K-ary N-cube Networks Research Articles

Embedding Hamiltonian Paths with Prescribed Linear Forests into k-ary n-Cube Networks

Graph embedding is a fundamental problem in computer science. Let [Formula: see text] (resp., [Formula: see text]) be the interconnection network for a parallel computer system [Formula: see text] (resp., [Formula: see text]). If [Formula: see text] could be embedded into [Formula: see text], then [Formula: see text] can simulate [Formula: see text]’s behavior. The [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] is a node-symmetric and link-symmetric recursive interconnection network for parallel computer systems. Let [Formula: see text] be a prescribed linear forest of [Formula: see text], and let [Formula: see text] and [Formula: see text] be any two distinct nodes in [Formula: see text] such that [Formula: see text] has no path with [Formula: see text] or [Formula: see text] as internal nodes, or both as end-nodes. This paper shows that there is a Hamiltonian path passing through [Formula: see text] between [Formula: see text] and [Formula: see text] in [Formula: see text] with [Formula: see text] and odd [Formula: see text] even if the number of links in [Formula: see text] is up to [Formula: see text].

Construction algorithms of fault-tolerant paths and disjoint paths in k-ary n-cube networks

With the increasing demand for information, the number of nodes in the network is increasing rapidly, followed by the problem of information loss and communication delay caused by insufficient communication. Therefore, how to effectively transmit data and minimize communication delay has been a key issue in networks. In this paper, we first propose a fault-tolerant path algorithm of the k-ary n-cube network Qnk to obtain a fault-free path between any two distinct fault-free nodes with a faulty node set F⊂V(Qnk) and |F|≤2n−1. Then, we design two algorithms to construct disjoint paths between any two distinct nodes of Qnk without any faults according to whether the two nodes are adjacent or not. Extensive evaluations show that the proposed algorithms has great advantages in average running time.

Correction to: The properties and t/s-diagnosability of k-ary n-cube networks

Correction to: The properties and t/s-diagnosability of k-ary n-cube networks

The properties and t/s-diagnosability of k-ary n-cube networks

The properties and t/s-diagnosability of k-ary n-cube networks

Exponential type of many-to-many edge disjoint paths on ternary n-cubes

One of most central issues in various interconnection networks of parallel and distributed systems is to find edge-disjoint paths concerned with an information transmission through links. It is universally acknowledged that an interconnection network can be modeled as an undirected simple graph G=(V,E) with processors and physical links between the processors represented as vertices and edges of the G, respectively. The studies on the edge-disjoint paths are closely related to the edge-connectivity. By the well-known Menger theorem, the maximum number of edge disjoint paths connecting any two disjoint connected subgraphs with g vertices in G can also define by the minimum modified edge-cut, called the g-extra edge-connectivity of Gλg(G). It is the cardinality of the minimum set of edges in G, if such a set exists, whose deletion disconnects G and leaves every remaining component with at least g vertices. The k-ary n-cube network is known as one of the most attractive interconnection networks for parallel and distributed systems. This article intends to show that for 3⌈n2⌉+r−⌊32r+e+12⌋≤g≤3⌈n2⌉+r, the g-extra edge-connectivity of ternary n-cube (also called 3-ary n-cube) (n≥3) exists a concentration behavior, and prove that these corresponding g-extra edge-connectivities of ternary n-cubes are the constants 2(⌊n2⌋−r)3⌈n2⌉+r, where r=0,1,2,⋯,⌊n2⌋−1, and e=1 if n is odd and e=0 if n is even. The above upper and lower bounds of g are sharp. As the integer g varies within the range between 3⌈n2⌉+r−⌊32r+e+12⌋ and 3⌈n2⌉+r, a necessary and sufficient condition of λg-optimality is found. Our results improve the several previous results.

Super Ck and Sub-Ck Connectivity of k-Ary n-Cube Networks

Let [Formula: see text] be an undirected graph. An [Formula: see text]-structure-cut (resp. [Formula: see text]-substructure-cut) of [Formula: see text] is a set of subgraphs of [Formula: see text], if any, whose deletion disconnects [Formula: see text], where the subgraphs deleted are isomorphic to a certain graph [Formula: see text] (resp. where for any [Formula: see text] of the subgraphs deleted, there is a subgraph [Formula: see text] of [Formula: see text], isomorphic to [Formula: see text], such that [Formula: see text] is a subgraph of [Formula: see text]). [Formula: see text] is super[Formula: see text]-connected (resp. super sub-[Formula: see text]-connected) if the deletion of an arbitrary minimum [Formula: see text]-structure-cut (resp. minimum [Formula: see text]-substructure-cut) isolates a component isomorphic to a certain graph [Formula: see text]. The [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] is one of the most attractive interconnection networks for multiprocessor systems. In this paper, we prove that [Formula: see text] with [Formula: see text] is super sub-[Formula: see text]-connected if [Formula: see text] and [Formula: see text] is odd, and super [Formula: see text]-connected if [Formula: see text] and [Formula: see text] is odd.

The reliability analysis of k-ary n-cube networks

With the development of scalability and application of multiprocessor system, the components of the system possibly become faulty. It is desirable to know the reliability of the system. However, the exact reliability of a complicated network system is usually difficult to determine. A typical approach is to decompose the system into smaller ones based on a graph-theory model in which nodes are assumed to fail independently with known probabilities to measure the subsystem-based reliability, which is defined as the probability that a fault-free subsystem of a specific size is still available when the system has faults. In this paper, we use the probability fault model to establish upper and lower bounds on the subsystem-based reliability of k-ary n-cube networks, by taking into account the intersection of no more than five or four subgraphs. In addition, some numerical simulations of the subsystem-based reliability of k-ary n-cube networks are conducted.

Structure fault tolerance of k-ary n-cube networks

Let G be a graph and H be a certain connected subgraph of G. The H-structure connectivity κ(G;H) (resp. H-substructure connectivity κs(G;H)) of G is the minimum number of a set of subgraphs F={H1,H2,⋯,Hm} (resp. F={H1′,H2′,⋯,Hm′}) such that Hi is isomorphic to H (resp. Hi′ is a connected subgraph of H) for every 1≤i≤m, and F's removal will disconnect G. For the k-ary n-cube Qnk, the κ(Qnk;H) and κs(Qnk;H) were determined for H∈{K1,K1,1,K1,2,K1,3}. In this paper, we show κ(Qnk;H) and κs(Qnk;H) for H∈{Pl,Cl} where 3≤l≤2n.

Structure Connectivity and Substructure Connectivity of $k$ -Ary $n$ -Cube Networks

We present new results on the fault tolerability of k-ary n-cube (denoted Q n k ) networks. Q n k is a topological model for interconnection networks that has been extensively studied since proposed, and this paper is concerned with the structure/substructure connectivity of Q n k networks, for paths and cycles, two basic yet important network structures. Let G be a connected graph and T a connected subgraph of G. The T-structure connectivity κ(G; T) of G is the cardinality of a minimum set of subgraphs in G, such that each subgraph is isomorphic to T, and the set's removal disconnects G. The T -substructure connectivity κ s (G; T) of G is the cardinality of a minimum set of subgraphs in G, such that each subgraph is isomorphic to a connected subgraph of T, and the set's removal disconnects G. In this paper, we study κ(Q n k ; T) and κ s (Q n k ; T) for T = P i , a path on i nodes (resp. T = C i , a cycle on i nodes). Lv et al. determined κ(Q n k ; T) and κ s (Q n k ; T) for T ∈ {P 1 , P 2 , P 3 }. Our results generalize the preceding results by determining κ(Q n k ; P i ) and κ s (Q n k ; P i ). In addition, we have also established κ(Q n k ; C i ) and κs(Q n k ; C i ).

A Nearly Optimal One-to-Many Routing Algorithm on k-ary n-cube Networks

A Nearly Optimal One-to-Many Routing Algorithm on k-ary n-cube Networks

Structure connectivity and substructure connectivity of k-ary n-cube networks

The k-ary n-cube is one of the most attractive interconnection networks for parallel and distributed computing system. In this paper, we investigate the fault-tolerant capabilities of k-ary n-cubes with respect to the structure connectivity and substructure connectivity. Let H is a connected graph. The structure connectivity of a graph G, denoted by κ(G; H), is the minimum cardinality of a set of connected subgraphs in G, whose deletion disconnects the graph G and every element in the set is isomorphic to H. The substructure connectivity of a graph G, denoted by κs(G; H), is the minimum cardinality of a set of connected subgraphs in G, whose deletion disconnects the graph G and every element in the set is isomorphic to a connected subgraph of H. We show κ(Qnk;H) and κs(Qnk;H) for each H ∈ {K1, K1, 1, K1, 2, K1, 3}.

Equal relation between the extra connectivity and pessimistic diagnosability for some regular graphs

Extra connectivity and the pessimistic diagnosis are two crucial subjects for a multiprocessor system's ability to tolerate and diagnose faulty processor. The pessimistic diagnosis strategy is a classic strategy based on the PMC model in which isolates all faulty vertices within a set containing at most one fault-free vertex. In this paper, the result that the pessimistic diagnosability tp(G) equals the extra connectivity κ1(G) of a regular graph G under some conditions are shown. Furthermore, the following new results are gotten: the pessimistic diagnosability tp(Sn2)=4n−9 for split-star networks Sn2; tp(Γn)=2n−4 for Cayley graphs generated by transposition trees Γn; tp(Γn(Δ))=4n−11 for Cayley graph generated by the 2-tree Γn(Δ); tp(BPn)=2n−2 for the burnt pancake networks BPn. As corollaries, the known results about the extra connectivity and the pessimistic diagnosability of many famous networks including the alternating group graphs, the alternating group networks, BC networks, the k-ary n-cube networks etc. are obtained directly.

The pessimistic diagnosability of three kinds of graphs

A system is t/t-diagnosable if, provided the number of faulty processors is bounded by t, all faulty processors can be isolated within a set of size at most t with at most one fault-free processor mistaken as a faulty one. The pessimistic diagnosability of a system G, denoted by tp(G), is the maximal number of faulty processors so that the system G is t/t-diagnosable. The pessimistic diagnosability of alternating group graphs AGn (Tsai, 2015); BC networks (Fan, 2005; Tsai, 2013); the k-ary n-cube networks Qnk, (Wang et al., 2012); regular graphs including the alternating group networks ANn (Hao et al., 2016) etc. But most of these results are about networks G with cn(G)≤2 (where cn(G) is the maximum number of common neighbors for any two distinct vertices). In this paper, we study the pessimistic diagnosability of three kinds of graphs which are (n,k)-arrangement graphs An,k, (n,k)-star graphs Sn,k and balanced hypercubes BHn, where cn(An,k)=cn(Sn,k)=n−k−1 and cn(BHn)=2n. We proved that tp(An,k)=(2k−1)(n−k)−1 for n≥k+2 and k≥3, tp(Sn,k)=n+k−3 for n≥k+2 and k≥3, and tp(BHn)=2n for n≥2. As corollaries, the pessimistic diagnosability of the known results about AGn and ANn is obtained directly.

The pessimistic diagnosability of bubble-sort star graphs and augmented k-ary n-cubes

ABSTRACTA system is t/t-diagnosable if, provided the number of faulty processors is bounded by t, all faulty processors can be isolated within a set of size at most t with at most one fault-free processor mistaken as a faulty one. The pessimistic diagnosability of a system G, denoted by , is the maximal number of faulty processors so that the system G is t/t-diagnosable. The known results about for alternating group graphs [Inform. Process. Lett. (2015), pp. 151–154]; BC networks [IEEE Trans. Comput. (2005), pp. 176–184]; the k-ary n-cube networks [IEEE Trans. Comput. (1991), pp. 232–237], [Int. J. Comput. Math. (2012), pp. 1–10] etc. have property that . In this paper, we study the pessimistic diagnosability of two kinds of graphs with , those are: bubble-sort star graphs and augmented k-ary n-cubes , and prove that for , for , and for and.

Relationship between conditional diagnosability and 2-extra connectivity of symmetric graphs

The conditional diagnosability and the 2-extra connectivity are two important parameters to measure ability of diagnosing faulty processors and fault-tolerance in a multiprocessor system. The conditional diagnosabilitytc(G) of G is the maximum number t for which G is conditionally t-diagnosable under the comparison model, while the 2-extra connectivityκ2(G) of a graph G is the minimum number k for which there is a vertex-cut F with |F|=k such that every component of G−F has at least 3 vertices. A quite natural problem is what is the relationship between the maximum and the minimum problem? This paper partially answers this problem by proving tc(G)=κ2(G) for a regular graph G with some acceptable conditions. As applications, the conditional diagnosability and the 2-extra connectivity are determined for some well-known classes of vertex-transitive graphs, including, star graphs, (n,k)-star graphs, alternating group networks, (n,k)-arrangement graphs, alternating group graphs, Cayley graphs obtained from transposition generating trees, bubble-sort graphs, k-ary n-cube networks, dual-cubes, pancake graphs and hierarchical hypercubes as well. Furthermore, many known results about these networks are obtained directly.

On the extraconnectivity of k-ary n-cube networks

ABSTRACTGiven a graph G and a non-negative integer g, the g-extraconnectivity of G is the cardinality of a minimum set of vertices in G, if such a set exists, whose deletion disconnects G and leaves every remaining component with more than g vertices. The 2-extraconnectivity of k-ary n-cubes is gotten by Hsieh and Chang [Extraconnectivity of k-ary n-cube networks. Theoret. Comput. Sci. 443 (2012) 63–69] for . This paper shows that the 3-extraconnectivity of the k-ary n-cubes is , where and.

Unchanging the diameter of k-ary n-cube networks with faulty vertices

The k-ary n-cube is one of the most commonly used interconnection networks for parallel and distributed systems. In this paper, for a k-ary n-cube , we show that if k is even and if k is odd, where is the maximum integer such that the diameter of remains unchanged when arbitrary vertices are faulty. Furthermore, we show that for even k, if the diameter of a faulty with 2n−1 faulty vertices is larger than its fault-free diameter, then all the faulty vertices are adjacent to a certain vertex and there is only one pair of vertices in this such that their distance is equal to the fault diameter. For k-ary n-cubes with odd k, similar results are given.

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.

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.

Off-chip communication architectures for high throughput network processors

In this paper, we propose a new interconnection mechanism for network line cards. We project that the packet storage needs for the next-generation networks will be much higher. Such that the number of memory modules required to store the packets will be more than that can be directly connected to the network processor (NPU). In other words, the NPU I/O pins are limited and they do not scale well with the growing number of memory modules and processing elements employed on the network line cards. As a result, we propose to explore more suitable off-chip interconnect and communication mechanisms that will replace the existing systems and that will provide extraordinary high throughput. In particular, we investigate if the packet-switched k-ary n-cube networks can be a solution. To the best of our knowledge, this is the first time, the k-ary n-cube networks are used on a board. We investigate multiple k-ary n-cube based interconnects and include a variation of 2-ary 3-cube interconnect called the 3D-mesh. All of the k-ary n-cube interconnects include multiple, highly efficient techniques to route, switch, and control packet flows in order to minimize congestion spots and packet loss within the interconnects. We explore the tradeoffs between implementation constraints and performance. Performance results show that k-ary n-cube topologies significantly outperform the existing line card interconnects and they are able to sustain higher traffic loads. Furthermore, the 3D-mesh reaches the highest performance results of all interconnects and allows future scalability to adopt more memories and/or processors to increase the line card’s processing power.