We propose a new probabilistic pattern formation algorithm for oblivious mobile robots that operates in the ASYNC model. Unlike previous work, our algorithm makes no assumptions about the local coordinate systems of robots (the robots do not share a common "North" nor a common "Right"), yet it preserves the ability from any initial configuration that contains at least 5 robots to form any general pattern (and not just patterns that satisfy symmetricity predicates). Our proposal also gets rid of the previous assumption (in the same model) that robots do not pause while moving (so, our robots really are fully asynchronous), and the amount of randomness is kept low -a single random bit per robot per Look-Compute-Move cycle is used. Our protocol consists in the combination of two phases, a probabilistic leader election phase, and a deterministic pattern formation one. As the deterministic phase does not use chirality, it may be of independent interest in the deterministic context. A noteworthy feature of our algorithm is the ability to form patterns with multiplicity points (except the gathering case due to impossibility results), a new feature in the context of pattern formation that we believe is an important asset of our approach.Robots operate in a 2-dimensional Euclidian space. Each robot has its own local coordinate system. For simplicity, we assume the existence (unknown from the robots) of a global coordinate system. Whenever it is clear from the context, we manipulate points in this global coordinate system, but each robot only sees the points in its local system. Two set of points A and B are similar, denoted A ≈ B, if B can be obtained from A by translation, scaling, rotation, or symmetry. A configuration P is a set of positions of robots at a given time. Each robot that looks at this configuration may see different (but similar) set of points.Each time a robot is activated it starts a Look/Compute/Move cycle. After the look phase, a robot obtains a configuration P representing the positions of the robots in its local coordinate system. After an arbitrary delay, the robot computes a path to a destination. Then, it moves toward the destination following the previously computed path. The duration of the move phase, and the delay between two phases, are chosen by an adversary and can be arbitrary long. The adversary decides when robots are activated assuming a fair scheduling i.e., in any configuration, all robots are activated within finite time. The adversary also controls the robots movement along their target path and can stop a robot before reaching its destination, but not before traveling at least a distance δ > 0 (δ being unknown to the robots).An execution of an algorithm is an infinite sequence P(0), P(1), . . . of configurations. An algorithm ψ forms a pattern F if, for any execution P(0), P(1), . . ., there exists a time t such that P(t) ≈ F and P(t ′ ) = P(t) for all t ′ ≥ t. In the sequel, the set of points F denotes the pattern to form. The coordinates of the points in F are given to the robots...

The key feature of wireless sensor networks is to aggregate data collected by individual sensors in an energy efficient manner. We consider two techniques to save energy. The first one is to avoid collisions due to simultaneous transmissions among neighboring nodes. Second, when a node receives data from multiple neighbors, it aggregates these with its own data. Then, one transmission is sufficient to transmit all consolidated data to another neighbor. If the overall delay has to be kept as low as possible, scheduling sensors to avoid collisions while aggregating data becomes challenging. The contribution of this paper is threefold. First, we give tight bounds for the complexity of data aggregation in static networks. In more details, we show that the problem remains NP-complete when the graph is of degree at most three. As it is trivial to solve the problem in static graphs of degree at most two, our result implies that the problem is intrinsically difficult for any practical setting. Second, we investigate the complexity of the same problem in a dynamic network, that is, a network whose topology can evolve through time. In the case of dynamic networks, we show that the problem is NP-complete even in the case where the graph is of degree at most two (and it is trivial to solve the problem when the graph is of degree at most one). Third, we give the first lower and upper bounds for the minimum data aggregation time in a dynamic graph. We observe that even in a well-connected evolving graphs, the optimal solution cannot be found by a distributed algorithm or by a centralized algorithm that does not know the future. Thus we finally give the first approximation algorithm (centralized that knows the future) whose approximation factor is T (n − 1) if there exists a bound T such that there is a journey (a path in a dynamic graph) for all pairs of nodes in every time interval [t, t + T ].

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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 [6], 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).

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