511 research outputs found
Long path and cycle decompositions of even hypercubes
We consider edge decompositions of the -dimensional hypercube into
isomorphic copies of a given graph . While a number of results are known
about decomposing into graphs from various classes, the simplest cases of
paths and cycles of a given length are far from being understood. A conjecture
of Erde asserts that if is even, and divides the number
of edges of , then the path of length decomposes . Tapadia et
al.\ proved that any path of length , where , satisfying these
conditions decomposes . Here, we make progress toward resolving Erde's
conjecture by showing that cycles of certain lengths up to
decompose . As a consequence, we show that can be decomposed into
copies of any path of length at most dividing the number of edges of
, thereby settling Erde's conjecture up to a linear factor
Embedding cube-connected cycles graphs into faulty hypercubes
We consider the problem of embedding a cube-connected cycles graph (CCC) into a hypercube with edge faults. Our main result is an algorithm that, given a list of faulty edges, computes an embedding of the CCC that spans all of the nodes and avoids all of the faulty edges. The algorithm has optimal running time and tolerates the maximum number of faults (in a worst-case setting). Because ascend-descend algorithms can be implemented efficiently on a CCC, this embedding enables the implementation of ascend-descend algorithms, such as bitonic sort, on hypercubes with edge faults. We also present a number of related results, including an algorithm for embedding a CCC into a hypercube with edge and node faults and an algorithm for embedding a spanning torus into a hypercube with edge faults
Constructing Two Edge-Disjoint Hamiltonian Cycles in Locally Twisted Cubes
The -dimensional hypercube network is one of the most popular
interconnection networks since it has simple structure and is easy to
implement. The -dimensional locally twisted cube, denoted by , an
important variation of the hypercube, has the same number of nodes and the same
number of connections per node as . One advantage of is that the
diameter is only about half of the diameter of . Recently, some
interesting properties of were investigated. In this paper, we
construct two edge-disjoint Hamiltonian cycles in the locally twisted cube
, for any integer . 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
Wildcard dimensions, coding theory and fault-tolerant meshes and hypercubes
Hypercubes, meshes and tori are well known interconnection networks for parallel computers. The sets of edges in those graphs can be partitioned to dimensions. It is well known that the hypercube can be extended by adding a wildcard dimension resulting in a folded hypercube that has better fault-tolerant and communication capabilities. First we prove that the folded hypercube is optimal in the sense that only a single wildcard dimension can be added to the hypercube. We then investigate the idea of adding wildcard dimensions to d-dimensional meshes and tori. Using techniques from error correcting codes we construct d-dimensional meshes and tori with wildcard dimensions. Finally, we show how these constructions can be used to tolerate edge and node faults in mesh and torus networks
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