Observe and Remain Silent Communication-less agent location discovery MFCS 2012

1. Observe and Remain SilentCommunication-less agent location discoveryMFCS 2012 Speaker: Thomas Gorry (University of Liverpool) Co-authors: Tom Friedetzky (Durham University) Leszek Gąsieniec Russell Martin (University of Liverpool)

2. Introduction • Swarms are large groups of entities (robots, agents) that can be deployed to perform an exploration or a monitoring task. • Many algorithms exist to deal with a variety of control problems in robot swarms.

3. Introduction • Most of these algorithms depend on access to the global picture of the network with each agent able to monitor the performance of all agents. • Several agent network exploration algorithms that mainly focus on network topology discovery. • Our work has focus on a distributed network model where communication and perception of agents are very limited.

4. Search/Coordination Problems • Agent Location Discovery • Network Search/Discovery • Rendezvous/Gathering • Network Patrolling

5. Network Model • The network is a ring with a unit circumference. • The ring is populated by n uniform anonymous agents. • Agents know n. • All agents travel with the uniform speed 1. • Agents perform actions in synchronised rounds, with each round being the time it would take an agent to walk the circumference of the circle if n = 1. 1

6. Network Model • At the start of each round agents are allowed to choose direction (clockwise, C, or anti-clockwise, A) randomly and independently. • Agents cannot overpass one another and upon collision instantly start moving with the same unit speed in the opposite direction. • Agents have zero visibility, cannot exchange messages or leave marks on the circle. • At the end of each round each agent gets some limited information about its trajectory adopted during this round that is stored for further analysis.

7. Problem and Results • Agents are initially located on the circle at arbitrary but distinct positions unknown to other agents. • THE TASK Each agent must discover the starting location of every other agent on the circle. • THE RESULT Our main result is a fully distributed randomised algorithm that solves the location discovery task whp in O(n log2 n) rounds.

8. Further details • As agents never overpass (exchange relative positions) we assume they are arranged cyclically from a0 to an-1. We also use pi to denote the original position of ai. (These labels are not disclosed to the agents). • In the algorithm we define a stage as the number of rounds it takes each ai to arrive at pi on the conclusion of the last round. • The agents can only choose their direction at the start of a stage. • Each stage is formed of at most n consecutive rounds of length 1.

9. Algorithm Overview • Throughout any stage the exact location and movement direction of agents depend solely on collisions with their neighbours. • A stage ends when, at the end of a round, the locations of all the agents coincide with their initial positions in the first round.

10. Baton Concept bi+1, ai+1 bi+1, ai • At the start of each stage every ai holds a unique virtual baton bithat upon a collision is swapped with the agent it collided with. • At the beginning of each round baton bi resides at pi. • Throughout the round, bi keeps moving in the same direction with the speed of 1. (No collisions) • Therefore bi must arrive at pi on the conclusion of the round. bi, ai bi, ai-1 After 1 round agents change positions but batons do not. Batons travel along the circumference and are unaffectedby other batons or agents. Agents travel along the circumference but are affected by the movement of other agents bi-1, ai-1 bi-1, ai+1

11. Batons and Locations • At the end of each round the initial position pi of the agent ai is occupied by the baton bi carried by some agent. • We observe that if ai resides at position pi+r for some integer r, since agents do not exchange relative positions ai-1 and ai+1 must reside at the respective positions pi+r-1 and pi+r+1.

12. Rotation Index During one round all agents are rotated along the initial starting positions by a rotation index of r = (nc– na). r depends only on the initial choice of random directions adopted by the agents. n = 5 nc = 1 na = 4 Rotation Index r = (1 – 4) = -3 3 1 2 4 5 Positions found = /5

13. Location Discovery Algorithm • If r = nc- nais relatively prime with n, denoted by gcd(nc- na, n) = 1, then the Stage will last exactly n rounds. • During such a stage, all initial positions of the agents are mutually discovered. • The stage like this is called successful since it concludes the discovery process.

14. Location Discovery Algorithm • Thus the main goal is to find out how quickly one can generate the successful stage by sending each agent to a random direction at the beginning of each stage. • If n is odd, one can prove that the choice of random directions leads to success with probability 1/log n • Therefore in order to generate a successful stage with high probability we need O(log2 n) stages • THEOREM: The location discovery problem can be solved in time O(n log2 n) with high probability.

15. Further Details • Even n – Still solvable in O(n log2n) time if extra information (point of the first collision in each round) is available to agents. • Unknown n – Solvable if the length of the circle is known. • Sense of Direction Agreement – Solvable when r ≠ 0. • Equidistant distribution and Patrolling – Achieved in one stage after location discovery is accomplished.

16. Summary • Agents can not communicate, move with constant speed, but must know in advance either n or the circumference of the circle. • Agents randomly choose a starting direction at the start of each stage that is concluded with agents’ arrival to their initial positions. • THEOREM: The location discovery problem can be solved in time O(n log2 n) with high probability.