We consider the following variant of the two dimensional gathering problem for swarms of robots: Given a swarm of n indistinguishable, point shaped robots on a two dimensional grid. Initially, the robots form a closed chain on the grid and must keep this connectivity during the whole process of their gathering. Connectivity means, that neighboring robots of the chain need to be positioned at the same or neighboring points of the grid. In our model, gathering means to keep shortening the chain until the robots are located inside a 2 × 2 subgrid. Our model is completely local (no global control, no global coordinates, no compass, no global communication or vision, . . . ). Each robot can only see its next constant number of left and right neighbors on the chain. This fixed constant is called the viewing path length. All its operations and detections are restricted to this constant number of robots. Other robots, even if located at neighboring or the same grid point cannot be detected. Only based on the relative positions of its detectable chain neighbors, a robot can decide to obtain a certain state. Based on this state and their local knowledge, the robots do local modifications to the chain by moving to neighboring grid points without breaking the chain. These modifications are performed without the knowledge whether they lead to a global progress or not. We assume the fully synchronous FSYN C model. For this problem, we present a gathering algorithm which needs linear time. This result generalizes the result from [KM09], where an open chain with specified distinguishable (and fixed) endpoints is considered.
In this paper, we solve the local gathering problem of a swarm of n indistinguishable, pointshaped robots on a two dimensional grid in asymptotically optimal time O(n) in the fully synchronous FSYN C time model. Given an arbitrarily distributed (yet connected) swarm of robots, the gathering problem on the grid is to locate all robots within a 2 × 2-sized area that is not known beforehand. Two robots are connected if they are vertical or horizontal neighbors on the grid. The locality constraint means that no global control, no compass, no global communication and only local vision is available; hence, a robot can only see its grid neighbors up to a constant L 1 -distance, which also limits its movements. A robot can move to one of its eight neighboring grid cells and if two or more robots move to the same location they are merged to be only one robot. The locality constraint is the significant challenging issue here, since robot movements must not harm the (only globally checkable) swarm connectivity. For solving the gathering problem, we provide a synchronous algorithm -executed by every robotwhich ensures that robots merge without breaking the swarm connectivity. In our model, robots can obtain a special state, which marks such a robot to be performing specific connectivity preserving movements in order to allow later merge operations of the swarm. Compared to the grid, for gathering in the Euclidean plane for the same robot and time model the best known upper bound is O(n 2 ) [DKL + 11].
Routing is a challenging problem for wireless ad hoc networks, especially when the nodes are mobile and spread so widely that in most cases multiple hops are needed to route a message from one node to another. In fact, it is known that any online routing protocol has a poor performance in the worst case, in a sense that there is a distribution of nodes resulting in bad routing paths for that protocol, even if the nodes know their geographic positions and the geographic position of the destination of a message is known. The reason for that is that radio holes in the ad hoc network may require messages to take long detours in order to get to a destination, which are hard to find in an online fashion.In this paper, we assume that the wireless ad hoc network can make limited use of long-range links provided by a global communication infrastructure like a cellular infrastructure or a satellite in order to compute an abstraction of the wireless ad hoc network that allows the messages to be sent along near-shortest paths in the ad hoc network. We present distributed algorithms that compute an abstraction of the ad hoc network in O log 2 n time using long-range links, which results in c-competitive routing paths between any two nodes of the ad hoc network for some constant c if the convex hulls of the radio holes do not intersect. We also show that the storage needed for the abstraction just depends on the number and size of the radio holes in the wireless ad hoc network and is independent on the total number of nodes, and this information just has to be known to a few nodes for the routing to work. ACM Subject Classification C.2.4 Distributed SystemsThroughout this paper, we consider V ⊂ R 2 to be a set of nodes in the Euclidean plane with unique IDs (e.g., phone numbers), where |V | = n. For any given pair of nodes u = (u x , u y ), v = (v x , v y ), we denote the Euclidean distance between u and v by ||uv|| =We model our cell phone network as a hybrid directed graph H = (V, E, E AH ) where the node set V represents the set of cell phones, an edge (v, w) is in E whenever v knows the phone number (or simply ID) of w, and an edge (v, w) ∈ E is also in the ad hoc edge set E AH whenever v can send a message to w using its Wifi interface. For all edges (v, w) ∈ E \ E AH , v can only use a long-range link to directly send a message to w. We adopt the unit disk graph model for the edges in E AH .Definition 1. For any point set V ⊆ R 2 the Unit Disk Graph of V , UDG (V ), is a bi-directed graph that contains all edges (u, v) with ||uv|| ≤ 1. C V I T 2 0 1 6
This work focuses on the following question related to the Gathering problem of n autonomous, mobile robots in the Euclidean plane: Is it possible to solve Gathering of robots that do not agree on any axis of their coordinate systems (disoriented robots) and see other robots only up to a constant distance (limited visibility) in o(n 2 ) fully synchronous rounds (the Fsync scheduler)? The best known algorithm that solves Gathering of disoriented robots with limited visibility in the OBLOT model (oblivious robots) needs Θ n 2 rounds [8]. The lower bound for this algorithm even holds in a simplified closed chain model, where each robot has exactly two neighbors and the chain connections form a cycle. The only existing algorithms achieving a linear number of rounds for disoriented robots assume robots that are located on a two dimensional grid [1] and [7]. Both algorithms make use of locally visible lights to communicate state information (the LUMIN OU S model).In this work, we show for the closed chain model, that n disoriented robots with limited visibility in the Euclidean plane can be gathered in Θ (n) rounds assuming the LUMIN OU S model. The lights are used to initiate and perform so-called runs along the chain. For the start of such runs, locally unique robots need to be determined. In contrast to the grid [1], this is not possible in every configuration in the Euclidean plane. Based on the theory of isogonal polygons by Branko Grünbaum, we identify the class of isogonal configurations in which -due to a high symmetry -no such locally unique robots can be identified. Our solution combines two algorithms: The first one gathers isogonal configurations; it works without any lights. The second one works for non-isogonal configurations; it identifies locally unique robots to start runs, using a constant number of lights. Interleaving these algorithms solves the Gathering problem in O (n) rounds.
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