Over the past decades, the Gathering problem, which asks to gather a group of robots in finite time given some restrictions, has been intensively studied. In this paper, we are given a group of n autonomous, dimensionless, deterministic, and anonymous robots, with bounded viewing range. Assuming a continuous time model, the goal is to gather these robots into one point in finite time. We introduce a simple convergence criterion that defines a new class of algorithms which perform gathering in $O(nd)$ time, where d is the diameter of the initial robot configuration. We show that some gathering algorithms in the literature belong to this class and propose two new algorithms that belong to this class and have quadratic running time, namely, Go-To-The-Relative-Center algorithm (GTRC) and Safe-Go-To-The-Relative-Center algorithm (S-GTRC). We prove that the latter can perform gathering without collision by using a slightly more complex robot model: non oblivious, chiral, and luminous (i.e. robots have observable external memory, as in [8]). We also consider a variant of the Gathering problem, the Near-Gathering problem, in which robots must get close to each other without colliding. We show that S-GTRC solves the Near-Gathering problem in quadratic time and assumes a weaker robot model than the one assumed in the current state-of-the-art.

在过去的几十年中，对集会问题进行了深入研究，该问题要求在一定的限制下在有限的时间内收集一组机器人。在本文中，我们给出了由n个自治，无量纲，确定性和匿名机器人组成的组，它们具有有限的可视范围。假设采用连续时间模型，则目标是在有限时间内将这些机器人聚集到一个点中。我们介绍了一个简单的收敛准则，该准则定义了一类新的算法，可以在$Ø（ñd）$时间，d是初始机器人配置的直径。我们证明了文献中的一些收集算法属于此类，并提出了两种属于此类并且具有二次运行时间的新算法，即相对中心相对移动算法（GTRC）和安全相对运行安全算法-相对中心算法（S-GTRC）。我们证明了后者可以通过使用稍微复杂一些的机器人模型来执行采集而不会发生碰撞：非遗忘，手性和发光（即，机器人具有可观察到的外部存储器，如[8]所示）。我们还考虑了“聚集”问题的一种变体，即“近聚集”问题，其中机器人必须彼此靠近而不发生碰撞。我们表明，S-GTRC在二次时间内解决了“近距离聚集”问题，并且假定的机器人模型比当前最新技术的模型要弱。