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.

In this paper, we consider the partial gathering problem of mobile agents in synchronous dynamic bidirectional ring networks. When k agents are distributed in the network, the partial gathering problem requires, for a given positive integer g (< k), that agents terminate in a configuration such that either at least g agents or no agent exists at each node. So far, the partial gathering problem has been considered in static graphs. In this paper, we start considering partial gathering in dynamic graphs. As a first step, we consider this problem in 1-interval connected rings, that is, one of the links in a ring may be missing at each time step. In such networks, focusing on the relationship between the values of k and g, we fully characterize the solvability of the partial gathering problem and analyze the move complexity of the proposed algorithms when the problem can be solved. First, we show that the g-partial gathering problem is unsolvable when k <= 2g. Second, we show that the problem can be solved with O(n log g) time and the total number of O(gn log g) moves when 2g + 1 <= k <= 3g - 2. Third, we show that the problem can be solved with O(n) time and the total number of O(kn) moves when 3g - 1 <= k <= 8g - 4. Notice that since k = O(g) holds when 3g - 1 <= k <= 8g - 4, the move complexity O(kn) in this case can be represented also as O(gn). Finally, we show that the problem can be solved with O(n) time and the total number of O(gn) moves when k >= 8g - 3. These results mean that the partial gathering problem can be solved also in dynamic rings when k >= 2g + 1. In addition, agents require a total number of \Omega(gn) moves to solve the partial (resp., total) gathering problem. Thus, when k >= 3g - 1, agents can solve the partial gathering problem with the asymptotically optimal total number of O(gn) moves.

An autonomous mobile robot system consisting of many mobile computational entities (called robots) attracts much attention of researchers, and to clarify the relation between the capabilities of robots and solvability of the problems is an emerging issue for a recent couple of decades. Generally, each robot can observe all other robots as long as there are no restrictions for visibility range or obstructions, regardless of the number of robots. In this paper, we provide a new perspective on the observation by robots; a robot cannot necessarily observe all other robots regardless of distances to them. We call this new computational model defected view model. Under this model, in this paper, we consider the gathering problem that requires all the robots to gather at the same point and propose two algorithms to solve the gathering problem in the adversarial ($N$,$N-2$)-defected model for $N \geq 5$ (where each robot observes at most $N-2$ robots chosen adversarially) and the distance-based (4,2)-defected model (where each robot observes at most 2 closest robots to itself) respectively, where $N$ is the number of robots. Moreover, we present an impossibility result showing that there is no (deterministic) gathering algorithm in the adversarial or distance-based (3,1)-defected model. Moreover, we show an impossibility result for the gathering in a relaxed ($N$, $N-2$)-defected model.