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Texte intégral

(1)

Self-stabilizing Deterministic Gathering

Yoann Dieudonné Franck Petit

Université de Picardie Jules Verne, France

Rhodes, ALGOSENSORS 2009

(2)

Gathering Problem

(3)

Model SSM

(1) (2) (1)

(2)

(3) (1)

(1) (1) (1)

(1) (5)

(1)

Assumption 1

Two or more robots can occupy the same point

Assumption 2

All the robots can determine the

number of robots on each point

(4)

1 Introduction

2 Model SSM

3 The Gathering Problem

4 Our protocol

5 Conclusions

(5)

Problem Definition

(1) (2) (1)

(2)

(3) (1)

(1) (1) (1)

(1) (5)

(1)

(18)

(6)

In literature

In literature

Mainly studied :

In a probabilistic way [Suzuki et al. 99, Gradinariu et al.

07,08,09]

Starting from a configuration where all the robots are

initially at distinct positions [Suzuki et al. 99, Cieliebak 03,

Flocchini et al. 03,...]

(7)

In literature

Starting from an arbitrary configuration and in a deterministic way, only the following result

Corollary (Suzuki and yamashita 99)

Starting from an arbitrary configuration the gathering problem

cannot be solved in SSM in a deterministic way if the number of

robots is even.

(8)

In literature

P1 P2

Impossibility of gathering

only converge in an infinite

number of steps

(9)

Results

we provide a deterministic protocol for solving the

gathering problem for an odd number of robots starting

from an arbitrary configuration

(10)

Results

we provide a deterministic protocol for solving the

gathering problem for an odd number of robots starting from an arbitrary configuration

The protocol is self-stabilizing because "regarless of the initial states", it is guaranteed to converge in a finite

number of steps.

(11)

1 Introduction

2 Model SSM

3 The Gathering Problem

4 Our protocol

5 Conclusions

(12)

Our Protocol

(1) (2)

(2)

(3) (1)

(1) (1) (1)

(1) (5)

(1)

(5)

Definition (Maximal Points)

A point p is maximal ⇐⇒ ∀

point p , |p| ≥ |p |

(13)

If there is only one maximal point

(14)

If there is only one maximal point

Only the closest robots are

allowed to move toward

the maximal point.

(15)

If there are only one maximal point

The unique maximal point remains unique

The number of robots located at the maximal point increases in finite time

Every robot becomes a close robot in finite time

(16)

If there are only one maximal point

The unique maximal point remains unique

The number of robots located at the maximal point increases in finite time

Every robot becomes a close robot in finite time Lemma

The deterministic gathering can be solved in SSM for an odd

number of robots if there exists only one maximal point.

(17)

If there are exactly 2 maximal points

(18)

If there are exactly 2 maximal points

Only the closest robots to

one of the two maximal

points are allowed to move

(19)

If there are exactly 2 maximal points

no other maximal point is created

number of robots on the two maximal points increases

exactly one maximal point in finite time because the

number of robots is odd

(20)

If there are exactly 2 maximal points

no other maximal point is created

number of robots on the two maximal points increases exactly one maximal point in finite time because the number of robots is odd

Lemma

The deterministic gathering can be solved in SSM for an odd

number of robots if there exist exactly 2 maximal points.

(21)

If there are at least 3 maximal points

(22)

If there are at least 3 maximal points

We consider the smallest

enclosing circle

(23)

If there are at least 3 maximal points

Theorem (Meggido 86)

The smallest circle enclosing a set of points is unique and can

be computed in linear time

(24)

If there is exactly 3 maximal points

2 cases to consider

At least one robot inside SEC No robot inside SEC

(25)

If there is exactly 3 maximal points

At least one robot inside SEC

the robots inside SEC ,

move to the center of SEC

(26)

If there is exactly 3 maximal points

At least one robot inside SEC

the robots inside SEC ,

move to the center of SEC

SEC remains invariant

(27)

If there is exactly 3 maximal points

At least one robot inside SEC

A unique maximal may

appear at the center of

SEC

(28)

If there is exactly 3 maximal points

At least one robot inside SEC

Only the maximal points

are allowed to move

towards the center

(29)

If there is exactly 3 maximal points

At least one robot inside SEC

A unique maximal may

appear at the center of

SEC

(30)

If there is exactly 3 maximal points

At least one robot inside SEC

(5)

(1)

(3)

(5)

(2) (2)

C

(2) (1)

C

(3)

(3) (1) (4)

(31)

If there is exactly 3 maximal points

Theorem (Chrystal 1885)

All the points inside the convex hull are inside SEC

(32)

If there is exactly 3 maximal points

Theorem (Chrystal 1885)

All the points inside the convex hull are inside SEC Lemma

All the robots inside SEC remains inside SEC even if SEC

changes while there exist at least three maximal points

(33)

If there is exactly 3 maximal points

the number of robots inside

SEC never increases

(34)

If there is exactly 3 maximal points

the number of robots inside

SEC never increases

(35)

If there is exactly 3 maximal points

the number of robots inside

SEC never increases

(36)

If there is exactly 3 maximal points

the number of robots inside

SEC never increases

(37)

If there is exactly 3 maximal points

Otherwise the number of robots inside SEC increases and

we obtain a unique maximal point in finite time.

(38)

If there is exactly 3 maximal points

Otherwise the number of robots inside SEC increases and we obtain a unique maximal point in finite time.

Lemma

The deterministic gathering can be solved in SSM for an odd

number of robots if there exist at least 3 maximal points and at

least one robot inside SEC.

(39)

If there is exactly 3 maximal points

No robot inside SEC

there are never some

robots inside SEC

(40)

If there is exactly 3 maximal points

No robot inside SEC

there are never some

robots inside SEC

(41)

If there is exactly 3 maximal points

No robot inside SEC

there are never some

robots inside SEC

(42)

If there is exactly 3 maximal points

No robot inside SEC

there are never some

robots inside SEC

(43)

Final result

Theorem

Starting from an arbitrary configuration the gathering problem

can be solved in SSM in a deterministic way ⇐⇒ the number of

robots is odd.

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