1. Stéphane Devismes VERIMAG UMR 5104 Univ. Joseph Fourier Grenoble, France Optimal Exploration of Small Rings Talk by Franck Petit, Univ. Pierre et Marie Curie - Paris 6, France

2. Context o A team of k “weak” robots evolving into a ring of n nodes 2WRAS 2010 o Autonomous: No central authority o Anonymous: Undistinguishable o Oblivious: No mean to know the past o Disoriented: No mean to agree on a common direction or orientation

3. Context o A team of k “weak” robots evolving into a ring of n nodes 3WRAS 2010 o Atomicity : In every configuration, each robot is located at exactly one node o Weak Multiplicity : In every configuration, each node may contain some robots (a robot cannot detect the exact number of robots located at each node but it is able to detect if there are zero, one, or more)

4. Context o A team of k “weak” robots evolving into a ring of n nodes 4WRAS 2010 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. Compute : Based on this observation, decides to either stay idle or move to one of the neighboring nodes 3. Move: Move toward its destination

5. Problem: Exploration o Exploration: Each node must be visited by at least one robot o Termination: Eventually, every robot stays idle 5WRAS 2010 o Performance: Number of robots (k<n) Starting from a configuration where no two robots are located at the same node:

6. Related works (Deterministic) o Tree networks Ω(n) robots are necessary in general A 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 coprime A deterministic algorithm for k ≥ 17 [Flocchini, Ilcinkas, Pelc, Santoro, OPODIS 07] 6WRAS 2010

7. Related works (Probabilistic) o Ring networks [Devismes, Petit, Tixeuil, SIROCCO 2010]  4 robots are necessary  For ring of size n>8, 4 robots are sufficient to solve the problem 7WRAS 2010

8. Contribution 8WRAS 2010 Question. Are 4 probabilistic robots necessary and sufficient to explore any ring of any size n ? Remark. The problem is not defined for n < 4 For n = 4, no algorithm required Contribution. Algorithm for 5 ≤ n ≤ 8 Corollary: 4 probabilistic robots are necessary and sufficient to explore any ring of any size n

9. Definitions 9F. Petit – SIROCCO 2009 Tower. A node with at least two robots. k ≥ 2

10. Definitions 10F. Petit – SIROCCO 2009 Segment. A maximal non-empty elementary path of occupied nodes. A 1-segment a 2-segment

11. Definitions 11F. Petit – SIROCCO 2009 Hole. A maximal non-empty elementary path of free nodes. 1 hole of length 3 A 1-hole

12. Definitions 12F. Petit – SIROCCO 2009 Arrow. A 1-segment, followed by a hole, a tower, and a 1-segment. 1 arrow Head of length 2 Tail

13. Definitions 13F. Petit – SIROCCO 2009 Arrow. A 1-segment, followed by a hole, a tower, and a 1-segment. Primary arrow

14. Definitions 14F. Petit – SIROCCO 2009 Arrow. A 1-segment, followed by a hole, a tower, and a 1-segment. final arrow

15. WRAS 2010 Algorithm: Overview o 3 main steps:  Phase I: Initial configuration  4-segment Invariant: no arrow  Phase II: 4-segment  primary arrow Invariant: 4-segment or primary arrow  Phase III: Primary arrow  final arrow Invariant: increasing arrow o (2 special cases) 15 Let start with phase II and III, it’s easier …

16. WRAS 2010 Algorithm: Phase II o Phase II: 4-segment  primary arrow  Invariant: 4-segment or primary arrow 16 Probabilistic moves

17. WRAS 2010 Algorithm: Phase II o Phase II: 4-segment  primary arrow  Invariant: 4-segment or primary arrow 17 Primary arrow

18. WRAS 2010 Algorithm: Phase III o Phase III: Primary arrow  final arrow  Invariant: increasing arrow 18 Deterministic move

19. WRAS 2010 Algorithm: Phase III o Phase III: Primary arrow  final arrow  Invariant: increasing arrow 19

20. WRAS 2010 Algorithm: Phase III o Phase III: Primary arrow  final arrow  Invariant: increasing arrow 20

21. WRAS 2010 Algorithm: Phase III o Phase III: Primary arrow  final arrow  Invariant: increasing arrow 21

22. WRAS 2010 Algorithm: Phase III o Phase III: Primary arrow  final arrow  Invariant: increasing arrow 22 Termination

23. WRAS 2010 Algorithm: Back to Phase I o Phase I: Initial configuration  4-segment  Invariant: no arrow o Principle:  No symmetry: Deterministic moves  Symmetry: Probabilistic or deterministic moves 23

24. WRAS 2010 Phase I: no symmetry o There exists a unique largest segment S:  move toward S following the shortest neighboring hole 24

25. WRAS 2010 Phase I: no symmetry o There exists a unique largest segment S:  move toward S following the shortest neighboring hole 25 Ambiguity: Decision taken by an adversary

26. WRAS 2010 Phase I: no symmetry o There exists a unique largest segment S:  move toward S following the shortest neighboring hole 26 Ambiguity: Decision taken by an adversary

27. WRAS 2010 Phase I: no symmetry o There exists a unique largest segment S:  move toward S following the shortest neighboring hole 27

28. WRAS 2010 Phase I: symmetry 28 Case by Case Study

29. WRAS 2010 Phase I: n = 5 o No symmetry  Initial configuration: a 4-segment  Phase I & II 29

30. WRAS 2010 Phase I: n = 6 o Only one symmetry is initially possible 30 Stop The 2 special cases

31. WRAS 2010 Phase I: n = 7 o Only one symmetry is initially possible 31

32. WRAS 2010 Phase I: n = 8 o Three symmetries are initially possible: 32 (a)(c) (b)

33. WRAS 2010 Phase I: n = 8, Case (a) 33 Case (c)

34. WRAS 2010 Phase I: n = 8, Case (b) 34 Case (c)

35. WRAS 2010 Phase I: n = 8, Case (c) 35 (c) o Really complex!!! o See the paper…

36. WRAS 2010 Conclusion o General Result:  4 probabilistic robots are necessary and sufficient to solve the exploration of any anonymous ring o Future works:  Convergence time (experimental result:O(n) moves)  Full asynchronous model  Other (regular) topologies 36

37. Conclusion 37WRAS 2010 Thank you.