Consider a team of $k \leq n$ autonomous mobile robots initially placed at a node of an arbitrary graph $G$ with $n$ nodes. The dispersion problem asks for a distributed algorithm that allows the robots to reach a configuration in which each robot is at a distinct node of the graph. If the robots are anonymous, i.e., they do not have any unique identifiers, then the problem is not solvable by any deterministic algorithm. However, the problem can be solved even by anonymous robots if each robot is given access to a fair coin which they can use to generate random bits. In this setting, it is known that the robots require $\Omega(\log{\Delta})$ bits of memory to achieve dispersion, where $\Delta$ is the maximum degree of $G$. On the other hand, the best known memory upper bound is $min \{\Delta, max\{\log{\Delta}, \log{D}\}\}$ ($D$ = diameter of $G$), which can be $\omega(\log{\Delta})$, depending on the values of $\Delta$ and $D$. In this paper, we close this gap by presenting an optimal algorithm requiring $O(\log{\Delta})$ bits of memory.

翻译：考虑一个由 $k\leq n n$ 自动自动移动机器人组成的团队, 最初被放置在任意图形节点上 $G$ 美元和美元节点。 分散问题要求使用分布式算法, 使机器人能够达到每个机器人在图形不同节点上的配置。 如果机器人是匿名的, 也就是说, 他们没有任何独特的识别符号, 那么问题就不会被任何确定性算法所解脱。 但是, 问题也可以通过匿名机器人来解决, 即使让每个机器人获得一个可以用来生成随机位数的公平硬币 。 在此设置中, 已知机器人需要 $\ omega (\ log\ Delta} 的记忆元来实现分散。 如果机器人是匿名的, 也就是说, $\ Delta$\ g$。 另一方面, 最知名的记忆上限是 $ ⁇ Delta, max max $G$ G$$@ data} 。