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

2024-11-15

arXiv.org

AI Summary

This paper addresses collision detection in population protocols: given $n$ agents, each initially holding a distinct integer from ${1,dots,n}$, the goal is to distributively determine whether the multiset of inputs contains duplicates, with all agents converging to a common output of $ exttt{true}$ (collision exists) or $ exttt{false}$ (all values are distinct). We present the first protocol that solves this problem with sublinear parallel time $O(n^{1-varepsilon})$ for some $varepsilon > 0$, polynomial state complexity $mathrm{poly}(n)$, and guaranteed correctness with probability one. Our approach operates under the standard pairwise random interaction model and integrates lightweight counting, randomized hashing, and a staged broadcast mechanism. This resolves an open problem posed at PODC 2021, overcoming the prior state-of-the-art trade-off—where all known exact solutions required either $Omega(n)$ parallel time or exponential state space—thereby significantly advancing the efficiency and scalability of distributed duplicate detection.

Technology Category

• Multiagent Systems: Distributed Problem Solving
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: The sharing economy

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.

Problem

• Detects duplicate input values among agents in population protocols.
• Uses polynomial states per agent for collision detection.
• Achieves sublinear parallel time with high probability.

Innovation

• Sublinear-time collision detection algorithm
• Polynomial number of states per agent
• High probability and expected sublinear parallel time