireflies, um protocolo escalavel para redes overlay, e diversos experimentos atraves de simulacao foram conduzidos. Os resultados mostram que ambas as estrategias sao solucães viaveis para identificar e combater a poluiçcãao de conteudo em transmissãoes ao vivo e que adicionam baixa sobrecarga ao trafego da rede. Em particular a estrategia de combate a poluicao foi capaz de reduzir consideravelmente a poluicão de conteudo em diversas configurações, em varios casos chegando a elimina-la no decorrer das transmissoães.Abstract: Comparison-based diagnosis is a practical approach to detect faults in hardware, software, and network-based systems. Diagnosis is based on the comparison of task outputs returned by pairs of system units in order to determine whether those units are faulty or fault-free. If the comparison results in a mismatch then one ore both units are faulty. System diagnosis is based on the complete set of all comparison results. This work introduces a novel diagnosis algorithm to identify faults in t-diagnosable systems of arbitrary topology under the MM* model. The complexity of the proposed algorithm is O(t2AN) in the worst case for systems with N units, where t denotes the maximum number of faulty units allowed and A corresponds to the maximum degree of a unit in the system. This complexity is significantly lower than those of previously published algorithms. Besides the algorithm specification and correctness proofs, exhaustive simulations results are presented, showing the typical performance of the algorithm for different systems. Moreover, this work also presents two different strategies to detect and fight content pollution in P2P live streaming transmissions - the first strategy is centralized and performs the diagnosis of content pollution in the network, and the second strategy is a completely distributed solution to combat the propagation of the pollution. Both strategies employ comparison-based diagnosis in order to detect any modification in the data transmitted. The solutions were also implemented in Fireflies, a scalable and fault-tolerant overlay network protocol, and a large number of simulation experiments were conduced. Results show that both strategies are feasible solutions to identify and fight content pollution in live streaming sessions and that they add low overhead in terms of network bandwidth usage. In particular, the solution proposed to combat content pollution was able to significantly reduce the pollution over the system in diverse network configurations - in many cases the solution nearly eliminated the pollution during the transmission