Edge-fault-tolerant vertex-pancyclicity of augmented cubes

The n-dimensional augmented cube, denoted as AQn, a variation of the hypercube, possesses some properties superior to those of the hypercube. In this paper, we show that every vertex in AQn lies on a fault-free cycle of every length from 3 to n², even if there are up to n−1 edge faults. We also show that our result is optimal.     

An Efficient Algorithm for Gray-to-Binary Permutation on Hypercubes

Both Gray code and binary code are frequently used in mapping arrays into hypercube architectures. While the former is preferred when communication between adjacent array elements is needed, the latter is preferred for FFT-type communication. When different phases of computations have different types of communication patterns, the need arises to remap the data. We give a nearly optimal algorithm for permuting data from a Gray code mapping to a binary code mapping on a hypercube with communication restricted to one input and one output channel per node at a time. Our algorithm improves over the best previously known algorithm (J. Parallel Distrib. Comput. 4, 2 (Apr. 1987), 133-172) by nearly a factor of two and is optimal to within a factor of n(n − 1) with respect to data transfer time on an n-cube. The expected speedup is confirmed by measurements on an Intel iPSCI2 hypercube.     

On embedding cycles into faulty dual-cubes

A dual-cube uses low-dimensional hypercubes as basic components such that keeps the main desired properties of the hypercube. Each hypercube component is referred as a cluster. A (n+1)-connected dual-cube DC(n) has ²2n+1 nodes and the number of nodes in a cluster is n². There are two classes with each class consisting of n² clusters. Each node is incident with exactly n+1 links where n is the degree of a cluster, one more link is used for connecting to a node in another cluster. In this paper, we show that every node of DC(n) lies on a cycle of every even length from 4 to ²2n+1 inclusive for n?3, that is, DC(n) is node-bipancyclic for n?3. Furthermore, we show that DC(n), n?3, is bipancyclic even if it has up to n−1 edge faults. The result is optimal with respect to the number of edge faults tolerant.     

Embedding the Complete Tree in the Hypercube

Let H_n be the n-dimensional boolean hypercube with 2ⁿ vertices labeled {0, 1, ... 2ⁿ − 1}, with an edge between two vertices whenever their Hamning distance is 1. We describe a spanning tree T_n of H_n with the following properties. T_n is complete for the first n − 2 levels with the remaining nodes on level n and n − 1 of the tree. Except for levels n and n − 1, there is a dilation 2 embedding of H_k on level k of T_n. T_n has minimum internal path length with respect to all binary spanning trees of H_n. Finally, each subtree of T_n is contained in the optimal sized subcube of H_n. This collection of almost complete binary trees is important for the implementation of tree-structured computation on hypercube configured multiprocessors.     

An Optimal Fault-Tolerant Nonredundant Broadcasting Scheme in Injured Hypercubes

We propose a nonredundant broadcasting scheme for injured hypercube multicomputers with direct-connection capability, that is, where each hypercube node has a separate router. The proposed scheme is first presented to cover faulty links only. Under the assumption that no more than n − 1 faulty links exist in an injured n-cube, the proposed scheme is optimal in the sense that it generates a spanning binomial tree from a given node whenever there exists such a tree originated from that node. The scheme is also extended to cover both node and link faults. Although the extended scheme is no longer optimal, we show that by using the extended scheme the maximum number of routing steps required from the source node s to any other node a is H(s, a) + 2, where H stands for the Hamming distance. The proposed scheme uses global network information. Evaluation studies show that the proposed scheme achieves optimal broadcasting a high percentage of times.     

Linear array and ring embeddings in conditional faulty hypercubes

The n-dimensional hypercube Q_n is a graph having 2ⁿ vertices labeled from 0 to 2ⁿ−1. Two vertices are connected by an edge if their binary labels differ in exactly one bit position. In this paper, we consider the faulty hypercube Q_n with n⩾3 that each vertex of Q_n is incident to at least two nonfaulty edges. Based on this requirement, we prove that Q_n contains a hamiltonian path joining any two different colored vertices even if it has up to 2n−5 edge faults. Moreover, we show that there exists a path of length 2ⁿ−2 between any two the same colored vertices in this faulty Q_n. Furthermore, we also prove that the faulty Q_n still contains a cycle of every even length from 4 to 2ⁿ inclusive.     

Conditional edge-fault-tolerant edge-bipancyclicity of hypercubes

The hypercube has been one of the most popular interconnection networks for parallel computer/communication systems. In this paper, we assume that each node is incident with at least two fault-free links. Under this assumption, we show that every fault-free edge lies on a fault-free cycle of every even length from 6 to 2ⁿ inclusive, even if it has up to 2n − 5 link faults. The result is optimal with respect to the number of link faults tolerated.     

Routing Permutations on Hypercube Machines with Half-Duplex Links

Algorithms are presented for realizing permutations on a less restrictive hypercube model called the S-MIMD (synchronous MIMD), which allows at most one data transfer on a given communication link at a given time instant, and where data movements are not restricted to a single dimension at a given time. First, an optimal algorithm for bit-permute permutations is developed from a very simple realization of the shuffle on a 3-cube; this algorithm needs 2⌊n/2⌋ routing steps on an n-dimensional hypercube. The technique is then extended to an optimal algorithm for bit-permute-complement permutations, one that needs n routing steps. Also, algorithms are sketched for routing permutations in the classes Ω and Ω⁻¹ in 3⌈n/2⌉ routing steps, yielding an off-line algorithm for routing arbitrary permutations in 3n steps.     

Vertex-pancyclicity of twisted cubes with maximal faulty edges

The n-dimensional twisted cube, denoted by TQ _n , a variation of the hypercube, possesses some properties superior to the hypercube. In this paper, we show that every vertex in TQ _n lies on a fault-free cycle of every length from 6 to 2 ⁿ , even if there are up to n?2 link faults. We also show that our result is optimal.     

Fault-free Hamiltonian cycles in crossed cubes with conditional link faults

The crossed cube, which is a variation of the hypercube, possesses some properties superior to the hypercube. In this paper, assuming that each node is incident with at least two fault-free links, we show that an n-dimensional crossed cube contains a fault-free Hamiltonian cycle, even if there are up to 2n − 5 link faults. The result is optimal with respect to the number of link faults tolerated. We also verify that the assumption is practically meaningful by evaluating its occurrence probability, which is very close to 1.     

Embedding Hypercubes and Related Networks into Mesh-Connected Processor Arrays

In this paper we study the problem of how computations programmed for hypercubes, and their bounded-degree relatives, the shuffle-exchange and cube-connected-cycles, can be efficiently emulated by mesh-connected arrays of processing elements. The emulations we present are implemented via graph embeddings. The graph embeddings we present are all optimal with respect to congestion, and are also noteworthy for they yield efficient emulation programs that are written in a SIMD style without the use of indirect addressing. The paper includes the three following emulation results, where each emulation algorithm requires no indirection. For any n ≥ 1 and fixed r ≥ 1, an n-sided r-dimensional mesh can emulate an n2ⁿ-node cube-connected-cycles network with slowdown Θ(2ⁿ/n^r−1), a 2ⁿ-node shuffle-exchange network with slowdown Θ(2ⁿ/n^r), and a 2ⁿ-node hypercube network operating in a weak model, with slowdown Θ(2ⁿ log n/n^r).     

Broadcasting Algorithms for the Star-Connected Cycles Interconnection Network

The star-connected cycles (SCC) graph was recently proposed as an alternative to the cube-connected cycles (CCC) graph, using a star graph to connect cycles of nodes rather than a hypercube. This paper presents an analysis of broadcasting algorithms for SIMD and MIMD SCCs, covering both one-port and multiple-port versions. We show that O(n log n) one-port broadcasting algorithms that have been proposed for the n-star cannot be efficiently extended to the case of the n-SCC graph. However, a simple but rather inefficient algorithm requiring O(n²) steps in the n-star graph can run in O(n) time in the n-SCC if a proper combination of parallelism and transmission rates in the links connecting the nodes is selected. The result is that broadcasting in the n-SCC can be accomplished efficiently, requiring a running time better than or equal to that of an n-star containing (n − 1) times fewer nodes.     

Broadcasting schemes for hypercubes with background traffic

Optimal broadcasting schemes for interconnection networks (INs) are most essential for the efficient interprocess communication amongst parallel computers. In this paper two novel broadcasting schemes are proposed for hypercube computers with bursty background traffic and a single-port mode of message passing communication. The schemes utilize a maximum entropy (ME) based queue-by-queue decomposition algorithm for arbitrary queueing network models (QNMs) [D.D. Kouvatsos, I. Awan, Perform. Eval. 51 (2003) 191] and are based on binomial trees and graph theoretic concepts. It is shown that the overall cost of the one-to-all broadcasting scheme is given by max{ω₁,ω₂,…,ω_2ⁿ/2}, where ω_i, i=1,2,…,2ⁿ/2 is the total weight at each leaf node of the binomial tree and n is the degree of the hypercube. Moreover, the upper bound of the total cost of the neighbourhood broadcasting scheme is determined by ∑_i=1^Fmax{ω_i}, where F is an upper bound of the number of steps and is equal to 1.33⌈log₂(n−1)⌉+1. Evidence based on empirical studies indicates the suitability of the schemes for achieving optimal broadcasting costs.     

Embedding a family of meshes into twisted cubes

The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. The twisted cube is an important variation of the hypercube. Let TQn denote the n-dimensional twisted cube. In this paper, we consider embedding a family of 2-dimensional meshes into a twisted cube. The main results obtained in this paper are: (1) For any odd integer n?1, there exists a mesh of size 2ײn−1 that can be embedded in the TQn with unit dilation and unit expansion. (2) For any odd integer n?5, there exists a mesh of size 4ײn−2 that can be embedded in the TQn with dilation 2 and unit expansion. (3) For any odd integer n?5, a family of two disjoint meshes of size 4ײn−3 can be embedded into the TQn with unit dilation and unit expansion. Results (1) and (3) are optimal in the sense that the dilations and expansions of the embeddings are unit values.     

Reconfiguration of Rings and Meshes in Faulty Hypercubes

In this paper we present schemes for reconfiguration of embedded task graphs in hypercubes. Previous results, which use either fault-tolerant embedding or an automorphism approach, can be expensive in terms of either the required number of spare nodes or reconfiguration time. Using the free dimension concept, we combine the above two approaches in our schemes which can tolerate about n faulty nodes under the worst case while keeping task migration time small. With expansion-2 initial embedding, three distributed reconfiguration schemes are presented in this paper. The first scheme, applied to chains and rings, can tolerate any ƒ ≤ n − 2 faulty nodes in an n-dimensional hypercube. The second and third schemes are applied to meshes or tori. For a mesh or torus of size 2^m₁ 1 ··· 1 2^m_d, the second scheme can tolerate any ƒ ≤ m_i − 1 faulty nodes, where m_i is the largest direction of the mesh and n = m₁ + ··· + m_d + 1. By embedding two copies of meshes or tori in cube, the third scheme can tolerate any ƒ ≤ n − 1 faulty nodes with the dilation of embedding after reconfiguration degraded to 2. The third scheme is quite general and can be applied to any task graph.     

Conditional edge-fault-tolerant Hamiltonicity of dual-cubes

The dual-cube is an interconnection network for linking a large number of nodes with a low node degree. It uses low-dimensional hypercubes as building blocks and keeps the main desired properties of the hypercube. A dual-cube DC(n) has n + 1 links per node where n is the degree of a cluster (n-cube), and one more link is used for connecting to a node in another cluster. In this paper, assuming each node is incident with at least two fault-free links, we show a dual-cube DC(n) contains a fault-free Hamiltonian cycle, even if it has up to 2n − 3 link faults. The result is optimal with respect to the number of tolerant edge faults.     

Highly fault-tolerant cycle embeddings of hypercubes

The hypercube Q_n is one of the most popular networks. In this paper, we first prove that the n-dimensional hypercube is 2n − 5 conditional fault-bipancyclic. That is, an injured hypercube with up to 2n − 5 faulty links has a cycle of length l for every even 4 ⩽ l ⩽ 2ⁿ when each node of the hypercube is incident with at least two healthy links. In addition, if a certain node is incident with less than two healthy links, we show that an injured hypercube contains cycles of all even lengths except hamiltonian cycles with up to 2n − 3 faulty links. Furthermore, the above two results are optimal. In conclusion, we find cycles of all possible lengths in injured hypercubes with up to 2n − 5 faulty links under all possible fault distributions.     

Embedding fault-free cycles in crossed cubes with conditional link faults

The crossed cube, which is a variation of the hypercube, possesses some properties that are superior to those of the hypercube. In this paper, we show that with the assumption of each node incident with at least two fault-free links, an n-dimensional crossed cube with up to 2n−5 link faults can embed, with dilation one, fault-free cycles of lengths ranging from 4 to 2 ⁿ . The assumption is meaningful, for its occurrence probability is very close to 1, and the result is optimal with respect to the number of link faults tolerated. Consequently, it is very probable that algorithms executable on rings of lengths ranging from 4 to 2 ⁿ can be applied to an n-dimensional crossed cube with up to 2n−5 link faults.

Cycles embedding in hypercubes with node failures

The hypercube has been widely used as the interconnection network in parallel computers. The n-dimensional hypercube Qn is a graph having n² vertices each labeled with a distinct n-bit binary strings. Two vertices are linked by an edge if and only if their addresses differ exactly in the one bit position. Let fv denote the number of faulty vertices in Qn. For n?3, in this paper, we prove that every fault-free edge and fault-free vertex of Qn lies on a fault-free cycle of every even length from 4 to n²−2fv inclusive even if fv?n−2. Our results are optimal.