Diagnosability of Bubble-Sort Star Graphs with Missing Edges

Star, bubble-sort, pancake, and Rotator-Faber-Moore (RFM) graphs are well-known interconnection networks that have node symmetric, maximum fault tolerance and hierarchical partition properties. These graphs are widely assumed to improve the network cost of a hypercube. This study proposes embedding methods for a star graph and its variations, and provides an analysis of the relevant costs. Results show that a bubble-sort graph can be embedded in a star graph with dilation 3, and in a RFM graph with dilation 2, while a star graph can be embedded in a pancake graph with dilation 4. The results suggest that the embedding method developed for the bubble-sort graph can be simulated in star graphs and RFM graphs in constant time O(1).

Characterization of component diagnosability of regular networks

The diagnosability of a multiprocessor system is of great significance in measuring the reliability and faulty tolerance of interconnection networks. In this paper, we firstly study the diagnosability of the [Formula: see text]-dimensional folded hypercube [Formula: see text] under the PMC model. We prove that [Formula: see text] keeps the strong local diagnosability property even if it has the set [Formula: see text] of [Formula: see text] faulty edges and [Formula: see text] is maximum number of faulty edges. Secondly, we study the diagnosability of [Formula: see text] [Formula: see text] and [Formula: see text] is odd) with conditional faulty edges under the PMC model. We prove that [Formula: see text] keeps strong local diagnosability property even if it has the set [Formula: see text] of [Formula: see text] faulty edges, provided that each vertex of [Formula: see text] is incident with at least two fault-free edges, where [Formula: see text] is maximum number of faulty edges. Finally, we prove that [Formula: see text] keeps strong local diagnosability property no matter how many edges are faulty, provided that each vertex of [Formula: see text] is incident with at least three fault-free edges.

If we only care about the status of a particular vertex, instead of doing global diagnosis, Hsu and Tan introduced the concept of local diagnosis and proposed an extended star structure to diagnose a vertex under comparison model. Usually, there is a gap between the local diagnosability and the lower bound guaranteed by the extended star structure mentioned above. In this paper, we propose a new testing structure and corresponding diagnosis algorithm to diagnose a vertex under comparison model to better evaluate the local diagnosability. The local diagnosability of a vertex is upper bounded by its degree in the system. If the local diagnosability of each vertex equals to its degree in the system then we say this system has the strong local diagnosability property. Based on the new structure, we show that the n-dimensional star graph \(S_n\) with faulty links has the strong local diagnosability property provided that each vertex connects to at least three fault-free links. Simulations are presented to show the performance of our tree structure.

Fault-tolerant maximal local-connectivity on Bubble-sort star graphs

ABSTRACTThe diagnosability of multiprocessor systems is an important research topic. In 2017, Wang et al. extended the k-ary n-cube and defined an expanded k-ary n-cube, which is a system topology structure with good properties. In this paper, we prove that the expanded k-ary n-cube has the strong local diagnosability property for and , and it keeps this strong property even if there exist missing edges in it under the comparison model.

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating functionGand a maximum principle which, we prove, is satisfied by every fixed point ofG. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

Freuder and Elfe (1996) introduced Neighborhood Inverse Consistency (NIC) as a strong local consistency property for binary CSPs. While enforcing NIC can significantly filter the variables domains, the proposed algorithm is too costly to be used on dense graphs or for lookahead during search. In this paper, we introduce and characterize Relational Neighborhood Inverse Consistency (RNIC) as a local consistency property that operates on the dual graph of a non-binary CSP. We describe and characterize a practical algorithm for enforcing it. We argue that defining RNIC on the dual graph unveils unsuspected opportunities to reduce the computational cost of our algorithm and increase its filtering effectiveness. We show how to achieve those effects by modifying the topology of the dual graph, yielding new variations the RNIC property. We also introduce an adaptive strategy to automatically select the appropriate property to enforce given the connectivity of the dual graph. We integrate the resulting techniques as full lookahead strategies in a backtrack search procedure for solving CSPs, and demonstrate the effectiveness of our approach for solving known difficult benchmark problems.

Excitation of hybridized multiple resonances can be an effective route for coherent light generation in irregular 1D systems larger than the localization length of light. Necklace states are often considered to have nonlocalized states. However, we propose that some hybridized/coupled states can have high-resonant tunneling with spatially extended fields. If strong localization properties are preserved in hybridized resonances, the excitation of such states allows for deposition of the excitation energy deep into the structure and spatial overlap with local gain regions. This result could allow for better utilizing hybridized resonances in biological or natural photonic systems.

The presence of 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 network. Edge-disjoint Hamiltonian cycles also provide the edge-fault tolerant hamiltonicity of an interconnection network. We will study the property of edge-disjoint Hamiltonian cycles in transposition networks in this article. The networks under study belong to a subclass of Cayley graphs whose generators are subsets of all possible transpositions. The transposition networks include other famous network topologies as their subgraphs, such as meshes, hypercubes, star graphs, and bubble-sort graphs. In this paper, we first introduce a novel decomposition of transposition networks. Using the proposed decomposition, we construct three edge-disjoint Hamiltonian cycles in 4-dimensional transposition network. We then show that n-dimensional transposition network with n ≥ 5 contains four edge-disjoint Hamiltonian cycles.

The generalized 3-connectivity of star graphs and bubble-sort graphs

This paper studies adaptive regularized methods for nonlinear least-squares problems
\nwhere the model of the objective function used at each iteration is either the 
\nEuclidean residual regularized by a quadratic term or the Gauss-Newton model regularized by a cubic term.
\nFor suitable choices of the regularization parameter the role of the regularization term is to 
\nprovide global convergence. 
\nIn this paper we investigate the impact of the regularization term on the local convergence rate of the methods and
\nestablish that, under the well-known error bound condition,
\nquadratic convergence to zero-residual solutions is enforced. 
\nThis result extends the existing analysis on the local convergence properties of adaptive regularized methods.
\nIn fact, the known results were derived under the standard full rank condition on the Jacobian at a zero-residual solution
\nwhile the error bound condition is weaker than the full rank condition and allows the solution set to be 
\nlocally nonunique.

We consider a quadratic programming-based method for nonlinear complementarity problems which allows inexact solutions of the quadratic subproblems. The main features of this method are that all iterates stay in the feasible set and that the method has some strong global and local convergence properties. Numerical results for all complementarity problems from the MCPLIB test problem collection are also reported.

Erratum to “Levenberg–Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints”: [J. Comput. Appl. Math. 173 (2005) 321–343

Diagnosability of a multiprocessor system is one important study topic. The bubble-sort graph n B is one of the attractive underlying topologies for the multiprocessor system. In this paper, the diagnosability of n B under the comparison diagnosis model has been studied. It is proved that n B is (n-1)-diagnosable under the comparison diagnosis model when n ≥ 4.