PhD Proposal

Observe and Remain SilentCommunication-less agent location discovery

MFCS 2012Speaker:Thomas Gorry(University of Liverpool)

Co-authors:Tom Friedetzky (Durham University)

Leszek GsieniecRussell Martin(University of Liverpool)

Introduction of the authors.

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IntroductionSwarms 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.

2/16Popularised topic of swarms: primitive cost effective groups of entitiesdeployed to perform a task in hard to reach or hostile environment.

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IntroductionMost 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.

3/16Majority: global pictureSeveral topology discovery in graph based networks or geometric settings.Our focus is on a distributed network modelExtremely limited communication and perception.3

Search/Coordination ProblemsNetwork Search/Discovery

Rendezvous/Gathering

Network Patrolling4/16Agent Location DiscoveryS&D: Evacuation problem, Treasure hunting, Cops and Robbers etc.

Rendezvous/Gathering Agents work together to meet up at some point in the network, perform some synchronised gathering task.

Network Patrolling Agent s work together to secure an area or boundary of the network either through regular patrols or surveillance monitoring.

All of these tasks generally require agents to know the location of each other at some point. This in itself is a non trivial task. Our work in this paper refers to such a problem.4Network ModelThe 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 = 1.5/161We adopt the distributed network model of a ring where agents can travel along only the circumference, this being a unit circumference and agents having the speed of 1.

There are n agents on the ring, each of witch are anonymous and uniform.

The agents are aware of n .

Synchronised rounds, round = Time it.5Network ModelAt 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.

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At the beginning of each round an agent chooses a direction of its move from C or AC and moves at unit speed.

When an agent collides with another (agent) it instantly starts moving with the same speed but in the opposite direction.

The agents cannot leave marks on the ring, they have zero visibility and cannot exchange messages at any time.

Upon the conclusion of a round each agent gets some limited information about where they are relative to where they started the round.6Problem and ResultsAgents 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.

7/16The setting is that we have n number of agents distributed at arbitrary and distinct positions along the circumference of the ring. The only position an agent is aware of is their own.

Our task is for each of the agents to figure out the starting location of the other agents on the ring.

When it comes to this task we must consider two cases, one being when n is even, and I shall talk more on this later on, but for now let us consider only the case when n is odd for witch we have a fully distributed randomised algorithm that performs the task with high probability in n log^2 n rounds.7As 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.

Further details8/16

Each agent wants to learn p0 to pn-1.

Expand on 3rd point.8Algorithm OverviewThroughout 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.9/16

The movement and collisions of the agents is one of a Periodic system.

So our algorithm is based mainly on two main concepts.9bi-1, ai+1bi+1, aibi, ai-1Baton ConceptAt the start of each stage every ai holds a unique virtual baton bi that 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.10/16After 1 round agents change positions but batons do not.

Batons travel along the circumference and are unaffected by other batons or agents.

Agents travel along the circumference but are affected by the movement of other agentsbi, aibi+1, ai+1bi-1, ai-1The first of these is the concept of batons.

This is a proof concept only as these batons do not actually exist but are virtual.

Much like in a relay race when a runner makes contact with their team mate they hand off a baton to them these agents hand off a baton to the other agent as well. This leads to batons traveling once around the circumference of the ring in one round.

For example here we have agents, shown as the large ball, and their respective batons, shown by the inner ball. This animation shows how the batons are unaffected by the movements of other batons or agents and returning to their original positions after one round.10Batons and LocationsAt 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+1.11/16As we show that each bi returns to pi at the end of each round we must also conclude that it was carried there by an Agent and so at the end of each round we have an agent in one of the positions, pi

Also as agents can not over pass then they have simply shifted their positions around the circumference of the ring while keeping this relative ordering.11Rotation Indexn = 5nc = 1na = 4

Rotation Index r = (1 4) = -3

Positions found = /512345During 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.

12/16On the basis of the previous observation

We notice that that the amount of shift that occurs each round is dependent on what we call the rotational index, shown here as r.

We get r by deducting the number of anticlockwise agents from the number of agents traveling clockwise. Recall each agent randomly and independently chose a direction at the start of the stage.

Lets take this red guy here as an example. He is the only one who chose to go clockwise initially. So as the others are anticlockwise he get shuved around the ring in this anticlockwise direction. Here r is -3 so this means that he would experience a shift of 3 starting positions anticlockwise during the round. (this would be clockwise is the sum produced a positive number).12Location Discovery AlgorithmIf r = nc- na is 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.

13/163nd point what we want is a successful stage.13Thus 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.Location Discovery Algorithm14/16The proof for the second point is non trivial but I will say here that we use the Prime number theorem, specifically the density of primes, to get this probability.

As each stage is dependent on rounds we get n log^2 n14Further DetailsEven n Still solvable in O(n log2 n) 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.

15/16We can compute this in the same time for when n is even as well however we need to change the information the agent receives after each round to include not only their relative location to their starting position but also where their first collision during that round occurred relative to this also.

If the agents are unaware of n they must know the circumference of the circle for it to still be solvable.

Having a uniform sense of direction is not necessary for this algorithm to work, however after the starting positions are found it may be that you want to perform some task that requires this. In this case the agents can agree on a common sense of direction during the first round.

A task you may want to perform with this information could be boundary patrolling. This would take one more round after completion of our algorithm for the agents to order themselves with equal spacing and patrol in the same direction with the same speed.15SummaryAgents 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.

16/16Just to wrap up, the scenario is there are n agents who can not communicate and they move with a constant speed. They know n or the circumference of the circle and they randomly choose a starting direction at the start of each stage. A stage ends when agents arrive back at their original starting positions at the end of some round.

This can be done with high probability on the order of n log^2 n using our algorithm.

Thank you very much for listening.16