K-ary N-cube Research Articles

Hamiltonian Properties of Augmented k-Ary n-Cubes with Faulty Edges

The augmented [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] is a recently proposed interconnection network that combines the structural characteristics of [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] and augmented cube [Formula: see text], with better features. In this paper, we investigate the Hamiltonian properties of augmented [Formula: see text]-ary [Formula: see text]-cubes. We prove that the augmented [Formula: see text]-ary [Formula: see text]-cube with any [Formula: see text] (resp. [Formula: see text]) faulty edges admits a fault-free Hamilton cycle (resp. Hamilton path).

Reliability assessment for k-ary n-cubes with faulty edges

The g-restricted edge connectivity is an important measurement to assess the reliability of networks. The g-restricted edge connectivity of a connected graph G is the minimum size of a set of edges in G, if it exists, whose deletion separates G and leaves every vertex in the remaining components with at least g neighbors. The k-ary n-cube is an extension of the hypercube network and has many desirable properties. It has been used to build the architecture of the Supercomputer Fugaku. This paper establishes that for g≤n, the g-restricted edge connectivity of 3-ary n-cubes is 3⌊g/2⌋(1+(gmod2))(2n−g), and the g-restricted edge connectivity of k-ary n-cubes with k≥4 is 2g(2n−g). These results imply that in Qn3 with at most 3⌊g/2⌋(1+(gmod2))(2n−g)−1 faulty edges, or Qnk(k≥4) with at most 2g(2n−g)−1 faulty edges, if each vertex is incident with at least g fault-free edges, then the remaining network is connected.

Paired 2-disjoint path covers of k-ary n-cubes under the partitioned edge fault model

The k-ary n-cube Qnk serves as an indispensable interconnection network in the design of data center networks, network-on-chips, and parallel computing systems since it possesses numerous attractive properties. In these parallel architectures, the paired (or unpaired) many-to-many m-disjoint path cover (m-DPC) plays a significant role in message transmission. Nevertheless, the construction of m-DPC is severely obstructed by large-scale edge faults due to the rapid growth of the system scale. In this paper, we investigate the existence of paired 2-DPC in Qnk under the partitioned edge fault (PEF) model, which is a novel fault model for enhancing the networks' fault-tolerance related to path embedding problem. We exploit this model to evaluate the edge fault-tolerance of Qnk when a paired 2-DPC is embedded into Qnk. Compared to the other known works, our results can help Qnk to achieve large-scale edge fault-tolerance.

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].

Arithmetic modeling of k-ary n-cubes and toroidal k-ary grids

The use of toroidal topologies offers many advantages in the computing field when it comes to both pattern spotting and path finding strategies. The former is covered by de Bruijn shapes, which permit to uniquely locate a single pattern throughout the shape. However, the latter is mainly carried out by k-ary n-cubes, which label node identifiers in a sequential order according to rows, columns, layers, and so on. This scheme facilitates the movement among nodes by just applying arithmetic operations, such as integer divisions and arithmetic modulo n. On the othe hand, toroidal k-ary grids are an alternative available in some specific cases, where determined patterns appear in all dimensions of each node, thus allowing to use those patterns to dictate paths to move among nodes. In this paper, the arithmetic bases of both path finding strategies have been presented and some pseudocode algorithms have been designed.

Diagnosability of multigraph composition networks

A common approach to scale up a network is by connecting several small-scale networks to form a larger network. A multigraph composition network is a typical network structure obtained by adding an edge set between several connected graphs in the same order. This paper investigates the diagnosability on two categories of multigraph composition networks, that is, multigraph alternating composition networks and multigraph 2-matching composition networks, under the PMC and MM⁎ models. As corollaries, the diagnosability of several known networks, such as alternating group graphs, k-ary n-cubes, and round matching composition networks, can be derived directly.

The t/k-diagnosability of m-ary n-cube networks

A crucial index used to evaluate the reliability of a multiprocessor system is diagnosability. The t/k-diagnosis strategy greatly improves the diagnosability of a system by allowing up to k fault-free processors to be misdiagnosed as faulty processors. Compared with the t-diagnosability and t1/t1-diagnosability, the t/k-diagnosability can better reflect the fault modes of a practical system. The m-ary n-cube Qnm (in order not to be confused with the “k” of the t/k-diagnosability, we replace k-ary n-cube Qnk with m-ary n-cube Qnm) is a significant and common network topology, which is used as the underlying network in the construction of many distributed memory multiprocessors. In this paper, we explore several useful lemmas and the t/k-diagnosability of Qnm. According to the sufficient condition for determining that a system is t/k-diagnosable, we show that for n≥2, m≥4, and 0≤k≤2n, Qnm is tk,n/k-diagnosable, where tk,n=2(k+1)n−12(k+1)(k+2)+1. Based on this, we provide a comparison among the t-diagnosability, the t1/t1-diagnosability, and the t/k-diagnosability of Qnm.

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.

The Strong Menger Connectivity of the Directed k-Ary n-Cube

A strong digraph [Formula: see text] is strongly Menger (arc) connected if, for [Formula: see text], [Formula: see text] can reach [Formula: see text] by min[Formula: see text] internally (arc) disjoint-directed paths. A digraph [Formula: see text] is [Formula: see text](-arc)-fault-tolerant strongly Menger([Formula: see text]-(A)FTSM, for short) (arc) connected if [Formula: see text] is strongly Menger (arc) connected for every [Formula: see text](respectively, [Formula: see text]) with [Formula: see text]. A digraph [Formula: see text] is [Formula: see text]-conditional (arc)-fault-tolerant strongly Menger ([Formula: see text]-C(A)FTSM, for short) (arc) connected if [Formula: see text] is strongly Menger (arc) connected for every [Formula: see text](respectively, [Formula: see text]) with [Formula: see text] and [Formula: see text]. The directed [Formula: see text]-ary [Formula: see text]-cube [Formula: see text] [Formula: see text] and [Formula: see text] is a digraph with vertex set [Formula: see text]. For two vertices [Formula: see text] and [Formula: see text], [Formula: see text] dominates [Formula: see text] if there exists an integer [Formula: see text], [Formula: see text], satisfying [Formula: see text]mod [Formula: see text] and [Formula: see text], when [Formula: see text]. In this paper, we show that [Formula: see text] [Formula: see text] is [Formula: see text]-AFTSM arc connected when [Formula: see text], [Formula: see text]-FTSM connected when [Formula: see text], [Formula: see text]-CAFTSM arc connected when [Formula: see text], and [Formula: see text]-CFTSM connected when [Formula: see text].

The partial diagnosability of interconnection networks under the Hybrid PMC model

System level diagnosis is a primary method to identify the faulty elements in a multiprocessor system. Different from the traditional system level fault diagnosis models, the new fault diagnosis model, hybrid PMC model (short for HPMC model), is suitable to the circumstances that the processor and link faults occur simultaneously. Under the HPMC model, the q-edge restricted diagnosability of a system H, denoted by tqe(H), is the maximum number p such that H can correctly identify all the faulty elements containing p faulty processors and at most q faulty links. Similarly, the p-vertex restricted edge-diagnosability of a system H, denoted by spv(H), is the maximum number q such that H can correctly identify all the faulty elements containing q faulty links and at most p faulty processors. These two diagnosability are collectively referred to as partial diagnosability. We investigate and determine the q-edge restricted diagnosability of a system H under the HPMC model for q≥η(H), where η(H)=max⁡{|N(u)∩N(v)||uv∈E(H)}. In addition, we investigate the relationship between tqe(H) and spv(H) under the HPMC model, and show that under some conditions, spv(H)=q if tqe(H)=p. Applying our results, the two kinds of partial diagnosability of k-ary n-cubes, bijective connection networks and undirected Kautz graphs are established.

Reliability of augmented k-ary n-cubes under the extra connectivity condition

Reliability of augmented k-ary n-cubes under the extra connectivity condition

Structure fault-tolerance of divide-and-swap k-ary n-cube

As generalizations of classic connectivity, F-structure connectivity κ(G;F) and F-substructure connectivity κs(G;F) are proposed to evaluate the reliability of networks. Let G be a graph, F be a connected subgraph of G and F={F1,F2,…,Fm} be a set of connected subgraphs of G. The F-structure connectivity (resp., F-substructure connectivity) of G, denoted by κ(G;F) (resp., κs(G;F)), is the cardinality of a minimum F and Fi is isomorphic to F (resp., a connected subgraph of F) for every 1≤i≤m, and G−F is disconnected. The divide-and-swap cube DSCn is one of the most popular interconnection networks. In this paper, we generalize this network to divide-and-swap k-ary n-cube DSCnk, and study its topological structure and properties. Furthermore, we show that κ(DSCnk;K1,1)=κs(DSCnk;K1,1)=d+2 for n=2d≥4 and even k≥4; κ(DSCnk;K1,m)=κs(DSCnk;K1,m)=⌈d+12⌉+1 with 2≤m≤d+2 for n=2d≥2 and even k≥4.

The t/m-diagnosis strategy of augmented k-ary n-cubes

As the scale of interconnected networks grows, so does the number of processor failures in the network. When a processor fails, the information transmitted through the failed processor is unreliable. In fact, the higher the network's diagnosability, the more reliable it is. The t/m-diagnosis strategy was proposed [20], the primary principle of which is to sacrifice less diagnostic accuracy to massively improve the diagnosability, with the goal of enhancing the reliability of the network. This paper presents the t/m-diagnosis algorithm and shows the t/m-diagnosability of the augmented k-ary n-cube AQn,k, which is an extension of k-ary n-cubes and augmented cubes. In detail, we show that AQn,k is [4n(1+m)−⌊5(1+m)22⌋]/m-diagnosable under the PMC model (n≥4, k≥4 and 0≤m≤n−2) and under the MM* model (n≥4, k≥4 and 1≤m≤n−2), respectively. Based on these results, we further propose the polynomial-time t/m-diagnosis algorithm on the node number of AQn,k under the PMC and MM* models respectively.

On k-ary n-cubes and isometric words

The k-ary n-cubes are a generalization of the hypercubes to alphabets of cardinality k, with k≥2. More precisely, a k-ary n-cube is a graph with kn vertices associated to the k-ary words of length n. Given a k-ary word f, the k-ary n-cube avoiding f is the subgraph obtained deleting those vertices which contain f as a factor. When such a subgraph is isometric to the cube, for any n≥1, the word f is said isometric. A binary word f is isometric if and only if it is Ham-isometric, i.e., for any pair of f-free binary words u and v, u can be transformed in v by complementing the bits on which they differ and generating only f-free words. The case of a k-ary alphabet, with k≥2, is here investigated. From k≥4, the isometricity in terms of cubes is no longer captured by the Ham-isometricity, but by the Lee-isometricity. Then, Ham-isometric and Lee-isometric k-ary words are characterized in terms of their overlaps with errors. The minimal length of two words which witness the non-isometricity of a word f is called its index. The index of f is bounded in terms of its length and the bounds are shown tight by examples.

Embedded edge connectivity of k-ary n-cubes

The t-embedded edge connectivity ηt(Gn) of an n-dimensional recursive network Gn is the minimum number of edges, if any, whose removal disconnects Gn and each vertex of the resultant network lies in a t-dimensional subnetwork of Gn. The k-ary n-cube is one of the most attractive interconnection networks for parallel computer systems. One of the main results in [15] showed that ηt(Qn3)=2(n−t)3t for 0≤t≤n−1. In this short paper, we generalize the above result and prove that ηt(Qnk)=2(n−t)kt for 0≤t≤n−1 and odd k≥3.

Induced subgraph and eigenvalues of some signed graphs

Hao Huang proved the Sensitivity Conjecture in [Induced graphs of the hypercube and a proof of the Sensitivity Conjecture, Annals of Mathematics, 190 (2019), 949-955] by signed graph spectral method. The main result of that paper is every -vertex induced subgraph of n-dimensional hypercube Qn has maximum degree at least by proving that Qn has just two distinct adjacency eigenvalues and with a signature. In this paper, we consider the eigenvalues of signed Cartesian product of bipartite graph and hypercube Qn , signed Cartesian product of complete graph Km and hypercube Qn which contain Huang’s result when or m = 2. Let f(G) be the minimum of the maximum degree of an induced subgraph of G on vertices where is the independent number of G. Huang also asked the following question: Given a graph G with high symmetry, what can we say about f(G)? We study this question for k-ary n-cubes where 2-ary n-cube is Qn . In this paper, we show that every -vertex induced subgraph of has maximum degree at least () by proving that has two eigenvalues which are either or with a signature.

The 3-path-connectivity of the [formula omitted]-ary [formula omitted]-cube

Let G be a connected simple graph with vertex set V(G). Let Ω be a subset with cardinality at least two of V(G). A path containing all vertices of Ω is said to be an Ω-path of G. Two Ω-paths T1 and T2 of G are internally disjointif V(T1)∩V(T2)=Ω and E(T1)∩E(T2)=∅. For an integer ℓ with 2≤ℓ, the ℓ-path-connectivityπℓ(G) is defined as πℓ(G)=min{πG(Ω)|Ω⊆V(G) and |Ω|=ℓ}, where πG(Ω) represents the maximum number of internally disjoint Ω-paths. In this paper, we completely determine 3-path-connectivity of the k-ary n-cube Qnk. By deeply exploring the structural proprieties of Qnk, we show that π3(Qnk)=⌊6n−14⌋ with n≥1 and k≥3.

Structure connectivity and substructure connectivity of the directed k-ary n-cube

Given a strongly connected digraph D and a connected subdigraph T of D, the T-structure connectivity of D is the cardinality of a minimum set of subdigraphs in D, whose removal results in a non-strongly connected digraph and . The T-substructure connectivity of D is the cardinality of a minimum set of subdigraphs in D, whose removal results in a non-strongly connected digraph and each element is isomorphic to a connected subdigraph of T. In this work, we study resp. for , and ; resp. for and ; and resp. for , , and , where is the directed k-ary n-cube, is the in-star on t + 1 vertices, and are, respectively, the directed path and cycle of length t.

The Diagnosability of Interconnection Networks with Missing Edges and Broken-Down Nodes Under the PMC and MM* Models

Abstract Diagnosability is often considered as an important factor for measuring the self-diagnostic ability of network systems. However, classic system-level diagnosis focuses only on processor faults and ignores the objective reality of communication faults. Under real circumstances, missing edges and node failures usually occur simultaneously in multiprocessor systems (called hybrid fault circumstances). Therefore, it is important to study the diagnosability of multiprocessor systems under hybrid fault circumstances. In this paper, we propose several diagnosabilities of interconnection networks with missing edges and faulty nodes. By exploring some important relationships between diagnosability and the minimum degree of a network under hybrid fault circumstances, we present and prove the diagnosability of several classic interconnection networks, including BC (bijective connection) networks, star graphs, folded hypercubes, exchanged hypercubes, exchanged crossed cubes, k-ary n-cubes, bubble-sort star graphs and balanced hypercubes, with missing edges and broken-down nodes under the PMC (Preparata, Metze and Chien) and MM* (Maeng and Malek) models.

The vertex Turán density in 3-ary [formula omitted]-cubes

The k-ary n-cube, denoted Qnk, has the vertex set Vnk={0,1,…,k−1}n, and two vertices (x1,x2,x3,…,xn) and (y1,y2,y3,…,yn) in Vnk are adjacent if and only if there is an integer j with 1≤j≤n such that xj≡yj±1 (mod k) and xi=yi for all i∈{1,2,…,n}∖{j}. Let λ(F) be the vertex Turán density of a graph G for a family of forbidden configurations F. Let F1={(0,0,0),(1,0,0),(0,1,0),(0,0,1)}, F2={(0,0,0),(1,1,0),(1,0,1),(0,1,1)}, F3={(0,0,0),(1,1,1)}, Fd,t,k={F⊆Vdk:|F|=t}, and μ(d,k)=max{t:λ(Fd,t,k)=0}. Johnson and Talbot (2010) determined λ(Fi) for i∈{1,2,3} and the bounds of μ(d,2) in Qn2 (where Qn2 is the n-dimensional hypercube Qn).In this paper, we derive the exact value λ(Fi) for i∈{1,2,3} in Qn3 and determine the bounds of μ(d,k) in Qnk for any k≥3. Furthermore, we consider the forbidden configurations Di={x∈V3k:|x|=0or|x|=i}. The bounds of λ(Di) in Qnk are derived for any i∈{1,2,3} and k≥2, and λ(Gd)=2/3 in Qn3 is derived, where Gd={x∈Vd3:|x|=0,(|x|=1,qsupp(x)=i[1])and(|x|=2,qsupp(x)={i[1],j[1]},i⁄≡j(mod2))} (see Definition 1 for qsupp(x)).