Fault-tolerant Embedding Research Articles

Fault-tolerant graph embeddings in Archimedean networks

Fault tolerance refers to the ability of a system to continue functioning even if some of its components fail. This concept is fundamental in several domains because reliability and continuous operation are essential in carrying out their functions. Graph embedding means mapping a graph into some other structure that preserves some of the properties or relations held by the original graph. Applications in machine learning, data mining, and network analysis make fault-tolerant embedding of graphs a crucial topic. In this paper, fault-tolerant embeddings are proposed in Archimedean tessellations. The path in the embeddings does not intersect or overlap.

Enabling high fault-tolerant embedding capability of alternating group graphs

The Hamiltonian path/cycle serves as a robust tool for transmitting messages within parallel and distributed systems. However, the prevalent device-intensive nature of these systems often leads to the occurrence of faults. Tackling the critical challenge of tolerating numerous faults when constructing Hamiltonian paths and cycles in these systems is of utmost significance. The alternating group graph AGn is a suitable interconnection network for building parallel and distributed systems due to its outstanding topological properties. Many studies have been dedicated to the fault-tolerant embedding of Hamiltonian paths and cycles in AGn, but they all fail to reach the desired level of fault-tolerance. In this paper, we aim to enhance the edge fault-tolerant embedding capability of AGn by leveraging a newly emerged fault model, called the Partitioned Edge Fault (PEF) model. We prove the existence of Hamiltonian paths/cycles tolerating large-scale edge faults in AGn under the PEF model. Additionally, we propose a fault-tolerant Hamiltonian path embedding algorithm for AGn under the PEF model and validate its effectiveness through experiments. To gauge the significance of each missed edge, we conduct experimental calculations for the average path length of every edge along the Hamiltonian path produced by the proposed embedding algorithm. Furthermore, the fault-tolerance comparison and experimental analysis reveal that our results improve the fault-tolerant embedding capability of AGn from a linear level to an exponential one. To our knowledge, this is the first time that a Hamiltonian path/cycle embedding approach is proposed and actually implemented for AGn with large-scale edge faults.

Fault-Tolerant Path-Embedding of Twisted Hypercube-Like Networks (THLNs)

It is known widely that an interconnection network can be denoted by a graph G = ( V , E ) , where V denotes the vertex set and E denotes the edge set. Investigating structures of G is necessary to design a suitable topological structure of interconnection network. One of the critical issues in evaluating an interconnection network is graph embedding, which concerns whether a host graph contains a guest graph as its subgraph. Linear arrays (i.e., paths) and rings (i.e., cycles) are two ordinary guest graphs (or basic networks) for parallel and distributed computation. In the process of large-scale interconnection network operation, it is inevitable that various errors may occur at nodes and edges. It is significant to find an embedding of a guest graph into a host graph where all faulty nodes and edges have been removed. This is named as fault-tolerant embedding. The twisted hypercube-like networks ( T H L N s ) contain several important hypercube variants. This paper is concerned with the fault-tolerant path-embedding of n-dimensional (n-D) T H L N s . Let G n be an n-D T H L N and F be a subset of V ( G n ) ∪ E ( G n ) with | F | ≤ n - 2 . We show that for two different arbitrary correct vertices u and v, there is a faultless path P u v of every length l with 2 n - 1 - 1 ≤ l ≤ 2 n - f v - 1 - α , where α = 0 if vertices u and v form a normal vertex-pair and α = 1 if vertices u and v form a weak vertex-pair in G n - F ( n ≥ 5 ).

Fault-tolerant embedding of starlike trees into restricted hypercube-like graphs

A d-starlike tree (or a d-quasistar) is a subdivision of a star tree of degree d. A family of hypercube-like interconnection networks, called restricted hypercube-like graphs, includes most non-bipartite hypercube-like networks found in the literature such as twisted cubes, crossed cubes, Möbius cubes, recursive circulant G(2m,4) of odd m, etc. In this paper, we prove that given an arbitrary fault-free vertex r in an m-dimensional restricted hypercube-like graph with a set F of faults (vertex and/or edge faults) and d positive integers, l1,l2,…,ld, whose sum is equal to the number of fault-free vertices minus one, there exists a d-starlike tree rooted at r, each of whose subtrees forms a fault-free path on li vertices for i∈{1,2,…,d}, provided |F|≤m−2 and |F|+d≤m. The bounds on |F| and |F|+d are the maximum possible.

Quality enhancement with fault tolerant embedding in video transmission over WMSNs in 802.11e WLAN

With the ever-growing demand in video transmissions, it is essential to design an efficient compression and powerful transmission scheme to transmit a large volume of data in wireless multimedia sensor networks (WMSNs). The main objective of this work is to maximize the network throughput and to maintain/support a better visual experience to different end-users. In this paper, we propose a predictive compression with fault-tolerant embedding based multi-path routing and concealment process to maintain the Quality of Service (QoS) and Quality of Experience (QoE) in video transmission. In order to reduce the bit rate in high-efficiency videos, a Scalable High-efficiency Inter-layer Prediction based Video coding (SHIPVC) is proposed with two layer predictions such as texture color, and motion with different quantization parameters to attain the high coding efficiency. After compression, a multi-path genetic algorithm establishes a routing and fault tolerance embedding during transmission to maintain the QoS. In decoding side, QoE is maintained by concealing the Intra frame (I-frame) and Predicted frame (P-frame) separately using the motion vector estimation. Experimental results show better performances through bit rate reduction, maximum fault tolerance, and minimum delay parameters. The PSNR calculation for concealment technique shows the better result when compared to the previous standards. Hence, the proposed method performs the effective video transmission with acceptable quality measurements in WMSNs.

The (t, k) -Diagnosability of Balanced Hypercube under the PMC Model

ABSTRACTThe network's diagnosability implies its self-diagnostic capability. Compared with classical sequential diagnosis that has been widely used to identify faulty nodes, the -diagnosis is considered as an improvement of sequential diagnosis, where . To be more specific, the -diagnosis can identify and replace at least k faulty nodes in each iteration until all t faulty nodes are found. Meanwhile, compared with the other hypercube graphs, the balanced hypercube (denoted ) has many good properties, such as a better fault-tolerant embedding capability and a smaller diameter. In this paper, we first propose a body discover algorithm (named BD) of , where a body consists of node sets that meet specific conditions. Based on bodies found by BD, we verify that the is -diagnosable. Then we prove that the -diagnosability of is under the PMC model.

Fault-tolerant embedding of complete binary trees in locally twisted cubes

The complete binary tree is an important network structure for parallel and distributed computing, which has many nice properties and is often used to be embedded into other interconnection architectures. The locally twisted cube LTQn is an important variant of the hypercube Qn. It has many better properties than Qn with the same number of edges and vertices. The main results obtained in this paper are: (1) The complete binary tree CBTn rooted at an arbitrary vertex of LTQn can be embedded with dilation 2 and congestion 1 into LTQn. (2) When there exists only one faulty node in LTQn, both the dilation and congestion will become 2 after reconfiguring CBTn. (3) When there exist two faulty nodes in LTQn, then both the dilation and congestion will become 3 after reconfiguring CBTn. (4) For any faulty set F of LTQn with 2<|F|≤2n−1, both the dilation and congestion will become 3 after reconfiguring CBTn under certain constraints.

Fault-tolerant cycles embedding in folded hypercubes

The generalized conditional fault-tolerant embedding is investigated, in which the n-dimensional folded hypercube networks (denoted by FQ n ) acts as the host graph, and the longest fault-free cycle represents the guest graph. Under the conditions looser than that of previous works, it is shown that FQ n has a cycle with length at least 2 n -2|F v | when the number of faulty vertices and non-critical edges is at most 2n - 4; where |F v | is the number of faulty vertices. It provides further theoretical evidence for the fact that FQ n has excellent node-fault-tolerance and edge-fault-tolerance when used as a topology of large scale computer networks.

Fault-tolerant embedding of cycles of various lengths in k-ary n-cubes

The k-ary n-cube has been one of the most popular interconnection networks for distributed-memory parallel systems. In this paper, we study the problem of embedding cycles of various lengths into faulty k-ary n-cubes. It is proved that a faulty k-ary n-cube with fv faulty vertices and fe faulty edges admits a fault-free cycle of every even length from 4 to kn−2fv if k⩾4 is even and fv+fe⩽2n−2. Furthermore, we show that every healthy edge of the faulty k-ary n-cube lies on a fault-free cycle of every even length from 4 to kn−2fv if k⩾4 is even and fv+fe⩽2n−3. The results are optimal with respect to the number of vertex and edge faults tolerated.

Extended Fault-Tolerant Cycle Embedding in Faulty Hypercubes

We consider fault-tolerant embedding, where an n-dimensional faulty hypercube, denoted by <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , acts as the host graph, and the longest fault-free cycle represents the guest graph. Let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> be a set of faulty nodes in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> . Also, let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> be a set of faulty edges in which at least one end-node of each edge is faulty, and let <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Fe</i> be a set of faulty edges in which the end-nodes of each edge are both fault-free. An edge in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> is said to be critical if it is either fault-free or in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> . In this paper, we prove that there exists a fault-free cycle of length at least 2 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -2| <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> | in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ges 3) with | <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> | les 2n-5, and | <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> |+| <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sub> | les 2n-4 , in which each node is incident to at least two critical edges. Our result improves on the previously best known results reported in the literature, where only faulty nodes or faulty edges are considered.

Fault-tolerant embedding of paths in crossed cubes

The crossed cube C Q n is an important variant of the hypercube Q n and possesses many desirable properties for interconnection networks. This paper shows that in C Q n with f v faulty vertices and f e faulty edges there exists a fault-free path of length ℓ between any two distinct fault-free vertices for each ℓ satisfying 2 n − 1 − 1 ≤ ℓ ≤ 2 n − f v − 1 provided that f v + f e ≤ n − 3 , where the lower bound of ℓ and the upper bound of f v + f e are tight for some n . Moreover, this result improves the known result that C Q n is ( n − 3 ) -Hamiltonian connected.

Optimal fault-tolerant embedding of paths in twisted cubes

The twisted cube is an important variation of the hypercube. It possesses many desirable properties for interconnection networks. In this paper, we study fault-tolerant embedding of paths in twisted cubes. Let TQ n ( V , E ) denote the n-dimensional twisted cube. We prove that a path of length l can be embedded between any two distinct nodes with dilation 1 for any faulty set F ⊂ V ( TQ n ) ∪ E ( TQ n ) with | F | ⩽ n - 3 and any integer l with 2 n - 1 - 1 ⩽ l ⩽ | V ( TQ n - F ) | - 1 ( n ⩾ 3 ). This result is optimal in the sense that the embedding has the smallest dilation 1. The result is also complete in the sense that the two bounds on path length l and faulty set size | F | for a successful embedding are tight. That is, the result does not hold if l ⩽ 2 n - 1 - 2 or | F | ⩾ n - 2 . We also extend the result on ( n - 3 ) -Hamiltonian connectivity of TQ n in the literature.

Optimizing the Fault-Tolerant Embedding of the Diagnostic Graph into the Toroidal Structures of Robust Computer Systems

For the diagnostic circulant graph which enables diagnosability without repair in the case of multiple system failures, consideration was given to the methods of its embedding into the toroidal (working) structure of the robust parallel computer system. The arcs of the embedded diagnostic graph constitute composite interconnection channels between the nonadjacent tested module and the testing processing modules of the system. The problem of optimizing the embedding of the diagnostic graph into the structures of computer systems was stated in formal terms. Criteria for optimization and indices of structural robustness were described. Structural robustness of the diagnostic graph vs. multiplicity of failures in the working graph was studied. A method of designing the optimal diagnostic graph was proposed and illustrated by examples.

On the fault-tolerant embeddings of complete binary trees in the mesh interconnection networks

In this article, several schemes are proposed for embedding complete binary trees (CBT) into meshes. All of the proposed methods outperform those in the previous studies. First, a link congestion 1 embedding is achieved. Its expansion ratio is at the lowest level as we know now. Except for this superiority, it also provides another capability for fault tolerance to resist abnormal system faults, thus the embedding structure can be more guaranteed and the node utilization is raised further. Second, a link congestion 1 embedding with no-bending constraint is obtained. This scheme provides efficient CBT embedding for both the optical mesh and general mesh at the same time while keeping those good properties as the previous scheme. The last one is an optimal embedding which is applied to a 3D cubic mesh, where the node is almost fully utilized and its link congestion is 2.

On the fault-tolerant embedding of complete binary trees in the pancake graph interconnection network

This paper studies the problem of embedding complete binary trees ( CBTs) into an n-dimensional pancake graph ( P n ) with fault-tolerant capability. First, a new embedding scheme is developed for mapping a source CBT with height ∑ m=2 n⌊ log m⌋ and dilation 2 onto the P n . This scheme not only embeds a CBT whose height is very close to the largest possible one, but also saves a lot of unused generators and generator products. Furthermore, these unused generators and generator products are used to recover faulty nodes and embed multiple CBTs. Maximally, near 2/3 nodes of the source CBT are allowed to be faulty at the same time and can be recovered by our scheme with dilation 4. Alternatively, a scheme which can embed a CBT with height ∑ m=2 n⌊ log m⌋−1 is also given. In this case, if all nodes in the CBT are faulty, they can be recovered in the smallest number of recovery steps and only with dilation 4.

On fault-tolerant embedding of Hamiltonian cycles, linear arrays and rings in a Flexible Hypercube

The Hamiltonian cycle of a Flexible Hypercube graph network is investigated in this paper. The result can readily be used in the optimal embedding of a linear array (or a ring) of processors in a faulty Flexible Hypercube. There are embedding algorithms proposed in this paper. These embedding algorithms show a ring with any number of nodes that can be embedded into a faulty Flexible Hypercube with load 1, congestion 1 and dilation 3 such that O( n 2−(⌊log 2 m⌋) 2) faults can be tolerated, where n is the dimension of the Flexible Hypercube and m is the number of nodes of the ring.

Embedding of generalized Fibonacci cubes in hypercubes with faulty nodes

The generalized Fibonacci cubes (abbreviated to GFCs) were recently proposed as a class of interconnection topologies, which cover a spectrum ranging from regular graphs such as the hypercube to semiregular graphs such as the second order Fibonacci cube. It has been shown that the kth order GFC of dimension n+k is equivalent to an n-cube for 0/spl les/n<k; and it is a proper subgraph of an n-cube for n/spl ges/k. Thus, a kth order GFC of dimension n+k can be obtained from the n-cube for all n/spl ges/k by removing certain nodes from an n-cube. This problem is very simple when no faulty node exists in an k-cube; but it becomes very complex if some faulty nodes appear in an n-cube. In this paper, we first consider the following open problem: How can a maximal (in terms of the number of nodes) generalized Fibonacci cube be distinguished from a faulty hypercube which can also be considered as a fault-tolerant embedding in hypercubes. Then, we shall show how to directly embed a GFC into a faulty hypercube and prove that if no more than three faulty nodes exist, then an [n/2]th order GFC of dimension n+[n/2] can be directly embedded into an n-cube in the worst case, for n=4 or n/spl ges/6.

The balanced hypercube: a cube-based system for fault-tolerant applications

In this paper, we present an interconnection structure, called the balanced hypercube, which is a variation of the standard hypercube with desirable properties of strong connectivity, regularity, and symmetry. The proposed structure is a special type of load balanced graph designed to tolerate processor failure. In balanced hypercubes, each processor has a backup (matching) processor that shares the same set of neighboring nodes. Therefore, tasks that run on a faulty processor can be reactivated in the backup processor to provide efficient system reconfiguration. Other properties of balanced hypercubes are examined. It is also shown that odd-dimensional balanced hypercubes have smaller diameters than that of standard hypercubes. As an application of balanced hypercubes, we show a fault-tolerant embedding of rings in balanced hypercubes.

Fault-tolerant embedding of linear arrays and rings in the star graph

Methods are presented to embed Hamiltonian paths (H-paths) and Hamiltonian cycles (H-cycles) in a star graph S n in a faulty environment. The models considered include single-processor failure, double-process failure, and multiple-processor failures. All three models are applied to an H-path/cycle, which is formed by visiting all the ( n! 4! ) S 4s in an S n in a particular order. An optimal embedding is obtained in the case of single-processor failure, wherein the length of the H-path/cycle is shown to be ( n! − 2). The multiple-processor failure model is developed based on single and double processor failure models. In this case the length of the H-cycle that can be embedded is shown to be ( n! − 2 f), where f ≤ n − 2 is the number of faults. Another case of multiple-failure scenario is investigated by assuming that all faults are contained in a single S m , m < n. The network in this case, is shown to reduce to a cluster-star graph. It is proven that it is always possible to formulate an H-cycle of length ( n! − m!) in such a network.

All-fault-tolerant embedding of a complete binary tree in a group of Cayley graphs

The paper solves the problems of fault-tolerant embeddings of a complete binary tree in a group of Cayley graphs. First, a complete binary tree (CBT) is embedded into a complete-transposition graph. Then, the derived result is used to further induce the CBT embeddings for the other Cayley graphs. The primary results are that a CBT with height k × (n – 2k + 1) + (k – 2) × 2k + 1, where k = ⌋log n⌊, can be embedded into an n-dimensional complete transposition graph (CTn), star graph (STn) and bubblesort graph (BSn) with dilations 1, 3, and 2n – 3, respectively. Furthermore, a fault-tolerant scheme is developed to recover multiple faults up to the size of the embedded CBT with the least recovery cost. The dilations after recovery become at most 3, 5, and 2n – 1 for the CTn, STn, and BSn, respectively.