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Optimal Probabilistic Ring Exploration by Semi-Synchronous Oblivious Robots

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.

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Optimal Probabilistic Ring Exploration by Semi-Synchronous Oblivious Robots

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  1. Optimal Probabilistic Ring Exploration by Semi-SynchronousOblivious Robots Franck Petit INRIA, LIP Lab.Univ. / ENS of LyonFrance Joint work withStéphaneDevismes, VERIMAG, Grenoble, France Sébastien Tixeuil,Univ. Pierre et Marie Curie - Paris 6, France

  2. Context • Autonomous • : No central authority • Anonymous • : Undistinguishable • Oblivious • : No mean to know the past • Disoriented • : No mean to agree on a common direction or orientation A team of k “weak” robots evolving into a ring of n nodes

  3. Context • Atomicity • : In every configuration, each robot is locatedatexactly one node • Multiplicity • : In every configuration, each node contains zero, one, or more than one robot • (every robot is able to detect it) A team of k “weak” robots evolving into a ring of n nodes

  4. Context • SSM • : In every configuration, k’ robots are activated (0 < k’ ≤ k) • The k’ activated robots execute the cycle: • Look • : Instantaneous snapshot with multiplicity detection • : Based on this observation, decides to either stay idle or move to one of the neighboring nodes • Compute • Move • : Move toward its destination A team of k “weak” robots evolving into a ring of n nodes

  5. Problem • Starting from a configuration where no two robotsare locatedatthe same node: • Performance:Number of robots (k<n) Exploration:Each node must be visited by at least one robot Termination:Eventually, every robot stays idle

  6. Relatedworks (Deterministic) Tree networksΩ(n) robots are necessary in generalA deterministicalgorithmwithO(log n/log log n) robots, assumingthatΔ ≤ 3[Flocchini, Ilcinkas, Pelc, Santoro, SIROCCO 08] Ring networksΘ(log n) robots are necessary and sufficient, providedthatn and k are coprimeA deterministicalgorithm for k ≥ 17[Flocchini, Ilcinkas, Pelc, Santoro, OPODIS 07]

  7. Contribution Theorem.4 probabilistic robots are necessary and sufficient, providedthatn > 8 n and k are not required to becoprime Exploration impossible withlessthan4 robots An algorithmworkingwith4probabilistic robots (n > 8)

  8. ObliviousRobots Termination Exploration Implicit memory At least one configuration thatcannotbe an initial configuration Remark.If n > k, any terminal configuration of anyprotocolcontainsat least one tower.

  9. Tower Definition.A node with at least two robots. k ≥ 2

  10. Tower Building Cannotbe a terminal configuration Can be an initial configuration

  11. Enabling Exploration k ≥ 3 Lemma.Every execution must contain a suffix of at least n–k+1configurations containing a tower of lessthank robots and anytwo of them are distinguishable.

  12. Enabling Exploration Two undistinguishableconfigurations Lemma.With 3 robots and a fixed tower of 2 robots, the maximum number of distinguishable configurations is equal to . Two other undistinguishableconfigurations

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

  14. Negativeresult Theorem.For every n ≥ 4, there exists no exploration protocol (even probabilistic) of a n-size ring with three robots. Proof : There exists no protocolwith3 robots in a 4-size ring with a distributedscheduler.

  15. Contribution Theorem.4 probabilistic robots are necessary and sufficient, providedthatn > 8 n and k are not required to becoprime Exploration impossible withlessthan4 robots Give an algorithmworkingwith4probabilistic robots (n > 8)

  16. Definitions Segment.A maximal non-empty elementary path of occupied nodes. a 2-segment 2 segments of length 1

  17. Definitions Hole.A maximal non-empty elementary path of free nodes. 1 hole of length 4 a 2-hole

  18. Definitions Arrow. A 1-segment, followed by a non-empty elementary path of free nodes, a tower, and a 1-segment. Tail Head 1 arrow of length 4

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

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

  21. Algorithm If I am an internalnode, then I tryto move on the otherinternalnode. 0 0 1 Initially, there is no tower Converge toward a 4-segment Build a tower Visit the ring and terminate

  22. Algorithm Primary arrow Initially, there is no tower Converge toward a 4-segment Build a tower Visit the ring and terminate

  23. Algorithm Primary arrow Final arrow Initially, there is no tower Converge toward a 4-segment Build a tower Visit the ring and terminate

  24. Algorithm Primary arrow Final arrow 3-segment If I amthe isolatednode, then I movethrough a shortesthole. Initially, there is no tower Converge toward a 4-segment Build a tower Visit the ring and terminate

  25. Algorithm Primary arrow Final arrow 3-segment a unique 2-segment If I am at the closest distance from the 2-segment, then I move toward the closest extremity. Initially, there is no tower Converge toward a 4-segment Build a tower Visit the ring and terminate

  26. Algorithm Primary arrow Final arrow 3-segment a unique 2-segment two 2-segments 0 1 If I am a neighbor of the longest hole, then I try to move toward the other 2-segment. Initially, there is no tower Converge toward a 4-segment Build a tower Visit the ring and terminate

  27. Algorithm Primary arrow Final arrow 3-segment a unique 2-segment two 2-segments four isolatednodes 1 L: length of the longest hole If 4 robots are neighbors of an L-hole, then I try to move through my longest neighboring hole. 0 Initially, there is no tower Converge toward a 4-segment Build a tower Visit the ring and terminate

  28. Algorithm Primary arrow Final arrow 3-segment a unique 2-segment two 2-segments four isolatednodes L: length of the longest hole If 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. Initially, there is no tower Converge toward a 4-segment Build a tower Visit the ring and terminate

  29. Algorithm Primary arrow Final arrow 3-segment a unique 2-segment two 2-segments four isolatednodes L: length of the longest hole If 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. Initially, there is no tower Converge toward a 4-segment Build a tower Visit the ring and terminate

  30. Phase 1, Summary

  31. Proof Lemma.No tower is created during Phase 1 in a n-ring with n > 8. Proof Base: Withn > 8 and 4 robots, therealwaysexists a hole of lengthgreaterthan1.

  32. Proof Lemma.No tower is created during Phase 1 in a n-ring with n > 8. Lemma.Startingfromany initial configuration, the system reaches in finiteexpected time a configurationcontaining 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.

  33. Conclusion • 4 probabilistic robots are necessary and sufficient, providedthatn > 8 • Future works: • Ad hoc solutions for n ≤ 8 (done) • Convergence time • Full asynchronousmodel

  34. Conclusion Thankyou.

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