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Asymptotic Analysis of a Leader Election Algorithm

Christian Lavault, Guy Louchard

TL;DR

The paper delivers a precise asymptotic analysis of the Itai–Rodeh leader election algorithm on a symmetric ring, deriving $M(\infty) \approx 2.441715879$ and the limiting second moment $M^{(2)}(\infty) \approx 8.794530817$, along with the limiting distribution $P(\infty,j)$. It develops a general, mechanically applicable asymptotic framework for recurrences that yields all moments and the full distribution by solving recurrences, exchanging limits with summations, and employing singularity analysis of generating functions. A key extension replaces the per-round participation probability with $t/n$, establishing a unique minimum of $M(\infty,t)$ on $(0,2)$ at $t^{*} \approx 1.065439$, with $M(\infty,t^{*}) \approx 2.434810964$, and demonstrating similar behavior for $t \ge 2$ across successive intervals. The results provide a robust methodology for asymptotic complexity measures of distributed algorithms, with potential for broad applicability beyond the specific algorithm studied.

Abstract

Itai and Rodeh showed that, on the average, the communication of a leader election algorithm takes no more than $LN$ bits, where $L \simeq 2.441716$ and $N$ denotes the size of the ring. We give a precise asymptotic analysis of the average number of rounds M(n) required by the algorithm, proving for example that $\dis M(\infty) := \lim\_{n\to \infty} M(n) = 2.441715879...$, where $n$ is the number of starting candidates in the election. Accurate asymptotic expressions of the second moment $M^{(2)}(n)$ of the discrete random variable at hand, its probability distribution, and the generalization to all moments are given. Corresponding asymptotic expansions $(n\to \infty)$ are provided for sufficiently large $j$, where $j$ counts the number of rounds. Our numerical results show that all computations perfectly fit the observed values. Finally, we investigate the generalization to probability $t/n$, where $t$ is a non negative real parameter. The real function $\dis M(\infty,t) := \lim\_{n\to \infty} M(n,t)$ is shown to admit \textit{one unique minimum} $M(\infty,t^{*})$ on the real segment $(0,2)$. Furthermore, the variations of $M(\infty,t)$ on thewhole real line are also studied in detail.

Asymptotic Analysis of a Leader Election Algorithm

TL;DR

The paper delivers a precise asymptotic analysis of the Itai–Rodeh leader election algorithm on a symmetric ring, deriving and the limiting second moment , along with the limiting distribution . It develops a general, mechanically applicable asymptotic framework for recurrences that yields all moments and the full distribution by solving recurrences, exchanging limits with summations, and employing singularity analysis of generating functions. A key extension replaces the per-round participation probability with , establishing a unique minimum of on at , with , and demonstrating similar behavior for across successive intervals. The results provide a robust methodology for asymptotic complexity measures of distributed algorithms, with potential for broad applicability beyond the specific algorithm studied.

Abstract

Itai and Rodeh showed that, on the average, the communication of a leader election algorithm takes no more than bits, where and denotes the size of the ring. We give a precise asymptotic analysis of the average number of rounds M(n) required by the algorithm, proving for example that , where is the number of starting candidates in the election. Accurate asymptotic expressions of the second moment of the discrete random variable at hand, its probability distribution, and the generalization to all moments are given. Corresponding asymptotic expansions are provided for sufficiently large , where counts the number of rounds. Our numerical results show that all computations perfectly fit the observed values. Finally, we investigate the generalization to probability , where is a non negative real parameter. The real function is shown to admit \textit{one unique minimum} on the real segment . Furthermore, the variations of on thewhole real line are also studied in detail.

Paper Structure

This paper contains 20 sections, 1 theorem, 82 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Algorithm scheme and notation
  3. Asymptotic analysis of the recurrence
  4. Asymptotic approximation of $\bm{M(n)}$
  5. Interchanging limit and summation
  6. Laplace method
  7. Lebesgue's dominated convergence method
  8. Asymptotic approximation of $\bm{M^{(2)}(n)}$
  9. Generalization
  10. Asymptotic approximation of $\bm{P(n,j)}$
  11. Asymptotic recurrence of $\bm{P(n,j)}$ ($\bm{n\to \infty}$)
  12. Asymptotic approximation of $\bm{P(\infty,j)}$ ($\bm{j\to \infty}$)
  13. Numerical results
  14. Is $\bm{1/n}$ the optimal probability?
  15. Optimal probability on the domain $\bm{(0,2)}$
  16. ...and 5 more sections

Key Result

Theorem 2.1

The asymptotic average number of rounds required by the algorithm to elect a leader is the constant $M(\infty)$. When $n\to \infty$, an asymptotic approximation of $M(n)$ writes The second moment of the discrete r.v. $X(n)$ is asymptotically and an asymptotic approximation of its variance ($n\to \infty$) yields where $S_1 =\; \sum_{k\ge 2} \frac{e^{-1}}{k!}\, M(k)$ and $S_2 =\; \sum_{k\ge 2} \f

Figures (5)

  • Figure 1: Probability $P(\infty,j)$, for $j = 1$,…, $10$
  • Figure 2: Convergence of $M(n)$ to $M(\infty)$, for $n = 250$,…, $300$
  • Figure 3: $P(\infty,j)$ and its asymptotic approximation in (\ref{['Pequivalent']}) for large $j$ ($j = 20$, …, $30$)
  • Figure 4: $M(\infty,t),\ t\in (0,2)$
  • Figure 5: $M(\infty,t),\ t\in [2,3)$

Theorems & Definitions (4)

  • Theorem 2.1
  • Remark 2.2
  • Remark 3.1
  • Remark 5.1