Scattering
1.
Scatter Problem
2.
No Deterministic Solution
3.
Randomized Solution
4.
Related Problems and Self-Stabilization
1. Gathering Problem
2. Pattern Formation Problems
5.
Conclusions
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
No central authority
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
Randomized Scatter
Compute the Voronoi Diagram If Random() = 0
then move arbitrarily in my cell
Randomized Scatter
Compute the Voronoi Diagram If Random() = 0
then move arbitrarily in my cell
Randomized Scatter
Compute the Voronoi Diagram If Random() = 0
then move arbitrarily in my cell
Randomized Scatter
Compute the Voronoi Diagram If Random() = 0
then move arbitrarily in my cell
Randomized Scatter
Compute the Voronoi Diagram If Random() = 0
then move arbitrarily in my cell
Randomized Scatter
Compute the Voronoi Diagram If Random() = 0
then move arbitrarily in my cell
Randomized Scatter
Compute the Voronoi Diagram If Random() = 0
then move arbitrarily in my cell
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
Deterministic 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
Compute the Voronoi Diagram If Random() = 0
then move arbitrarily in my cell
How to make them quiescent?
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 Gathering
If there exists at least two positions with more than one robot
then Scatter else Gather
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
Self-Stabilizing Gathering
If there exists at least one position with more than one robot
then Scatter
else Pattern Formation
Self-Stabilizing Gathering
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
