Optimal Probabilistic Ring Exploration by Asynchronous Oblivious Robots

We consider a team of $k$ identical, oblivious, asynchronous mobile robots that are able to sense (\emph{i.e.}, view) their environment, yet are unable to communicate, and evolve on a constrained path. Previous results in this weak scenario show that initial symmetry yields high lower bounds when problems are to be solved by \emph{deterministic} robots. In this paper, we initiate research on probabilistic bounds and solutions in this context, and focus on the \emph{exploration} problem of anonymous unoriented rings of any size. It is known that $\Theta(\log n)$ robots are necessary and sufficient to solve the problem with $k$ deterministic robots, provided that $k$ and $n$ are coprime. By contrast, we show that \emph{four} identical probabilistic robots are necessary and sufficient to solve the same problem, also removing the coprime constraint. Our positive results are constructive

Optimal byzantine resilient convergence in oblivious robot networks

Given a set of robots with arbitrary initial location and no agreement on a global coordinate system, convergence requires that all robots asymptotically approach the exact same, but unknown beforehand, location. Robots are oblivious-- they do not recall the past computations -- and are allowed to move in a one-dimensional space. Additionally, robots cannot communicate directly, instead they obtain system related information only via visual sensors. We draw a connection between the convergence problem in robot networks, and the distributed \emph{approximate agreement} problem (that requires correct processes to decide, for some constant $\epsilon$, values distance $\epsilon$ apart and within the range of initial proposed values). Surprisingly, even though specifications are similar, the convergence implementation in robot networks requires specific assumptions about synchrony and Byzantine resilience. In more details, we prove necessary and sufficient conditions for the convergence of mobile robots despite a subset of them being Byzantine (i.e. they can exhibit arbitrary behavior). Additionally, we propose a deterministic convergence algorithm for robot networks and analyze its correctness and complexity in various synchrony settings. The proposed algorithm tolerates f Byzantine robots for (2f+1)-sized robot networks in fully synchronous networks, (3f+1)-sized in semi-synchronous networks. These bounds are optimal for the class of cautious algorithms, which guarantee that correct robots always move inside the range of positions of the correct robots

The Random Bit Complexity of Mobile Robots Scattering

We consider the problem of scattering $n$ robots in a two dimensional continuous space. As this problem is impossible to solve in a deterministic manner, all solutions must be probabilistic. We investigate the amount of randomness (that is, the number of random bits used by the robots) that is required to achieve scattering. We first prove that $n \log n$ random bits are necessary to scatter $n$ robots in any setting. Also, we give a sufficient condition for a scattering algorithm to be random bit optimal. As it turns out that previous solutions for scattering satisfy our condition, they are hence proved random bit optimal for the scattering problem. Then, we investigate the time complexity of scattering when strong multiplicity detection is not available. We prove that such algorithms cannot converge in constant time in the general case and in $o(\log \log n)$ rounds for random bits optimal scattering algorithms. However, we present a family of scattering algorithms that converge as fast as needed without using multiplicity detection. Also, we put forward a specific protocol of this family that is random bit optimal ($n \log n$ random bits are used) and time optimal ($\log \log n$ rounds are used). This improves the time complexity of previous results in the same setting by a $\log n$ factor. Aside from characterizing the random bit complexity of mobile robot scattering, our study also closes its time complexity gap with and without strong multiplicity detection (that is, $O(1)$ time complexity is only achievable when strong multiplicity detection is available, and it is possible to approach it as needed otherwise)

Optimal Torus Exploration by Oblivious Robots

International audienceWe consider autonomous robots that are endowed with motion actuators and visibility sensors. The robots we consider are weak, i.e., they are anonymous, uniform, unable to explicitly communicate, and oblivious (they do not remember any of their past actions). In this paper, we propose an optimal (w.r.t. the number of robots) solution for the terminating exploration of a torus-shaped network by a team of $k$ such robots. In more details, we first show that it is impossible to explore a simple torus of arbitrary size with (strictly) less than four robots, even if the algorithm is probabilistic. If the algorithm is required to be deterministic, four robots are also insufficient. This negative result implies that the only way to obtain an optimal algorithm (w.r.t. the number of robots participating to the algorithm) is to make use of probabilities. Then, we propose a probabilistic algorithm that uses four robots to explore all simple tori of size $\ell \times L$, where $7 \leq \ell \leq L$. Hence, in such tori, four robots are necessary and sufficient to solve the (probabilistic) terminating exploration. As a torus can be seen as a 2-dimensional ring, our result shows, perhaps surprisingly, that increasing the number of possible symmetries in the network (due to increasing dimensions) does not come at an extra cost w.r.t. the number of robots that are necessary to solve the problem

Optimal torus exploration by oblivious robots

International audienceWe deal with a team of autonomous robots that are endowed with motion actuators and visibility sensors. Those robots are weak and evolve in a discrete environment. By weak, we mean that they are anonymous, uniform, unable to explicitly communicate, and oblivious. We first show that it is impossible to solve the terminating exploration of a simple torus of arbitrary size with less than 4 or 5 such robots, respectively depending on whether the algorithm is probabilistic or deterministic. Next, we propose in the SSYNC model a probabilistic solution for the terminating exploration of torus-shaped networks of size ℓ×L, where 7≤ℓ≤L, by a team of 4 such weak robots. So, this algorithm is optimal w.r.t. the number of robots