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Dynamic Size Counting in the Population Protocol Model

Dominik Kaaser, Maximilian Lohmann

TL;DR

The paper addresses dynamic size counting in the population protocol model under adversarial changes to the population size. It introduces a uniform loosely-stabilizing protocol that uses the maximum of geometrically distributed random variables to estimate $\log n$, and a CHVP-based synchronization mechanism to realize a dynamic, loosely-stabilizing phase clock. Theoretical results show convergence from arbitrary configurations to a valid estimate of $\log n$ in $O(\log \hat{n} + \log n)$ parallel time, with holding time polynomial, and memory usage of $O(\log s + \log \log n)$ bits per agent. The work also provides empirical simulations up to $10^6$ agents demonstrating practical effectiveness and robustness to population changes. The protocol improves space efficiency over prior dynamic-size solutions while trading some convergence speed, and opens avenues for composing dynamic size counting with other uniform non-uniform workflows.

Abstract

The population protocol model describes collections of distributed agents that interact in pairs to solve a common task. We consider a dynamic variant of this prominent model, where we assume that an adversary may change the population size at an arbitrary point in time. In this model we tackle the problem of counting the population size: in the dynamic size counting problem the goal is to design an algorithm that computes an approximation of $\log n$. This estimate can be used to turn static, non-uniform population protocols, i.e., protocols that depend on the population size $n$, into dynamic and loosely-stabilizing protocols. Our contributions in this paper are three-fold. Starting from an arbitrary initial configuration, we first prove that the agents converge quickly to a valid configuration where each agent has a constant-factor approximation of $\log n$, and once the agents reach such a valid configuration, they stay in it for a polynomial number of time steps. Second, we show how to use our protocol to define a uniform and loosely-stabilizing phase clock for the population protocol model. Finally, we support our theoretical findings by empirical simulations that show that our protocols work well in practice.

Dynamic Size Counting in the Population Protocol Model

TL;DR

The paper addresses dynamic size counting in the population protocol model under adversarial changes to the population size. It introduces a uniform loosely-stabilizing protocol that uses the maximum of geometrically distributed random variables to estimate , and a CHVP-based synchronization mechanism to realize a dynamic, loosely-stabilizing phase clock. Theoretical results show convergence from arbitrary configurations to a valid estimate of in parallel time, with holding time polynomial, and memory usage of bits per agent. The work also provides empirical simulations up to agents demonstrating practical effectiveness and robustness to population changes. The protocol improves space efficiency over prior dynamic-size solutions while trading some convergence speed, and opens avenues for composing dynamic size counting with other uniform non-uniform workflows.

Abstract

The population protocol model describes collections of distributed agents that interact in pairs to solve a common task. We consider a dynamic variant of this prominent model, where we assume that an adversary may change the population size at an arbitrary point in time. In this model we tackle the problem of counting the population size: in the dynamic size counting problem the goal is to design an algorithm that computes an approximation of . This estimate can be used to turn static, non-uniform population protocols, i.e., protocols that depend on the population size , into dynamic and loosely-stabilizing protocols. Our contributions in this paper are three-fold. Starting from an arbitrary initial configuration, we first prove that the agents converge quickly to a valid configuration where each agent has a constant-factor approximation of , and once the agents reach such a valid configuration, they stay in it for a polynomial number of time steps. Second, we show how to use our protocol to define a uniform and loosely-stabilizing phase clock for the population protocol model. Finally, we support our theoretical findings by empirical simulations that show that our protocols work well in practice.
Paper Structure (29 sections, 17 theorems, 26 equations, 4 figures, 2 algorithms)

This paper contains 29 sections, 17 theorems, 26 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Results in a Nutshell
  3. Background and Related Work
  4. Approximate Counting.
  5. Dynamic Size Counting
  6. Phase Clocks.
  7. Detection.
  8. Model and Results
  9. Simplified Algorithm
  10. Our Results
  11. Dynamic Size Counting Protocol
  12. Geometrically Distributed Random Variables
  13. Analysis
  14. Preliminaries
  15. Global Maximum.
  16. ...and 14 more sections

Key Result

theorem 1

For any constant $k \geq 2$, and largest initial size estimate $\log \hat{n}$, our algorithm is a $\left(O(\log \hat{n} + \log n), \Theta(n^{k-1} \log n)\right)$-loosely-stabilizing protocol solving the size counting problem w.h.p. When $s$ denotes the largest value initially stored in any of the ag

Figures (4)

  • Figure 1: Phase transitions
  • Figure 2: Size estimate in a system of $10^6$ agents
  • Figure 3: Relative error for different values of $n$
  • Figure 5: Estimates for different population sizes initialized with an estimate of $60$

Theorems & Definitions (31)

  • theorem 1
  • theorem 2
  • lemma 1: restate=restatekngrvs,label=lemma:kn_grvs
  • lemma 2
  • lemma 3: restate=restatechvpmax,label=lemma:chvp_max
  • lemma 4: restate=restatechvpmin,label=lemma:chvp_min
  • lemma 5
  • proof
  • lemma 6
  • proof
  • ...and 21 more