We consider a repulsion actuator located in an n-sided convex environment full of point particles. When the actuator is activated, all the particles move away from the actuator. We study the problem of gathering all the particles to a point. We give an $O(n^2)$ time algorithm to compute all the actuator locations that gather the particles to one point with one activation, and an $O(n)$ time algorithm to find a single such actuator location if one exists. We then provide an $O(n)$ time algorithm to place the optimal number of actuators whose sequential activation results in the gathering of the particles when such a placement exists.

中文翻译：

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排斥相聚

我们考虑了位于具有点粒子的n侧凸环境中的推斥致动器。激活执行器后，所有粒子都将离开执行器。我们研究了将所有粒子聚集到一个点的问题。我们给$Ø（{ñ}^2）$ 时间算法，通过一次激活计算将粒子聚集到一个点的所有执行器位置，然后执行一次 $Ø（ñ）$时间算法来找到一个这样的执行器位置（如果存在）。然后，我们提供$Ø（ñ）$ 时间算法来放置最佳数量的执行器，当存在这种执行器时，其顺序激活会导致粒子聚集。