Faulty Processors Research Articles

Optimistic crash recovery without rolling back nonfaulty processors

The problem of recovering from transient processor failures in distributed computing systems using optimistic message logging is considered, and two crash recovery algorithms are presented. Both algorithms have three advantages: 1. (1) no nonfaulty processor is forced to roll back 2. (2) output messages may be committed immediately, and 3. (3) a nonfaulty processor is not disrupted if none of its neighbors is faulty. The first algorithm uses acknowledgments, permits at most two neighboring processors to fail at the same time, and does not block the receiver or the sender of each message. The second algorithm copes with the simultaneous failure of any number of nodes, but it requires that the sender id of each message (not its content) be logged to stable storage before the message is processed. Though the second algorithm blocks the receiver, the blocking time can be substantially reduced. In both cases, the state of a faulty processor immediately before its failure is recreated. Thus, after either of the two crash recovery algorithms is run, the distributed system restarts from a consistent state.

Computational arrays with flexible redundancy

Different multiple redundancy schemes for fault detection and correction in computational arrays are proposed and analyzed. The basic idea is to embed a logical array of nodes onto a processor/switch array such that d processors, 1/spl les/d/spl les/4, are dedicated to the computation associated with each node. The input to a node is directed to the d processors constituting that node, and the output of the node is computed by taking a majority vote among the outputs of the d processors. The proposed processor/switch array (PSVA) is versatile in the sense that it may be configured as a nonredundant system or as a system which supports double, triple or quadruple redundancy. It also allows for spares to be distributed in the PSVA in a way that permits spare sharing among nodes, thus enhancing the overall system reliability. In addition to choosing the required degree of redundancy, the flexibility of the PSVA architecture allows for the embedding of redundant arrays onto defective PSVA's and for run-time reconfiguration to avoid faulty processors and switches. Different embedding and reconfiguration algorithms are presented and analyzed using Markov chain techniques, using probability arguments, and via simulation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Pipelined Diagnosis of Wafer-Scale Linear Arrays

We present a comparison-based algorithm for identifying faulty and fault-free elements in a wafer-scale linear array of processors (or other logic elements). Only nearest neighbor communication is assumed to be possible between the processors in the array. Because the algorithm is simple and requires no storage of test vectors or test outcomes, it is ideally suited for implementation on the wafer to provide the capability for built-in production (or post production) testing. We show that, surprisingly, this method achieves high accuracy of diagnosis over a wide range of yields even though the diagnosis may be based on a high proportion of results produced by faulty processors. We also propose an improvement to the above algorithm which uses a processor diagnosed as fault-free by the basic algorithm as the starting point in improving the accuracy with which faulty processors are identified. Quantitative and qualitative reasoning validate the efficiency of these schemes.

A New Measure in System-Level Diagnosis of Hypercubes

In this paper a new measure in system-level diagnosis for hypercube multicomputer systems is defined. One-step diagnosis of hypercubes which involves only one testing phase during which processors test each other is discussed. Two different kinds of one-step diagnosis are considered: one called one-step diagnosis and studied earlier by other researchers and the other called one-step diagnosis with the forbidden faulty sets (one-step-FFS) and studied first here in the context of application to hypercubes. One of the two main results presented here is a proof that the degree of diagnosability of the n-dimensional hypercube (for short, n-cube), where n ≥ 4 increases from n to 2n − 2 as the diagnosis strategy changes from the one-step strategy to the one-step-FFS strategy. The other main result is the proof that if the fault bound; i.e., the upper bound on the possible number of faulty processors, is kept to the same number n in both cases of one-step and one-step-FFS diagnosis, then the one-step-FFS strategy requires [n/2] + 1 testing links per processor, whereas the one-step strategy requires n testing links per processor. An algorithm for selecting ([n/2] + 1)n/2 (2-way) links in an n-cube for use as testing links is presented.

Efficient agreement using fault diagnosis

We give an extremely simple Byzantine agreement protocol that uses O (t2) processors, min (f + 2, t + 1) rounds of communication, O (n ċ t ċ f ċ log |V|) total message bits, and O (log |V|) maximum message size, where n is the total number of processors that actually participate in the protocol, t is an upper bound on the number of faulty processors, f is the number of processors that actually fail in a given execution, and V is the set of possible inputs. This protocol uses roughly the same resources as a more complex protocol due to Dolev, Reischuk, and Strong. By adding explicit fault diagnosis to our first protocol, we produce a somewhat more complicated protocol that uses O(t1.5) processors, min (f + 2, t + 1) rounds, O(n ċ t2 ċ f ċ log|V|) total message bits, and O(tċ log|V|) maximum message size.

Optimal and efficient probabilistic distributed diagnosis schemes

The distributed self-diagnosis of a multiprocessor/multicomputer system based on interprocessor tests with imperfect fault coverage that permits intermittently faulty processors is addressed. Focusing on probabilistic diagnosis methods, the authors define several different categories of probabilistic diagnosis based on the type of fault syndrome information used in the diagnosis. Rigorous probabilistic analysis is then used to derive diagnosis algorithms optimal in terms of diagnostic accuracy for the diagnosis categories introduced. Analysis and simulations are used to evaluate the performance of the diagnosis algorithms introduced.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

DISTRIBUTED FAULT DIAGNOSIS OF A RING OF PROCESSORS

Based on the theory of system level diagnosis pioneered by Preparata, Metze and Chien [1], we study fault diagnosis of a ring of processors. We use the comparison based model of Chwa and Hakimi [4]. It is shown that the processors in a ring can be diagnosed uniquely if, and only if, the faults are distributed in a certain manner. We present algorithms to diagnose the faulty processors when these requirements are satisfied. These algorithms permit distributed implementation on both unidirectional and bidirectional rings.

Self-diagnosability of multiprocessor systems with hybrid faults: diagnosis by comparison approach

Considers the diagnosis of multiprocessor systems where a fault is detected by comparing the responses of two processors for the same tasks. The system-level diagnosability of such systems in the presence of permanent and intermittent faults is analyzed, assuming that the number of fault processors is no more than t and the number of intermittently faulty processors is no more than i. Characterization of a system under this model for i-diagnosability is presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Efficient algorithms for system diagnosis with both processor and comparator faults

For the comparison-based self-diagnosis of multiprocessor systems, an extended model that considers both processor and comparator faults is presented. It is shown that in this model the system diagnosability is t<or=Z delta /2Z, where delta is the minimum vertex degree of the system graph. However, if the number of faulty comparators is assumed not to exceed the number of faulty processors, the diagnosability of the model reaches t<or= delta . An optimal O( mod E mod ) algorithm, where E is the set of comparators, is given for identifying all faulty processors and comparators, provided that the total number of faulty components does not exceed the system diagnosability, and an O( mod E mod )/sup 2/ algorithm for the case t<or= delta is also presented. These efficient algorithms determine the faulty processors by calculating each processor's weight, which is mainly defined by the number of adjacent relative tests stating 'agreement'. After sorting the processors according to their weights, the algorithms determine all faulty components by separating the sorted processor list.<<ETX>>

A polynomial time algorithm for reconfiguring multiple-track models

A polynomial time algorithm for solving the combinatorial problem that underlies the reconfiguration issues in the m1/2-track-m-spare model, for any arbitrary m, is discussed. The following combinatorial problem is solved: Given a set of points in a two-dimensional grid, find a set of noninteracting straight lines such that every line starts at a point and connects to one of the boundaries of the grid, there are no more than m lines overlapping in any row or column of the grid, and there are no near-miss situations. The time complexity of the algorithm is shown to be O(m mod F mod /sup 2/), where mod F is the number of faulty processors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Fault-Tolerant Permutation Routing in Hypercubes

This paper introduces a deterministic mechanism for fault-tolerant permutation routing on an n-cube with less than n processor and/or link faults, using O( n) steps and constant queue size. The basic approach is to modify an existing route of a given permutation to avoid the faulty processors and/or faulty links, yet only incurring a constant factor slowdown in communication by ensuring that, at each step of the routing, every message either stays where it is or is sent to a nonfaulty neighbor, each link has at most one message traversing it in each direction, and the routing uses constant queue size. The existing route used in this paper is Benes routes. However, our method can be applied to any routing method where, in each step, all messages use links across the same dimension. The same approach can also be applied to online routing based on Batcher′s bitonic sorting to avoid faults.

Common knowledge and consistent simultaneous coordination

There is a very close relationship between common knowledge and simultaneity in synchronous distributed systems. The analysis of several well-known problems in terms of common knowledge has led to round-optimal protocols for these problems, including Reliable Broadcast, Distributed Consensus, and the Distributed Firing Squad problem. These problems require that the correct processors coordinate their actions in some way but place no restrictions on the behaviour of the faulty processors. In systems with benign processor failures, however, it is reasonable to require that the actions of a faulty processor be consistent with those of the correct processors, assuming it performs any action at all. We consider problems requiring consistent, simultaneous coordination. We then analyze these problems in terms of common knowledge in several failure models. The analysis of these stronger problems requires a stronger definition of common knowledge, and we study the relationship between these two definitions. In many cases, the two definitions are actually equivalent, and simple modifications of previous solutions yield round-optimal solutions to these problems. When the definitions differ, however, we show that such problems cannot be solved, even in failure-free executions.

Subcube Fault-Tolerance in Hypercubes

We consider the problem of determining the minimum number of faulty processors, K ( n , m ), and of faulty links, λ( n , m ), in an n -dimensional hypercube computer so that every m -dimensional subcube is faulty. Best known lower bounds for K ( n , m ) and λ( n , m ) are proved, several new recursive inequalities and new upper bounds are established, their asymptotic behavior for fixed m and for fixed n − m is analyzed, and their exact values are determined for small n and m . Most of the methods employed show how to construct sets of faults attaining the bounds. An extensive survey of related work is also included, showing connections to resource allocation, k -independent sets, and exhaustive testing.

Combinatorial analysis of the fault-diameter of the n-cube

It is shown that the diameter of an n-dimensional hypercube can only increase by an additive constant of 1 when (n-1) faulty processors are present. Based on the concept of forbidden faulty sets, which guarantees the connectivity of the cube in the presence of up to (2n-3) faulty processors. It is shown that the diameter of the n-cube increases to (n-2) as a result of (2n-3) processor failures. It is also shown that only those nodes whose Hamming distance is (n-2) have the potential to be located at two ends of the diameter of the damaged cube. It is proven that all the n-cubes with (2n-3) faulty processors and a fault-diameter of (n+2) are isomorphic. A generalization to the subject study is presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Multiprocessor fault diagnosis under local constraints

The authors study the fault diagnosis of multiprocessor systems when fault constraints in the local domain of each processor are specified. They use the comparison-based model. A multiprocessor system S is t-in-L diagnosable if, given a syndrome, all faulty processors can be uniquely identified provided there are at most t faulty processors in the local domain L(u/sub i/) union (u/sub i/) of every processor, u/sub i/ in S, where L (u/sub i/) denotes the set of processors adjacent to u/sub i/. Certain basic results that lead to efficient conditions for unique diagnosis of a system when certain fault constraints are satisfied in the local domain of each processor in the system are presented. The t-in-L diagnosability of certain regular interconnected systems is examined under the assumption that less than half of the processors in the system are faulty. Diagnosis algorithms for these systems are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Diagnosis and repair in multiprocessor systems

Diagnosis of multiprocessor systems in which faulty processors can be replaced by spares or repaired is known as sequential diagnosis. A generalization is considered of classical sequential diagnosis, referred to as diagnosis and repair, under a probabilistic model for the faults and test outcomes in a system. It is shown that correct diagnosis and repair of all faulty processors can be achieved with high probability in a large class of systems including, for example, rings, grids, meshes, tori, and hypercubes. These results show, without restrictive assumptions on the behavior of faulty processors, that correct diagnosis can be achieved in these widely used, low-degree systems when a fixed percentage of the processors in the system are faulty.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Diagnosability and diagnosis of algorithm-based fault-tolerant systems

Parallel processing architectures are commonly used for signal processing and other computationally intensive applications. These applications are characterized by high throughput and long processing periods. Such characteristics decrease the reliability of high-performance architectures. The erroneous data produced by faulty processors could have damaging consequences, particularly in critical real-time applications. It is therefore desirable that any erroneous data produced by the system be detected and located as quickly as possible. Algorithm-based fault tolerance (ABFT) is a low-cost system-level concurrent error detection and fault location scheme. Methods used in the analysis of multiprocessor systems using system-level diagnosis are applied to the analysis of ABFT systems. A new algorithm for analyzing an ABFT system for its fault diagnosability is developed using these methods. Based on this work, a fault diagnosis algorithm is developed for ABFT systems. >

Reconfiguring processor arrays using multiple-track models: the 3-track-1-spare-approach

Present new results on systematic procedures for reconfiguring processor arrays in the presence of faulty processors. In particular, the authors consider models that use multiple tracks along every channel and a single spare row (or column) of processing elements (PEs) along each boundary of the array. In the presence of faulty PEs the general methodology for reconfiguration involves replacing every faulty PE logically (rather than physically) by a spare PE through a sequence of logical substitutions; these sequences of substitutions are referred to as compensation paths. The authors show that if there exists a set of compensation paths subjected only to the constraints of continuity and nonintersection, then routing channels with three tracks are enough for the reconfiguration of the array. They refer to the underlying model as a S-track-l-spare model; this is done to distinguish it from other models that not only use multiple tracks but also multiple spare rows (or columns) along each boundary. An efficient algorithm for reconfiguration in our 3-track-1-spare model is presented and its performance evaluated. Experimental results show that the 3-track-1-spare model has much higher reconfiguration probability than other models that use considerably more spare processors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Design of self-diagnostic boards by multiple signature analysis

A new design approach, based on multiple signature analysis, for self-diagnostic boards is presented. For this approach, test responses from all chips on the board are compressed into space-time signatures using nonbinary multiple error-correcting codes, and faulty chips are identified by analyzing relations between distortions in these signatures. This approach results in a considerable reduction of a hardware overhead, required for diagnostics, as compared with the straightforward approach where separate signatures are computed for each chip on the board. The diagnostic approach presented can also be used for identification of faulty boards in a system or for faulty processors in a multiprocessor environment. >

On finding non-intersecting straightline connections of grid points to the boundary

We consider the problem of determining whether it is possible to connect a given set of N points in an ( m × n) rectangular 2D grid to the grid's boundary using N disjoint straight (horizontal or vertical) lines. If this is possible, we find such a set of lines. We provide an algorithm with either O( m + n) or O( N log N) complexity. In higher dimensions, the problem is NP-complete. We then extend our results to accommodate an additional constraint, namely forbidding connections in opposite directions that run next to one another. A solution to this problem can be used to provide a set of processor substitutions which reconfigure a fault-tolerant rectangular array of processing elements to avoid the faulty processors while retaining its important properties.