Observe and Remain Silent (Communication-Less Agent Location Discovery)

We study a randomised distributed communication-less coordination mechanism for

n

uniform anonymous agents located on a circle with unit circumference. We assume the agents are located at arbitrary but distinct positions, unknown to other agents. The agents perform actions in synchronised rounds. At the start of each round an agent chooses the direction of its movement (

clockwise

or

anticlockwise

), and moves at unit speed during this round. Agents are not allowed to overpass, i.e., when an agent collides with another it instantly starts moving with the same speed in the opposite direction. Agents cannot leave marks on the ring, have zero vision and cannot exchange messages. However, on the conclusion of each round each agent has access to (some, not necessarily all) information regarding its trajectory during this round. This information can be processed and stored by the agent for further analysis.

The

location discovery task

to be performed by each agent is to determine the initial position of every other agent and eventually to stop at its initial position, or proceed to another task, in a fully synchronised manner. Our primary motivation is to study distributed systems where agents collect the minimum amount of information that is necessary to accomplish this location discovery task.

Our main result is a fully distributed randomised (Las Vegas type) algorithm, solving the

location discovery problem

w.h.p.

in

O

(

n

log

2

n

) rounds (assuming the agents collect sufficient information). Note that our result also holds if initially the agents do not know the value of

n

and they have no coherent sense of direction.