Unidirectional Hypercubes Research Articles

The 1-good-neighbor diagnosability of unidirectional hypercubes under the PMC model

The hypercubes are a famous class of networks for multiprocessor systems and the unidirectional hypercubes are hypercube interconnection topologies with simplex unidirectional links. Under the classic PMC model, each processor in a multiprocessor system tests a subset of its neighbors. The collection of tests in this system can be modeled by a directed graph. The diagnosability of a system is the maximum number of faulty processors that the system may identify according to the outcomes of the tests, and the g-good-neighbor diagnosability is a more accurate indicator than the diagnosability. In this paper, we first determine the 1-good-neighbor connectivity of unidirectional hypercubes and then determine the diagnosability and 1-good-neighbor diagnosability of hypercube networks when unidirectional hypercubes are used as the collection of tests under the PMC model.

Restricted Arc Connectivity of Unidirectional Hypercubes and Unidirectional Folded Hypercubes

Unidirectional hypercubes and unidirectional folded hypercubes are generalizations of hypercubes and folded hypercubes to digraphs. The super-$\lambda$ property of a digraph is a index for network reliability, which can be measured by the restricted arc-connectivity quantitatively. In this paper, we first show that the restricted arc-connectivity of the $n$-dimensional unidirectional hypercube is $n-1$ when $n$ is even and is $n-2$ when $n$ is odd, and then we show that the restricted arc-connectivity of the $n$-dimensional unidirectional folded hypercube is $n-1$ when $n$ is even and is $n$ when $n$ is odd. As a consequence, we prove that both unidirectional hypercube and unidirectional folded hypercube are super-$\lambda$.

On Hamiltonian properties of unidirectional hypercubes

The study of fault Hamiltonicity is an important topic in studying the structures of interconnection networks. Indeed, many advanced results have been obtained for undirected interconnection networks. However, much less is known for the directed counterparts. In the note, we consider the fault Hamiltonicity for unidirectional hypercubes.

Routing and wavelength assignment for hypercube in array-based WDM optical networks

Hypercube is one of the most versatile and efficient communication patterns shared by a large number of computational problems. As the number of edges in hypercube grows logarithmically with the size of networks, the complexity of network topologies can be significantly reduced to realize hypercube in optical networks by taking advantage of the parallel transmission characteristic of optical fibers. In this paper, we study the routing and wavelength assignment for realizing hypercube on WDM optical networks including linear arrays and rings with the consideration of communication directions. Specifically, we analyze this problem for both bidirectional and unidirectional hypercubes. For each case, we identify a lower bound on the number of wavelengths required, and design the embedding scheme and wavelength assignment algorithm that uses a provably near-optimal number of wavelengths. In addition, we extend the results to meshes and tori. By our embedding schemes, many algorithms, originally designed based on hypercubes, can be applied to optical networks, and the wavelength requirements can be easily derived using our obtained results.

Uni-Directional Alternating Group Graphs

A class of uni-directional Cayley graphs based on alternating groups is proposed in this paper. It is shown that this class of graphs is strongly connected and recursively scalable. The analysis of the shortest distance between any pair of nodes in a graph of this class is also given. Based on the analysis, we develop a polynomial time routing algorithm which yields a path distance at most one more than the theoretic lower bound. Furthermore, comparisons among uni-directional hypercubes, uni-directional star graphs, and uni-directional alternating group graphs are given. These observations validate the superiority of uni-directional alternating group graphs among known uni-directional topologies.

On container length and connectivity in unidirectional hypercubes

On container length and connectivity in unidirectional hypercubes

On container length and connectivity in unidirectional hypercubes

In this paper, we show that the two unidirectional binary n-cubes, namely, Q1(n) and Q2(n), proposed as high-speed networking schemes by Chou and Du are maximum fault-tolerant, that is, from vertex a to vertex b in the network, we can determine a set of ζ(a, b) vertex-disjoint routing paths, where ζ(a, b) = ⌈n/2⌉, if a has even parity and b has odd parity and ⌊n/2⌋ otherwise. Furthermore, the container problem as defined by Hsu for the two maximum fault-tolerant, unidirectional topologies is studied. In particular, we show that the smallest possible length for any maximum fault-tolerant container from a to b is at most (1) l + 4, where l is the shortest path length in Q1(n) from a to b, and (2) l + 5, where l is the shortest path length in Q2(n) (n is odd) (i) from a to b when a and b have the same leading-bit values and (ii) from a to b′ (b′ and b only differ at leading-bit positions) when otherwise. © 1998 John Wiley & Sons, Inc. Networks 32: 307–317, 1998

Hamiltonicity, vertex symmetry, and broadcasting of uni-directional hypercubes

In the paper, the authors show that the two uni-directional hypercubes, namely UHC1n and UHC2n, proposed by Chou and Du as interconnection schemes, are Hamiltonian. In addition, the authors show that if n is even, both architectures are vertex symmetric, and that if n is odd, both architectures have exactly two vertex-symmetric components. Furthermore, the study of the symmetry leads to an effective analysis of the maximum delay of one-port one-to-all broadcasting for either architecture which is at most ⌈1.5n⌉.