K-ary N-cube Research Articles

Many-to-many disjoint path covers in [formula omitted]-ary [formula omitted]-cubes

The class of k-ary n-cubes represents the most commonly used interconnection topology for parallel and distributed computing systems. Embedding of disjoint paths has attracted much attention in the parallel processing. In disjoint path problems, the many-to-many disjoint path problem is the most generalized one. This paper considers the problem of many-to-many disjoint path covers in the k-ary n-cube Qnk with even k≥4, and obtains the following result. Let m be an integer with 1≤m≤2n−1. For any two sets S and T of m vertices in different partite sets, Qnk has m disjoint (S,T)-paths containing all vertices of Qnk and our result is optimal.

The Conditional Diagnosability of k-Ary n-Cubes under the Comparison Diagnosis Model

Processor fault diagnosis plays an important role in measuring the reliability of multiprocessor systems and diagnosing many well-known interconnection networks. Conditional diagnosability is a novel measure of diagnosability that adds the additional condition that any faulty set cannot contain all of the neighbors of any vertex in a system. This study investigates some topological properties of k-ary n-cubes, where k ≥ 4 and n ≥ 4, and shows that the conditional diagnosability of k-ary n-cubes under the comparison diagnosis model is 6n - 5.

Parallel Implementation of the Gauss-Seidel Algorithm on <i>k</i>-Ary <i>n</i>-Cube Machine

In this paper, we present parallel implementation of the Gauss-Seidel (GS) iterative algorithm for the solution of linear systems of equations on a k-ary n-cube parallel machine using Open MPI as a parallel programming environment. The proposed algorithm is of O(N3/kn) computational complexity and uses O(nN) communication time to decompose a matrix of order N on the a k-ary n-cube provided N ≥ kn-1. The incurred communication time is better than the best known results for hypercube, O(N log n!), and the mesh, O(N n!), each with approximately n! nodes. The numerical results show that speedup improves as number of processors increased and the proposed algorithm has approximately 80% parallel efficiency.

Adaptive Routing Algorithms and Implementation for TESH Network

The Tori-connected mESH (TESH) Network is a k-ary n-cube networks of multiple basic modules, in which the basic modules are 2D-mesh networks that are hierarchically interconnected for higher level k-ary n-cube networks. Many adaptive routing algorithms for k-ary n-cube networks have already been proposed. Thus, those algorithms can also be applied to TESH network. We have proposed three adaptive routing algorithms—channel-selection, link-selection, and dynamic dimension reversal—for the efficient use of network resources of a TESH network to improve dynamic communication performance. In this paper, we implement these routers using VHDL and evaluate the hardware cost and delay for the proposed routing algorithms and compare it with the dimension order routing. The delay and hardware cost of the proposed adaptive routing algorithms are almost equal to that and slightly higher than that of dimension order routing, respectively. Also we evaluate the communication performance with hardware implementation. It is found that the communication performance of a TESH network using these adaptive algorithms is better than when the dimension-order routing algorithm is used.

Fault tolerance in [formula omitted]-ary [formula omitted]-cube networks

The k-ary n-cube Qnk is one of the most commonly used interconnection topologies for parallel and distributed computing systems. Let f(n,m) be the minimum number of faulty nodes that make every (n−m)-dimensional subcube Qn−mk faulty in Qnk under node-failure models. In this paper, we prove that f(n,0)=1, f(n,1)=k for odd k≥3, f(n,n−1)=kn−1 for odd k≥3, and km≤f(n,m)≤(n−1m−1)km−(n−2m−1)km−1 for odd k≥3.

K-Pancyclicity of k-ary n-Cube Networks under the Conditional Fault Model

The k-ary n-cube is one of the most popular interconnection networks for parallel and distributed systems. We prove that a k-ary n-cube with at most in 5 faulty edges but where every vertex is incident with at least two healthy edges is k-pancyclic and bipancyclic for n ≥ 3 and odd k ≥ 3.

A general technique to establish the asymptotic conditional diagnosability of interconnection networks

We develop a general and demonstrably widely applicable technique for determining the asymptotic conditional diagnosability of interconnection networks prevalent within parallel computing under the comparison diagnosis model. We apply our technique to replicate (yet extend) existing results for hypercubes and k-ary n-cubes before going on to obtain new results as regards folded hypercubes, pancake graphs and augmented cubes. In particular, we show that the asymptotic conditional diagnosability of: folded hypercubes {FQn} is 3n−2, pancake graphs {Pn} is 3n−7, and augmented cubes {AQn} is 6n−17. We demonstrate how our technique is independent of structural properties of the interconnection network G in question and essentially only dependent upon the minimal size of the neighbourhood of a path of length 2 in G, the number of neighbours any two distinct vertices of G have in common, and the minimal degree of any vertex in G.

An Improved Construction Method for MIHCs on Cycle Composition Networks

Many well-known interconnection networks, such as kary n-cubes, recursive circulant graphs, generalized recursive circulant graphs, circulant graphs and so on, are shown to belong to the family of cycle composition networks. Recently, various studies about mutually independent hamiltonian cycles, abbreviated as MIHC’s, on interconnection networks are published. In this paper, using an improved construction method, we obtain MIHC’s on cycle composition networks with a much weaker condition than the known result. In fact, we established the existence of MIHC’s in the cycle composition networks and the result is optimal in the sense that the number of MIHC’s we constructed is maximal. Keywords—Hamiltonian cycle, k-ary n-cube, cycle composition networks, mutually independent.

Matching preclusion and conditional matching preclusion problems for tori and related Cartesian products

The matching preclusion number of an even graph is the minimum number of edges whose deletion results in a graph that has no perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. The conditional matching preclusion number of an even graph was introduced to look for obstruction sets beyond those induced by a single vertex. It is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices and no perfect matchings. In this paper we study this problem for the tori by proving matching preclusion and conditional matching preclusion results of the Cartesian products of graphs involving cycles. Our results generalize the one given for k-ary n-cube by Wang et al. (2010) [10] as well as provide classification for optimal conditional matching preclusion sets for these graphs.

Extraconnectivity of [formula omitted]-ary [formula omitted]-cube networks

Given a graph G and a non-negative integer g, the g-extraconnectivity of G is the minimum cardinality of a set of vertices in G, if such a set exists, whose deletion disconnects G and leaves every remaining component with more than g vertices. This study shows that the 2-extraconnectivity of a k-ary n-cube Qnk for k≥4 and n≥5 is equal to 6n−5.

ON THE COMPUTATIONAL COMPLEXITY OF ROUTING IN FAULTY K-ARY N-CUBES AND HYPERCUBES

We equate a routing algorithm in a (faulty) interconnection network whose underlying graph is a k-ary n-cube or a hypercube, that attempts to route a packet from a fixed source node to a fixed destination node, with the sub-digraph of (healthy) links potentially usable by this routing algorithm as it attempts to route the packet. This gives rise to a naturally defined problem, parameterized by this routing algorithm, relating to whether a packet can be routed from a given source node to a given destination node in one of our interconnection networks in which there are (possibly exponentially many) faulty links. We show that there exist such problems that are PSPACE-complete (all are solvable in PSPACE) but that there are (existing and popular) routing algorithms for which the computational complexity of the corresponding problem is significantly easier (yet still computationally intractable).

Efficient implementation of globally-aware network flow control

Network flow control mechanisms that are aware of global conditions potentially can achieve higher performance than flow control mechanisms that are only locally aware. Owing to high implementation overhead, globally-aware flow control mechanisms in their purest form are seldom adopted in practice, leading to less efficient simplified implementations. In this paper, we propose an efficient implementation of a globally-aware flow control mechanism, called Critical Bubble Scheme, for k-ary n-cube networks. This scheme achieves near-optimal performance with the same minimal buffer requirements of globally-aware flow control and can be further generalized to implement the general class of buffer occupancy-based network flow control. We prove deadlock freedom of the proposed scheme and exploit its use in handling protocol-induced deadlocks in on-chip environments. We evaluate the proposed scheme using both synthetic traffic and real application loads. Simulation results show that the proposed scheme can reduce the buffer access component of packet latency by as much as 62% over locally-aware flow control, and improve average packet latency by 18.8% and overall execution time by 7.2% in full system simulation.

The diagnosability of the k-ary n-cubes using the pessimistic strategy

The pessimistic strategy, also called the t 1/t 1-diagnosis strategy, allows to contain all faulty vertices and at most one fault-free vertex. However, the degree of the diagnosability of a system increases quickly using the pessimistic strategy. In this paper, we consider the k-ary n-cube, which is an important hypercube variant, and show that the 3-ary n-cube and k-ary n-cube with k≥4 are (4n−3)/(4n−3)-diagnosable and (4n−2)/(4n−2)-diagnosable, respectively. Finally, we obtain the degree of the diagnosability of the tori using the pessimistic strategy.

Routing and wavelength assignment for 3-ary n-cube in array-based optical network

The k-ary n-cube, denoted Q n k , is one of the popular communication patterns of parallel algorithms. This paper addresses the routing and wavelength assignment for Q n 3 communication pattern in array-based WDM optical network. By using congestion estimation and giving a routing and wavelength assignment strategy, we prove that the optimal number of wavelengths is 3 n − 1 .

Embedding long cycles in faulty k-ary 2-cubes

The class of k-ary n-cubes represents the most commonly used interconnection topology for distributed-memory parallel systems. Given an even k ⩾ 4, let ( V 1, V 2) be the bipartition of the k-ary 2-cube, f v1 , f v2 be the numbers of faulty vertices in V 1 and V 2, respectively, and f e be the number of faulty edges. In this paper, we prove that there exists a cycle of length k 2 − 2max{ f v1 , f v2 } in the k-ary 2-cube with 0 ⩽ f v1 + f v2 + f e ⩽ 2. This result is optimal with respect to the number of faults tolerated.

Bipancyclicity in k-Ary n-Cubes with Faulty Edges under a Conditional Fault Assumption

We prove that a k-ary 2-cube Q_2^k with three faulty edges but where every vertex is incident with at least two healthy edges is bipancyclic, if k\ge 3, and k-pancyclic, if k\ge 5 is odd (these results are optimal). We go on to show that when k\ge 4 is even and n\ge 3, any k-ary n-cube Q_n^k with at most 4n-5 faulty edges so that every vertex is incident with at least two healthy edges is bipancyclic, and that this result is optimal.

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 2 n − 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 2 n − 1 prescribed edges if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths.

A RELIABLE MULTIPATH ROUTING PROTOCOL FOR MOBILE AD-HOC NETWORKS: ADAPTING TECHNIQUES FROM INTERCONNECTION NETWORKS

This paper illustrates how known methods and techniques for solving problems in the area of wired interconnection networks can be adapted to solve problems for wireless mobile networks. We make use of a known construction of disjoint paths in the k-ary n-cube interconnection network to design a reliable multipath routing protocol for mobile ad-hoc networks (MANETs). With the help of node positioning, node mobility is masked and the problem of routing between mobile nodes is transformed to a problem of routing between fixed cells of a logical 3-dimensional grid. Analytical performance evaluation results for the proposed protocol are obtained showing its high reliability. To the best of our knowledge, the proposed protocol is the first multipath source routing protocol for MANETs. We believe other wireless communication problems can be solved using a similar approach based on adapting solutions from wired interconnection networks.

Mutually independent Hamiltonian cycles in k-ary n-cubes when k is even

The k-ary n-cube has been used as the underlying topology for many practical multicomputers, and has been extensively studied in the past. In this article, we will prove that any k-ary n-cube Q n k , where n ⩾ 2 is an integer and k ⩾ 4 is an even integer, contains 2 n mutually independent Hamiltonian cycles. More specifically, let N = | V ( Q n k ) | , v ( i ) ∈ V ( Q n k ) for 1 ⩽ i ⩽ N , and 〈 v ( 1 ) , v ( 2 ) , … , v ( N ) , v ( 1 ) 〉 be a Hamiltonian cycle of Q n k . We prove that Q n k contains 2 n Hamiltonian cycles, denoted by C l = 〈 v ( 1 ) , v l ( 2 ) … , v l ( N ) , v ( 1 ) 〉 for all 0 ⩽ l ⩽ 2 n - 1 , such that v l ( i ) ≠ v l ′ ( i ) for all 2 ⩽ i ⩽ N whenever l ≠ l ′ . The result is optimal since each vertex of Q n k has exactly 2 n neighbors.

Embedding hamiltonian paths in [formula omitted]-ary [formula omitted]-cubes with conditional edge faults

It is well known that the k-ary n-cube has been one of the most efficient interconnection networks for distributed-memory parallel systems. A k-ary n-cube is bipartite if and only if k is even. In this paper, we consider the faulty k-ary n-cube with even k≥4 and n≥2 such that each vertex of the k-ary n-cube is incident with at least two healthy edges. Based on this requirement, we prove that the k-ary n-cube contains a hamiltonian path joining every pair of vertices which are in different parts, even if it has up to 4n−6 edge faults.