Scatter Problem

Scatter Problem

Scatter Problem

!  Convergence

Regardless of the initial positions of the

matter clusters – mobile robots, entities, or agents -,

no two such things are eventually located at the same position.

!  Closure

Starting from a configuration where non two

matter clusters – mobile robots, entities, or agents -, are located at the same position,

no two such things are located at the same position thereafter.

Model

!  Autonomous

Weak!robots

Move on the plane

Undistinguishable, same program No past

Observation of positions

Detect that more than one robot at a given position Common coordinate system

!  Mobile

!  Anonymous

!  Oblivious

!  No Explicit Communications

!  Multiplicity Detection

!  Localization Knowledge

Model

!  Time Infinite sequence of time instants

!  At each time instant t

" 

Each robot is either active or inactive

" 

Each active robot

! 

Observes

! 

Computes

! 

Moves

" 

At least one robot is activated

" 

Every robot is infinitely often activated

(fairness)

Deterministic Scatter

Weak!robots

Impos sible

Randomized Scatter

Compute the Voronoi Diagram If Random() = 0

then move arbitrarily in my cell

Related Problems

!  Gathering Problem (GP)

Regardless of the initial positions on the plane of the mobile robots, make them gather at one point in a finite number of time instants.

[Ando et al. 99] [ Suzuki Yamashita 99] [Flocchini et al. 01]

[Cieliebak Prencipe 02] [Cieliebak et al. 03]

!  Pattern Formation Problem (PFP)

Regardless of the initial positions on the plane of the mobile robots, make them to form a desired pattern in a finite number of time

instants.

For instance: regular n-gon (circle formation problem)

[ Suzuki Yamashita 99] [Flocchini et al. 99] [Flocchini et al. 01]

[Defago Konagaya 02] [Katreniak 05] [Dieudonné et al. 06]

[Dieudonné Petit 07]

Related Problems

!  Gathering Problem (GP)

!  Pattern Formation Problem (PFP)

!  Convergence: Starting from an arbitrary configuration, the robots eventually form a desirable configuration

!  Closure: Starting from a configuration where the robots are in the desirable

configuration, they remain in a desirable configuration thereafter

Common Requirements for

specification:

Self-Stabilization

!  Convergence

Regardless of the initial state of the system, a self-stabilizing

system is guaranteed to converge to the indented behavior in a finite number of time instants

!  Closure

If the system behaves according to its specifications, then it behaves according to its specification forever

Convergence phase

Expected behavior

closure

with a probability equal to 1

Probabilistic

Related Problems

!  Gathering Problem (GP)

!  Pattern Formation Problem (PFP)

!  Convergence and Closure

Common Requirements for specification:

!  In the initial configuration, no two robots are located at the same position

No Deterministic solution for GP or PFP is

«!truly!» self-stabilizing

!

Probabilistic Self-Stabilization

Combination of our scatter algorithm and any algorithm for the GP or PFP

… but unfortunately it does not work yet!

Randomized Scatter

If there exists a position with more than one robot

then Compute the Voronoi Diagram If Random() = 0

then move arbitrarily in my cell

How to make them quiescent?

Termination detection of the

scatter algorithm

Self-Stabilizing Gathering

If there exists at least two positions with more than one robot

then Scatter

else Gather

Self-Stabilizing

Pattern Formation

If there exists at least one position with more than one robot

then Scatter

else Pattern Formation

Conclusion

!  Scatter Problem

" 

No Deterministic Solution

" 

Randomized Solution

!  Randomized Self-Stabilizing Solutions

" 

Gathering Problem

" 

Pattern Formation Problems