A swarm of anonymous oblivious mobile robots, operating in deterministic Look-Compute-Move cycles, is confined within a circular track. All robots agree on the clockwise direction (chirality), they are activated by an adversarial semi-synchronous scheduler (SSYNCH), and an active robot always reaches the destination point it computes (rigidity). Robots have limited visibility: each robot can see only the points on the circle that have an angular distance strictly smaller than a constant $\vartheta$ from the robot's current location, where $0<\vartheta\leq\pi$ (angles are expressed in radians). We study the Gathering problem for such a swarm of robots: that is, all robots are initially in distinct locations on the circle, and their task is to reach the same point on the circle in a finite number of turns, regardless of the way they are activated by the scheduler. Note that, due to the anonymity of the robots, this task is impossible if the initial configuration is rotationally symmetric; hence, we have to make the assumption that the initial configuration be rotationally asymmetric. We prove that, if $\vartheta=\pi$ (i.e., each robot can see the entire circle except its antipodal point), there is a distributed algorithm that solves the Gathering problem for swarms of any size. By contrast, we also prove that, if $\vartheta\leq \pi/2$, no distributed algorithm solves the Gathering problem, regardless of the size of the swarm, even under the assumption that the initial configuration is rotationally asymmetric and the visibility graph of the robots is connected. The latter impossibility result relies on a probabilistic technique based on random perturbations, which is novel in the context of anonymous mobile robots. Such a technique is of independent interest, and immediately applies to other Pattern-Formation problems.

中文翻译：

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匿名遗忘机器人聚集在一个能见度有限的圆圈上

一群匿名的、不经意的移动机器人，以确定性的 Look-Compute-Move 循环运行，被限制在一个圆形轨道内。所有机器人都同意顺时针方向（手性），它们由对抗性半同步调度程序（SSYNCH）激活，并且主动机器人总是到达它计算的目标点（刚性）。机器人的能见度有限：每个机器人只能看到圆上与机器人当前位置的角距离严格小于常数 $\vartheta$ 的点，其中 $0<\vartheta\leq\pi$（角度表示为弧度）。我们研究这样一群机器人的聚集问题：即所有机器人最初都在圆上的不同位置，它们的任务是在有限的圈数内到达圆上的同一点，无论调度程序以何种方式激活它们。请注意，由于机器人的匿名性，如果初始配置是旋转对称的，则此任务是不可能的；因此，我们必须假设初始配置是旋转不对称的。我们证明，如果$\vartheta=\pi$（即每个机器人都可以看到除其对映点以外的整个圆），则存在一种分布式算法可以解决任何规模的群体的聚集问题。相比之下，我们还证明了，如果 $\vartheta\leq\pi/2$，无论集群的大小如何，即使假设初始配置是旋转不对称且可见性机器人的图形是连接的。后一种不可能结果依赖于基于随机扰动的概率技术，这在匿名移动机器人的背景下是新颖的。这种技术具有独立的意义，并立即适用于其他模式形成问题。