Convergence of some leader election algorithms

The Annals of Probability, 2017

Motivated by the open problem of finding the asymptotic distributional behavior of the number of collisions in a Poisson-Dirichlet coalescent, the following version of a stochastic leaderelection algorithm is studied. Consider an infinite family of persons, labeled by 1, 2, 3,. . ., who generate iid random numbers from an arbitrary continuous distribution. Those persons who have generated a record value, that is, a value larger than the values of all previous persons, stay in the game, all others must leave. The remaining persons are relabeled by 1, 2, 3,. .. maintaining their order in the first round, and the election procedure is repeated independently from the past and indefinitely. We prove limit theorems for a number of relevant functionals for this procedure, notably the number of rounds T (M) until all persons among 1,. .. , M , except the first one, have left (as M → ∞). For example, we show that the sequence (T (M) − log * M) M ∈N , where log * denotes the iterated logarithm, is tight, and study its weak subsequential limits. We further provide an appropriate and apparently new kind of normalization (based on tetrations) such that the original labels of persons who stay in the game until round n converge (as n → ∞) to some random non-Poissonian point process and study its properties. The results are applied to describe all subsequential distributional limits for the number of collisions in the Poisson-Dirichlet coalescent, thus providing a complete answer to the open problem mentioned above.

Journal of Applied Probability

A class of games for finding a leader among a group of candidates is studied in detail. This class covers games based on coin tossing and rock-paper-scissors as special cases and its complexity exhibits similar stochastic behaviors: either of logarithmic mean and bounded variance or of exponential mean and exponential variance. Many applications are also discussed.

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.

We present an efficient randomized algorithm for leader election in large-scale distributed systems. The proposed algorithm is optimal in message complexity (O(n) for a set of n total processes), has round complexity logarithmic in the number of processes in the system, and provides high probabilistic guarantees on the election of a unique leader. The algorithm relies on a balls and bins abstraction and works in two phases. The main novelty of the work is in the first phase where the number of contending processes is reduced in a controlled manner. Probabilistic quorums are used to determine a winner in the second phase. We discuss, in detail, the synchronous version of the algorithm, provide extensions to an asynchronous version and examine the impact of failures.

2018

We present a fast loosely-stabilizing leader election protocol in the population protocol model. It elects a unique leader in a poly-logarithmic time and holds the leader for a polynomial time with arbitrarily large degree in terms of parallel time, i.e, the number of steps per the population size. 2012 ACM Subject Classification Theory of computation → Self-organization

Theoretical Computer Science, 2019

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.

We consider the problem of electing a leader among nodes in a highly dynamic network where the adversary has unbounded capacity to insert and remove nodes (including the leader) from the network and change connectivity at will. We present a randomized algorithm that (re)elects a leader in O(D log n) rounds with high probability, where D is a bound on the dynamic diameter of the network and n is the maximum number of nodes in the network at any point in time. We assume a model of broadcast-based communication where a node can send only 1 message of O(log n) bits per round and is not aware of the receivers in advance. Thus, our results also apply to mobile wireless adhoc networks, improving over the optimal (for deterministic algorithms) O(Dn) solution presented at FOMC 2011. We show that our algorithm is optimal by proving that any randomized Las Vegas algorithm takes at least Ω(D log n) rounds to elect a leader with high probability, which shows that our algorithm yields the best possible (up to constants) termination time.

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.

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...

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].

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.

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...

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.

Electronic Communications in Probability, 2004

Consider a two-person zero-sum game played on a random n × n-matrix where the entries are iid normal random variables. Let Z be the number of rows in the support of the optimal strategy for player I given the realization of the matrix. (The optimal strategy is a.s. unique and Z a.s. coincides with the number of columns of the support of the optimal strategy for player II.) Faris an Maier [4] make simulations that suggest that as n gets large Z has a distribution close to binomial with parameters n and 1/2 and prove that P (Z = n) ≤ 2 −(k−1). In this paper a few more theoretically rigorous steps are taken towards the limiting distribution of Z: It is shown that there exists a < 1/2 (indeed a < 0.4) such that P (1 2 − a)n < Z < (1 2 + a)n → 1 as n → ∞. It is also shown that EZ = (1 2 + o(1))n. We also prove that the value of the game with probability 1 − o(1) is at most Cn −1/2 for some C < ∞ independent of n. The proof suggests that an upper bound is in fact given by f (n)n −1 , where f (n) is any sequence such that f (n) → ∞, and it is pointed out that if this is true, then the variance of Z is o(n 2) so that any a > 0 will do in the bound on Z above.

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.