Bijective Connection Research Articles

Adaptive System-Level Fault Diagnosis of Bijective Connection Networks

Adaptive System-Level Fault Diagnosis of Bijective Connection Networks

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.

GMM-IL: Image Classification Using Incrementally Learnt, Independent Probabilistic Models for Small Sample Sizes

When deep-learning classifiers try to learn new classes through supervised learning, they exhibit catastrophic forgetting issues. In this paper we propose the Gaussian Mixture Model - Incremental Learner (GMM-IL), a novel two-stage architecture that couples unsupervised visual feature learning with supervised probabilistic models to represent each class. The key novelty of GMM-IL is that between images and labels there is a bijective connection. New classes can be incrementally learnt using a small set of annotated images with no requirement for any previous training data. This enables the incremental addition of classes to a database, that can be indexed by visual features and reasoned over based on perception. Using Gaussian Mixture Models to represent the independent classes, we outperform a benchmark of an equivalent network with a Softmax head, obtaining increased accuracy for sample sizes smaller than 12 and increased weighted F1 score for 3 imbalanced class profiles in that sample range. This novel method enables new classes to be added to a system with only access to a few annotated images of the new class.

A parallel algorithm to construct edge independent spanning trees on the line graphs of conditional bijective connection networks

The line graphs of some well-known graphs, such as the generalized hypercube and the crossed cube, have been adopted for the logic graphs of data center networks (DCNs). Conditional bijective connection networks (conditional BC networks) are a class of networks which have been proved to include hypercubes, crossed cubes, locally twisted cubes and Möbius cubes, etc. Hence, the researches on the line graphs of conditional BC networks can be applied to DCNs. Edge independent spanning trees (EISTs) on a graph have received extensive attention because of their applications in reliable communication, fault-tolerant broadcasting and secure message distribution. Since the line graph of an n-dimensional conditional BC network, denoted as L(XCn), is (2n−2)-regular, whether there exist 2n−2 EISTs on L(XCn) is an open question. In this paper, we first propose a parallel algorithm to construct 2n−2 EISTs rooted at an arbitrary node on L(XCn), where n≥2. Then, we prove the correctness of our algorithm. Finally, we give a simulation result of our algorithm.

The a-average Degree Edge-Connectivity of Bijective Connection Networks

Abstract The conditional edge-connectivity is an important parameter to evaluate the reliability and fault tolerance of multi-processor systems. The $n$-dimensional bijective connection networks $B_{n}$ contain hypercubes, crossed cubes, Möbius cubes and twisted cubes, etc. The conditional edge-connectivity of a connected graph $G$ is the minimum cardinality of edge sets, whose deletion disconnects $G$ and results in each remaining component satisfying property $\mathscr{P}$. And let $F$ be the edge set as desired. For a positive integer $a$, if $\mathscr{P}$ denotes the property that the average degree of each component of $G-F$ is no less than $a$, then the conditional edge-connectivity can be called the $a$-average degree edge-connectivity $\overline{\lambda }_{a}(G)$. In this paper, we determine that the exact value of the $a$-average degree edge-connectivity of an $n$-dimensional bijective connection network $\overline{\lambda }_{a}(B_{n})$ is $(n-a)2^a$ for each $0\leq a \leq n-1 $ and $n\geq 1$. 1

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.

Set-to-Set Disjoint Path Routing in Bijective Connection Graphs

The bijective connection graph encompasses a family of cube-based topologies, and <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-dimensional bijective connection graphs include the hypercube and almost all of its variants with the order <inline-formula> <tex-math notation="LaTeX">$2^{n}$ </tex-math></inline-formula> and the degree <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. Hence, it is important to design and implement algorithms that work in bijective connection graphs. The set-to-set disjoint paths problem is as follows: given a set of source nodes <inline-formula> <tex-math notation="LaTeX">$S=\{ \boldsymbol s _{1}, \boldsymbol s _{2},\ldots, \boldsymbol s _{p}\}$ </tex-math></inline-formula> and a set of destination nodes <inline-formula> <tex-math notation="LaTeX">$D=\{ \boldsymbol d _{1}, \boldsymbol d _{2},\ldots, \boldsymbol d _{p}\}$ </tex-math></inline-formula> in a <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-connected graph <inline-formula> <tex-math notation="LaTeX">$G=(V,E)$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$p\le k$ </tex-math></inline-formula>, construct <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> paths <inline-formula> <tex-math notation="LaTeX">$P_{i}$ </tex-math></inline-formula>: <inline-formula> <tex-math notation="LaTeX">$\boldsymbol s _{i}\leadsto \boldsymbol d _{j_{i}}$ </tex-math></inline-formula> (<inline-formula> <tex-math notation="LaTeX">$1\le i\le p$ </tex-math></inline-formula>) such that <inline-formula> <tex-math notation="LaTeX">$\{j_{1},j_{2},\ldots,j_{p}\}=\{1,2,\ldots,p\}$ </tex-math></inline-formula> and the paths <inline-formula> <tex-math notation="LaTeX">$P_{i}$ </tex-math></inline-formula> are node-disjoint. Finding a solution to this problem is an important issue in parallel and distributed computation as well as the node-to-node disjoint paths problem and the node-to-set disjoint paths problem. In this paper we propose an algorithm that constructs <inline-formula> <tex-math notation="LaTeX">$p~(\le n)$ </tex-math></inline-formula> disjoint paths between any pair of node sets in <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-dimensional bijective connection graphs in polynomial-order time of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. We give a proof of correctness of the algorithm as well as the estimates of the time complexity <inline-formula> <tex-math notation="LaTeX">$O(n^{3}p^{4})$ </tex-math></inline-formula> and the maximum path length <inline-formula> <tex-math notation="LaTeX">$n+p-1$ </tex-math></inline-formula>. According to a computer experiment in a locally twisted cube as an example of a bijective connection graph to construct <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> disjoint paths, the average time complexity of the algorithm is <inline-formula> <tex-math notation="LaTeX">$O(n^{2})$ </tex-math></inline-formula>, and the average maximum path is <inline-formula> <tex-math notation="LaTeX">$0.6333n-0.266$ </tex-math></inline-formula>.

A Parallel Algorithm to Construct Edge Independent Spanning Trees on the Line Graphs of Conditional Bijective Connection Networks

A Parallel Algorithm to Construct Edge Independent Spanning Trees on the Line Graphs of Conditional Bijective Connection Networks

A short proof of two identities using techniques of complex numbers

The sweeping development of mathematics is largely due to the introduction of complex numbers. Although complex numbers are just absurd notations for most people, complex numbers play essential roles in engineering fields. When mathematicians began to take an investigation of complex numbers, human beings enter a marvelous world. Complex numbers convince scientists that our world is magical, full of wonderful insights and even miraculous. In this paper, I first review several basic properties of complex numbers. The set of complex numbers is a group not only under the operation of the multiplication but under the operation of addition. Then, I show a visualisation of complex numbers through building a bijective connection between complex numbers and points on the complex plane. I also give several alternative expression forms of complex numbers, namely the trigonometric form and the general form. By invoking the arithmetic properties of complex numbers, I prove two trigonometric identities.

The Relationship Between theg-Extra Connectivity and theg-Extra Diagnosability of Networks Under the MM* Model

AbstractMotivated by $g$-extra connectivity, the $g$-extra diagnosability is proposed as a better and more realistic measurement for fault diagnosis of interconnection networks, which is defined as the maximum number of faulty vertices that can be identified when each remaining component has no fewer $g+1$ vertices. Under the MM* model, a variety of interconnection networks’ $g$-extra diagnosability have been investigated, such as hypercube, folded hypercube, $(n,k)$-star network, alternating group graph, etc. These results mostly share similar derivation processes to derive the $g$-extra diagnosability of involved networks by using the $g$-extra connectivity. Therefore, a general approach to derive the $g$-extra diagnosability of a network from its $g$-extra connectivity was investigated in (Wang, S. Y. and Wang, M. (2019) The $g$-good-neighbor and $g$-extra diagnosability of networks. Theor. Comput. Sci., 773, 107–114) and (Huang, Y. Z., Lin, L. M. and Xu, L. (2020) A new proof for exact relationship between extra connectivity and extra diagnosability of regular connected graphs under MM* model. Theor. Comput. Sci., 828–829, 70–80). However, there are some shortcomings in both references. By summarizing the existing shared practices, we propose a new relationship between the $g$-extra connectivity and the $g$-extra diagnosability of networks under the MM* model. As applications, we derive the $g$-extra diagnosability of bijective connection networks and $(n,k)$-star graphs.

Reliability and conditional diagnosability of hyper bijective connection networks

The g-extra connectivity and diagonalisability are two important metrics to fault-tolerance and robustness of a multiprocessor system whose network structure is modelled by a graph. In this work, we explore the reliability of a newly proposed network called hyper bijective connection networks (HBC, for short), which is an extension of the family of the well-known interconnection networks, such as hypercube and its variants. We prove that 2-extra vertex connectivity and 3-extra vertex connectivity of n-dimensional HBC are 3n + m−6 for and and 4n + m−8 for and , respectively. Using its desirable fault-tolerance, we show that the conditional diagonalizabilities of n-dimensional HBC under the PMC model are m + 4n−7 (resp., 4n−5) for and (resp., m = 3 and ) and its conditional diagonalizability under MM model is m + 3n−6 for and .

Node-Disjoint Paths Problems in Directed Bijective Connection Graphs

Node-Disjoint Paths Problems in Directed Bijective Connection Graphs

Reliability Analysis of the Bijective Connection Networks for Components

Connectivity is a critical parameter that can measure the reliability of networks. Let Q ⊆ V ( G ) be a vertex set. If G − Q is disconnected and every component of G − Q contains at least k + 1 vertices, then Q is an extra-cut. The number of vertices in the smallest extra-cut is the extraconnectivity κ k ( G ) . Suppose ω ( G ) is the number of components of G and W ⊆ V ( G ) ; if ω ( G − W ) ≥ t , then w is a t-component cut of G. The number of vertices in the least t-component cut is the t-component connectivity c κ t ( G ) of G. The t-component edge connectivity c λ t ( G ) is defined similarly. In this note, we study the BC networks and obtain the t-component (edge) connectivity of bijective connection networks for some t.

A note on minimum linear arrangement for BC graphs

A linear arrangement is a labeling or a numbering or a linear ordering of the vertices of a graph. In this paper, we solve the minimum linear arrangement problem for bijective connection graphs (for short BC graphs) which include hypercubes, Möbius cubes, crossed cubes, twisted cubes, locally twisted cube, spined cube, [Formula: see text]-cubes, etc., as the subfamilies.

Reliability and Diagnosability Analysis of Hyper Bijective Connection Networks

Bijective connection (BC) networks, including a family of interconnection networks of multiprocessor systems, have been studied extensively due to its desirable properties, such as lower diameter, high reliability, and diagnosability. To meet the demand of processing integrating tasks with large-scale and complex architectures, it is significant to explore alternative interconnection networks for multiprocessor configuration. To this end, we propose a novel framework called hyper bijective connection network (HBC network) as an extension of BC networks, which allows to study the properties of other potential interconnection networks in unity rather than in individual. We prove that when n ≥ 3, m ≥ 2, every n-dimensional HBC network H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> (m) has (edge) connectivity m + n - 2, super connectivity 2n + m - 4, and super edge-connectivity 2n + 2m - 6, and is super-connected and super-edge-connected. These results indicate the high reliability of HBC networks. Moreover, we analyze three classic diagnos abilities of a HBC network, including tp-, t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> /t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -, and t/k-diagnosability. We show that when n ≥ 3 and m ≥ 2, an n-dimensional HBC network H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> (m) is (m + n - 2)-diagnosable, (2n + m - 4)/(2n + m - 4)-diagnosable, and t(m, n, k)/k-diagnosable, where 0 ≤ k ≤ m + n - 2 and t(m, n, k) = (k + 1)n + (m - 2) - ((k + 1)(k + 2)/2) + 1. Besides, it is shown that the corresponding properties for BC networks can be derived naturally as special cases of that for HBC networks.

Bijections for Weyl Chamber walks ending on an axis, using arc diagrams and Schnyder woods

In the study of lattice walks there are several examples of enumerative equivalences which amount to a trade-off between domain and endpoint constraints. We present a family of such bijections for simple walks in Weyl chambers which use arc diagrams in a natural way. One consequence is a set of new bijections for standard Young tableaux of bounded height. A modification of the argument in two dimensions yields a bijection between Baxter permutations and walks ending on an axis, answering a recent question of Burrill et al. (2016). Some of our arguments (and related results) are proved using Schnyder woods. Our strategy for simple walks extends to any dimension and yields a new bijective connection between standard Young tableaux of height at most 2k and certain walks with prescribed endpoints in the k-dimensional Weyl chamber of type D.

Strong Fault-Hamiltonicity for the Crossed Cube and Its Extensions

Fault-Hamiltonicity is an important measure of robustness for interconnection networks. Given a graph G = (V, E). The goal is to ensure that G − F remains Hamiltonian for every F ⊆ V ∪E such that |F| ≤ k for some k. Obviously, one wants the best k possible. For many interconnection networks, in particular, for the crossed cube, the optimal result is known. There is a natural bound for k as the resulting graph cannot have a vertex of degree at most 1. Thus, if r is the minimum degree in G, then k ≤ r − 2. Since interconnection networks are modeled after computer networks, it is reasonable to assume that the vertex/edge faults occur following certain probability distributions. As such, it is reasonable to assume that it is unlikely for a vertex to have degree at most 1 even if r − 1 vertices/edges are deleted. Thus we may be able to delete more vertices and/or vertices if we assume that all the faults are “not clustered” at a vertex. To be precise, the goal is to find the best possible k in which G − F remains Hamiltonian for every F ⊆ V ∪E such that |F| ≤ k and the minimum degree of G − F is at least 2. In this paper, we study this problem for the crossed cube and then extend the result to cover a large subclass of bijective connection networks as well as the more general matching composition networks.

Constructing completely independent spanning trees in crossed cubes

The Completely Independent Spanning Trees (CISTs) are a very useful construct in a computer network. It can find applications in many important network functions, especially in reliable broadcasting, i.e., guaranteeing broadcasting operation in the presence of faulty nodes. The question for the existence of two CISTs in an arbitrary network is an NP-hard problem. Therefore most research on CISTs to date has been concerning networks of specific structures. In this paper, we propose an algorithm to construct two CISTs in the crossed cube, a prominent, widely studied variant of the well-known hypercube. The construction algorithm will be presented, and its correctness proved. Based on that, the existence of two CISTs in a special Bijective Connection network based on crossed cube is also discussed.

Dimensional-Permutation-Based Independent Spanning Trees in Bijective Connection Networks

In recent years, there are many new findings on independent spanning trees (ISTs for short) in hypercubes, crossed cubes, locally twisted cubes, and Mobius cubes, which all belong to a more general network category called bijective connection networks (BC networks). However, little progress has been made for ISTs in general BC networks. In this paper, we first propose the definitions of conditional BC networks and V -dimensional-permutation. We then give a linear parallel algorithm of ISTs rooted at an arbitrary vertex in conditional BC networks, which include hypercubes, crossed cubes, locally twisted cubes, and Mobius cubes, based on the ascending circular dimensional-permutation, where the ISTs are all isomorphic to the binomial-like tree. In addition, we show that there exists an efficient algorithm to construct a spanning tree rooted at an arbitrary vertex in any BC network X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , and all V-dimensional-permutations can be used to construct spanning trees isomorphic to the n-level binomial tree and rooted at an arbitrary vertex in X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> .

Reliability Evaluation of BC Networks in Terms of the Extra Vertex- and Edge-Connectivity

Reliability evaluation of interconnection network is important to the design and maintenance of multiprocessor systems. The extra connectivity and the extra edge-connectivity are two important parameters for the reliability evaluation of interconnection networks. The <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\mbi {n}}$</tex> </formula> -dimensional bijective connection network (in brief, BC network) includes several well known network models, such as, hypercubes, Möbius cubes, crossed cubes, and twisted cubes. In this paper, we explore the extra connectivity and the extra edge-connectivity of BC networks, and discuss the structure of BC networks with many faults. We obtain a sharp lower bound of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${{g}}$</tex> </formula> -extra edge-connectivity of an <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\mbi {n}}$</tex> </formula> -dimensional BC network for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${{n}} \geq 4$</tex> </formula> and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$1 \leq { {g}} \leq {2^{[{{{n}} \over 2}]}}$</tex> </formula> . We also obtain a sharp lower bound of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${ {g}}$</tex> </formula> -extra connectivity of an <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${{n}}$</tex> </formula> -dimensional BC network for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${{n}} \geq 4$</tex> </formula> and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$1 \leq { {g}} \leq 2{ {n}}$</tex> </formula> which improves the result in [“Reliability evaluation of BC networks,” IEEE Trans. Computers, DOI: 10.1109/tc.2012.106.] for <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$1 \leq { {g}} \leq { {n}} - 3$</tex> </formula> . Furthermore, we give a remark about exploring the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${ {g}}$</tex> </formula> -extra edge-connectivity of BC networks for the more general <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\mbi {g}}$</tex> </formula> , and we also characterize the structure of BC networks with many faulty nodes or links. As an application, we obtain several results on the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\mbi {g}}$</tex> </formula> -extra (edge-) connectivity and the structure of faulty networks on hypercubes, Möbius cubes, crossed cubes, and twisted cubes.