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Problem Silent Self-stabilizing Leader Election Model: Locally - PowerPoint PPT Presentation

Feb 19, 2024 •21 likes •1.83k views

Self-Stabilizing Leader Election in Polynomial Steps 1 Karine Altisen Alain Cournier Stphane Devismes Anas Durand Franck Petit September 29, 2014 1This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) and

  1. Abnormal Trees � 3 , 1 � � 3 , 2 � KinshipOk 2 5 � 3 , 0 � � 1 , 2 � � 1 , 0 � 3 4 6 7 8 � 3 , 1 � � 1 , 1 � Key: � idR , level � Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 10 / 23

  2. Abnormal Trees Abnormal root � 3 , 1 � � 3 , 2 � KinshipOk 2 5 � 3 , 0 � � 1 , 2 � � 1 , 0 � 3 4 6 7 8 � 3 , 1 � � 1 , 1 � Key: � idR , level � Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 10 / 23

  3. Abnormal Trees � 3 , 1 � � 3 , 2 � 2 5 Abnormal root � 3 , 0 � � 1 , 2 � � 1 , 0 � 3 4 6 7 8 � 3 , 1 � � 1 , 1 � Key: � idR , level � Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 10 / 23

  4. Abnormal Trees � 3 , 1 � T 1 � 3 , 2 � 2 5 � 3 , 0 � � 1 , 2 � � 1 , 0 � 3 4 6 T 2 T 3 7 8 � 3 , 1 � � 1 , 1 � Key: � idR , level � Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 10 / 23

  5. Abnormal Trees � 3 , 1 � T 1 � 3 , 2 � 2 5 � 3 , 0 � � 1 , 2 � � 1 , 0 � 3 4 6 T 2 T 3 7 8 � 3 , 1 � � 1 , 1 � Key: � idR , level � Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 10 / 23

  6. Abnormal Trees � 3 , 1 � T 1 � 3 , 2 � 2 5 � 3 , 0 � � 1 , 2 � � 1 , 0 � 3 4 6 T 2 T 3 7 8 � 3 , 1 � � 1 , 1 � Key: � idR , level � Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 10 / 23

  7. Cleaning C � idR , level � Key: Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

  8. Cleaning EB -action � 1 , 0 � 6 2 8 C � 1 , 1 � � 1 , 1 � � idR , level � Key: Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

  9. Cleaning EB -action � 1 , 0 � 6 2 8 C � 1 , 1 � � 1 , 1 � � idR , level � Key: Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

  10. Cleaning EB � idR , level � Key: Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

  11. Cleaning � 1 , 5 � 7 EB 3 � 1 , 6 � EF -action � idR , level � Key: Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

  12. Cleaning � 1 , 5 � 7 EB 3 � 1 , 6 � EF -action � idR , level � Key: Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

  13. Cleaning EF � idR , level � Key: Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

  14. Cleaning R -action � 1 , 0 � 6 2 8 EF � 1 , 1 � � 1 , 1 � � idR , level � Key: Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

  15. Cleaning � 6 , 0 � R -action 6 2 8 � 1 , 1 � � 1 , 1 � EF EF � idR , level � Key: Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 11 / 23

  16. Stabilization Time in Rounds No alive abnormal tree created Height of an abnormal tree: at most n Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 12 / 23

  17. Stabilization Time in Rounds No alive abnormal tree created Height of an abnormal tree: at most n Cleaning: ◮ EB-wave : n ◮ EF-wave : n ◮ R-wave : n Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 12 / 23

  18. Stabilization Time in Rounds No alive abnormal tree created Height of an abnormal tree: at most n Cleaning: ◮ EB-wave : n ◮ EF-wave : n ◮ R-wave : n Building of the Spanning Tree: D Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 12 / 23

  19. Stabilization Time in Rounds No alive abnormal tree created Height of an abnormal tree: at most n Cleaning: ◮ EB-wave : n ◮ EF-wave : n ◮ R-wave : n Building of the Spanning Tree: D O ( 3 n + D ) rounds Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 12 / 23

  20. Lower Bound on the Worst Case Stabilization Time in Rounds � 0 , n -1 � 1 n � 0 , n -2 � 3 � 0 , 1 � k links . j = k + 3 . � 0 , 0 � 2 4 � 0 , 2 � . D = n − k j 5 � 0 , j -2 � � 0 , 3 � . . . Key: � idR , level � Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

  21. Lower Bound on the Worst Case Stabilization Time in Rounds � 0 , n -1 � 1 n � 0 , n -2 � 3 � 0 , 1 � EB -wave k links . j = k + 3 . � 0 , 0 � 2 4 � 0 , 2 � . D = n − k j 5 � 0 , j -2 � � 0 , 3 � . . . Key: � idR , level � Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

  22. Lower Bound on the Worst Case Stabilization Time in Rounds � 0 , n -1 � 1 n � 0 , n -2 � 3 � 0 , 1 � k links . j = k + 3 . � 0 , 0 � 2 4 � 0 , 2 � . D = n − k j 5 � 0 , j -2 � � 0 , 3 � . . . Key: � idR , level � Clean EBroadcast EFeedback n Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

  23. Lower Bound on the Worst Case Stabilization Time in Rounds � 0 , n -1 � 1 n � 0 , n -2 � 3 � 0 , 1 � EF -wave k links . j = k + 3 . � 0 , 0 � 2 4 � 0 , 2 � . D = n − k j 5 � 0 , j -2 � � 0 , 3 � . . . Key: � idR , level � Clean EBroadcast EFeedback n Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

  24. Lower Bound on the Worst Case Stabilization Time in Rounds � 0 , n -1 � 1 n � 0 , n -2 � 3 � 0 , 1 � k links . j = k + 3 . � 0 , 0 � 2 4 � 0 , 2 � . D = n − k j 5 � 0 , j -2 � � 0 , 3 � . . . Key: � idR , level � Clean EBroadcast EFeedback n + n Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

  25. Lower Bound on the Worst Case Stabilization Time in Rounds � 0 , n -1 � 1 n � 0 , n -2 � 3 � 0 , 1 � R -wave k links . j = k + 3 . � 0 , 0 � 2 4 � 0 , 2 � . D = n − k j 5 � 0 , j -2 � � 0 , 3 � . . . Key: � idR , level � Clean EBroadcast EFeedback n + n Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

  26. Lower Bound on the Worst Case Stabilization Time in Rounds � 1 , 0 � 1 n � 2 , n - k � 3 � 2 , 1 � k links . j = k + 3 . � 2 , 0 � 2 4 � 2 , 1 � . D = n − k j 5 � 2 , 1 � � 2 , 1 � . . . Key: � idR , level � Clean EBroadcast EFeedback n + n + n Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

  27. Lower Bound on the Worst Case Stabilization Time in Rounds � 1 , 0 � 1 n � 2 , n - k � 3 � 2 , 1 � Building k links . j = k + 3 . � 2 , 0 � 2 4 � 2 , 1 � . D = n − k j 5 � 2 , 1 � � 2 , 1 � . . . Key: � idR , level � Clean EBroadcast EFeedback n + n + n Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

  28. Lower Bound on the Worst Case Stabilization Time in Rounds � 1 , 0 � 1 n � 1 , 1 � 3 � 1 , n - k � k links . j = k + 3 . � 1 , n - k -1 � 2 4 � 1 , n - k � . D = n − k j 5 � 1 , n - k -2 � � 1 , n - k � . . . Key: � idR , level � Clean EBroadcast EFeedback n + n + n +( n − k ) Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

  29. Lower Bound on the Worst Case Stabilization Time in Rounds � 1 , 0 � 1 n � 1 , 1 � 3 � 1 , n - k � k links . j = k + 3 . � 1 , n - k -1 � 2 4 � 1 , n - k � . D = n − k j 5 � 1 , n - k -2 � � 1 , n - k � . . . Key: � idR , level � Clean EBroadcast EFeedback n + n + n +( n − k ) = exactly 3 n + D rounds Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 13 / 23

  30. Stabilization Time in Steps A segment Another segment Death of an abnormal tree Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 14 / 23

  31. Stabilization Time in Steps A segment Another segment Death of an abnormal tree At most n alive abnormal trees + No alive abnormal tree created Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 14 / 23

  32. Stabilization Time in Steps A segment Another segment Death of an abnormal tree At most n alive abnormal trees + No alive abnormal tree created − → At most n + 1 segments Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 14 / 23

  33. Stabilization Time in Steps A segment Another segment Death of an abnormal tree At most n alive abnormal trees + No alive abnormal tree created − → At most n + 1 segments In a segment J -action J -action J -action EB -action EF -action R -action J -action idR : 7 5 3 2 7 3 Death of an abnormal tree = End of the segment Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 14 / 23

  34. Stabilization Time in Steps A segment Another segment Death of an abnormal tree At most n alive abnormal trees + No alive abnormal tree created − → At most n + 1 segments In a segment J -action J -action J -action EB -action EF -action R -action J -action idR : 7 5 3 2 7 3 Death of an abnormal tree = End of the segment • n − 1 J -actions • 1 EB -action • 1 EF -action • 1 R -action Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 14 / 23

  35. Stabilization Time in Steps A segment Another segment Death of an abnormal tree At most n alive abnormal trees + No alive abnormal tree created − → At most n + 1 segments In a segment J -action J -action J -action EB -action EF -action R -action J -action idR : 7 5 3 2 7 3 Death of an abnormal tree = End of the segment • n − 1 J -actions • 1 EB -action • 1 EF -action • 1 R -action ⇒ O ( n ) actions per process Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 14 / 23

  36. Stabilization Time in Steps A segment Another segment Death of an abnormal tree At most n alive abnormal trees + No alive abnormal tree created − → At most n + 1 segments In a segment J -action J -action J -action EB -action EF -action R -action J -action idR : 7 5 3 2 7 3 Death of an abnormal tree = End of the segment • n − 1 J -actions • 1 EB -action • 1 EF -action • 1 R -action ⇒ O ( n ) actions per process O ( n 3 ) steps 2 n 2 − 11 2 + 2 n 2 + n n 3 n 3 6 + 5 Lower Bound: 3 n + 2 steps Upper Bound: 2 + 1 steps Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 14 / 23

  37. Lower Bound on the Worst Case Stabilization Time in Steps Build � n -1 , 0 � Reset 2 n -1 � n -2 , 0 � n +1 � 1 , 0 � 2 n -2 � 2 n , 0 � 2 n . . � n -3 , 0 � 2 n -3 . 2 n -4 � n -4 , 0 � Key: � idR , level � Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

  38. Lower Bound on the Worst Case Stabilization Time in Steps Build � n -1 , 0 � Reset 2 n -1 � n -2 , 0 � n +1 � 1 , 0 � 2 n -2 � 2 n , 0 � 2 n . . � n -3 , 0 � 2 n -3 . 2 n -4 � n -4 , 0 � Key: � idR , level � Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

  39. Lower Bound on the Worst Case Stabilization Time in Steps Build � n -1 , 0 � Reset 2 n -1 � n -2 , 0 � n +1 � 1 , 0 � 2 n -2 � 2 n , 0 � 2 n . . � n -3 , 0 � 2 n -3 . 2 n -4 � n -4 , 0 � Key: � idR , level � Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

  40. Lower Bound on the Worst Case Stabilization Time in Steps � n -4 , 3 � 2 n -1 � n -4 , 2 � n +1 � 1 , 0 � 2 n -2 � 2 n , 0 � 2 n . . � n -4 , 1 � 2 n -3 . 2 n -4 � n -4 , 0 � Key: � idR , level � Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

  41. Lower Bound on the Worst Case Stabilization Time in Steps � n -4 , 3 � 2 n -1 � n -4 , 2 � n +1 � 1 , 0 � 2 n -2 � 2 n , 0 � 2 n . . � n -4 , 1 � 2 n -3 . 2 n -4 � n -4 , 0 � Key: � idR , level � Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

  42. Lower Bound on the Worst Case Stabilization Time in Steps � n -4 , 3 � 2 n -1 � n -4 , 2 � n +1 � 1 , 0 � 2 n -2 � 2 n , 0 � 2 n . . � n -4 , 1 � 2 n -3 . 2 n -4 � 2 n -4 , 0 � Key: � idR , level � Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

  43. Lower Bound on the Worst Case Stabilization Time in Steps � n -4 , 3 � 2 n -1 � n -4 , 2 � n +1 � 1 , 0 � 2 n -2 � 2 n , 0 � 2 n . . � 2 n -3 , 0 � 2 n -3 . 2 n -4 � 2 n -4 , 0 � Key: � idR , level � Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

  44. Lower Bound on the Worst Case Stabilization Time in Steps � n -4 , 3 � 2 n -1 � 2 n -2 , 0 � n +1 � 1 , 0 � 2 n -2 � 2 n , 0 � 2 n . . � 2 n -3 , 0 � 2 n -3 . 2 n -4 � 2 n -4 , 0 � Key: � idR , level � Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

  45. Lower Bound on the Worst Case Stabilization Time in Steps Case of the reset of 2 n − 4 . . . processes: 2 n − 1 2 n − 2 2 n − 3 2 n − 4 � 2 n -1 , 0 � 2 n -1 � 2 n -2 , 0 � n +1 � 1 , 0 � 2 n -2 � 2 n , 0 � 2 n . . . . . � 2 n -3 , 0 � 2 n -3 . 2 n -4 � 2 n -4 , 0 � Key: � idR , level � Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

  46. Lower Bound on the Worst Case Stabilization Time in Steps Case of the reset of 2 n − 4 . . . processes: 2 n − 1 2 n − 2 2 n − 3 2 n − 4 idR = 2 n -2 � 2 n -2 , 1 � 2 n -1 � 2 n -2 , 0 � n +1 � 1 , 0 � 2 n -2 � 2 n , 0 � 2 n . . . . . � 2 n -3 , 0 � 2 n -3 . 2 n -4 � 2 n -4 , 0 � Key: � idR , level � Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

  47. Lower Bound on the Worst Case Stabilization Time in Steps Case of the reset of 2 n − 4 . . . processes: 2 n − 1 2 n − 2 2 n − 3 2 n − 4 idR = 2 n -2 � 2 n -2 , 1 � 2 n -1 idR = 2 n -3 � 2 n -3 , 1 � n +1 � 1 , 0 � 2 n -2 � 2 n , 0 � 2 n . . . . . � 2 n -3 , 0 � 2 n -3 . 2 n -4 � 2 n -4 , 0 � Key: � idR , level � Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

  48. Lower Bound on the Worst Case Stabilization Time in Steps Case of the reset of 2 n − 4 . . . processes: 2 n − 1 2 n − 2 2 n − 3 2 n − 4 idR = 2 n -2 � 2 n -3 , 2 � 2 n -1 idR = 2 n -3 idR = 2 n -3 � 2 n -3 , 1 � n +1 � 1 , 0 � 2 n -2 � 2 n , 0 � 2 n . . . . . � 2 n -3 , 0 � 2 n -3 . 2 n -4 � 2 n -4 , 0 � Key: � idR , level � Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

  49. Lower Bound on the Worst Case Stabilization Time in Steps Case of the reset of 2 n − 4 . . . processes: 2 n − 1 2 n − 2 2 n − 3 2 n − 4 idR = 2 n -2 � 2 n -3 , 2 � 2 n -1 idR = 2 n -3 idR = 2 n -3 � 2 n -3 , 1 � n +1 � 1 , 0 � 2 n -2 idR = 2 n -4 � 2 n , 0 � 2 n . . . . . � 2 n -4 , 1 � 2 n -3 . 2 n -4 � 2 n -4 , 0 � Key: � idR , level � Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

  50. Lower Bound on the Worst Case Stabilization Time in Steps Case of the reset of 2 n − 4 . . . processes: 2 n − 1 2 n − 2 2 n − 3 2 n − 4 idR = 2 n -2 � 2 n -3 , 2 � 2 n -1 idR = 2 n -3 idR = 2 n -3 � 2 n -4 , 2 � n +1 � 1 , 0 � 2 n -2 idR = 2 n -4 idR = 2 n -4 � 2 n , 0 � 2 n . . . . . � 2 n -4 , 1 � 2 n -3 . 2 n -4 � 2 n -4 , 0 � Key: � idR , level � Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

  51. Lower Bound on the Worst Case Stabilization Time in Steps Case of the reset of 2 n − 4 . . . processes: 2 n − 1 2 n − 2 2 n − 3 2 n − 4 idR = 2 n -2 � 2 n -4 , 3 � 2 n -1 idR = 2 n -3 idR = 2 n -3 � 2 n -4 , 2 � n +1 � 1 , 0 � 2 n -2 idR = 2 n -4 idR = 2 n -4 idR = 2 n -4 � 2 n , 0 � 2 n . . . . . � 2 n -4 , 1 � 2 n -3 . 2 n -4 � 2 n -4 , 0 � Key: � idR , level � Clean EBroadcast EFeedback Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

  52. Lower Bound on the Worst Case Stabilization Time in Steps Case of the reset of 2 n − 4 . . . processes: 2 n − 1 2 n − 2 2 n − 3 2 n − 4 idR = 2 n -2 � 2 n -4 , 3 � 2 n -1 idR = 2 n -3 idR = 2 n -3 j = 4 � 2 n -4 , 2 � n +1 � 1 , 0 � � j − 1 2 n -2 i = 1 i idR = 2 n -4 idR = 2 n -4 idR = 2 n -4 � 2 n , 0 � 2 n . . . . . � 2 n -4 , 1 � 2 n -3 . 2 n -4 � 2 n -4 , 0 � j − 1 n Key: � idR , level � i ⇒ Θ( n 3 ) steps � � Θ( n ) reset ⇒ Clean EBroadcast EFeedback j = 1 i = 1 Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 15 / 23

  53. Analytical Study of Datta et al , 2011 3 3Datta, Larmore, and Vemula. Self-stabilizing Leader Election in Optimal Space under an Arbitrary Scheduler. 2011 Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 16 / 23

  54. Principles Join a tree � 1 , 0 � 1 � 1 , 1 � � 1 , 1 � 4 3 � 2 , 0 � � 6 , 0 � 2 6 Key: � idR , level � Can be joined Cannot be joined Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 17 / 23

  55. Principles Join a tree � 1 , 0 � 1 � 1 , 1 � � 1 , 1 � 4 3 � 1 , 2 � � 6 , 0 � 2 6 Key: � idR , level � Can be joined Cannot be joined Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 17 / 23

  56. Principles Change of color � 1 , 2 � � 1 , 2 � 4 4 � 1 , 3 � � 1 , 3 � 7 7 � 1 , 4 � � 1 , 4 � � 1 , 4 � � 1 , 4 � 2 5 2 5 Key: � idR , level � Can be joined Cannot be joined Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 18 / 23

  57. Principles Change of color � 1 , 2 � � 1 , 2 � 4 4 � 1 , 3 � � 1 , 3 � 7 7 � 1 , 4 � � 1 , 4 � � 1 , 4 � � 1 , 4 � 2 5 2 5 Key: � idR , level � Can be joined Cannot be joined Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 18 / 23

  58. Principles Change of color � 1 , 2 � � 1 , 2 � 4 4 � 1 , 3 � � 1 , 3 � 7 7 � 1 , 4 � � 1 , 4 � � 1 , 4 � � 1 , 4 � 2 5 2 5 Key: � idR , level � Can be joined Cannot be joined Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 18 / 23

  59. Principles Color Waves Absorption Normal tree Abnormal tree Key: � idR , level � Can be joined Cannot be joined Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 19 / 23

  60. Principles Color Waves Absorption Normal tree Abnormal tree Key: � idR , level � Can be joined Cannot be joined Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 19 / 23

  61. Principles Color Waves Absorption Normal tree Abnormal tree Key: � idR , level � Can be joined Cannot be joined Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 19 / 23

  62. Datta et al, 2011 Execution in Ω( n 4 ) steps: β = n 8 . . . 1,1 1,2 1, β . . . 2, β 2,1 2,2 . . . 3,1 3,2 3, β . . . 4,1 4,2 4, β Key: ( i , j ) . ID = ( i − 1 ) β + j . . . 5,1 5,2 5, β ( i , j ) . idR = 0 Can be joined . . . 6,1 6,2 6, β Cannot be joined . . . 7,1 7,2 7, β . . . 8,1 8,2 8, β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

  63. Datta et al, 2011 Execution in Ω( n 4 ) steps: β = n 8 . . . 1,1 1,2 1, β . . . 2, β 2,1 2,2 . . . 3,1 3,2 3, β . . . 4,1 4,2 4, β Key: ( i , j ) . ID = ( i − 1 ) β + j . . . 5,1 5,2 5, β ( i , j ) . idR = 0 Can be joined . . . 6,1 6,2 6, β Cannot be joined . . . 7,1 7,2 7, β . . . 8,1 8,2 8, β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

  64. Datta et al, 2011 Execution in Ω( n 4 ) steps: β = n 8 . . . 1,1 1,2 1, β . . . 2, β 2,1 2,2 . . . 3,1 3,2 3, β . . . 4,1 4,2 4, β Key: ( i , j ) . ID = ( i − 1 ) β + j . . . 5,1 5,2 5, β ( i , j ) . idR = 0 Can be joined . . . 6,1 6,2 6, β Cannot be joined . . . 7,1 7,2 7, β . . . 8,1 8,2 8, β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

  65. Datta et al, 2011 Execution in Ω( n 4 ) steps: β = n 8 . . . 1,1 1,2 1, β . . . 2, β 2,1 2,2 . . . 3,1 3,2 3, β . . . 4,1 4,2 4, β Key: ( i , j ) . ID = ( i − 1 ) β + j . . . 5,1 5,2 5, β ( i , j ) . idR = 0 Can be joined . . . 6,1 6,2 6, β Cannot be joined . . . 7,1 7,2 7, β . . . 8,1 8,2 8, β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

  66. Datta et al, 2011 Execution in Ω( n 4 ) steps: β = n 8 . . . 1,1 1,2 1, β . . . 2, β 2,1 2,2 . . . 3,1 3,2 3, β . . . 4,1 4,2 4, β Key: ( i , j ) . ID = ( i − 1 ) β + j . . . 5,1 5,2 5, β ( i , j ) . idR = 0 Can be joined . . . 6,1 6,2 6, β Cannot be joined . . . 7,1 7,2 7, β . . . 8,1 8,2 8, β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

  67. Datta et al, 2011 Execution in Ω( n 4 ) steps: β = n 8 . . . 1,1 1,2 1, β . . . 2, β 2,1 2,2 . . . 3,1 3,2 3, β . . . 4,1 4,2 4, β Key: ( i , j ) . ID = ( i − 1 ) β + j . . . 5,1 5,2 5, β ( i , j ) . idR = 0 Can be joined . . . 6,1 6,2 6, β Cannot be joined . . . 7,1 7,2 7, β . . . 8,1 8,2 8, β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

  68. Datta et al, 2011 Execution in Ω( n 4 ) steps: β = n 8 . . . 1,1 1,2 1, β . . . 2, β 2,1 2,2 . . . 3,1 3,2 3, β . . . 4,1 4,2 4, β Key: ( i , j ) . ID = ( i − 1 ) β + j . . . 5,1 5,2 5, β ( i , j ) . idR = 0 Can be joined . . . 6,1 6,2 6, β Cannot be joined . . . 7,1 7,2 7, β . . . 8,1 8,2 8, β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

  69. Datta et al, 2011 Execution in Ω( n 4 ) steps: β = n 8 . . . 1,1 1,2 1, β . . . 2, β 2,1 2,2 . . . 3,1 3,2 3, β . . . 4,1 4,2 4, β Key: ( i , j ) . ID = ( i − 1 ) β + j . . . 5,1 5,2 5, β ( i , j ) . idR = 0 Can be joined . . . 6,1 6,2 6, β Cannot be joined . . . 7,1 7,2 7, β . . . 8,1 8,2 8, β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

  70. Datta et al, 2011 Execution in Ω( n 4 ) steps: β = n 8 . . . 1,1 1,2 1, β . . . 2, β 2,1 2,2 . . . 3,1 3,2 3, β . . . 4,1 4,2 4, β Key: ( i , j ) . ID = ( i − 1 ) β + j . . . 5,1 5,2 5, β ( i , j ) . idR = 0 Can be joined . . . 6,1 6,2 6, β Cannot be joined . . . 7,1 7,2 7, β . . . 8,1 8,2 8, β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

  71. Datta et al, 2011 Execution in Ω( n 4 ) steps: β = n 8 . . . 1,1 1,2 1, β . . . 2, β 2,1 2,2 β . . . 3,1 3,2 3, β . . . 4,1 4,2 4, β Key: ( i , j ) . ID = ( i − 1 ) β + j . . . 5,1 5,2 5, β ( i , j ) . idR = 0 Can be joined . . . 6,1 6,2 6, β Cannot be joined . . . 7,1 7,2 7, β . . . 8,1 8,2 8, β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

  72. Datta et al, 2011 Execution in Ω( n 4 ) steps: β = n 8 . . . 1,1 1,2 1, β . . . 2, β 2,1 2,2 β . . . 3,1 3,2 3, β . . . 4,1 4,2 4, β Key: ( i , j ) . ID = ( i − 1 ) β + j . . . 5,1 5,2 5, β ( i , j ) . idR = 0 Can be joined . . . 6,1 6,2 6, β Cannot be joined . . . 7,1 7,2 7, β . . . 8,1 8,2 8, β Anaïs Durand (VERIMAG) Self-Stabilizing Leader Election September 29, 2014 20 / 23

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