In this paper, we revisit the problem of classical \textit{meeting times} of random walks in graphs. In the process that two tokens (called agents) perform random walks on an undirected graph, the meeting times are defined as the expected times until they meet when the two agents are initially located at different vertices. A key feature of the problem is that, in each discrete time-clock (called \textit{round}) of the process, the scheduler selects only one of the two agents, and the agent performs one move of the random walk. In the adversarial setting, the scheduler utilizes the strategy that intends to \textit{maximizing} the expected time to meet. In the seminal papers \cite{collisions,israeli1990token,tetali1993simult}, for the random walks of two agents, the notion of \textit{atomicity} is implicitly considered. That is, each move of agents should complete while the other agent waits. In this paper, we consider and formalize the meeting time of \textit{non-atomic} random walks. In the non-atomic random walks, we assume that in each round, only one agent can move but the move does not necessarily complete in the next round. In other words, we assume that an agent can move at a round while the other agent is still moving on an edge. For the non-atomic random walks with the adversarial schedulers, we give a polynomial upper bound on the meeting times.

翻译：本文重新审视了经典图随机游走中的“相遇时间”问题。在双智能体（称为代理）于无向图上执行随机游走的过程中，相遇时间定义为：当两个代理初始位于不同顶点时，它们相遇所需时间的期望值。该问题的关键特征在于，在每个离散时间时钟（称为“轮次”）中，调度器仅选择两个代理中的一个，使其执行一次随机游走移动。在对抗性设置下，调度器采用旨在最大化期望相遇时间的策略。在开创性论文中（Collisions，Israeli 1990 Token，Tetali 1993 Simult.），双代理随机游走隐含考虑了“原子性”概念，即每个代理的移动必须在另一代理等待时完成。本文考虑并形式化了“非原子”随机游走的相遇时间。在非原子随机游走中，假设每一轮仅有一个代理可以移动，但该移动不一定在下一轮完成。换言之，我们允许代理在一轮中移动时，另一代理仍在边上移动。针对对抗性调度器下的非原子随机游走，我们给出了相遇时间的多项式上界。