Asynchronous Rendezvous of Anonymous Deterministic Mobile Automata in the Plane

Abstract

Two mobile agents, modeled as points moving in the plane, have to meet at some point. Computationally, agents are identical deterministic finite automata. Each agent has a compass showing the cardinal directions. Agents start at two different points, chosen by the adversary. Each agent makes a series of moves. Before each move it takes a snapshot, which is the disc of radius 1 centered at the current position of the agent. This snapshot is an input that causes the automaton to possibly change state and make the next move in a chosen direction at a chosen distance. Moves of the agents are asynchronous: the adversary controls the possibly variable speed of an agent during each move. Without the possibility of leaving marks, meeting is often impossible, e.g. if agents start simultaneously at a distance larger than 1 and move at the same speed. Hence we allow the agents to use movable pebbles. All pebbles used by an agent are identical and they differ between the agents. A pebble of an agent can be dropped by it, and later possibly picked up again.
Our main result shows that, using a constant number of pebbles, deterministic rendezvous is always possible, regardless of the actions of the asynchronous adversary. The cost of a rendezvous algorithm executed by the agents is the worst-case length of the trajectory of both agents, over all adversary’s decisions. We show that our rendezvous algorithm has cost O(D²), if the initial positions of the agents are at a distance at most D. This complexity is optimal.
As a by-product, we obtain the solution of the leader election problem between two anonymous agents modeled as automata asynchronously navigating in the plane.