Asymptotic analysis of a leader election algorithm

The Annals of Applied Probability, 1996

We investigate the duration of an elimination process for identifying a winner by coin tossing, or, equivalently, the height of a random incomplete trie. Applications of the process include the election of a leader in a computer network. Using direct probabilistic arguments we obtain exact expressions for the discrete distribution and the moments of the height. Elementary approximation techniques then yield asymptotics for the distribution. We show that no limiting distribution exists, as the asymptotic expressions exhibit periodic uctuations.

gwu.edu

We consider a serialized coin-tossing leader election algorithm that proceeds in rounds until a winner is chosen, or all contestants are eliminated. The analysis allows for either biased or fair coins. We find the exact distribution for the duration of any fixed contestant; asymptotically it turns out to be a geometric distribution. Rice's method (an analytic technique) shows that the moments of the duration contain oscillations, which we give explicitly for the mean and variance. We also use convergence in the Wasserstein metric space to show that the distribution of the total number of coin flips (among all participants), suitably normalized, approaches a normal limiting random variable.

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...

2020

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(logn), where n is the number of processors. It elects a leader in O(D + logn) 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 + logn). This substantially improves upon the best known algorithm whose bit round complexity is O(D logn). 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 int...

2018

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...

Algorithmica

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].

Discret. Math. Theor. Comput. Sci., 2008

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 \emph{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 $\al$ of the players survive in each round. We prove a kind of convergence in distribution for $X_n - \log_{1/\!\alpha}(n)$; 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 $d\l(X_n, \lceil Z + \log_{1/\!\alpha} (n)\rceil\r) \to 0$, where $d$ is either the total variation distance or the Wasserstein distance. Applications of the general result include the leader election algorithm where players a...

Journal of Computer and System Sciences, 2001

In the leader election problem, n players wish to elect a random leader. The difficulty is that some coalition of players may conspire to elect one of its own members. We adopt the perfect information model: all communication is by broadcast, and the bad players have unlimited computational power. Within a round, they may also wait to see the inputs of the good players. A protocol is called resilient if a good leader is elected with probability bounded away from 0. We give a simple, constructive leader election protocol that is resilient against coalitions of size βn, for any β < 1=2. Our protocol takes log n + O(1) rounds, each player sending at most log n bits per round. For any constant k, our protocol can be modified to take k rounds and be resilient against coalitions of size εn=(log (k) n) 3 , where ε is a small enough constant and log (k) denotes the logarithm iterated k times. This is constructive for k 3.

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.

International Conference of Distributed Computing and Networking, 2018

Leader election is one of the fundamental problems in distributed computing. In its implicit version, only the leader must know who is the elected leader. This paper focuses on studying the message complexity of leader election in synchronous distributed networks, in particular, in networks of diameter two. Kutten et al. [JACM 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., diameter 1), Kutten et al. [TCS 2015] established a tight bound ofΘ(√ n) 1 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 diametertwo 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 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) messages deterministic algorithm that takes O (log n) rounds (but needs knowledge of n); we show that this message complexity is tight for deterministic algorithms. Our results show that leader election can be solved in diametertwo graphs in (essentially) linear (in n) message complexity and thus the Ω(m) lower bound does not apply to diameter-two graphs. 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.