ONE BY ONE EMBEDDING THE CROSSED HYPERCUBE INTO PANCAKE GRAPH

Let G and H be two simple undirected graphs. An embedding of the graph G into the graph H is an injective mapping f from vertices of G to the vertices of H. The dilation of embedding is the maximum distance between f(u), f(v) taken over edges (u, v) of G. The Pancake graph is one as viable interconnection scheme for parallel computers, which has been examined by a number of researchers. The Pancake was proposed as alternatives to the hypercube for interconnecting processors in parallel computer. Some good attractive properties of this interconnection network include: vertex symmetry, small degree, a sub-logarithmic diameter, extendability, and high connectivity (robustness), easy routing and regularity of topology, fault tolerance, extensibility and embeddability of others topologies. In this paper, we give a construction of one by one embedding of dilation 5 of crossed hypercube into Pancake graph

Fault-free longest paths in star networks with conditional link faults

AbstractThe star network, which belongs to the class of Cayley graphs, is one of the most versatile interconnection networks for parallel and distributed computing. In this paper, adopting the conditional fault model in which each node is assumed to be incident with two or more fault-free links, we show that an n-dimensional star network can tolerate up to 2n−7 link faults, and be strongly (fault-free) Hamiltonian laceable, where n≥4. In other words, we can embed a fault-free linear array of length n!−1 (n!−2) in an n-dimensional star network with up to 2n−7 link faults, if the two end nodes belong to different partite sets (the same partite set). The result is optimal with respect to the number of link faults tolerated. It is already known that under the random fault model, an n-dimensional star network can tolerate up to n−3 faulty links and be strongly Hamiltonian laceable, for n≥3

Paired 2-disjoint path covers of burnt pancake graphs with faulty elements

The burnt pancake graph $BP_n$ is the Cayley graph of the hyperoctahedral group using prefix reversals as generators. Let $\{u,v\}$ and $\{x,y\}$ be any two pairs of distinct vertices of $BP_n$ for $n\geq 4$. We show that there are $u-v$ and $x-y$ paths whose vertices partition the vertex set of $BP_n$ even if $BP_n$ has up to $n-4$ faulty elements. On the other hand, for every $n\ge3$ there is a set of $n-2$ faulty edges or faulty vertices for which such a fault-free disjoint path cover does not exist.Comment: 14 pages, 4 figure

Interconnection networks for parallel and distributed computing

Parallel computers are generally either shared-memory machines or distributed- memory machines. There are currently technological limitations on shared-memory architectures and so parallel computers utilizing a large number of processors tend tube distributed-memory machines. We are concerned solely with distributed-memory multiprocessors. In such machines, the dominant factor inhibiting faster global computations is inter-processor communication. Communication is dependent upon the topology of the interconnection network, the routing mechanism, the flow control policy, and the method of switching. We are concerned with issues relating to the topology of the interconnection network. The choice of how we connect processors in a distributed-memory multiprocessor is a fundamental design decision. There are numerous, often conflicting, considerations to bear in mind. However, there does not exist an interconnection network that is optimal on all counts and trade-offs have to be made. A multitude of interconnection networks have been proposed with each of these networks having some good (topological) properties and some not so good. Existing noteworthy networks include trees, fat-trees, meshes, cube-connected cycles, butterflies, Möbius cubes, hypercubes, augmented cubes, k-ary n-cubes, twisted cubes, n-star graphs, (n, k)-star graphs, alternating group graphs, de Bruijn networks, and bubble-sort graphs, to name but a few. We will mainly focus on k-ary n-cubes and (n, k)-star graphs in this thesis. Meanwhile, we propose a new interconnection network called augmented k-ary n- cubes. The following results are given in the thesis.1. Let k ≥ 4 be even and let n ≥ 2. Consider a faulty k-ary n-cube Q(^k_n) in which the number of node faults f(_n) and the number of link faults f(_e) are such that f(_n) + f(_e) ≤ 2n - 2. We prove that given any two healthy nodes s and e of Q(^k_n), there is a path from s to e of length at least k(^n) - 2f(_n) - 1 (resp. k(^n) - 2f(_n) - 2) if the nodes s and e have different (resp. the same) parities (the parity of a node Q(^k_n) in is the sum modulo 2 of the elements in the n-tuple over 0, 1, ∙∙∙ , k - 1 representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang, Tan and Hsu, and by Fu. Furthermore, we extend known results, obtained by Kim and Park, for the case when n = 2.2. We give precise solutions to problems posed by Wang, An, Pan, Wang and Qu and by Hsieh, Lin and Huang. In particular, we show that Q(^k_n) is bi-panconnected and edge-bipancyclic, when k ≥ 3 and n ≥ 2, and we also show that when k is odd, Q(^k_n) is m-panconnected, for m = (^n(k - 1) + 2k - 6’ / ‘_2), and (k -1) pancyclic (these bounds are optimal). We introduce a path-shortening technique, called progressive shortening, and strengthen existing results, showing that when paths are formed using progressive shortening then these paths can be efficiently constructed and used to solve a problem relating to the distributed simulation of linear arrays and cycles in a parallel machine whose interconnection network is Q(^k_n) even in the presence of a faulty processor.3. We define an interconnection network AQ(^k_n) which we call the augmented k-ary n-cube by extending a k-ary n-cube in a manner analogous to the existing extension of an n-dimensional hypercube to an n-dimensional augmented cube. We prove that the augmented k-ary n-cube Q(^k_n) has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented k-ary n-cube Q(^k_n) - is a Cayley graph (and so is vertex-symmetric); has connectivity 4n - 2, and is such that we can build a set of 4n - 2 mutually disjoint paths joining any two distinct vertices so that the path of maximal length has length at most max{{n- l)k- (n-2), k + 7}; has diameter [(^k) / (_3)] + [(^k - 1) /( _3)], when n = 2; and has diameter at most (^k) / (_4) (n+ 1), for n ≥ 3 and k even, and at most [(^k)/ (_4) (n + 1) + (^n) / (_4), for n ^, for n ≥ 3 and k odd.4. We present an algorithm which given a source node and a set of n - 1 target nodes in the (n, k)-star graph S(_n,k) where all nodes are distinct, builds a collection of n - 1 node-disjoint paths, one from each target node to the source. The collection of paths output from the algorithm is such that each path has length at most 6k - 7, and the algorithm has time complexity O(k(^3)n(^4))

Properties and algorithms of the (n, k)-star graphs

The (n, k)-star interconnection network was proposed in 1995 as an attractive alternative to the n-star topology in parallel computation. The (n, k )-star has significant advantages over the n-star which itself was proposed as an attractive alternative to the popular hypercube. The major advantage of the (n, k )-star network is its scalability, which makes it more flexible than the n-star as an interconnection network. In this thesis, we will focus on finding graph theoretical properties of the (n, k )-star as well as developing parallel algorithms that run on this network. The basic topological properties of the (n, k )-star are first studied. These are useful since they can be used to develop efficient algorithms on this network. We then study the (n, k )-star network from algorithmic point of view. Specifically, we will investigate both fundamental and application algorithms for basic communication, prefix computation, and sorting, etc. A literature review of the state-of-the-art in relation to the (n, k )-star network as well as some open problems in this area are also provided

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Edge-Fault-Tolerant Path/Cycle Embedding on Three Classes of Cayley Graphs

當前VLSI技術的進步，使得建造數千甚至數萬顆處理器的超大型平行分散式系統已經可以實現了。而在這些平行分散式系統當中，其中一個重要的步驟就是決定各個處理器之間連結的拓樸性質（稱之為連結網路）。網路的拓樸性質影響了平行分散式系統的硬體層面和軟體層面的各種設計。有許多的連結網路已經被提出來研究，其中有一類被稱為凱利圖的網路引起廣泛的注意和興趣。於連結網路來說，有一類重要的問題，是在某一個網路上模擬另外一個網路。這個問題稱為「嵌入網路」問題。在這當中，線性陣列和環是對於平行與分散式計算的連結網路來說，兩種最基本的網路。它們適合發展簡單而有效的演算法，並且同時只需花費相當低的通訊代價。線性陣列和環在實際層面上的廣泛運用給了我們在圖上做「嵌入路徑」和「嵌入圈」問題的動機。先前在「嵌入路徑」和「嵌入圈」問題上的研究多半聚焦於在圖上找出最長的路徑或最長的圈，另外有一類問題則聚焦於在圖上找出所有可能長度的圈。一個系統當中，處理器和它們彼此之間的鏈結都有可能發生錯誤，因此考慮一個連結網路的容錯能力是非常重要的。也就是說，連結網路在發生某些處理器錯誤或者鏈結錯誤時，這個網路仍能保有原先的性質。一般來說，有兩種考量連結網路容錯能力的模型：隨機錯誤(random fault)與條件錯誤(conditional fault)。若假設錯誤會毫無限制的隨機發生，就稱之為隨機錯誤。反之，若假設錯誤的發生會滿足某些條件，則稱之為條件錯誤。這本論文裡，我們研究在邊錯誤下的其中三類凱利圖上的無錯嵌入路徑、嵌入圈問題：星網絡(star graph)，交替群圖(alternating group graph)，與煎餅網絡(pancake graph)。我們證明了在隨機錯誤模型中，一個n維交替群圖在至多2n-6個鏈結損壞的情形下，仍然可以找到所有可能長度的環。以能容忍鏈結損壞的角度來看，這個結果是最佳的。另一方面，我們也在條件錯誤模型中討論了三個問題。假設每個節點至少和兩個健康的鏈結相連的情況下，我們證明了一個n維星網絡在至多2n-7個鏈結損壞的情形下，仍能在任意兩點之間找到最長的路徑；一個n維交替群圖在至多4n-13個鏈結損壞的情形下，仍然可以找到最長的環；一個n維煎餅網絡在至多2n-7個鏈結損壞的情形下，仍然可以找到最長的環。上述三個在條件錯誤模型的結果中，在星網絡與交替群圖的兩個結果，以能容忍鏈結損壞的角度來看，這兩個結果是最佳的。後，我們在條件錯誤模型中，針對這三個不同的圖另外找出三個小性質，做為對上述主要問題的補充與對照。此外，針對我們在條件錯誤模型中的假設（假設每個節點至少和兩個健康的鏈結相連），我們也透過計算機率的方式來說明該假設是有實質應用上的意義的。Advances in technology, especially the advent of VLSI circuit technology, have made it possible to build a large parallel and distributed system involving thousands or even tens of thousands of processors. One crucial step on designing a large-scale parallel and distributed system is to determine the topology of the interconnection network (network for short). The network topology not only affects the hardware architecture but also the nature of theystem software that can be used in a parallel and distributed system. A number of network topologies have been proposed. Among them, a class of graphs called Cayley graphs is of interest for the design and analysis of interconnection networks.or interconnection networks, the problem of simulating one network by another is modelled as a network embedding problem. Linear arrays and rings, which are two of theost fundamental interconnection networks for parallel and distributed computation, are suitable to develop simple and efficient algorithms with low communication costs. The wide applications of linear arrays and rings motivate us to investigate path and cycle embedding in networks. Some of previous researches on path or cycle embedding focused on finding longest paths or cycles (i.e., the Hamiltonian problem, in terms of graph theory). Some others focused on finding cycles of all possible lengths (i.e., the pancycle problem, in terms of graph theory).ince node faults and/or link faults may occur to networks, it is significant to consider faulty networks. Fault tolerance ability is an important consideration for interconnection network topology. That is, the network is still functional when some node faults and/or link faults occur. Among them, two fault models were adopted; one is the random fault model, and the other is the conditional fault model. The random fault model assumed that the faults might occur everywhere without any restriction, whereas the conditional fault model assumed that the distribution of faults must satisfy some properties. It is more difficult to solve problems under the conditional fault model than the random fault model.n this dissertation, we investigate fault-free path/cycle embedding problems with edge faults on three instances of Cayley graphs: star graphs, alternating group graphs, and pancake graphs. If the random fault model is adopted, we show that an n-dimensional alternating group graph is (2n−6)-edge-fault-tolerant pancyclic. This result is optimal with respect to the number of edge faults tolerated.n the other hand, under the conditional fault model and with the assumption of at least two non-faulty links incident to each node, we show that an n-dimensional star graph is (2n−7)-edge-fault-tolerant strongly Hamiltonian laceable and (n−4)-edge-fault-tolerant hyper Hamiltonian laceable, an n-dimensional alternating group graph is (4n−13)-edge-fault-tolerant Hamiltonian and (2n−7)-edge-fault-tolerant Hamiltonian connected, and an n-dimensional pancake graph is (2n−7)-edge-fault-tolerant Hamiltonian and (n−4)-edge-fault-tolerant Hamiltonian connected. The results on star graphs and alternating group graphs are optimal with respect to the number of edge faults tolerated. We also verify that the assumption is meaningful in practical by evaluating its probability of occurrence, which is very close to 1, even if n is small.1 Introduction 1.1 Path/Cycle Embedding 3.2 Fault Tolerance 4.2.1 The Random Fault Model 4.2.2 The Conditional Fault Model 5.3 Thesis Organization 7 Preliminaries 8.1 Definitions and Notations 8.2 Cayley Graphs 10.2.1 Star Graphs 11.2.2 Alternating Group Graphs 13.2.3 Pancake Graphs 15 Edge-Fault-Tolerant Hamiltonicity of Star Graphs 18.1 Properties of Star Graphs 18.2 Fault-Free Longest Paths under the Conditional Fault Model 22.3 Probability and Optimality 31 Edge-Fault-Tolerant Hamiltonicity of Alternating Group Graphs 34.1 Properties of Alternating Group Graphs 34.2 Fault-Free Cycles under the Random Fault Model 36.3 Fault-Free Hamiltonian Cycles under the Conditional Fault Model 42.4 Probability and Optimality 48 Edge-Fault-Tolerant Hamiltonicity of Pancake Graphs 51.1 Properties of Pancake Graphs 51.2 Fault-Free Hamiltonian Cycles under the Conditional Fault Model 58 Discussion and Conclusion 68.1 Contribution 68.2 Remarks 71.3 Further Research 72ibliography 75ppendix A Source Codes for Lemma 3.6 83ppendix B Source Codes for Theorem 4.2 89ppendix C Source Codes for Lemma 5.5 9