Randomized Leader Election Research Articles

Leader Election in Well-Connected Graphs

In this paper, we look at the problem of randomized leader election in synchronous distributed networks with a special focus on the message complexity. We provide an algorithm that solves the implicit version of leader election (where non-leader nodes need not be aware of the identity of the leader) in any general network with \(O(\sqrt{n} \log ^{7/2} n \cdot t_{mix})\) messages and in \(O(t_{mix}\log ^2 n)\) time, where n is the number of nodes and \(t_{mix}\) refers to the mixing time of a random walk in the network graph G. For several classes of well-connected networks (that have a large conductance or alternatively small mixing times e.g., expanders, hypercubes, etc), the above result implies extremely efficient (sublinear running time and messages) leader election algorithms. Correspondingly, we show that any substantial improvement is not possible over our algorithm, by presenting an almost matching lower bound for randomized leader election. We show that \(\varOmega (\sqrt{n}/\phi ^{3/4})\) messages are needed for any leader election algorithm that succeeds with probability at least \(1-o(1)\), where \(\phi \) refers to the conductance of a graph. To the best of our knowledge, this is the first work that shows a dependence between the time and message complexity to solve leader election and the connectivity of the graph G, which is often characterized by the graph’s conductance \(\phi \). Apart from the \(\varOmega (m)\) bound in Kutten et al. (J ACM 62(1):7:1–7:27, 2015) (where m denotes the number of edges of the graph), this work also provides one of the first non-trivial lower bounds for leader election in general networks.

Exploiting Spontaneous Transmissions for Broadcasting and Leader Election in Radio Networks

We study two fundamental communication primitives: broadcasting and leader election in the classical model of multi-hop radio networks with unknown topology and without collision detection mechanisms. It has been known for almost 20 years that in undirected networks with n nodes and diameter D , randomized broadcasting requires Ω( D log n / D + log 2 n ) rounds, assuming that uninformed nodes are not allowed to communicate (until they are informed). Only very recently, Haeupler and Wajc (PODC'2016) showed that this bound can be improved for the model with spontaneous transmissions, providing an O ( D log n log log n /log D + log O (1) n )-time broadcasting algorithm. In this article, we give a new and faster algorithm that completes broadcasting in O ( D log n /log D + log O (1) n ) time, succeeding with high probability. This yields the first optimal O ( D )-time broadcasting algorithm whenever n is polynomial in D . Furthermore, our approach can be applied to design a new leader election algorithm that matches the performance of our broadcasting algorithm. Previously, all fast randomized leader election algorithms have used broadcasting as a subroutine and their complexity has been asymptotically strictly larger than the complexity of broadcasting. In particular, the fastest previously known randomized leader election algorithm of Ghaffari and Haeupler (SODA'2013) requires O ( D log n / D min {log log n , log n / D } + log O (1) n )-time, succeeding with high probability. Our new algorithm again requires O ( D log n /log D + log O (1) n ) time, also succeeding with high probability.

The complexity of leader election in diameter-two networks

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 $$\varOmega (m)$$ (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 of $$\tilde{\varTheta }(\sqrt{n})$$ 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 of $$\tilde{\varTheta }(n)$$ 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 $$\varOmega (n)$$ 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-a-vis the graph diameter.

Sublinear bounds for randomized leader election

This paper concerns randomized leader election in synchronous distributed networks. A distributed leader election algorithm is presented for complete n-node networks that runs in O(1) rounds and (with high probability) uses only O(nlog3/2⁡n) messages to elect a unique leader (with high probability). When considering the “explicit” variant of leader election where eventually every node knows the identity of the leader, our algorithm yields the asymptotically optimal bounds of O(1) rounds and O(n) messages. This algorithm is then extended to one solving leader election on any connected non-bipartite n-node graph G in O(τ(G)) time and O(τ(G)nlog3/2⁡n) messages, where τ(G) is the mixing time of a random walk on G. The above result implies highly efficient (sublinear running time and messages) leader election algorithms for networks with small mixing times, such as expanders and hypercubes. In contrast, previous leader election algorithms had at least linear message complexity even in complete graphs. Moreover, super-linear message lower bounds are known for time-efficient deterministic leader election algorithms. Finally, we present an almost matching lower bound for randomized leader election, showing that Ω(n) messages are needed for any leader election algorithm that succeeds with probability at least 1/e+ε, for any small constant ε>0. We view our results as a step towards understanding the randomized complexity of leader election in distributed networks.

Randomized leader election

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.

MDVM System Concept, Paging Latency and Round-2 Randomized Leader Election Algorithm in SG

The future trend in the computing paradigm is marked by mobile computing based on mobile-client/server architecture connected by wireless communication network. However, the mobile computing systems have limitations because of the resource-thin mobile clients operating on battery power. The MDVM system allows the mobile clients to utilize memory and CPU resources of Server-Groups (SG) to overcome the resource limitations of clients in order to support the high-end mobile applications such as, m-commerce and virtual organization (VO). In this paper the concept of MDVM system and the architecture of cellular network containing the SG are discussed. A round-2 randomized distributed algorithm is proposed to elect a unique leader and co-leader of the SG. The algorithm is free from any assumption about network topology, buffer space limitations and is based on dynamically elected coordinators eliminating single point of failure. The algorithm is implemented in distributed system setup and the network-paging latency values of wired and wireless networks are measured experimentally. The experimental results demonstrate that in most cases the algorithm successfully terminates in first round and the possibility of second round execution decreases significantly with the increase in the size of SG (|<I>Na</I>|). The overall message complexity of the algorithm is O(|<I>Na</I>|). The comparative study of network-paging latencies indicates that 3G/4G mobile communication systems would support the realization of MDVM system.

On the distribution for the duration of a randomized leader election algorithm

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 fluctuations. In many similar problems associated with digital trees, no such exact expressions can be derived. We therefore outline a powerful general approach, based on the analytic techniques of Mellin transforms, Poissonization and de-Poissonization, from which distributional asymptotics for the height can also be derived. In fact, it was this complex variables approach that led to our original discovery of the exact distribution. Complex analysis methods are indispensable for deriving asymptotic expressions for the mean and variance, which also contain periodic terms of small magnitude.

The Bit Complexity of Randomized Leader Election on a Ring

The inherent bit complexity of leader election on asynchronous unidirectional rings of processors is examined under various assumptions about global knowledge of the ring. If processors have unique identities with a maximum of m bits, then the expected number of communication bits sufficient to elect a leader with probability 1, on a ring of (unknown) size n is $O(nm)$. If the ring size is known to within a multiple of 2, then the expected number of communication bits sufficient to elect a leader with probability 1 is $O(n\log n)$. These upper bounds are complemented by lower bounds on the communication complexity of a related problem called solitude verification that reduces to leader election in $O(n)$ bits. If processors have unique identities chosen from a sufficiently large universe of size s, then the average, overall choices of identities, of the communication complexity of verifying solitude is $\Omega (n\log s)$ bits. When the ring size is known only approximately, then $\Omega (n\log n)$ bits are required for solitude verification. The lower bounds address the complexity of certifying solitude. This is modelled by the best-case behaviour of nondeterministic solitude-verification algorithms.