A bivalency proof of the lower bound for uniform consensus

Bivalency argument is a widely-used technique that employs forward induction to show impossibility results and lower bounds related to consensus. However, for a synchronous distributed system of n processes with up to t potential and f actual crash failures, applying bivalency argument to prove the lower bound for reaching uniform consensus is still an open problem. In this paper, we address this problem by presenting a bivalency proof that the lower bound for reaching uniform consensus is (f+2)-rounds where 0?f?t−2.     

A constructive proof for FLP

We present a simple elementary proof for the result of Fischer, Lynch, and Paterson (FLP) [J. ACM 32 (2) (April 1985) 374-382] that the consensus problem cannot be solved deterministically in an asynchronous system where a single process may fail by crashing. Our proof is, in contrast to the original, constructive in its crucial lemma, showing not only that a non-terminating execution does exist but also how it can be constructed. Our proof is based on the new notion of non-uniformity of a configuration. Non-uniformity is different from bivalency, which is the central notion in the original proof as well as in proofs of related results.     

Implementing uniform reliable broadcast with binary consensus in systems with fair-lossy links

When implementing multivalued consensus using binary consensus, previous algorithms assume the availability of uniform reliable broadcast, which is not implementable in systems with fair-lossy links. In this paper, we show that with binary consensus we can implement uniform reliable broadcast directly in systems with fair-lossy links, and thus the separate assumption of the availability of uniform reliable broadcast is not necessary. We further prove that, when binary consensus instances are available only as black boxes, any implementation of uniform reliable broadcast in the fair-lossy link model requires the invocation of an infinite number of binary consensus instances even if no process ever broadcasts any messages, and this is true even when multivalued consensus is used.     

Lower bounds for asynchronous consensus

Impossibility results and best-case lower bounds are proved for the number of message delays and the number of processes required to reach agreement in an asynchronous consensus algorithm that tolerates non-Byzantine failures. General algorithms exist that achieve these lower bounds in the normal case, when the response time of non-faulty processes and the transmission delay of messages they send to one another are bounded. Our theorems allow algorithms to do better in certain exceptional cases, and such algorithms are presented. Two of these exceptional algorithms may be of practical interest.     

Easy impossibility proofs for distributed consensus problems

Easy proofs are given, of the impossibility of solving several consensus problems (Byzantine agreement, weak agreement, Byzantine firing squad, approximate agreement and clock synchronization) in certain communication graphs.It is shown that, in the presence ofm faults, no solution to these problems exists for communication graphs with fewer than 3m+1 nodes or less than 2m+1 connectivity. While some of these results had previously been proved, the new proofs are much simpler, provide considerably more insight, apply to more general models of computation, and (particularly in the case of clock synchronization) significantly strengthen the results.Michael J. Fischer is currently Professor of Computer Science at Yale University, New Haven, CT, where he heads the Theory of Computation Group. He is also Editor-in-Chief of the Journal of the Association for Computing Machinery. His research interests include theory of distributed systems, cryptographic protocols, and computational complexity.Dr. Fischer received the B. S. degree in matheamtics from the University of Michigan, Ann Arbor, in 1963, and the M. A. and Ph. D. degrees in applied mathematics from Harvard University, Cambridge, MA, in 1965 and 1968, respectively. He has taught previously at Carnegie-Mellon University, the Massachusetts Institute of Technology, and University of Washington.Nancy Lynch is currently Associate professor of Computer Science at M.I.T., and heads the Theory of Distributed Systems group in M.I.T.'s Laboratory for Computer Science. Her interests are in all aspects of distributed computing theory, including formal models, algorithms, analysis, and correctness proofs. Dr. Lynch received the B.S. degree in mathematics from Brooklyn College in 1968 and the Ph. D. degree in mathematics from M.I.T. in 1972. She has served on the faculty of Tufts University, the University of Southern California, Florida International University, Georgia Tech.Michael Merritt is currently a member of the technical staff with AT&T Bell Laboratories. During the 1984 –85 academic year, he was a visiting lecturer at M.I.T., sponsered by Bell Labs. His research interests include distributed computation, cryptography and security. Dr. Merritt received the B. S. degree in computer science and philosophy from Yale in 1978 and the M. Sc. and Ph. D. degrees in 1980 and 1983, respectively, both in information and computer science from Georgia Tech. He is a member of SIGACT and of Computer Professionals for Social Responsibility.This paper has appeared in the ACM Conference Proceedings of PODC 1985. © 1985, Association for Computing Machinery, reprinted by permission     

On the complexity of global computation in the presence of link failures: the general case

Summary This paper presents (m logn) and (mn) messages lower bounds on the problem of computing a gobal sensitive function in biderectional networks with link failures (i.e., dynamically changing topology), wheren andm are the total number of nodes and links in the network. The (m logn) lower bound is under the assumption thatn is a-priori known to the nodes, while the second bound is for the case in which such knowledge is not available. A global sensitive function ofn variables is a function that may not be computed without the knowledge of the values of all then variables (e.g. maximum, sum, etc). Thus, computing such a function at one node of a distributed network requires this node to communicate with every other node in the network. Though lower bounds higher than (m) messages are known for this problem in the context of link failures, none holds for dense bidirectional networks. Moreover, we are not aware of any other nontrivial lower bound higher than (m) for dense bidirectional networks. Yehuda Afek received a B.Sc. in Electrical Engineering from the Technion and an M.Sc. and Ph.D. in Computer Science from the University of California, Los-Angeles. In 1985 he joined the Distributed Systems Research Department in AT&T Bell Laboratories as a Member of Technical Staff. In 1988 he joined the Computer Science Department in Tel-Aviv University, where he now holds a permanent position. From 1989 to 1994 he was also a consultant for AT&T Bell Laboratories. His interests include communication protocols, distributed computing and asynchronous shared memory systems. Danny Hendler was born in Kiryat-Haim near Haifa, Israel, on April 17th 1961. He received his B.Sc. and M.Sc. in Computer Science from Tel-Aviv University, Israel, in 1986 and 1993, respectively. In the past 8 years he has worked as a free lance software-consultant, specializing mainly in communication, telephony and voice-mail applications.     

The weakest failure detector to solve nonuniform consensus

We determine the weakest failure detector to solve nonuniform consensus in any environment, i.e., regardless of the number of faulty processes. Together with previous results, this closes all aspects of the following question: What is the weakest failure detector to solve (uniform or nonuniform) consensus in any environment?     

Communication-efficient failure detection and consensus in omission environments

Failure detectors have been shown to be a very useful mechanism to solve the consensus problem in the crash failure model, for which a number of communication-efficient algorithms have been proposed. In this paper we deal with the definition, implementation and use of communication-efficient failure detectors in the general omission failure model, where processes can fail by crashing and by omitting messages when sending and/or receiving. We first define a new failure detector class for this model in terms of completeness and accuracy properties. Then we propose an algorithm that implements a failure detector of the proposed class in a communication-efficient way, in the sense that only a linear number of links are used to send messages forever. We also explain how the well-known consensus algorithm of Chandra and Toueg can be adapted in order to use the proposed failure detector.     

A simple and communication-efficient Omega algorithm in the crash-recovery model

This paper presents a new algorithm implementing the Omega failure detector in the crash-recovery model. Contrary to previously proposed algorithms, this algorithm does not rely on the use of stable storage and is communication-efficient, i.e., eventually only one process (the elected leader) keeps sending messages. The algorithm relies on a nondecreasing local clock associated with each process. Since stable storage is not used to keep the identity of the leader in order to read it upon recovery, unstable processes, i.e., those that crash and recover infinitely often, output a special ⊥ value upon recovery, and then agree with correct processes on the leader after receiving a first message from it.     

Bounded cost algorithms for multivalued consensus using binary consensus instances

In this paper, we present two bounded cost algorithms that solve multivalued consensus using binary consensus instances. Our first algorithm uses log₂n number of binary consensus instances where n is the number of processes, while our second algorithm uses at most binary consensus instances, where is the maximum length of the binary representation of all proposed values in the run. Both algorithms are significant improvements over the previous algorithm in [A. Mostefaoui, M. Raynal, F. Tronel, From binary consensus to multivalued consensus in asynchronous message-passing systems, Information Processing Letters 73 (5–6) (2000) 207–212], where the number of binary consensus instances needed to solve one multivalued consensus is unbounded.     

A new lower bound for curriculum-based course timetabling

In this paper, we propose a new method to compute lower bounds for curriculum-based course timetabling (CTT), which calls for the best weekly assignment of university course lectures to rooms and time slots. The lower bound is obtained by splitting the objective function into two parts, considering one separate problem for each part of the objective function, and summing up the corresponding optimal values (or, in some cases, lower bounds on these values), found by formulating the two parts as Integer Linear Programs (ILPs). The solution of one ILP is obtained by using a column generation procedure. Experimental results show that the proposed lower bound is often better than the ones found by the previous methods in the literature, and also much better than those found by other new ILP formulations illustrated in this paper. The proposed approach is able to obtain improved lower bounds on real-world benchmark instances from the literature, used in the international timetabling competitions ITC2002 and ITC2007, proving for the first time that some of the best-known heuristic solutions are indeed optimal (or close to the optimal ones).     

Fast and simple distributed consensus

Summary The problem of fault-tolerant agreement is fundamental to distributed computing. When agreement is to be reached in spite of arbitrary behavior by faulty processors, this problem is calledDistributed Consensus. By requiring that the number of faulty processors be , wheren is the number of processors in the system, we are able to derive two new protocols forDistributed Consensus. Both are simple and use messages that are only one bit in length, and both provide forearly stopping: the fewer failures there are, the fewer rounds of communication are required. One protocol is optimal with respect to the number of rounds of communication required, and the other is asymptotically optimal with respect to the total number of message bits exchanged. James E. Burns received the B.S. degree in mathematics from the California Institute of Technology, the M.B.I.S. degree from Georgia State University, and the M.S. and Ph.D. degrees in information and computer science from the Georgia Institute of Technology. He served on the faculty of Computer Science at Indiana University and the College of Computing at the Georgia Institute of Technology before joining Bellcore in 1993. He is currently a Member of Technical Staff in the Network Control Research Department, where he is studying the telephone control network with special interest in behavior when faults occur. He also has research interests in theoretical issues of distributed and parallel computing, especially relating to problems of synchronization and fault tolerance. Gil Neiger was born on February 19, 1957 in New York, New York. In June 1979, he received an A.B. in Mathematics and Psycholinguistics from Brown University in Proidence, Rhode Island. In February 1985, he spent two weeks picking cotton in Nicaragua in a brigade of international volunteers. In January 1986, he received an M.S. in Computer Science from Cornell University in Ithaca, New York and, in August 1988, he received a Ph.D. in Computer Science, also from Cornell University. On August 20, 1988, Dr. Neiger married Hilary Lombard in Lansing, New York. Since August 1988, he has been an Assistant Professor in the College of Computing (formely School of Information and Computer Science) at the Georgia Institute of Technology in Atlanta, Georgia. Dr. Neiger is a member of the editorial board of theChicago Journal of Theoretical Computer Science and theJournal of Parallel and Distributed Computing.This author was supported in part by the National Science Foundation under grants CCR-8909663, CCR-9106627, and CCR-9301454.     

A tight lower bound for job scheduling with cancellation

The Job Scheduling with Cancellation problem is a variation of classical scheduling problems in which jobs can be cancelled while waiting for execution. In this paper we prove a tight lower bound of 5 for the competitive ratio of any deterministic online algorithm for this problem, for the case where all jobs have the same processing time.     

Using asynchrony and zero degradation to speed up indulgent consensus protocols

Existing consensus protocols suffer from slowdowns caused by the failures of processes and the mistakes made by the underlying oracles. In this paper, we propose two novel techniques to circumvent such slowdowns in failure-detector-based consensus protocols. The first technique guarantees the Round-Zero-Degradation (RZD) property (an extension of the Zero-Degradation property) in order to avoid the slowdown caused by a failed coordinator process. The second technique, named “Look-Ahead”, helps speed up the execution of the consensus protocol by making use of the messages delivered before their receivers enter the corresponding phases or rounds. The first technique is effective only when the underlying failure detector makes no or few mistakes, while the second technique always works well regardless of the performance of the failure detector. Moreover, Look-Ahead is a general technique and can be applied to consensus protocols based on any kind of oracle. By applying the two proposed techniques, several consensus protocols are developed. The simulation results show that the RZD technique is effective even if the error rate of the failure detector reaches about 15%, while the Look-Ahead technique can always improve the performance in all cases.     

On Byzantine generals with alternative plans

This paper proposes a variation of the Byzantine generals problem (or Byzantine consensus). Each general has a set of good plans and a set of bad plans. The problem is to make all loyal generals agree on a good plan proposed by a loyal general, and never on a bad plan.     

New lower bound for the Capacitated Arc Routing Problem

We present a new lower bound, the Multiple Cuts Node Duplication Lower Bound, for the undirected Capacitated Arc Routing Problem. We prove that this new bound dominates the existing bounds for the problem. Computational results are also provided.     

The complexity of synchronous iterative Do-All with crashes

The ability to cooperate on common tasks in a distributed setting is key to solving a broad range of computation problems ranging from distributed search such as SETI to distributed simulation and multi-agent collaboration. Do-All, an abstraction of such cooperative activity, is the problem of performing N tasks in a distributed system of P failure-prone processors. Many distributed and parallel algorithms have been developed for this problem and several algorithm simulations have been developed by iterating Do-All algorithms. The efficiency of the solutions for Do-All is measured in terms of work complexity where all processing steps taken by all processors are counted. Work is ideally expressed as a function of N, P, and f, the number of processor crashes. However the known lower bounds and the upper bounds for extant algorithms do not adequately show how work depends on f. We present the first non-trivial lower bounds for Do-All that capture the dependence of work on N, P and f. For the model of computation where processors are able to make perfect load-balancing decisions locally, we also present matching upper bounds. We define the r-iterative Do-All problem that abstracts the repeated use of Do-All such as found in typical algorithm simulations. Our f-sensitive analysis enables us to derive tight bounds for r-iterative Do-All work (that are stronger than the r-fold work complexity of a single Do-All). Our approach that models perfect load-balancing allows for the analysis of specific algorithms to be divided into two parts: (i) the analysis of the cost of tolerating failures while performing work under free load-balancing, and (ii) the analysis of the cost of implementing load-balancing. We demonstrate the utility and generality of this approach by improving the analysis of two known efficient algorithms. We give an improved analysis of an efficient message-passing algorithm. We also derive a tight and complete analysis of the best known Do-All algorithm for the synchronous shared-memory model. Finally we present a new upper bound on simulations of synchronous shared-memory algorithms on crash-prone processors.Received: 15 May 2002, Accepted: 15 June 2003, Published online: 6 February 2004This work is supported in part by the NSF TOC Grants 9988304 and 0311368, and the NSF ITR Grant 0121277. The work of the second author is supported in part by the NSF CAREER Award 0093065. The work of the third author is supported in part by the NSF CAREER Award 9984778.     

Transaction commit in a realistic timing model

Animportant problem in the construction of fault-tolerant distributed database systems is the design of nonblocking transaction commit protocols. This problem has been extensively studied for synchronous systems (i.e., systems where no messages ever arrive late). In this paper, the synchrony assumption is relaxed. A new partially synchronous timing model is described. Developed for this model is a new nonblocking randomized transaction commit protocol, which incorporates an agreement protocol of Ben-Or. The new protocol works as long as fewer than half the processors fail. A matching lower bound is proved, showing that the number of processor faults tolerated is optimal. If half or more of the processors fail, the protocol degrades gracefully: it blocks, but no processor produces a wrong answer. A notion of asynchronous round is defined, and the protocol is shown to terminate in a small constant expected number of asynchronous rounds. In contrast it is shown that no protocol in this model can guarantee that a processor terminates in a bounded expected number of its own steps, even if processors are synchronous. Brian A. Coan received the B.S.E. degree in electrical engineering and computer science from Princeton University, Princeton, New Jersey, in 1977; the M.S. degree in computer engineering from Stanford University, Stanford, California, in 1979; and the Ph.D. degree in computer science from the Massachusetts Institute of Technology, Cambridge, Massachusetts, in 1987. He has worked for Amdahl Corporation and AT & T Bell Laboratories. Currently he is a member of the technical staff at Bellcore. His main research interest is fault tolerance in distributed systems. Jennifer Lundelius Welch received her B.A. in 1979 from the University of Texas at Austin, and her S.M. and Ph.D. from the Massachusetts Institute of Technology in 1984 and 1988 respecively. She was a member of technical staff at GTE Laboratories Incorporated in Waltham, Massachusetts, from 1988 to 1989. She is currently an assistant professor at the University of North Carolina in Chapel Hill. Her research interests include algorithms and lower bounds for distributed computing.The authors were with the MIT Laboratory for Computer Science when the bulk of this work was done. This work was supported in part by the Advanced Research Projects Agency of the Department of Defense under Contract N00014-83-K-0125, the National Science Foundation under Grant DCR-83-02391, the Office of Army Research under Contract DAAG29-84-K-0058, and the Office of Naval Research under Contract N00014-85-K-0168. A preliminary version of this paper appears in theProceedings of the Fifth Annual ACM Symposium on Principles of Distributed Computing [2]     

A weakest failure detector-based asynchronous consensus protocol for f<n

has been shown to be the weakest realistic failure detector class needed to solve the consensus problem in an asynchronous distributed system prone to f<n process crashes in which communication is by message-passing. However, the only protocol that is known to meet this bound is based on three layers of protocol construction, and is therefore not efficient. This paper presents a surprisingly simple and very efficient direct message-passing implementation of a -based consensus protocol, and proves its correctness.