The problem of dispersion of mobile robots on a graph asks that n robots initially placed arbitrarily on the nodes of an n-node anonymous graph, autonomously move to reach a final configuration where each node has at most one robot on it. This problem is of significant interest due to its relationship to other fundamental robot coordination problems, such as exploration, scattering, load balancing, relocation of self-driving electric cars to recharge stations, etc. The robots have unique IDs, typically in the range $[1,poly(n)]$ and limited memory, whereas the graph is anonymous, i.e., the nodes do not have identifiers. The objective is to simultaneously minimize two performance metrics: (i) time to achieve dispersion and (ii) memory requirement at each robot. This problem has been relatively well-studied when robots are non-faulty.

In this paper, we introduce the notion of Byzantine faults to this problem, i.e., we formalize the problem of dispersion in the presence of up to f Byzantine robots. We then study the problem on a ring while simultaneously optimizing the time complexity of algorithms and the memory requirement per robot. Specifically, we design deterministic algorithms that attempt to match the time lower bound ($\mathrm{\Omega }(n)$ rounds) and memory lower bound ($\mathrm{\Omega }(\log n)$ bits per robot).

Our main result is a deterministic algorithm that is both time and memory optimal, i.e., $O(n)$ rounds and $O(\log n)$ bits of memory required per robot, subject to certain constraints. We subsequently provide results that require less assumptions but are either only time or memory optimal but not both. We also provide a primitive, utilized often, that takes robots initially gathered at a node of the ring and disperses them in a time and memory optimal manner without additional assumptions required.

移动机器人在图上的分散问题要求n 个机器人最初任意放置在n节点匿名图的节点上，自主移动以达到每个节点上最多有一个机器人的最终配置。由于该问题与其他基本的机器人协调问题（例如探索、分散、负载平衡、自动驾驶电动汽车重新定位到充电站等）的关系，该问题具有重要意义。机器人具有唯一的 ID，通常在范围内$[1,磷○升是(n)]$和有限的内存，而图是匿名的，即节点没有标识符。目标是同时最小化两个性能指标：（i）实现分散的时间和（ii）每个机器人的内存需求。当机器人没有故障时，这个问题已经得到了相对充分的研究。

在本文中，我们将拜占庭故障的概念引入到这个问题中，即，我们将在最多f 个拜占庭机器人存在的情况下的分散问题形式化。然后我们在环上研究问题，同时优化算法的时间复杂度和每个机器人的内存需求。具体来说，我们设计了尝试匹配时间下限的确定性算法（$\mathrm{\Omega }(n)$ 轮数）和内存下限（$\mathrm{\Omega }(\mathrm{日志}n)$ 每个机器人的位数）。

我们的主要结果是一个确定性算法，它是时间和内存最优的，即， $哦(n)$ 轮次和 $哦(\mathrm{日志}n)$每个机器人所需的内存位，受某些限制。我们随后提供的结果需要较少的假设，但只是时间或内存最佳，但不是两者兼而有之。我们还提供了一个经常使用的原语，它采用最初聚集在环节点的机器人，并以时间和内存最佳的方式分散它们，而无需额外的假设。