[Paper Review] On Leader Green Election

Jacek Cichoń, Rafał Kapelko+1 more|arXiv (Cornell University)|Apr 30, 2016

Advanced Combinatorial Mathematics15 references3 citations

TL;DR

This paper analyzes the Leader Green Election (LGE) algorithm for distributed leader election using probabilistic methods, focusing on the expected number of survivors after the initial phase. It applies Rice's method to derive precise formulas for survivor probabilities and establishes that LGE should be used only once to reduce contenders to a small set, followed by a secondary protocol. The key contribution is a tight upper bound on survivors and a lower bound on required rounds for leader election protocols based on urns and balls models.

ABSTRACT

We investigate the number of survivors in the Leader Green Election (LGE) algorithm introduced by P. Jacquet, D. Milioris and P. Muhlethaler in 2013. Our method is based on the Rice method and gives quite precise formulas. We derive upper bounds on the number of survivors in this algorithm and we propose a proper use of LGE. Finally, we discuss one property of a general urns and balls problem and show a lower bound for a required number of rounds for a large class of distributed leader election protocols.

Motivation & Objective

• To analyze the number of survivors in the Leader Green Election (LGE) algorithm using probabilistic methods.
• To derive precise formulas for the expected number of survivors and their distribution using Rice's method.
• To establish upper bounds on the number of survivors in LGE and propose its proper use as a pre-election phase.
• To investigate the connection between LGE and the general urns and balls problem, particularly the existence of singletons.
• To derive a lower bound on the number of rounds required for a large class of distributed leader election protocols.

Proposed method

• Uses Rice's method to analyze the distribution of the number of survivors in LGE, providing exact formulas for survival probabilities.
• Models the LGE process as a geometrically distributed selection of random numbers followed by a binary key transmission phase.
• Applies the maximum of geometric random variables (MGeo distribution) to model the time slots used in LGE.
• Introduces the WMGeo distribution to model the number of contenders selecting the maximum value.
• Analyzes the urns and balls model with non-uniform probabilities to study the existence of singletons, crucial for leader election success.
• Uses the function f(𝐩) = Σ(Q=2 to n) Pr[S𝐩,Q]/Q to derive bounds on the maximal success probability of singleton existence.

Experimental results

Research questions

• RQ1 What is the expected number of survivors in the LGE algorithm under geometric distribution assumptions?
• RQ2 How can Rice's method be applied to derive exact formulas for survivor probabilities in LGE?
• RQ3 What is the tightest upper bound on the number of survivors in LGE for a given number of contenders and parameter p?
• RQ4 What is the minimal number of rounds required for a large class of distributed leader election protocols to succeed with high probability?
• RQ5 Can the existence of a singleton in an urns and balls model be guaranteed with high probability when the number of balls is unknown and bounded?

Key findings

• The expected number of survivors in LGE is given by E[Wₙ,ₚ] = (np/q) × Σ(b=0 to n−1) [(-1)ᵇ × C(n−1,b) / (1 − qᵇ⁺¹)], where q = 1−p.
• The probability that exactly a contenders survive is Pr[Wₙ,ₚ = a] = C(n,a) × pᵃ × Σ(b=0 to n−a) [C(n−a,b) × (-1)ᵇ / (1 − qᵃ⁺ᵇ)].
• An upper bound on the tail probability is Pr[M > C × (ln n)/ln(1/(1−p))] ≤ 1/n^(C−1), where M ~ MGeo(n,p).
• The maximal success probability MSP(L,n) of having at least one singleton in L urns with up to n balls is strictly less than (L−1)/(Hₙ−1), where Hₙ is the nth harmonic number.
• If L ≤ ½ ln n + (1+γ)/2, then MSP(L,n) < ½, meaning that with high probability, no unique choice exists in such models.
• For M ≤ log₂(½ ln n + (1+γ)/2), the probability of selecting a unique leader using M-bit sequences is less than ½, indicating insufficient randomness for reliable selection.