J. Beauquier, J. Durand-Lose, M. Gradinariu, and C. Johnen. Token based self-stabilizing uniform algorithms. Journal of Parallel and Distributed Computing, 62(5):899-921, 2002.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....with probability 1. Proving convergence of randomized self stabilizing algorithms has always been done so far using the same basic principle: one exhibits a measure over the set of con gurations that decreases with non null probability at each step of execution until L is reached (see, e.g. [1,5,7]) Finding such a measure often requires however a deep knowledge of the algorithm under study. We propose here instead a new and general method that is inspired by concepts used in statistical physics. In statistical physics, many systems can be regarded indeed as self stabilizing randomized ....

J. Beauquier, J. Durand-Lose, M. Gradinariu, and C. Johnen. Token based selfstabilizing uniform algorithms. J. Parallel and Distributed Computing, 62(5), 2002.

....is a mechanism that selects one enabled process at each step. The distributed system corresponds to repeated application of transition rules according to the philosopher chosen by the scheduler at each step. Given a scheduler A, we are interested in proving the following convergence property (see [7, 2]) No matter which initial con guration x 0 one starts from, the probability that under A reaches a legitimate con guration in a nite number of transitions is 1. This will be written: P r(x 0 L) See [4] for a formal de nition. 2.2. Lehmann Rabin s algorithm We present Lehmann Rabin s ....

....holds with no fairness assumption on the scheduler, i.e. Theorem 1 For any arbitrary scheduler A and every x 2 fH; W; S; Dg P r(x ) 1. 4. Convergence of R 4.1. Scheme of the proof We are going to prove Theorem 1 by using the following property proved in [4] cf Theorem 1 of [2] and Theorem 5 of [3] Theorem 2 Suppose that there exists a measure and an ordering such that: 8x 62 L 8i 2 E(x) 9x (x ( x ) x) x ) Then, for any (central) scheduler A: 8x P r(x ) 1. More precisely, we will nd an appropriate rewriting strategy for ....

J. Beauquier, J. Durand-Lose, M. Gradinariu, and C. Johnen. Token based selfstabilizing uniform algorithms. J. of Parallel and Distributed Systems, To appear.

....L E(x) 6= It is easy to show that Lehmann Rabin s algorithm satis es the no deadlock property on non legitimate con gurations. Actually, for every con guration x 62 L, each process P i is always enabled. Given a scheduler A, we are interested in proving the following convergence property (see [6,2]) No matter which initial (non legitimate) con guration one starts from, the probability that under A reaches a legitimate con guration in a nite number of transitions is 1. Formally, let T L be the tree constructed from T (A; x 0 ) by cutting the edges going out of vertices corresponding to ....

....Theorem 1. Suppose that there exist a measure and an ordering such that Prop (R; L) 8x 62 L 8i 2 E(x) 9x 0 (x i R x 0 ( x 0 ) x) x 0 2 L) Then, for any central scheduler A: 8x P r(x A R L) 1. Theorem 1 can be seen as a special version of Theorem 1 of [2] (see also theorem 5 of [4] In the next section, we will show how (a variant of) Lehmann Rabin s algorithm satis es Prop . More precisely, we will nd an appropriate rewriting strategy for probabilistic rule R0 (i.e. a xed choice of the new letter W or W , depending on the context of ....

J. Beauquier, J. Durand-Lose, M. Gradinariu, and C. Johnen. Token based self-stabilizing uniform algorithms. J. of Parallel and Distributed Systems, To appear.

.... often means that there exists always at least one token (8x (x) 1) The set L of legitimate con gurations then actually corresponds to con gurations with exactly one token (i.e. x 2 L i (x) c = 1) Given a scheduler A, we are interested in proving the following convergence property (see [13,3]) No matter which initial con guration one starts from, the probability that under A reaches a legitimate con guration in a nite number of transitions is 1. Formally, let T L be the tree constructed from T (A; x 0 ) by cutting the edges going out of vertices corresponding to legitimate ....

....(x i y ( x) y) D(x) D(y) Then, for any central scheduler A: 8x P r(x A L) 1. The proof is given in appendix A. The result extends to the distributed case in the natural way (by replacing E(x) with 2 E(x) Theorem 10 can be seen as a restricted version of Theorem 1 of [3] (see also theorem 5 of [7] In [8] we prove the convergence of Kakugawa Yamashita s algorithm [13] using Theorem 10. For lack of space, we present below an application to a simpler algorithm. Example 11. Let us consider Israeli Jalfon s algorithm [11] The scheduler is central and arbitrary. A ....

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J. Beauquier, J. Durand-Lose, M. Gradinariu, and C. Johnen. Token based selfstabilizing uniform algorithms. J. of Parallel and Distributed Systems, To appear.

....ins ere un message dans l anneau uniquement lorsqu il aucun paquet message circulant dans l anneau. Ce protocole est le seul protocole supposant que les el ements du r eseaux communiquent par messages. Dans [BGJ99a] un protocole auto stabilisant de LE sur un anneau est pr esent e. Dans [DIM97b, BDLGJ02] des protocoles auto stabilisants de LE pour des r eseaux de topologie quelconque sont pr esent es. Il a et e prouv e que l espace m emoire n ecessaire aux protocoles [BGJ99a, BDLGJ02] est minimal. 5.2 Protocoles a vagues Les protocoles a vagues sont des outils de synchronisation a ....

....par messages. Dans [BGJ99a] un protocole auto stabilisant de LE sur un anneau est pr esent e. Dans [DIM97b, BDLGJ02] des protocoles auto stabilisants de LE pour des r eseaux de topologie quelconque sont pr esent es. Il a et e prouv e que l espace m emoire n ecessaire aux protocoles [BGJ99a, BDLGJ02] est minimal. 5.2 Protocoles a vagues Les protocoles a vagues sont des outils de synchronisation a l origine de nombreuses solutions dans la r esolution de probl emes de contr ole du r eseau. Une vague est en fait une ex ecution du protocole impliquant tous les processeurs du r eseau et se ....

J. Beauquier, J. Durand-Lose, M. Gradinariu, and C. Johnen. Token based self-stabilizing uniform algorithms. Journal of Parallel and Distributed Computing, 62(5):899-921, May 2002.

....under unfair distributed schedulers is presented. BGJ99b] presents a space optimal token circulation protocol on unidirectional rings that self stabilizes under unfair distributed schedulers. An adaptation of this protocol that has a better stabilization time is presented in [Ros00] In [BDLGJ02], the protocol of [BGJ99b] is extended in order to manage any anonymous unidirectional networks. The protocols of [Her90, BCD95, KY97, BGJ99b, Ros00, DGT00] are all based on the same technique: to randomly retard the token circulation . A processor having a token randomly decides to pass or not ....

J. Beauquier, J. Durand-Lose, M. Gradinariu, and C. Johnen. Token based selfstabilizing uniform algorithms. Journal of Parallel and Distributed Computing, 62(5):899--921, May 2002.

....under unfair distributed schedulers is designed. Beauquier and al in [5] presents a space optimal token circulation protocol on unidirectional rings that is self stabilizes under unfair distributed schedulers. An adaptation of this protocol that has a better stabilization time is given in [32] In [4], the protocol of [5] is extended in order to manage any anonymous unidirectional networks. The previous protocols on unidirectional anonymous networks under distributed schedulers [19, 2, 25, 5, 32, 9] are all based on the same technique: to randomly retard the token circulation . A processor ....

J. Beauquier, J. Durand-Lose, M. Gradinariu, and C. Johnen. Token based self-stabilizing uniform algorithms. Journal of Parallel and Distributed Computing, 62(5):899--921, May 2002.

....Scheduler transf. 7] general networks, bidir. with id central to unfair [1] general networks, bidir. with id central to unfair [1] general networks, bidir. with id X 1 bounded to unfair [3] rings, unidir. anonymous X 1 bounded to unfair [6] rings, unidir. anonymous alternating to central [2] general networks, unidir. anonymous X 2 bounded to unfair X 1 = n Gamma 1; and X 2 = n:MaxOut Diam where MaxOut is the maximal network out degree and Diam is the network diameter. The procotols [1] and [7] are working in the id based networks. In the case of anonymous networks an algorithm ....

Beauquier J., Durand-Lose J., Gradinariu M., Johnen C.: Token based selfstabilizing uniform algorithms. tech. Rep. no. 1250, LRI, Universit'e Paris-Sud; to appear in The Chicago Journal of Theoretical Computer Science (2000)

....type Scheduler transf. GH99] general networks, bidirectional with id central to unfair [BDGM00] general networks, bidirectional with id central to unfair [BDGM00] general networks, bidirectional with id X 1 bounded to unfair [BGJ99a] rings, unidirectional anonymous X 1 bounded to unfair [BDLGJ] general networks, unidirectional anonymous X 2 bounded to unfair X 1 = n Gamma 1; and X 2 = n:MaxOut Diam where MaxOut is the maximal network out degree and Diam is the network diameter. The procotols [BDGM00] and [GH99] are working in the id based networks. In the case of anonymous ....

J. Beauquier, J. Durand-Lose, M. Gradinariu, and C. Johnen. Token based selfstabilizing uniform algorithms. Raport technique no.1250, LRI, Universit'e Paris Sud; `a paraitre dans The Chicago Journal of Theoretical Computer Science.

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J. Beauquier, J. Durand-Lose, M. Gradinariu, and C. Johnen. Token based self-stabilizing uniform algorithms. Journal of Parallel and Distributed Computing, 62(5):899-921, 2002.

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