Introduction Model SSM The Gathering Problem Our protocol Conclusions

Self-stabilizing Deterministic Gathering

Yoann Dieudonné Franck Petit

Université de Picardie Jules Verne, France

Rhodes, ALGOSENSORS 2009

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

Gathering Problem

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

Model SSM

(1)

(1)(2)

(2)

(3)

(1)

(1)(1) (1)

(1)

(5)

(1) Assumption 1Two or more robots can occupythe same point

Assumption 2All the robots can determine thenumber of robots on each point

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

1 Introduction

2 Model SSM

3 The Gathering Problem

4 Our protocol

5 Conclusions

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

Problem Definition

(1)

(1)(2)

(2)

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(1)

(1)(1) (1)

(1)

(5)

(1)

(18)

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

In literature

In literatureMainly studied :

In a probabilistic way [Suzuki et al. 99, Gradinariu et al.07,08,09]Starting from a configuration where all the robots areinitially at distinct positions [Suzuki et al. 99, Cieliebak 03,Flocchini et al. 03,...]

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

In literature

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

Corollary (Suzuki and yamashita 99)Starting from an arbitrary configuration the gathering problemcannot be solved in SSM in a deterministic way if the number ofrobots is even.

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

In literature

P1 P2

Impossibility of gatheringonly converge in an infinitenumber of steps

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

Results

we provide a deterministic protocol for solving thegathering problem for an odd number of robots startingfrom an arbitrary configuration

Yoann Dieudonné, Franck Petit Rhodes, ALGOSENSORS 2009 17 / 49

Introduction Model SSM The Gathering Problem Our protocol Conclusions

Results

we provide a deterministic protocol for solving thegathering problem for an odd number of robots startingfrom an arbitrary configurationThe protocol is self-stabilizing because "regarless of theinitial states", it is guaranteed to converge in a finitenumber of steps.

Yoann Dieudonné, Franck Petit Rhodes, ALGOSENSORS 2009 17 / 49

Introduction Model SSM The Gathering Problem Our protocol Conclusions

1 Introduction

2 Model SSM

3 The Gathering Problem

4 Our protocol

5 Conclusions

Yoann Dieudonné, Franck Petit Rhodes, ALGOSENSORS 2009 18 / 49

Introduction Model SSM The Gathering Problem Our protocol Conclusions

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′|

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is only one maximal point

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is only one maximal point

Only the closest robots areallowed to move towardthe maximal point.

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there are only one maximal point

The unique maximal point remains uniqueThe number of robots located at the maximal pointincreases in finite timeEvery robot becomes a close robot in finite time

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there are only one maximal point

The unique maximal point remains uniqueThe number of robots located at the maximal pointincreases in finite timeEvery robot becomes a close robot in finite time

LemmaThe deterministic gathering can be solved in SSM for an oddnumber of robots if there exists only one maximal point.

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there are exactly 2 maximal points

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there are exactly 2 maximal points

Only the closest robots toone of the two maximalpoints are allowed to move

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there are exactly 2 maximal points

no other maximal point is creatednumber of robots on the two maximal points increasesexactly one maximal point in finite time because thenumber of robots is odd

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there are exactly 2 maximal points

no other maximal point is creatednumber of robots on the two maximal points increasesexactly one maximal point in finite time because thenumber of robots is odd

LemmaThe deterministic gathering can be solved in SSM for an oddnumber of robots if there exist exactly 2 maximal points.

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there are at least 3 maximal points

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there are at least 3 maximal points

We consider the smallestenclosing circle

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there are at least 3 maximal points

Theorem (Meggido 86)The smallest circle enclosing a set of points is unique and canbe computed in linear time

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

2 cases to consider

At least one robot inside SEC No robot inside SEC

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

At least one robot inside SEC

the robots inside SEC,move to the center of SEC

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

At least one robot inside SEC

the robots inside SEC,move to the center of SECSEC remains invariant

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

At least one robot inside SEC

A unique maximal mayappear at the center ofSEC

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

At least one robot inside SEC

Only the maximal pointsare allowed to movetowards the center

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

At least one robot inside SEC

A unique maximal mayappear at the center ofSEC

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

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)

(4)

(4)

(1)

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

Theorem (Chrystal 1885)All the points inside the convex hull are inside SEC

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

Theorem (Chrystal 1885)All the points inside the convex hull are inside SEC

LemmaAll the robots inside SEC remains inside SEC even if SECchanges while there exist at least three maximal points

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

the number of robots insideSEC never increases

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

the number of robots insideSEC never increases

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

the number of robots insideSEC never increases

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

the number of robots insideSEC never increases

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

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

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

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

LemmaThe deterministic gathering can be solved in SSM for an oddnumber of robots if there exist at least 3 maximal points and atleast one robot inside SEC.

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

No robot inside SEC

there are never somerobots inside SEC

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

No robot inside SEC

there are never somerobots inside SEC

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

No robot inside SEC

there are never somerobots inside SEC

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

If there is exactly 3 maximal points

No robot inside SEC

there are never somerobots inside SEC

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Introduction Model SSM The Gathering Problem Our protocol Conclusions

Final result

TheoremStarting from an arbitrary configuration the gathering problemcan be solved in SSM in a deterministic way⇐⇒ the number ofrobots is odd.

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