Optimal grid exploration by asynchronous oblivious robots

We consider a team of {\em autonomous weak robots} that are endowed with visibility sensors and motion actuators. Autonomous means that the team cannot rely on any kind of central coordination mechanism or scheduler. By weak we mean that the robots are devoid of (1) any (observable) IDs allowing to differentiate them (anonymous), (2) means of communication allowing them to communicate directly, and (3) any way to remember any previous observation nor computation performed in any previous step (oblivious). Robots asynchronously operate in cycles of three phases: Look, Compute, and Move. Furthermore, the network is an anonymous unoriented grid. In such settings, the robots must collaborate to solve a collective task, here the terminating grid exploration (exploration for short), despite being limited with respect to input from the environment, asymmetry, memory, etc. Exploration requires that robots explore the grid and stop when the task is complete. We propose optimal (w.r.t. the number of robots) solutions for the deterministic terminating exploration of a grid shaped network by a team of $k$ asynchronous oblivious robots in the fully asynchronous and non-atomic model, so called CORDA. In more details, we first assume the ATOM model in which each Look-Compute-Move cycle execution is executed atomically, ie every robot that is activated at instant t instantaneously executes a full cycle between t and t+1. ATOM being strictly stronger than CORDA, all impossibility results in ATOM also hold in CORDA. We show that it is impossible to explore a grid of at least three nodes with less than three robots in ATOM. (This first result holds for both deterministic and probabilistic settings.) Next, we show that it is impossible to deterministically explore a (2,2)-Grid with less than 4 robots, and a (3,3)-Grid with less than 5 robots, respectively. Then, we propose deterministic algorithms in CORDA to exhibit the optimal number of robots allowing to explore of a given grid. Our results show that except in two particular cases, 3 robots are necessary and sufficient to deterministically explore a grid of at least three nodes. The optimal number of robots for the two remaining cases is: 4 for the (2,2)-Grid and 5 for the (3,3)-Grid.

💡 Research Summary

The paper investigates the problem of terminating exploration of an anonymous, unoriented grid by a team of asynchronous, oblivious robots that have no identifiers, no direct communication, and no memory of past actions. The robots operate in Look‑Compute‑Move cycles and can only perceive the positions and multiplicities of other robots on the nodes. Two execution models are considered: ATOM, where each activated robot executes a full L‑C‑M cycle atomically, and CORDA, the more general non‑atomic model where Look, Compute and Move may be interleaved arbitrarily by an adversarial scheduler.

First, the authors establish impossibility results in the stronger ATOM model, which automatically apply to CORDA. They prove that any grid containing at least three nodes cannot be explored with two or fewer robots, regardless of whether the algorithm is deterministic or probabilistic. For the specific (2, 2) grid, at least four robots are required, and for the (3, 3) grid, at least five robots are necessary for deterministic exploration. The proofs rely on symmetry arguments: because robots are anonymous and oblivious, symmetric initial configurations lead to indistinguishable views, causing the robots to make identical decisions and thus remain trapped in a symmetric pattern that never visits new nodes. Multiplicity detection is assumed to be strong for the impossibility proofs, ensuring that the adversary cannot hide the presence of multiple robots on a node.

Having identified tight lower bounds, the paper proceeds to construct optimal deterministic algorithms in the CORDA model that meet these bounds. For any grid of size i × j with i·j ≥ 3 (excluding the trivial case k = n), three robots suffice. The algorithm works as follows:

1. Initial positioning – robots move (using their local view) until each occupies a distinct corner of the grid. Because corners have degree two, they are uniquely identifiable from any view.
2. Boundary definition – each robot determines a “search frontier” based on the current multiplicities of neighboring nodes; the frontier is essentially the set of unvisited rows or columns adjacent to the robot’s current location.
3. Frontier traversal – robots advance along the frontier in a consistent direction (e.g., clockwise). When a robot reaches a corner, it turns and continues along the next edge, thereby sweeping the perimeter.
4. Interior coverage – after the perimeter is covered, robots systematically move inward, each time selecting the next unvisited cell that lies on the current frontier. The decision rule depends only on the current view, guaranteeing correctness under arbitrary delays between Look and Move.
5. Termination – once every node has been visited, all robots converge on a single corner, forming a tower (multiple robots on the same node). The algorithm specifies that a robot that perceives a tower stays idle forever, ensuring that the system reaches a terminal configuration.

For the (2, 2) grid, the algorithm is adapted to use four robots: two robots first form a tower on one corner, while the remaining two robots sequentially visit the other two corners. For the (3, 3) grid, five robots are employed. The initial placement deliberately breaks symmetry by placing a “leader” robot on a corner with higher multiplicity; the leader’s presence guides the other robots, which follow the leader’s trail to explore the remaining cells. The leader does not need a compass; it only uses the relative positions of other robots to decide when to turn or move inward.

The correctness of all algorithms is proved by two main arguments. Coverage is guaranteed because at any moment there exists at least one robot whose view indicates an adjacent unvisited node, and the robot’s deterministic rule forces it to move toward that node. Termination is guaranteed because the only configuration in which no robot decides to move is the tower configuration; the algorithm explicitly halts in that state, and the adversarial scheduler cannot force an infinite sequence of moves once the tower is formed. The proofs hold for any possible activation schedule, including the extreme cases where the scheduler activates robots one at a time (sequential execution) or all at once (synchronous execution).

The results demonstrate that, contrary to the ring topology where deterministic exploration requires at least five robots (or four in probabilistic settings), the grid topology is fundamentally easier for oblivious robots: three robots are sufficient for all but two small grids. Moreover, the paper shows that the optimal number of robots does not depend on number‑theoretic conditions such as coprimality between the number of robots and the number of nodes, which are required in many ring‑exploration algorithms.

In conclusion, the authors provide a complete characterization of the minimal number of oblivious, anonymous, asynchronous robots needed to deterministically explore and terminate on any finite grid. They deliver matching lower‑bound impossibility proofs and constructive upper‑bound algorithms, thereby closing the gap for this class of distributed robotic systems. The work opens several avenues for future research, including probabilistic algorithms that might reduce the robot count further, extensions to grids with obstacles or dynamic changes, and investigations of weaker sensing models (limited visibility, no multiplicity detection).