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.

翻译：本文研究群体协议中的碰撞检测问题。网络由称为智能体的状态机构成。在每个时间步，系统均匀随机选择恰好一对智能体进行交互，从而改变两者的状态。碰撞检测问题要求每个智能体以$1$至$n$之间的整数作为输入（其中$n$为智能体总数），并判定所有智能体中是否存在重复的输入值。具体而言，当所有输入值互异时，所有智能体应输出假；否则应输出真。本文提出一种算法，该算法要求每个智能体仅需多项式数量的状态，即能以高概率及期望意义下的亚线性并行时间，以概率一解决碰撞检测问题。据我们所知，这是首个在亚线性并行时间内使用多项式数量状态解决碰撞检测问题的算法，首次对Burman、Chen、Chen、Doty、Nowak、Severson和Xu在[PODC 2021]中提出的问题给出了肯定性解答。