Hamiltonian paths and cycles with prescribed edges in the 3-ary n-cube

The k-ary n-cube has been one of the most popular interconnection networks for massively parallel systems. Given a set P of at most 2n − 2 (n ? 2) prescribed edges and two vertices u and v, we show that the 3-ary n-cube contains a Hamiltonian path between u and v passing through all edges of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths, none of them having u or v as internal vertices or both of them as end-vertices. As an immediate result, the 3-ary n-cube contains a Hamiltonian cycle passing through a set P of at most 2n − 1 prescribed edges if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths.     

Many-to-many disjoint paths in faulty hypercubes

This paper considers the problem of many-to-many disjoint paths in the hypercube Q_n with f_v faulty vertices and f_e faulty edges, and obtains the following result. For any integer k with 1?k?n-1, any two sets S and T of k fault-free vertices in different parts, if f_v+f_e?n-k-1, then there exist k disjoint fault-free (S,T)-paths in Q_n which contains at least 2ⁿ-2f_v vertices. This result is optimal in the worst case.     

Fault-tolerant routing over shortest node-disjoint paths in hypercubes

This paper presented a routing algorithm that finds n disjoint shortest paths from the source node s to target node d in the n-dimensional hypercube. Fault-tolerant routing over all shortest node-disjoint paths has been investigated to overcome the failure encountered during routing in hypercube networks. In this paper, we proposed an efficient approach to provide fault-tolerant routing which has been investigated on hypercube networks. The proposed approach is based on all shortest node-disjoint paths concept in order to find a fault-free shortest path among several paths provided. The proposed algorithm is a simple uniform distributed algorithm that can tolerate a large number of process failures, while delivering all n messages over optimal-length disjoint paths. However, no distributed algorithm uses acknowledgement messages (acks) for fault tolerance. So, for dealing the faults, acknowledgement messages (acks) are included in the proposed algorithm for routing messages over node-disjoint paths in a hypercube network.     

An efficient construction of one-to-many node-disjoint paths in folded hypercubes

A folded hypercube is basically a hypercube with additional links augmented, where the additional links connect all pairs of nodes with longest distance in the hypercube. In an n$n$-dimensional folded hypercube, it has been shown that n+1$n+1$ node-disjoint paths from one source node to other n+1$n+1$ (mutually) distinct destination nodes, respectively, can be constructed in O(n⁴)$O\left(n^4\right)$ time so that their maximal length is not greater than ⌈n/2⌉+1$\lceil n/2\rceil +1$, where n+1$n+1$ is the connectivity and ⌈n/2⌉$\lceil n/2\rceil$ is the diameter. Besides, their maximal length is minimized in the worst case. In this paper, we further show that by minimizing the computations of minimal routing functions, these node-disjoint paths can be constructed in O(n³)$O\left(n^3\right)$ time, which is more efficient, and is hard to be reduced because it must take O(n³)$O\left(n^3\right)$ time to compute a minimal routing function by solving a corresponding maximum weighted bipartite matching problem with the best known algorithm.     

Hamiltonian paths and cycles passing through a prescribed path in hypercubes

Assume that P is any path in a bipartite graph G of length k with 2?k?h, G is said to be h-path bipancyclic if there exists a cycle C in G of every even length from 2k to |V(G)| such that P lies in C. In this paper, the following result is obtained: The n-dimensional hypercube Qn with n?3 is (2n−3)-path bipancyclic but is not (2n−2)-path bipancyclic, moreover, a path P of length k with 2?k?2n−3 lies in a cycle of length 2k−2 if and only if P contains two edges of the same dimension. In order to prove the above result we first show that any path of length at most 2n−1 is a subpath of a Hamiltonian path in Qn with n?2, moreover, the upper bound 2n−1 is sharp when n?4.     

Embedding paths of variable lengths into hypercubes with conditional link-faults

Faults in a network may take various forms such as hardware failures while a node or a link stops functioning, software errors, or even missing of transmitted packets. In this paper, we study the link-fault-tolerant capability of an n-dimensional hypercube (n-cube for short) with respect to path embedding of variable lengths in the range from the shortest to the longest. Let F be a set consisting of faulty links in a wounded n-cube Q_n, in which every node is still incident to at least two fault-free links. Then we show that Q_n-F has a path of any odd (resp. even) length in the range from the distance to 2ⁿ-1 (resp. 2ⁿ-2) between two arbitrary nodes even if |F|=2n-5. In order to tackle this problem, we also investigate the fault diameter of an n-cube with hybrid node and link faults.     

Pancyclicity and bipancyclicity of conditional faulty folded hypercubes

A graph is said to be pancyclic if it contains cycles of every length from its girth to its order inclusive; and a bipartite graph is said to be bipancyclic if it contains cycles of every even length from its girth to its order. The pancyclicity or the bipancyclicity of a given network is an important factor in determining whether the network’s topology can simulate rings of various lengths. An n-dimensional folded hypercube FQ_n is an attractive variant of an n-dimensional hypercube Q_n that is obtained by establishing some extra edges between the vertices of Q_n. FQ_n for any odd n is known to be bipartite. In this paper, we explore the pancyclicity and bipancyclicity of FQ_n. For any FQ_n (n ? 2) with at most 2n − 3 faulty edges, where each vertex is incident to at least two fault-free edges, we prove that there exists a fault-free cycle of every even length from 4 to 2ⁿ; and when n ? 2 is even, we prove there also exists a fault-free cycle of every odd length from n + 1 to 2ⁿ − 1. The result is optimal with respect to the number of faulty edges tolerated.     

Node-disjoint paths in hierarchical hypercube networks

The hierarchical hypercube network is suitable for massively parallel systems. One of its appealing properties is the low number of connections per processor, which can facilitate the VLSI design and fabrication. Other alluring features include symmetry and logarithmic diameter, which can derive easy and fast algorithms for communication. In this paper, a maximal number of node-disjoint paths are constructed between every two distinct nodes of the hierarchical hypercube network. Their maximal length is not greater than max{2^m+1+2m+1,2^m+1+m+4}, where 2^m+1 is the diameter. The effectiveness of node-disjoint paths is further verified by experiments.     

Node-disjoint paths in incomplete WK-recursive networks

The incomplete WK-recursive networks have been recently proposed to relieve the restriction on the sizes of the WK-recursive networks. In this paper, a maximal set of node-disjoint paths is constructed between arbitrary two nodes of an incomplete WK-recursive network. The effectiveness of the constructed paths is verified by both theoretic analysis and extensive experiments. A tight upper bound on the maximal length is suggested. On the other hand, experimental results show that for arbitrary two nodes, the expected maximal length is not greater than twice their distance and about equal to the diameter. When the two nodes are the farthest pair, the maximal length is not greater than twice the diameter and the expected maximal length is not greater than 1.5 times the diameter.     

A note on embeddings among folded hypercubes,even graphs and odd graphs

In this short note, we establish topological relationships among the folded hypercube FQ _n , the even graph E _k and the odd graphs O _d via embedding. Such embeddings are measured via dilation.     

具有自适应性的star网络容错寻径策略研究

大规模并行处理机系统中寻径算法对互连网络的通信性能和系统性能起着至关重要的作用,而star互连网络作为超立方体网络的最好替代之一,其寻径问题的解决变得非常重要。在有条件的容错模型基础上,对寻径时的规则进行了研究,提出了一种基于自适应规则的容错寻径算法。对算法的正确性以及容错性进行了分析。经仿真实验证明了该算法具有较高的成功概率。在边失效独立的情况下,对star网络终端对间通信可靠性进行了分析,推导出了其约束下界,并给予了证明。     

Designing A Disjoint Paths Interconnection Network with Fault Tolerance and Collision Solving

In fault-tolerant interconnection designs, many prior researches suggest good use of disjoint paths to improve the reliability of interconnection networks. Although disjoint paths increase reliability, they always cost the throughput penalty. To address the problems of both performance and fault-tolerant capability, the following issues should be carefully considered: (1) guarantee of at least two disjoint paths, (2) easy rerouting between disjoint paths, (3) keep low rerouting hops, (4) solve the occurrences of packets’ collision. In this paper, we consider these issues to design a fault-tolerant network called CSMIN (Combining Switches Multistage Interconnection Network). CSMIN provides two disjoint paths to guarantee one fault-tolerant and can dynamically reroute packets between these two paths to solve the collision situation. In other words, to switch packets between these two disjoint paths easily, CSMIN causes these two disjoint paths to have regular distances at each stage. Accordingly, a packet can be dynamically sent to the other disjoint path if it encounters a faulty or busy element. In addition, CSMIN presents low rerouting hops (an average of one rerouting hop) to maintain a low collision ratio. From the simulation result, CSMIN performs with a better arrival ratio than Gamma and other related disjoint paths networks do.This research was supported by the National Science Council NSC-91-2218-E-324-006.     

Long paths in hypercubes with conditional node-faults

Let F be a set of f?2n-5 faulty nodes in an n-cube Q_n such that every node of Q_n still has at least two fault-free neighbors. Then we show that Q_n-F contains a path of length at least 2ⁿ-2f-1 (respectively, 2ⁿ-2f-2) between any two nodes of odd (respectively, even) distance. Since the n-cube is bipartite, the path of length 2ⁿ-2f-1 (or 2ⁿ-2f-2) turns out to be the longest if all faulty nodes belong to the same partite set. As a contribution, our study improves upon the previous result presented by [J.-S. Fu, Longest fault-free paths in hypercubes with vertex faults, Information Sciences 176 (2006) 759-771] where only n-2 faulty nodes are considered.     

Longest fault-free paths in hypercubes with vertex faults

The hypercube is one of the most versatile and efficient interconnection networks (networks for short) so far discovered for parallel computation. Let f denote the number of faulty vertices in an n-cube. This study demonstrates that when f ? n − 2, the n-cube contains a fault-free path with length at least 2ⁿ − 2f − 1 (or 2ⁿ − 2f − 2) between two arbitrary vertices of odd (or even) distance. Since an n-cube is a bipartite graph with two partite sets of equal size, the path is longest in the worst-case. Furthermore, since the connectivity of an n-cube is n, the n-cube cannot tolerate n − 1 faulty vertices. Hence, our result is optimal.     

Fault-free cycles in folded hypercubes with more faulty elements

Let FFv (respectively, FFe) be the set of faulty vertices (respectively, faulty edges) in an n-dimensional folded hypercube FQn. In this paper, we show that FQn−FFv−FFe contains a fault-free cycle with length at least n²−2|FFv| if |FFe|+|FFv|?2n−4 and |FFe|?n−1, where n?3. Our result improves the previously known result of [S.-Y. Hsieh, A note on cycle embedding in folded hypercubes with faulty elements, Information Processing Letters (2008), in press, doi:10.1016/j.ipl.2008.04.003] where |FFe|+|FFv|?n−1 and n?4.     

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.     

Fault-Tolerant Embedding of Pairwise Independent Hamiltonian Paths on a Faulty Hypercube with Edge Faults

A Hamiltonian path in G is a path which contains every vertex of G exactly once. Two Hamiltonian paths P ₁=〈u ₁,u ₂,…,u _n 〉 and P ₂=〈v ₁,v ₂,…,v _n 〉 of G are said to be independent if u ₁=v ₁, u _n =v _n , and u _i ≠v _i for all 1<i<n; and both are full-independent if u _i ≠v _i for all 1≤i≤n. Moreover, P ₁ and P ₂ are independent starting at u ₁, if u ₁=v ₁ and u _i ≠v _i for all 1<i≤n. A set of Hamiltonian paths {P ₁,P ₂,…,P _k } of G are pairwise independent (respectively, pairwise full-independent, pairwise independent starting at u ₁) if any two different Hamiltonian paths in the set are independent (respectively, full-independent, independent starting at u ₁). A bipartite graph G is Hamiltonian-laceable if there exists a Hamiltonian path between any two vertices from different partite sets. It is well known that an n-dimensional hypercube Q _n is bipartite with two partite sets of equal size. Let F be the set of faulty edges of Q _n . In this paper, we show the following results:

Unpaired many-to-many vertex-disjoint path covers of a class of bipartite graphs

Let Wn denote any bipartite graph obtained by adding some edges to the n-dimensional hypercube Qn, and let S and T be any two sets of k vertices in different partite sets of Wn. In this paper, we show that the graph Wn has k vertex-disjoint (S,T)-paths containing all vertices of Wn if and only if k=²n−1 or the graph Wn−(S∪T) has a perfect matching. Moreover, if the graph Wn−(S∪T) has a perfect matching M, then the graph Wn has k vertex-disjoint (S,T)-paths containing all vertices of Wn and all edges in M. And some corollaries are given.     

Near-optimal message routing and broadcasting in faulty hypercubes

A distributed routing algorithm for faulty hypercubes is described. This algorithm uses a directed depth-first approach to find a path between the sender and receiver of a message whenever at least one non-faulty path exists. We show that, when an arbitrary number of elements of the hypercube can be faulty, the algorithm always routes messages using fewer than 2N hops, whereN is the number of nodes in the hypercube. This performance is shown to be within a factor of two of the optimal worst-case routing efficiency. Through foult simulations, we show that, even when up to half of the elements in the cube are faulty, complete the analysis, we prove that our algorithm is deadlock-free. Finally, we present two extensions of the algorithm. The first uses local storage to reduce the overhead of the algorithm while the second allows reliable broadcasting in the presence of an arbitrary number of faults.Supported in part by the National Science Foundation under Grant CCR-9010547.Supported in part by the National Science Foundation Instrumentation Grant CDA-8820627.