Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sublinear-time Collision Detection with a Polynomial Number of States in Population Protocols

Takumi Araya, Yuichi Sudo

TL;DR

This work resolves whether collision detection in population protocols can be achieved with polynomial-state per agent in sublinear parallel time. It introduces a sublinear-time collision-detection algorithm (CDWB) that uses square-root decomposition and a phase clock, augmented by an epidemic mechanism to spread the detection signal. The protocol stabilizes in $Oigl(n^{3/2} ext{ } ilde{O}( ext{log}^{3/2} n)igr)$ time with high probability and $Oigl(n^{3/2} ext{ } ext{log}^{1/2} nigr)$ in expectation, using $ ilde{O}(n)$ states per agent (excluding the input rank, which requires $O(n)$ states). It also handles a robust derandomization and a version without assumptions by coupling with an Approximate leader-estimation protocol. Overall, the paper answers Burman et al.'s open question by showing polynomial-state, sublinear-time collision detection is achievable, while leaving open the possibility of sublinear-time self-stabilizing ranking with polynomial state space.

Abstract

This paper addresses the collision detection problem in population protocols. The network consists of state machines called agents. At each time step, exactly one pair of agents is chosen uniformly at random to have an interaction, changing the states of the two agents. The collision detection problem involves each agent starting with an input integer between $1$ and $n$, where $n$ is the number of agents, and requires those agents to determine whether there are any duplicate input values among all agents. Specifically, the goal is for all agents to output false if all input values are distinct, and true otherwise. In this paper, we present an algorithm that requires a polynomial number of states per agent and solves the collision detection problem with probability one in sub-linear parallel time, both with high probability and in expectation. To the best of our knowledge, this algorithm is the first to solve the collision detection problem using a polynomial number of states within sublinear parallel time, affirmatively answering the question raised by Burman, Chen, Chen, Doty, Nowak, Severson, and Xu [PODC 2021] for the first time.

Sublinear-time Collision Detection with a Polynomial Number of States in Population Protocols

TL;DR

This work resolves whether collision detection in population protocols can be achieved with polynomial-state per agent in sublinear parallel time. It introduces a sublinear-time collision-detection algorithm (CDWB) that uses square-root decomposition and a phase clock, augmented by an epidemic mechanism to spread the detection signal. The protocol stabilizes in time with high probability and in expectation, using states per agent (excluding the input rank, which requires states). It also handles a robust derandomization and a version without assumptions by coupling with an Approximate leader-estimation protocol. Overall, the paper answers Burman et al.'s open question by showing polynomial-state, sublinear-time collision detection is achievable, while leaving open the possibility of sublinear-time self-stabilizing ranking with polynomial state space.

Abstract

This paper addresses the collision detection problem in population protocols. The network consists of state machines called agents. At each time step, exactly one pair of agents is chosen uniformly at random to have an interaction, changing the states of the two agents. The collision detection problem involves each agent starting with an input integer between and , where is the number of agents, and requires those agents to determine whether there are any duplicate input values among all agents. Specifically, the goal is for all agents to output false if all input values are distinct, and true otherwise. In this paper, we present an algorithm that requires a polynomial number of states per agent and solves the collision detection problem with probability one in sub-linear parallel time, both with high probability and in expectation. To the best of our knowledge, this algorithm is the first to solve the collision detection problem using a polynomial number of states within sublinear parallel time, affirmatively answering the question raised by Burman, Chen, Chen, Doty, Nowak, Severson, and Xu [PODC 2021] for the first time.

Paper Structure

This paper contains 12 sections, 8 theorems, 4 equations, 1 figure, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Model
  4. With High Probability
  5. Collision Detection Problem
  6. Uniform Protocols
  7. Tools
  8. One-way Epidemic
  9. Phase Clock with a Leader
  10. Collision Detection
  11. Collision Detection with a Leader and Rough knowledge of $n$
  12. Collision Detection without Assumptions

Key Result

Lemma 1

Suppose that an execution of the protocol ${\normalfont\textsc{Epidemic}}\xspace(\mathtt{var})$ starts from a configuration $C$ where $M = \max_{v \in V} v.\mathtt{var}$. Then, for any fixed $\eta > 0$, there exists a constant $d$ such that for sufficiently large population size $n$, with probabilit

Figures (1)

  • Figure 1: Segments of the second part of ${\normalfont\textsc{CDWB}}\xspace(n_{L},n_{U})$

Theorems & Definitions (16)

  • Definition 1: with high probability
  • Definition 2: Collision Detection Problem
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 6 more