Hypermeshes have been given much attention as a versatile interconnection network of parallel computers. A hypermesh is obtained from a mesh by replacing each linear connection with a hyperedge. In this paper, we show how to embed a butterfly or multiple copies of a butterfly into a hypermesh. First, a butterfly B(s) of (s + 1)2s nodes is embedded into a 2s × X hypermesh where X = 2⌊ log 2 s ⌋+ 1. Second, the butterfly B(s) is embedded into a square hypermesh. Third, multiple copies of the butterfly B(s) are embedded into a hypermesh of variable aspect ratio. The efficiency of these embeddings is measured by alignment cost, congestion, and expansion. The alignment cost of all of these embeddings is optimal. The congestion of the first and third embedding is optimal. The expansion of the first and third embedding is one if s = 2k - 1 for some integer k, otherwise, less than two. The expansion of the second embedding is 2 + ∊ (s) where ∊(s) = (2 log (s + 1) + 2)/(s + 1).
A Disjoint Path Cover (DPC for short) of a graph is a set of pairwise (internally) disjoint paths that altogether cover every vertex of the graph. Given a set S of k sources and a set T of k sinks, a many-to-many k-DPC between S and T is a disjoint path cover each of whose paths joins a pair of source and sink. It is classified as paired if each source of S must be joined to a designated sink of T , or unpaired if there is no such constraint. In this paper, we show that every m-dimensional restricted hypercube-like graph with at most m − 3 faulty vertices and/or edges being removed has a paired (and unpaired) 2-DPC joining arbitrary two sources and two sinks where m ≥ 5. The bound m − 3 on the number of faults is optimal for both paired and unpaired types.
SUMMARYWe analyse and compare a protection mechanism based on load distribution with a typical protection mechanism in an multiprotocol label switching (MPLS) network. The protection mechanism based on load distribution is modelled as a fully shared mechanism (FSM) and the typical protection mechanism is a partially shared mechanism (PSM). By comparing the FSM and the PSM, we numerically analyse the effect of load distribution in path protection of MPLS. The comparison is based on numerical equations representing the relationship between service reliability and resource utilization. From the equations, we show that both the FSM and the PSM have a tradeoff between service reliability and resource utilization. In addition, we provide solutions for the FSM and the PSM to determine the amount of bandwidth occupied according to the requested service reliability. The comparison of the FSM and the PSM shows that the PSM cannot provide greater service reliability than the FSM under the same utilization. In addition, the PSM is not capable of accommodating greater resource utilization than the FSM for the same level of service reliability. However, an appropriate choice of the number of protection paths allows the PSM to provide the same level of service reliability as the FSM. The choice is the maximum among the possible numbers of protection paths of the PSM. In short, the typical protection mechanism is as good as the FSM in terms of service reliability and resource utilization even though the FSM is an attractive alternative to the typical protection mechanism.
W e present two embedding methods of the full binary tree into the hypercube when the tree has greater number of nodes than the hypercube. Both methods map the tree edges onto the edge-disjoint paths o f t h e hypercube(each hypercube edge being considered as two anti-parallel directed edges), and distribute the same level tree nodes evenly t o the h:ypercube nodes. One embedding method with the optimal dilation has a load factor greater than the optimal tlalue by one. And the other embedding method with the optimal load factor has a dilation greater than the optimal value by one.Furthermore, we show that both the dilation and load factor can not be optimized sim.ultaneously when the same level tree nodes are evenly distributed.
Restricted Hypercube-Like (RHL) graphs are a graph class that widely includes useful interconnection networks such as crossed cube, Möbius cube, Mcube, twisted cube, locally twisted cube, multiply twisted cube, and generalized twisted cube. In this paper, we show that for an m-dimensional RHL graph G, m≥4, with an arbitrary faulty edge set F⊂E(G), |F|≤m-2, graph G∖F has a hamiltonian path between any distinct two nodes s and t if dist(s, V(F))≠1 or dist(t, V(F))≠1. Graph G∖F is the graph G whose faulty edges are removed. Set V(F) is the end vertex set of the edges in F and dist(v, V(F)) is the minimum distance between vertex v and the vertices in V(F).
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