We consider a swarm of $n$ robots in \mathbb{R}^d. The robots are oblivious, disoriented (no common coordinate system/compass), and have limited visibility (observe other robots up to a constant distance). The basic formation task gathering requires that all robots reach the same, not predefined position. In the related near-gathering task, they must reach distinct positions such that every robot sees the entire swarm. In the considered setting, gathering can be solved in $\mathcal{O}(n + Δ^2)$ synchronous rounds both in two and three dimensions, where $Δ$ denotes the initial maximal distance of two robots. In this work, we formalize a key property of efficient gathering protocols and use it to define $λ$-contracting protocols. Any such protocol gathers $n$ robots in the $d$-dimensional space in $\mathcal{O}(Δ^2)$ synchronous rounds. Moreover, we prove a corresponding lower bound stating that any protocol in which robots move to target points inside of the local convex hulls of their neighborhoods -- $λ$-contracting protocols have this property -- requires $Ω(Δ^2)$ rounds to gather all robots. Among others, we prove that the $d$-dimensional generalization of the GtC-protocol is $λ$-contracting. Remarkably, our improved and generalized runtime bound is independent of $n$ and $d$. The independence of $d$ answers an open research question. We also introduce an approach to make any $λ$-contracting protocol collisionfree to solve near-gathering. The resulting protocols maintain the runtime of $Θ(Δ^2)$ and work even in the semi-synchronous model.

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 short 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 $\mathcalO łeft(łog ^2 n\right)$ 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 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 2x2 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, can a robot 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 FSYNC model. For this problem, we present a gathering algorithm which needs linear time. This result generalizes the result from [1], 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, point-shaped robots on a two-dimensional grid in asymptotically optimal time O(n) in the fully synchronous FSYNC 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 x 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 see its grid neighbors only up to a constant L1-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 robot -- which 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(n2).