Probabilistic Asynchronous Arbitrary Pattern Formation (Short Paper)

Abstract

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...

Asynchronous Arbitrary Pattern Formation: the effects of a rigorous approach

“…In our study, as well as in [22], patterns may allow multiplicities, and this deeply affects the design of resolution algorithms. -Other approaches found in the literature to solve PF are probabilistic, see [4,5,32]. In particular, in [4,5] the authors claim to solve APF (and hence PF) with a strategy that is divided into two main phases: the first phase is probabilistic and is used to make asymmetric the input configuration; the second phase is deterministic and solves PF from asymmetric configurations even without assuming any form of multiplicity detection.…”

“…If removing such a case, no configuration might be declared stationary during an execution. Similarly, the definition of static robot from [5] does not describe for instance the case where a robot is not moving but has already performed the Look phase.…”

“…We now redefine the concept of clockwise (and hence of counter-clockwise) direction for a multiset of points P so as to make it independent of a global coordinate system. 4 If P is asymmetric and p = min_view(P \ c(P)), then the direction used to compute V (p) during the analysis of all the rays starting from c(P) is the clockwise direction of P. If P is symmetric, there might be many multipoints of minimum 3 When chirality can be exploited, reflections can be ignored as V + (p) can be always discriminated from V − (q). 4 Indeed, this is not necessary when chirality is assumed.…”

“…In this section we provide fundamental arguments affecting the correctness of [22], [4,5] and [16]. It is worth to remark 6 The technique adopted by means of Procedure STATIONARYMOVE is similar to the so-called cautious move in [20,25].…”

Arbitrary Pattern Formation on Infinite Grid by Asynchronous Oblivious Robots

“…[14] first studied the problem with limited visibility. Randomized pattern formation algorithms were studied in [2,15]. In [5], Das et al investigated the problem of forming a sequence of patterns in a given order.…”