## Randomized registers and iterative algorithms (2005)

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Venue: | Distributed Computing |

Citations: | 4 - 1 self |

### BibTeX

@ARTICLE{Lee05randomizedregisters,

author = {Hyunyoung Lee and Jennifer L. Welch},

title = {Randomized registers and iterative algorithms},

journal = {Distributed Computing},

year = {2005},

volume = {17},

pages = {2005}

}

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### Abstract

We present three different specifications of a read-write register that may occasionally return out-of-date values — namely, a (basic) random register, a P-random register, and a monotone random register. We show that these specifications are implemented by the probabilistic quorum algorithm of Malkhi, Reiter, Wool, and Wright, and we illustrate how to program with such registers in the framework of Bertsekas, using the notation of Üresin and Dubois. Consequently, existing iterative algorithms for a significant class of problems (including solving systems of linear equations, finding shortest paths, constraint satisfaction, and transitive closure) will converge with high probability if executed in a system in which the shared data is implemented with registers satisfying the new specifications. Furthermore, the algorithms in this framework will inherit positive attributes concerning load and fault-tolerance from the underlying register implementation. The expected convergence time for iterative algorithms using the monotone implementation is analyzed and shown experimentally to improve on that of the original implementation. The message complexity for iterative algorithms using the monotone probabilistic quorum implementation is shown to improve on that of non-probabilistic implementations in a quantifiable situation.

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Citation Context ... Related Work A number of consistency conditions for shared memory have been proposed over the years, including safety, regularity and atomicity [14, 15], sequential consistency [13], linearizability =-=[11]-=-, causal consistency [3] and hybrid consistency [5]. These definitions have all been deterministic with little or no regard to the possibility of incorrect return values. Afek et al. [2] and Jayanti e... |

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Citation Context ...ition is a probabilistic variation on the concept of regularity from Lamport’s paper [15]. We show that our definition of a random register can be implemented by the probabilistic quorum algorithm of =-=[19, 18]-=-, which has the advantages mentioned above, that the load on the busiest replica server is limited and the fault-tolerance in the face of server crashes is high. Next we show how registers satisfying ... |

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Citation Context ... propose a formal definition of a basic random read-write register. The consistency condition provided by our definition is a probabilistic variation on the concept of regularity from Lamport’s paper =-=[15]-=-. We show that our definition of a random register can be implemented by the probabilistic quorum algorithm of [19, 18], which has the advantages mentioned above, that the load on the busiest replica ... |

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Citation Context ...ead of producing a wrong answer, the difficulty in specifying the shared object is lessened, since any values returned will satisfy the deterministic specification. Examples of this situation include =-=[24, 23, 9]-=-, discussed below. Shavit and Zemach have implemented novel randomized synchronization mechanisms called combining funnels [24] and diffracting trees [23] over simpler shared objects. In these algorit... |

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Citation Context ...s limited and the fault-tolerance in the face of server crashes is high. Next we show how registers satisfying our definition can be used to program iterative algorithms in the framework of Bertsekas =-=[6]-=-, presented by Üresin and Dubois [27]. The implication is that we can use existing iterative algorithms for a significant class of problems (including solving systems of linear equations, finding shor... |

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Citation Context ...of further research. 2 Related Work A number of consistency conditions for shared memory have been proposed over the years, including safety, regularity and atomicity [14, 15], sequential consistency =-=[13]-=-, linearizability [11], causal consistency [3] and hybrid consistency [5]. These definitions have all been deterministic with little or no regard to the possibility of incorrect return values. Afek et... |

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Citation Context ... consistency conditions for shared memory have been proposed over the years, including safety, regularity and atomicity [14, 15], sequential consistency [13], linearizability [11], causal consistency =-=[3]-=- and hybrid consistency [5]. These definitions have all been deterministic with little or no regard to the possibility of incorrect return values. Afek et al. [2] and Jayanti et al. [12] have studied ... |

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Citation Context ...earizability [11], causal consistency [3] and hybrid consistency [5]. These definitions have all been deterministic with little or no regard to the possibility of incorrect return values. Afek et al. =-=[2]-=- and Jayanti et al. [12] have studied a shared memory model in which a fixed set of the shared objects might return incorrect values, while the others never do. This model differs from the one we are ... |

16 |
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(Show Context)
Citation Context ... the face of server crashes is high. Next we show how registers satisfying our definition can be used to program iterative algorithms in the framework of Bertsekas [6], presented by Üresin and Dubois =-=[27]-=-. The implication is that we can use existing iterative algorithms for a significant class of problems (including solving systems of linear equations, finding shortest paths, constraint satisfaction, ... |

10 | Overcoming the Majority Barrier in Large-Scale Systems
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(Show Context)
Citation Context ...load of Θ √n has only 2 One mechanism would be to use a failure detector to eliminate all faulty servers, and then choose a quorum at random among all quorums that do not contain a faulty server. See =-=[29]-=- for another approach to finding a live quorum. 11sO ( √ n) fault-tolerance. Malkhi et al. [19] showed that using probabilistic quorums breaks � this � 1 trade-off and achieves simultaneously high fau... |

9 |
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(Show Context)
Citation Context ...ead of producing a wrong answer, the difficulty in specifying the shared object is lessened, since any values returned will satisfy the deterministic specification. Examples of this situation include =-=[24, 23, 9]-=-, discussed below. Shavit and Zemach have implemented novel randomized synchronization mechanisms called combining funnels [24] and diffracting trees [23] over simpler shared objects. In these algorit... |

8 |
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(Show Context)
Citation Context ... compares each process’s local copy of the row for which that process is responsible, against the precomputed correct answer for that row. The simulation completes when each comparison is equal. (Cf. =-=[7, 28]-=- for discussions of the issues involved in detecting termination for iterative algorithms.) For each of the four combinations of monotone/non-monotone and synchronous/asynchronous, seven runs of the s... |

6 | Simulating Shared Memory in Real Time: On the Computation Power of Reconfigurable Architectures
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(Show Context)
Citation Context ...ead of producing a wrong answer, the difficulty in specifying the shared object is lessened, since any values returned will satisfy the deterministic specification. Examples of this situation include =-=[24, 23, 9]-=-, discussed below. Shavit and Zemach have implemented novel randomized synchronization mechanisms called combining funnels [24] and diffracting trees [23] over simpler shared objects. In these algorit... |

3 |
Yet Another Application of Binomial Recurrence
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(Show Context)
Citation Context ...andom variable X = max(Y1,1, Y1,2, . . . , Yp,m). Essentially the same problem of computing the expectation of the maximum of a number of 21sgeometrically distributed random variables, was studied in =-=[26]-=-, resulting in the expression given in the statement of the theorem. For completeness, we give a more elementary derivation here. We have E[X] = � ∞ r=0 Pr[X > r] since X is a non-negative integer-val... |

2 |
J.L.: Randomized shared queues applied to distributed optimization algorithms
- Lee, Welch
- 2001
(Show Context)
Citation Context ... blocks in stronger register implementations. It would be interesting to investigate how to define, implement, and use other randomized data types. For example, randomized queues have been studied in =-=[17]-=-. This paper has addressed the fault-tolerance of replica servers for applications running on top of quorum implementations for shared data. In contrast, the issue of fault-tolerance of clients for as... |

1 |
Programs to compute the probability distribution of outdatedness for the probabilistic quorum implementation. Available from http://www.cs.du.edu/~hlee/Research/Programs
- Lee
(Show Context)
Citation Context ...ity of being ℓ-outdated. We do not have a closed form to compute p(ℓ). However, the preceding analysis gives an algorithm to compute p(ℓ), the implementation of which can be found and downloaded from =-=[16]-=-. We ran the program in [16] with several sets of parameters. The plot in Figure 2 shows the result of the executions of the program with 34 replicas. Up to 34 outdatedness was computed for each of th... |