## Leader Election in Anonymous Rings: Franklin Goes Probabilistic

### BibTeX

@MISC{Bakhshi_leaderelection,

author = {Rena Bakhshi and Wan Fokkink and Jun Pang and Jaco Van De Pol},

title = {Leader Election in Anonymous Rings: Franklin Goes Probabilistic},

year = {}

}

### OpenURL

### Abstract

Abstract. We present a probabilistic leader election algorithm for anonymous, bidirectional, asynchronous rings. It is based on an algorithm from Franklin [22], augmented with random identity selection, hop counters to detect identity clashes, and round numbers modulo 2. As a result, the algorithm is finite-state, so that various model checking techniques can be employed to verify its correctness, that is, eventually a unique leader is elected with probability one. We also sketch a formal correctness proof of the algorithm for rings with arbitrary size. 1

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Citation Context ...em prover Confcheck [43] from µCRL. We also experimented with an on-the-fly τ-reduction [10, 39]. It is based on Tarjan’s algorithm for decomposition of a graph into its strongly connected components =-=[45]-=-. In this reduction, for each state a representative state is computed, which it can reach by means of confluent τ-transitions. To compute the representative of a state, a depth-first search traversal... |

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Citation Context ... instance the case in the leader election algorithm used within the IEEE 1394 (FireWire) standard, see [38]. In a so-called anonymous (or uniform) network, processes do not carry an identity. Angluin =-=[1]-=- showed that there does not exist a terminating deterministic algorithm for electing a leader in an anonymous, asynchronous network. Itai and Rodeh [32, 33] studied how to break the symmetry in anonym... |

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Citation Context ...s essential; see e.g. [46, Sect. 9.4.1].Leader Election in Anonymous Rings: Franklin Goes Probabilistic 3 algorithm. They analysed the resulting algorithm using the probabilistic model checker PRISM =-=[29]-=-. In Sect. 2, we present a probabilistic leader election algorithm for anonymous bidirectional rings, based on Franklin’s algorithm. As in the Itai-Rodeh algorithm, it is assumed that all processes kn... |

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Citation Context ...a leader oracle Ω?, which for some point onwards is guaranteed to return the same leader to all processes (see also Sect. 7).4 Rena Bakhshi, Wan Fokkink, Jun Pang, and Jaco van de Pol Several papers =-=[11, 30, 34, 18]-=- present leader election algorithms for anonymous rings of prime size, in the presence of a central demon, which acts as a scheduler. 3 Leader election is related to token circulation for solving mutu... |

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Citation Context ...d version of the µCRL toolset [7] to store the generated state space over a cluster of computers. Moreover, we sketched a formal correctness proof for the algorithm in Sect. 4. The model checker CADP =-=[17]-=- provided counter-examples to show that: (1) round numbers cannot be omitted from the probabilistic Franklin algorithm altogether (see Sect. 3), and (2) in case of a probabilistic version of the Dolev... |

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Citation Context ...ork, processes do not carry an identity. Angluin [1] showed that there does not exist a terminating deterministic algorithm for electing a leader in an anonymous, asynchronous network. Itai and Rodeh =-=[32, 33]-=- studied how to break the symmetry in anonymous networks using probabilistic algorithms. They presented a probabilistic algorithm, based on the Chang-Roberts algorithm, to elect a leader in an anonymo... |

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Citation Context ...ariables model. Both algorithms are based on token circulation. Several other papers present self-stabilizing token circulation algorithms for anonymous, unidirectional rings: the algorithm of Herman =-=[27]-=- works on synchronous rings of odd size; Duchon, Hanusse and Tixeuil [16] present algorithms for synchronous rings of arbitrary size; Beauquier, Gradinariu and Johnen [4] and Datta, Gradinariu and Tix... |

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Citation Context ...s, and in return it receives such messages from these neighbours. An active process only progresses to the next election round if its own identity is larger than the two incoming identities. Peterson =-=[42]-=- and Dolev, Klawe and Rodeh [15] independently adapted Franklin’s algorithm for unidirectional rings. Sometimes the processes in a network cannot be distinguished by means of unique identities. Firstl... |

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Citation Context ...er, Ofek, Ostrovsky and Yung [40] show that on an anonymous ring, leader election is equivalent to providing a self-stabilizing round-robin token management scheme. Angluin, Aspnes, Fischer and Jiang =-=[2]-=- construct a self-stabilizing leader election algorithm for anonymous, unidirectional, asynchronous rings of odd size in the framework of their model of population protocols. Beauquier, Gradinariu and... |

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Citation Context ...ies) with the “normal” state space generation strategy.10 Rena Bakhshi, Wan Fokkink, Jun Pang, and Jaco van de Pol For more efficient state space generation, we applied symbolic confluence reduction =-=[10]-=-. To this end, a theorem prover can be used to automatically detect and mark confluent τ’s, i.e. internal transitions and hidden communications that are not causally related, for instance, because the... |

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Citation Context ...nt a leader election algorithm for anonymous, unidirectional rings of known size; their algorithm is similar to the algorithm of Itai and Rodeh, augmented with a time-out mechanism. Fischer and Jiang =-=[19]-=- give a self-stabilizing leader election algorithm for anonymous, unidirectional rings, based on a leader oracle Ω?, which for some point onwards is guaranteed to return the same leader to all process... |

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Citation Context ...a leader oracle Ω?, which for some point onwards is guaranteed to return the same leader to all processes (see also Sect. 7).4 Rena Bakhshi, Wan Fokkink, Jun Pang, and Jaco van de Pol Several papers =-=[11, 30, 34, 18]-=- present leader election algorithms for anonymous rings of prime size, in the presence of a central demon, which acts as a scheduler. 3 Leader election is related to token circulation for solving mutu... |

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Citation Context ...erators are eliminated by the linearisation algorithm from [25]. Next it is symbolically reduced by static analysis: constant propagation (replace provably constant parameters by their initial value) =-=[24]-=-, and dead variable analysis (reset variables that are not used anymore to a default value). The number of states and transitions of the state space that have been generated in this way are presented ... |

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Citation Context ...er. This reduction keeps only the confluent τ’s going out of a state, and all the other transitions going out of the state are removed. This symbolic prioritization is implemented in the Confelm tool =-=[6]-=- from µCRL. We used it to remove confluent τ-summands, marked by the theorem prover Confcheck [43] from µCRL. We also experimented with an on-the-fly τ-reduction [10, 39]. It is based on Tarjan’s algo... |

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Citation Context ...s on synchronous rings of odd size; Duchon, Hanusse and Tixeuil [16] present algorithms for synchronous rings of arbitrary size; Beauquier, Gradinariu and Johnen [4] and Datta, Gradinariu and Tixeuil =-=[13]-=- use several types of tokens and assume the synchronous communication model of shared variables, while the algorithm of Rosaz [44] uses the same idea in asynchronous message passing systems; the algor... |

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Citation Context ...Computer Science j.c.vandepol@ewi.utwente.nl Abstract. We present a probabilistic leader election algorithm for anonymous, bidirectional, asynchronous rings. It is based on an algorithm from Franklin =-=[22]-=-, augmented with random identity selection, hop counters to detect identity clashes, and round numbers modulo 2. As a result, the algorithm is finite-state, so that various model checking techniques c... |

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Citation Context ...tity v again, and sends messages (v, 1) in both direction (Fig. 3(c)). Upon the receipt of messages (v, 1) and (w, 1), the processes with identity w become passive (Fig. 3(d)). Finally, the 4 Lamport =-=[37]-=- actually advocates that all distributed algorithms should be model checked before publication.Leader Election in Anonymous Rings: Franklin Goes Probabilistic 7 outdated messages (u, 1) make the proc... |

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Citation Context ...odel of shared variables, while the algorithm of Rosaz [44] uses the same idea in asynchronous message passing systems; the algorithms of Kakugawa and Yamashita [36] for asynchronous rings and Johnen =-=[35]-=- for shared memory settings run under unfair distributed schedulers. Of these papers, [4, 36, 35] require knowledge of ring size. Mayer, Ostrovsky and Yung [41] give a randomized compiler for anonymou... |

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Citation Context ...tions are not abstracted away), while maintaining the branching structure of the state space. In case of the distributed version of the µCRL toolset, we applied a distributed reduction algorithm from =-=[9]-=-. (a) State space for two identities # Procs States Transitions Table 1. State space generation statistics 2 657 1,368 3 15,445 43,968 4 380,609 1,396,512 5 9,819,065 44,242,920 6 260,753,105 1,393,96... |

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Citation Context ...h messages from these neighbours. An active process only progresses to the next election round if its own identity is larger than the two incoming identities. Peterson [42] and Dolev, Klawe and Rodeh =-=[15]-=- independently adapted Franklin’s algorithm for unidirectional rings. Sometimes the processes in a network cannot be distinguished by means of unique identities. Firstly, there is no concept of identi... |

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Citation Context ... assume the synchronous communication model of shared variables, while the algorithm of Rosaz [44] uses the same idea in asynchronous message passing systems; the algorithms of Kakugawa and Yamashita =-=[36]-=- for asynchronous rings and Johnen [35] for shared memory settings run under unfair distributed schedulers. Of these papers, [4, 36, 35] require knowledge of ring size. Mayer, Ostrovsky and Yung [41] ... |

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Citation Context ...eauquier, Gradinariu and Johnen [4] and Datta, Gradinariu and Tixeuil [13] use several types of tokens and assume the synchronous communication model of shared variables, while the algorithm of Rosaz =-=[44]-=- uses the same idea in asynchronous message passing systems; the algorithms of Kakugawa and Yamashita [36] for asynchronous rings and Johnen [35] for shared memory settings run under unfair distribute... |

9 |
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Citation Context ...is implemented in the Confelm tool [6] from µCRL. We used it to remove confluent τ-summands, marked by the theorem prover Confcheck [43] from µCRL. We also experimented with an on-the-fly τ-reduction =-=[10, 39]-=-. It is based on Tarjan’s algorithm for decomposition of a graph into its strongly connected components [45]. In this reduction, for each state a representative state is computed, which it can reach b... |

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Citation Context ... [36] for asynchronous rings and Johnen [35] for shared memory settings run under unfair distributed schedulers. Of these papers, [4, 36, 35] require knowledge of ring size. Mayer, Ostrovsky and Yung =-=[41]-=- give a randomized compiler for anonymous rings that transforms a self-stabilizing algorithm based on bidirectional communication to one that requires unidirectional, synchronous communication. 2 Fran... |

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Citation Context |

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Citation Context ...h fresh identities are chosen at the start of each election round, and one in which fresh identities are only chosen at the detection of an identity clash (see Sect. 6). Related Work Higham and Myers =-=[28]-=- present a leader election algorithm for anonymous, unidirectional rings of known size; their algorithm is similar to the algorithm of Itai and Rodeh, augmented with a time-out mechanism. Fischer and ... |

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Citation Context ...d. To overcome this problem, successive election rounds are numbered, and each process and message is supplied with a round number. Thus an old message can be recognized and ignored. Fokkink and Pang =-=[20, 21]-=- showed that in case of FIFO channels, round numbers can be omitted from the Itai-Rodeh 2 The latter assumption is essential; see e.g. [46, Sect. 9.4.1].Leader Election in Anonymous Rings: Franklin G... |

6 |
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Citation Context ...ther papers present self-stabilizing token circulation algorithms for anonymous, unidirectional rings: the algorithm of Herman [27] works on synchronous rings of odd size; Duchon, Hanusse and Tixeuil =-=[16]-=- present algorithms for synchronous rings of arbitrary size; Beauquier, Gradinariu and Johnen [4] and Datta, Gradinariu and Tixeuil [13] use several types of tokens and assume the synchronous communic... |

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Citation Context ...ies cannot always be sent around the network, for instance for reasons of efficiency; this is for instance the case in the leader election algorithm used within the IEEE 1394 (FireWire) standard, see =-=[38]-=-. In a so-called anonymous (or uniform) network, processes do not carry an identity. Angluin [1] showed that there does not exist a terminating deterministic algorithm for electing a leader in an anon... |

5 |
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Citation Context ...elected. For ring size five, in case of a domain of three process identities, and for ring size six, in case of a domain of two process identities, we used the distributed version of the µCRL toolset =-=[7]-=- to store the generated state space over a cluster of computers. Moreover, we sketched a formal correctness proof for the algorithm in Sect. 4. The model checker CADP [17] provided counter-examples to... |

4 | Variations on Itai-Rodeh leader election for anonymous rings and their analysis in PRISM
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(Show Context)
Citation Context ...d. To overcome this problem, successive election rounds are numbered, and each process and message is supplied with a round number. Thus an old message can be recognized and ignored. Fokkink and Pang =-=[20, 21]-=- showed that in case of FIFO channels, round numbers can be omitted from the Itai-Rodeh 2 The latter assumption is essential; see e.g. [46, Sect. 9.4.1].Leader Election in Anonymous Rings: Franklin G... |

2 |
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Citation Context ...ns going out of the state are removed. This symbolic prioritization is implemented in the Confelm tool [6] from µCRL. We used it to remove confluent τ-summands, marked by the theorem prover Confcheck =-=[43]-=- from µCRL. We also experimented with an on-the-fly τ-reduction [10, 39]. It is based on Tarjan’s algorithm for decomposition of a graph into its strongly connected components [45]. In this reduction,... |

1 |
de Pol. µCRL specification of probabilistic Franklin leader election algorithm. http://www.few.vu.nl/~rbakhshi/ alg/franklin.mcrl
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Citation Context ...nd its correctness has been verified for rings with a size up to six processes. The input language for µCRL is based on process algebra and abstract data types. Our µCRL specification is available at =-=[3]-=-. Tables 1(a) and 1(b) provide state space generation results for domains of two and three identities, respectively. To carry out the verification for six processes (in case of two identities) and fiv... |