Analysis of leader election algorithms
Christian Lavault
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Abstract
We start with a set of n players. With some probability $P(n, k)$, we kill $n−k$ players; the other ones stay alive, and we repeat with them. What is the distribution of the number $X_n$ of phases (or rounds) before getting only one player? We present a probabilistic analysis of this algorithm under some conditions on the probability distributions $P(n, k)$, including stochastic monotonicity and the assumption that roughly a fixed proportion $\alpha$ of the players survive in each round. We prove a kind of convergence in distribution for $\lceilX_n−\log_{1/\alpha}(n)\rceil$; as in many other similar problems there are oscillations and no true limit distribution, but suitable subsequences converge, and there is an absolutely continuous random variable $Z$ such that $d(Xn, \lceilZ+\log_{1/\alpha}(n)\rceil) \to 0$, where d is either the total variation distance or the Wasserstein distance. Applications of the general result include the leader election algorithm where players are elimin...
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gwu.edu
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Deterministic Leader Election Takes $$\Theta (D + \log n)$$Θ(D+logn) Bit Rounds
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Leader election is, together with consensus, one of the most central problems in distributed computing. This paper presents a distributed algorithm, called ST T , for electing deterministically a leader in an arbitrary network, assuming processors have unique identifiers of size O(log n), where n is the number of processors. It elects a leader in O(D + log n) rounds, where D is the diameter of the network, with messages of size O(1). Thus it has a bit round complexity of O(D + log n). This substantially improves upon the best known algorithm whose bit round complexity is O(D log n). In fact, using the lower bound by Kutten et al. (2015) and a result of Dinitz and Solomon (2007), we show that the bit round complexity of ST T is optimal (up to a constant factor), which is a significant step forward in understanding the interplay between time and message optimality for the election problem. Our algorithm requires no knowledge on the graph such as n or D, and the pipelining technique we introduce to break the O(D log n) barrier is general. This research has been partially supported by ANR projects DESCARTES and ESTATE (resp. ANR-16-CE40-0023 and ANR-16-CE25-0009-03). A preliminary subset of this work appeared in the proceedings of DISC 2016 [12].
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International Conference of Distributed Computing and Networking, 2018
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A loosely-stabilizing leader election protocol with polylogarithmic convergence time in the population protocol model is presented in this paper. In the population protocol model, which is a common abstract model of mobile sensor networks, it is known to be impossible to design a self-stabilizing leader election protocol. Thus, in our prior work, we introduced the concept of loose-stabilization, which is weaker than self-stabilization but has similar advantage as selfstabilization in practice. Following this work, several loosely-stabilizing leader election protocols are presented. The loosely-stabilizing leader election guarantees that, starting from an arbitrary configuration, the system reaches a safe configuration with a single leader within a relatively short time, and keeps the unique leader for an sufficiently long time thereafter. The convergence times of all the existing loosely-stabilizing protocols, i.e., the expected time to reach a safe configuration, are polynomial in n where n is the number of nodes (while the holding times to keep the unique leader are exponential in n). In this paper, a loosely-stabilizing protocol with polylogarithmic convergence time is presented. Its holding time is not exponential, but arbitrarily large polynomial in n.
Deterministic Leader Election Takes Θ(D + n) Bit Rounds
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Leader election is, together with consensus, one of the most central problems in distributed computing. This paper presents a distributed algorithm, called , for electing deterministically a leader in an arbitrary network, assuming processors have unique identifiers of size O( n), where n is the number of processors. It elects a leader in O(D + n) rounds, where D is the diameter of the network, with messages of size O(1). Thus it has a bit round complexity of O(D + n). This substantially improves upon the best known algorithm whose bit round complexity is O(D n). In fact, using the lower bound by Kutten et al. (2015) and a result of Dinitz and Solomon (2007), we show that the bit round complexity of is optimal (up to a constant factor), which is a significant step forward in understanding the interplay between time and message optimality for the election problem. Our algorithm requires no knowledge on the graph such as n or D, and the pipelining technique we introduce to break the O...
Deterministic Leader Election in $$O(D+\log n)$$ Time with Messages of Size O(1)
Lecture Notes in Computer Science, 2016
This paper presents a distributed algorithm, called ST T , for electing deterministically a leader in an arbitrary network, assuming processors have unique identifiers of size O(log n), where n is the number of processors. It elects a leader in O(D + log n) rounds, where D is the diameter of the network, with messages of size O(1). Thus it has a bit round complexity of O(D + log n). This substantially improves upon the best known algorithm whose bit round complexity is O(D log n). In fact, using the lower bound by Kutten et al. [13] and a result of Dinitz and Solomon [8], we show that the bit round complexity of ST T is optimal (up to a constant factor), which is a step forward in understanding the interplay between time and message optimality for the election problem. Our algorithm requires no knowledge on the graph such as n or D.
Logarithmic Expected-Time Leader Election in Population Protocol Model
Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing
In this paper, we present a leader election protocol in the population protocol model that stabilizes within O(log n) parallel time in expectation with O(log n) states per agent, where n is the number of agents. Given a rough knowledge m of the population size n such that m ≥ log 2 n and m = O(log n), this protocol guarantees that exactly one leader is elected and the unique leader is kept forever thereafter.
Leader Election Requires Logarithmic Time in Population Protocols
Parallel Processing Letters
This paper shows that every leader election protocol requires logarithmic stabilization time both in expectation and with high probability in the population protocol model. This lower bound holds even if each agent has knowledge of the exact size of a population and is allowed to use an arbitrarily large number of agent states. This lower bound concludes that the protocol given in [Sudo et al., SSS 2019] is time-optimal in expectation.
The complexity of leader election in diameter-two networks
Distributed Computing, 2019
This paper focuses on studying the message complexity of implicit leader election in synchronous distributed networks of diameter two. Kutten et al. (J ACM 62(1):7:1-7:27, 2015) showed a fundamental lower bound of Ω(m) (m is the number of edges in the network) on the message complexity of (implicit) leader election that applied also to Monte Carlo randomized algorithms with constant success probability; this lower bound applies for graphs that have diameter at least three. On the other hand, for complete graphs (i.e., graphs with diameter one), Kutten et al. (Theor Comput Sci 561(Part B):134-143, 2015) established a tight bound ofΘ(√ n) on the message complexity of randomized leader election (n is the number of nodes in the network). For graphs of diameter two, the complexity was not known. In this paper, we settle this complexity by showing a tight bound ofΘ(n) on the message complexity of leader election in diameter-two networks. We first give a simple randomized Monte-Carlo leader election algorithm that with high probability (i.e., probability at least 1 − n −c , for some fixed positive constant c) succeeds and uses O(n log 3 n) messages and runs in O(1) rounds; this algorithm works without knowledge of n (and hence needs no global knowledge). We then show that any algorithm (even Monte Carlo randomized algorithms with large enough constant success probability) needs Ω(n) messages (even when n is known), regardless of the number of rounds. We also present an O(n log n) message deterministic algorithm that takes O(log n) rounds (but needs knowledge of n); we show that this message complexity is tight for deterministic algorithms. Together with the two previous results of Kutten et al., our results fully characterize the message complexity of leader election vis-à-vis the graph diameter.
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