Fault-free Path Research Articles

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.

BCCC Disjoint Path Construction Algorithm and Fault-Tolerant Routing Algorithm under Restricted Connectivity

Connectivity in large-scale data center networks is a critical indicator to evaluate network state. A feasible and performance-guaranteed algorithm enables us to find disjoint paths between network vertices to ensure effective data transfer and to maintain the normal operation of network in case of faulty nodes. As an important data center network, BCube Connected Crossbars (BCCC) has many excellent properties that have been widely studied. In this paper, we first propose a vertex disjoint path algorithm with the time complexity of O(nk) in BCCC, where n denotes a switch connected to n servers and k denotes dimension. Then, we prove that the restricted connectivity of BCCC(n,k). Finally, we present an O(knκ1(G)) fault-free algorithm in BCCC, where κ1(G) is the restricted connectivity. This algorithm can obtain a fault-free path between any two distinct fault-free vertices under the condition that each vertex has at least one fault-free neighbor in the BCCC and a set of faulty vertices F with |F|<κ1(G).

The bipanconnectivity of bipartite hypercube-like networks

Bipanconnectivity is an important parameter in bipartite networks related on the embedding problem of linear arrays and rings. In this paper, we study the fault-tolerant bipanconnectivity of bipartite n-dimensional hypercube-like networks, denoted as Bn′. We show that for any n-dimensional bipartite hypercube-like network G∈Bn′ with f faulty elements (edges and/or vertices), including fv faulty vertices such that f≤n−2, for each pair of fault-free vertices of distance d in G, there exists a fault-free path of length l linking them, where 2n−4≤l≤2n−2fv−1 and l−d≡0 (mod 2). Our result is optimal when considering the number of faulty elements.

Cluster-Fault Tolerant Routing in a Torus.

The number of Internet-connected devices grows very rapidly, with even fears of running out of available IP addresses. It is clear that the number of sensors follows this trend, thus inducing large sensor networks. It is insightful to make the comparison with the huge number of processors of modern supercomputers. In such large networks, the problem of node faults necessarily arises, with faults often happening in clusters. The tolerance to faults, and especially cluster faults, is thus critical. Furthermore, thanks to its advantageous topological properties, the torus interconnection network has been adopted by the major supercomputer manufacturers of the recent years, thus proving its applicability. Acknowledging and embracing these two technological and industrial aspects, we propose in this paper a node-to-node routing algorithm in an -dimensional -ary torus that is tolerant to faults. Not only is this algorithm tolerant to faulty nodes, it also tolerates faulty node clusters. The described algorithm selects a fault-free path of length at most with an worst-case time complexity with the set of faulty nodes induced by the faulty clusters.

Terminal reliability analysis of multistage interconnection networks

Terminal reliability of multistage interconnection network (MIN) is the probability of at least one fault free path between any source–destination nodes pair. MIN is used to interconnect processors and memory modules in multi-processor system. In this paper, terminal reliability is estimated and validated for existing MINs (i.e., SEN, SEN+, SEN+2, GIN, CGIN, PCGIN, 3D-GIN, 4D-GIN-1, 4D-GIN-2, 4D-GIN-3, and SEGINs). The comparison and performance analysis of MINs based on various indices are also presented.

Fault-Tolerant Panconnectivity of Augmented Cubes AQn

The augmented cube [Formula: see text] is a variation of the hypercube [Formula: see text]. This paper considers the fault-tolerant Panconnectivity of [Formula: see text]. Assume that [Formula: see text] and [Formula: see text]. We prove that for any two fault-free vertices [Formula: see text] and [Formula: see text] with distance [Formula: see text] in [Formula: see text], there exists a fault-free path [Formula: see text] of each length from [Formula: see text] to [Formula: see text] in [Formula: see text] if [Formula: see text], where [Formula: see text] is the number of faulty vertices in [Formula: see text]. Moreover, the bound is sharp.

Strongly Menger connectedness of data center network and (n,k)-star graph

A graph G=(V,E) is F-strongly Menger-vertex-connected (resp. Menger-edge-connected), if for a subgraph G−F of G with a conditional faulty vertex (resp. edge) set F⊆V (resp. F⊆E), each pair of vertices u and v are connected by min{degG−F(u),degG−F(v)} internally disjoint (resp. edge-disjoint) fault-free paths in G−F. Where degG−F(u) and degG−F(v) are the degrees of u and v in G−F, respectively. He et al. (2018) [7] introduced some sufficient conditions of F-strongly Menger (edge) connected for k-regular triangle-free graphs. In this paper, we consider the strongly Menger connectedness of two kinds of graphs with triangles: data center network Dk,n and (n,k)-star graph Sn,k. We prove Dk,n is (k−1)-strongly Menger-vertex-connected, (n+k−3)-strongly Menger-edge-connected of order 1, (2n+2k−8)-strongly Menger-edge-connected of order 2, and (3n+3k−15)-strongly Menger-edge-connected of order 3. Furthermore, we obtain Sn,k is (k−2)-strongly Menger-vertex-connected, (n−3)-strongly Menger-edge-connected of order 1, (2n−8)-strongly Menger-edge-connected of order 2, and (3n−15)-strongly Menger-edge-connected of order 3.

Fault diagnosis based on identified discrete-event models

Fault diagnosis of Discrete-Event Systems consists of detecting and isolating the occurrence of faults within a bounded number of event occurrences. Recently, a new model for discrete-event system identification with the aim of fault detection, called Deterministic Automaton with Outputs and Conditional Transitions (DAOCT), has been proposed in the literature. The model is computed from observed fault-free paths, and represents the fault-free system behavior. In order to obtain compact models, loops are introduced in the model, which implies that sequences that are not observed can be generated leading to an exceeding language. This exceeding language is associated with possible non-detectable faults, and must be reduced in order to use the model for fault detection. After detecting the fault occurrence, its isolation is carried out by analyzing residuals. In this paper, we present a fault diagnosis scheme based on the DAOCT model. We show that the proposed fault diagnosis scheme is more efficient than other approaches proposed in the literature, in the sense that the exceeding language can be drastically reduced, reducing the number of non-detectable fault occurrences, and, in some cases, reducing also the delay for fault diagnosis. A practical example, consisting of a plant simulated by using a 3D simulation software controlled by a Programmable Logic Controller, is used to illustrate the results of the paper.

Discrete event system identification with the aim of fault detection

In this paper, we present a method for discrete event system identification with the aim of fault detection. The method is based on a new model called Deterministic Automaton with Outputs and Conditional Transitions (DAOCT), which is computed from observed fault-free paths, and represents the fault-free system behavior. In practice, a trade-off between size and accuracy of the identified automaton has to be found. In order to obtain compact models, loops are introduced in the model, which implies that sequences that are not observed can be generated by the model leading to an exceeding language. This exceeding language is associated with possible non-detectable faults, and must be reduced in order to use the model for fault detection. We show, in this paper, that the exceeding language generated by the DAOCT is smaller than the exceeding language generated by another model proposed in the literature, reducing, therefore, the number of possible non-detectable faults. We also show that if the identified DAOCT does not have cyclic paths, then the exceeding language is empty, and the model represents all and only all observed fault-free sequences generated by the system. In order to illustrate the results of the paper, a physical system is simulated by using a 3D simulation software controlled by a Programmable Logic Controller (PLC). The main idea is to use a virtual digital system to simulate the fault-free behavior of a physical system, captured by the sequences of input and output signals of the PLC, and then use the method proposed in the paper to obtain the DAOCT model of the plant.

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.

A Hierarchical and Distributed Fault Tolerant Proposal for NoC-Based MPSoCs

Aggressive scaling of CMOS process technology allows the fabrication of highly integrated chips such as NoC-based MPSoCs. However, fault probability increases when devices’ size reduces. Hence, fault tolerant design has an important role in current nanometric technologies, leading to research on fault mitigation techniques for NoC-based MPSoCs. Most of the state-of-the-art papers present partial solutions to design a fault tolerant MPSoC, i.e., they present fault tolerant mechanisms for either NoCs or processing elements (PEs). The goal of this paper is to propose a comprehensive integration of previously defined recovery mechanisms. The main novelty is the system-level integration itself, which is organized in a hierarchical and distributed manner, ensuring the correct execution of applications in the presence of multiple transient or permanent faults in both the NoC and/or the PEs. The combination of both NoC and PE recovery methods enable the proposed system to tolerate a very severe number of faults. Depending on the severity of the fault in the NoC, it may operate in degraded mode or require the search of fault-free paths. In both cases, the communication is reestablished in less than 50 microseconds. Faults detected into the PEs fire a lightweight and fast task relocation protocol, which executes in less than one millisecond.

Enhanced discrete event model for system identification with the aim of fault detection

In this paper, we present a new model for discrete-event system identification that is suitable for fault detection, called Deterministic Automaton with Outputs and Conditional Transitions (DAOCT). The model is computed from observed fault-free paths, and represents the fault-free system behavior. In practice, a trade-off between size and accuracy of the identified automaton has to be found. In order to obtain compact models, loops are introduced in the model, which implies that sequences that are not observed can be generated by the model leading to an exceeding language. This exceeding language is associated with possible non-detectable faults, and must be reduced in order to use the model for fault detection. We show, in this paper, that the exceeding language generated by the DAOCT is smaller than the exceeding language generated by other models proposed in the literature, reducing, therefore, the number of possible non-detectable faults. We also show that if the identified DAOCT does not have cyclic paths, then the exceeding language is empty, and the model represents all and only all observed fault-free sequences generated by the system. A practical example is used to illustrate the results of the paper.

Disjoint path covers with path length constraints in restricted hypercube-like graphs

A disjoint path cover of a graph is a set of pairwise vertex-disjoint paths that altogether cover every vertex of the graph. In this paper, we prove that given k sources, s1, …, sk, in an m-dimensional restricted hypercube-like graph with a set F of faults (vertices and/or edges), associated with k positive integers, l1, …, lk, whose sum is equal to the number of fault-free vertices, there exists a disjoint path cover composed of k fault-free paths, each of whose paths starts at si and contains li vertices for i∈{1,…,k}, provided |F|+k≤m−1. The bound, m−1, on |F|+k is the best possible.

A note on path embedding in crossed cubes with faulty vertices

In this note, we investigate the problem of embedding paths of various lengths into crossed cubes with faulty vertices. In Park et al. (2007) [14] showed that, for any hypercube-like interconnection network of 2n vertices with a set F of faulty vertices and/or edges, there exists a fault-free path of length ℓ between any two distinct fault-free vertices for each integer ℓ satisfying 2n−3⩽ℓ⩽2n−|F|−1. In this note, we show that, for crossed cubes CQn with n⩾5, the range of ℓ can be extended to [2n−5,2n−|F|−1]. Moreover, we also show that the vertices of CQ5 can be partitioned into two symmetric groups.

Enhanced Algorithms for Fault Nodes Recovery in Wireless Sensors Network

An integration of sensing environment with the numerous deployments of sensor nodes in Wireless Sensor Network (WSN) causes the severe security threats and hence the trust assurance mechanisms are required. For the large scale WSN, the existence of a number of intermediate nodes is responsible for the data forwarding to the sink node. Due to the battery operated sensors, the recharge and replace mechanisms suffer from the energy conservation and minimum network lifetime. The identification of fault nodes on the transmission path plays the major role in energy conservation. With the dense deployment of sensor nodes, the failures in node and link are high that disrupts the entire communication. This paper proposes the suitable alternative fault-free path prediction model to perform the communication among the nodes. Initially, the sensor nodes are deployed in the WSN environment. Once the initialization of source and destination nodes are over, the path between them is predicted through the Hamiltonian path prediction model. During the failure, scenario, this paper estimates the node and link parameters such as Received Signal Strength Indicator (RSSI), queue size, response time, and bandwidth are individually estimated and group them into the Quality Factor (QF). Based on the QF, the proposed work predicts the fault-free link to alleviate the unnecessary transmissions to the fault node and reduces the energy consumption. The comparison between the proposed Hamiltonian Path-based Hyper Cube (HPHC) network with the existing fault detection mechanisms regarding the performance measures such as Packet Delivery Ratio (PDR), fault node detection rate, throughput and end-to-end delay assures the effectiveness of HPHC in WSN communication.

Many-to-many disjoint paths in hypercubes with faulty vertices

This paper considers the problem of many-to-many disjoint paths in the hypercube Qn with f faulty vertices and obtains the following result. For any integer k with 1≤k≤n−1 and any two sets S and T of k fault-free vertices in different partite sets of Qn(n≥2), if f≤2n−2k−2 and each fault-free vertex has at least two fault-free neighbors, then there exist k fully disjoint fault-free paths linking S and T which contain at least 2n−2f vertices. A linear algorithm for finding such disjoint paths is also given. This result improves some known results in a sense.

A runtime fault-tolerant routing algorithm based on region flooding in NoCs

Aggressive scaling of the CMOS process technology allows the fabrication of highly integrated chips, and enables the design of the network-on-chip (NoC). However, it also leads to widespread reliability problems. A reliable NoC system must operate normally even in the face of a lot of transistor failures. Aiming towards permanent faults on communication links, we introduce a fault-tolerant MPI-like communication protocol. It detects the link failure if there exist unresponsive requests and automatically starts the new path exploration. The region flooding algorithm is proposed to search for a fault-free path and reroute packets to avoid system stalls. The experimental result shows our approach significantly reduces the latency compared with the basic flooding algorithm. The maximum latency reduction is 25% under the bit complement traffic pattern. Also, it brings only 2% fault tolerance loss.

Thermal-Aware Small-Delay Defect Testing in Integrated Circuits for Mitigating Overkill

At-speed testing of deep-submicrometer or nano-scale integrated circuits (ICs) consumes excessive power and creates hotspots and temperature gradient in the chip-under-test. The problem worsens for 3-D ICs, where heat dissipation across layers is more unbalanced. These hotspots in a circuit often cause severe degradation of performance and reliability, as a rise in temperature can introduce an extra delay along paths. As a result, the delay of an otherwise fault-free path may exceed the functional clock period. Such thermal emergencies can thus lead to over-detection and undue yield loss during testing. Their effects will be more severe for small-delay defects (SDDs), which target to sensitize the long paths in a circuit. In this paper, we quantify, for the first time, the impact of thermal emergencies on SDDs and provide a solution to mitigate them. The proposed method is based on: 1) a new thermal-aware (TA) path-selection method, 2) a TA test-ordering method, and 3) an effective scan architecture and a test-application scheme. Experimental results on benchmarks demonstrate that the new method can significantly reduce the number of over-detections of SDDs.

Conditional fault-tolerant routing of (n, k)-star graphs

As a generalization of the star graph, the -star graph is one of the most important interconnection networks for parallel and distributed computing system. In this paper, we investigate the fault-tolerant routing problem of -star graphs under the conditional fault assumption (CFA), where all the neighbours of any fault-free node cannot be faulty simultaneously. We show that, under the CFA, the-star graph can tolerate up to faulty nodes, and there is a fault-free path of length at most the diameter of the -star graph plus 6 between any two fault-free nodes.

Fault-tolerant routing over shortest node-disjoint paths in hypercubes

This paper presented a routing algorithm that finds n disjoint shortest paths from the source node s to target node d in the n-dimensional hypercube. Fault-tolerant routing over all shortest node-disjoint paths has been investigated to overcome the failure encountered during routing in hypercube networks. In this paper, we proposed an efficient approach to provide fault-tolerant routing which has been investigated on hypercube networks. The proposed approach is based on all shortest node-disjoint paths concept in order to find a fault-free shortest path among several paths provided. The proposed algorithm is a simple uniform distributed algorithm that can tolerate a large number of process failures, while delivering all n messages over optimal-length disjoint paths. However, no distributed algorithm uses acknowledgement messages (acks) for fault tolerance. So, for dealing the faults, acknowledgement messages (acks) are included in the proposed algorithm for routing messages over node-disjoint paths in a hypercube network.