Optimal Probabilistic Ring Exploration by Semi-Synchronous Oblivious Robots

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Franck Petit INRIA, LIP Lab. Univ. / ENS of Lyon France Optimal Probabilistic Ring Exploration by Semi-Synchronous Oblivious Robots Joint work with Stéphane Devismes, VERIMAG, Grenoble, France Sébastien Tixeuil , Univ. Pierre et Marie Curie - Paris 6, France

description

Optimal Probabilistic Ring Exploration by Semi-Synchronous Oblivious Robots. Franck Petit INRIA, LIP Lab. Univ . / ENS of Lyon France. Joint work with Stéphane Devismes , VERIMAG, Grenoble, France Sébastien Tixeuil , Univ . Pierre et Marie Curie - Paris 6, France. Context. Autonomous. - PowerPoint PPT Presentation

Transcript of Optimal Probabilistic Ring Exploration by Semi-Synchronous Oblivious Robots

Page 1: Optimal  Probabilistic  Ring Exploration by  Semi-Synchronous Oblivious  Robots

Franck PetitINRIA, LIP Lab.

Univ. / ENS of LyonFrance

Optimal Probabilistic Ring Exploration by Semi-Synchronous Oblivious Robots

Joint work withStéphane Devismes, VERIMAG, Grenoble, France

Sébastien Tixeuil, Univ. Pierre et Marie Curie - Paris 6, France

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Context

o A team of k “weak” robots evolving into a ring of n nodes

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o Autonomous

: No central authorityo Anonymous: Undistinguishable o Oblivious : No mean to know the past o Disoriented : No mean to agree on a

common direction or orientation

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Context

o A team of k “weak” robots evolving into a ring of n nodes

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o Atomicity

: In every configuration, each robot is located at exactly one node o Multiplicit

y: In every configuration, each node contains zero, one, or more than one robot (every robot is able to detect it)

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Context

o A team of k “weak” robots evolving into a ring of n nodes

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o SSM : In every configuration, k’ robots are activated (0 < k’ ≤ k)

1. Look : Instantaneous snapshot with multiplicity detection

o The k’ activated robots execute the cycle: 2. Comput

e: Based on this observation, decides to either stay idle or move to one of the neighboring nodes

3. Move : Move toward its destination

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Problem

o Exploration:Each node must be visited by at least one robot

o Termination:Eventually, every robot stays idle

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o Performance: Number of robots (k<n)

Starting from a configuration where no two robots are located at the same node:

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Related works (Deterministic)o Tree networks

Ω(n) robots are necessary in generalA deterministic algorithm with O(log n/log log n) robots, assuming that Δ ≤ 3[Flocchini, Ilcinkas, Pelc, Santoro, SIROCCO 08]

o Ring networks Θ(log n) robots are necessary and sufficient, provided that n and k are coprimeA deterministic algorithm for k ≥ 17[Flocchini, Ilcinkas, Pelc, Santoro, OPODIS 07] 6F. Petit – SIROCCO 2009

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Contribution

o n and k are not required to be coprime

1. Exploration impossible with less than 4 robots

2. An algorithm working with 4 probabilistic robots (n > 8)

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Theorem. 4 probabilistic robots are necessary and sufficient, provided that n > 8

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Oblivious Robots

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At least one configuration that cannot be an initial configuration

Remark. If n > k, any terminal configuration of any protocol contains at least one tower.

Termination Exploration

Implicit memory

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Tower

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Definition. A node with at least two robots.

k ≥ 2

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Tower Building

10F. Petit – SIROCCO 2009 Can be an initial configuration

Cannot be a terminal configuration

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Enabling Exploration

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k ≥ 3 Lemma. Every execution must contain a suffix of at least n–k+1 configurations containing a tower of less than k robots and any two of them are distinguishable.

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Enabling Exploration

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Two undistinguishable

configurationsTwo other undistinguishable

configurations

Lemma. With 3 robots and a fixed tower of 2 robots, the maximum number of distinguishable configurations is equal to .

n2 ⎢ ⎣ ⎢

⎥ ⎦ ⎥

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Enabling Exploration

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Lemma. For every n > 4, there exists no exploration protocol (even probabilistic) of a n-size ring with 3 robots.

Proof :

n2 ⎢ ⎣ ⎢

⎥ ⎦ ⎥≥ n − k +1⇒ n ≤ 4

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Negative result

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Theorem. For every n ≥ 4, there exists no exploration protocol (even probabilistic) of a n-size ring with three robots.

Proof :There exists no protocol with 3 robots in a 4-size ring with a distributed scheduler.

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Contribution

o n and k are not required to be coprime

1. Exploration impossible with less than 4 robots

2. Give an algorithm working with 4 probabilistic robots (n > 8)

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Theorem. 4 probabilistic robots are necessary and sufficient, provided that n > 8

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Definitions

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Segment. A maximal non-empty elementary path of occupied nodes.

2 segments of length 1

a 2-segment

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Definitions

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Hole. A maximal non-empty elementary path of free nodes.

1 hole of length 4

a 2-hole

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Definitions

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Arrow. A 1-segment, followed by a non-empty elementary path of free nodes, a tower, and a 1-segment.

1 arrow

Head

of length 4

Tail

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Definitions

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Arrow. A 1-segment, followed by a non-empty elementary path of free nodes, a tower, and a 1-segment.

final arrow

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Definitions

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Arrow. A 1-segment, followed by a non-empty elementary path of free nodes, a tower, and a 1-segment.

Primary arrow

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Algorithm

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o Initially, there is no tower

1. Converge toward a 4-segment2. Build a tower3. Visit the ring and terminate

0

0

If I am an internal node, then I try to move on the other internal node.

1

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Algorithm

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o Initially, there is no tower

1. Converge toward a 4-segment2. Build a tower3. Visit the ring and terminate

Primary arrow

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Algorithm

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o Initially, there is no tower

1. Converge toward a 4-segment2. Build a tower3. Visit the ring and terminate Final arrow

Primary arrow

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Algorithm

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o Initially, there is no tower

1. Converge toward a 4-segment2. Build a tower3. Visit the ring and terminate

a) 3-segment

Final arrow

Primary arrow

If I am the isolated node, then I move through a shortest hole.

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Algorithm

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o Initially, there is no tower

1. Converge toward a 4-segment2. Build a tower3. Visit the ring and terminate

a) 3-segmentb) a unique 2-segment

Final arrow

Primary arrow

If I am at the closest distance from the 2-segment, then I move toward the closest extremity.

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Algorithm

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o Initially, there is no tower

1. Converge toward a 4-segment2. Build a tower3. Visit the ring and terminate

a) 3-segmentb) a unique 2-segmentc) two 2-segments

Final arrow

Primary arrow

If I am a neighbor of the longest hole, then I try to move toward the other 2-segment.

10

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Algorithm

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o Initially, there is no tower

1. Converge toward a 4-segment2. Build a tower3. Visit the ring and terminate

a) 3-segmentb) a unique 2-segmentc) two 2-segmentsd) four isolated nodes

Final arrow

Primary arrow

L: length of the longest holeIf 4 robots are

neighbors of an L-hole, then I try to move through my longest neighboring hole.

1

0

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Algorithm

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o Initially, there is no tower

1. Converge toward a 4-segment2. Build a tower3. Visit the ring and terminate

a) 3-segmentb) a unique 2-segmentc) two 2-segmentsd) four isolated nodes

Final arrow

Primary arrow

L: length of the longest holeIf 3 robots are

neighbors of an L-hole, then if I am one of this 3 robots and a neighbor of a smaller hole h, then I move through h.

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Algorithm

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o Initially, there is no tower

1. Converge toward a 4-segment2. Build a tower3. Visit the ring and terminate

a) 3-segmentb) a unique 2-segmentc) two 2-segmentsd) four isolated nodes

Final arrow

Primary arrow

L: length of the longest holeIf 2 robots are

neighbors of an L-hole, then if I am neighbor of the L-hole, then I move through the other neighboring hole.

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Phase 1, Summary

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Proof

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Lemma. No tower is created during Phase 1 in a n-ring with n > 8.

Proof Base:With n > 8 and 4 robots, there always exists a hole of length greater than 1.

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Proof

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Lemma. No tower is created during Phase 1 in a n-ring with n > 8.

Lemma. Starting from any initial configuration, the system reaches in finite expected time a configuration containing a 4-segment.

Theorem. The algorithm (Phases 1 to 3) is a probabilistic exploration protocol for 4 robots in a ring of n > 8 nodes.

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Conclusion

o 4 probabilistic robots are necessary and sufficient, provided that n > 8

o Future works: Ad hoc solutions for n ≤ 8 (done) Convergence time Full asynchronous model

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Conclusion

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Thank you.