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Self-Stabilizing Silent Disjunction in an Anonymous Network Ajoy K. Datta Stphane Devismes Lawrence L. Larmore University of Nevada Las Vegas Universit Joseph Fourier, Grenoble Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction

  1. Arbitrary Start: Output = 1: No Problem! Some Input Bits 1. Red = Enabled to Reset. Green = Enabled to Join. Clusterheads Reset. Flooding Moves at Speed 1. 1 0 1 6 1 1 0 1 8 1 1 0 1 1 5 1 1 1 6 1 1 1 8 1 0 1 1 1 1 VII 7 7 15 Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 7 / 17

  2. Arbitrary Start: Output = 1: No Problem! Some Input Bits 1. Red = Enabled to Reset. Green = Enabled to Join. Clusterheads Reset. Flooding Moves at Speed 1. 0 1 1 8 1 1 1 1 2 1 1 0 1 1 2 1 1 1 2 1 1 1 2 1 0 1 1 1 1 VII 7 7 2 Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 7 / 17

  3. Arbitrary Start: Output = 1: No Problem! Some Input Bits 1. Red = Enabled to Reset. Green = Enabled to Join. Clusterheads Reset. Flooding Moves at Speed 1. Convergence within 1 + Diam Rounds . 1 1 3 1 1 2 1 1 1 1 1 0 1 1 2 1 1 1 1 2 1 1 2 1 0 1 1 1 3 3 1 VII 2 Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 7 / 17

  4. Arbitrary Start: Output = 0: Serious Problem! VIII Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 8 / 17

  5. Arbitrary Start: Output = 0: Serious Problem! All Input Bits 0. VIII Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 8 / 17

  6. Arbitrary Start: Output = 0: Serious Problem! All Input Bits 0. Output Bits Inside Circles. 1 1 3 4 1 1 2 5 0 8 0 VIII 8 Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 8 / 17

  7. Arbitrary Start: Output = 0: Serious Problem! All Input Bits 0. Output Bits Inside Circles. Red = Enabled to Reset. 1 1 3 4 1 1 2 5 0 8 0 VIII 8 Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 8 / 17

  8. Arbitrary Start: Output = 0: Serious Problem! All Input Bits 0. Output Bits Inside Circles. Red = Enabled to Reset. Green = Enabled to Join. 1 1 3 4 1 1 2 5 0 8 0 VIII 8 Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 8 / 17

  9. Arbitrary Start: Output = 0: Serious Problem! All Input Bits 0. Output Bits Inside Circles. Red = Enabled to Reset. Green = Enabled to Join. Chain Deletes at Head End. But Recruits at Tail (Leaf) End. 1 1 3 4 0 1 8 5 1 6 0 VIII 8 Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 8 / 17

  10. Arbitrary Start: Output = 0: Serious Problem! All Input Bits 0. Output Bits Inside Circles. Red = Enabled to Reset. Green = Enabled to Join. Chain Deletes at Head End. But Recruits at Tail (Leaf) End. Keeps Going Around! 0 1 8 4 0 1 8 5 1 6 1 7 VIII Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 8 / 17

  11. Arbitrary Start: Output = 0: Serious Problem! All Input Bits 0. Output Bits Inside Circles. Red = Enabled to Reset. Green = Enabled to Join. Chain Deletes at Head End. But Recruits at Tail (Leaf) End. Keeps Going Around! 0 0 8 8 1 1 8 5 1 6 1 7 VIII Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 8 / 17

  12. Arbitrary Start: Output = 0: Serious Problem! All Input Bits 0. Output Bits Inside Circles. Red = Enabled to Reset. Green = Enabled to Join. Chain Deletes at Head End. But Recruits at Tail (Leaf) End. Keeps Going Around! 1 0 9 8 1 0 8 8 1 6 1 7 VIII Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 8 / 17

  13. Arbitrary Start: Output = 0: Serious Problem! All Input Bits 0. Output Bits Inside Circles. Red = Enabled to Reset. Green = Enabled to Join. Chain Deletes at Head End. But Recruits at Tail (Leaf) End. Keeps Going Around! 1 1 9 10 1 0 8 8 0 8 1 7 VIII Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 8 / 17

  14. Arbitrary Start: Output = 0: Serious Problem! All Input Bits 0. Output Bits Inside Circles. Red = Enabled to Reset. Green = Enabled to Join. Chain Deletes at Head End. But Recruits at Tail (Leaf) End. Keeps Going Around. Return to First Configuration , Except for Levels. 1 1 9 10 1 1 8 11 0 8 0 VIII 8 Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 8 / 17

  15. Arbitrary Start: Output = 0: Serious Problem! All Input Bits 0. Output Bits Inside Circles. Red = Enabled to Reset. Green = Enabled to Join. Chain Deletes at Head End. But Recruits at Tail (Leaf) End. Keeps Going Around. Return to First Configuration, Except for Levels. Endless! 1 1 9 10 0 1 8 11 1 12 0 VIII 8 Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 8 / 17

  16. Color Wave Algorithm IX Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 9 / 17

  17. Color Wave Algorithm Self-Stabilizing and Silent IX Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 9 / 17

  18. Color Wave Algorithm Self-Stabilizing and Silent Complexities IX Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 9 / 17

  19. Color Wave Algorithm Self-Stabilizing and Silent Complexities Arbitrary Initial Configuration. IX Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 9 / 17

  20. Color Wave Algorithm Self-Stabilizing and Silent Complexities Arbitrary Initial Configuration. 3 Diam + O ( 1 ) Rounds to Achieve Legitimacy if Output = 1. IX Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 9 / 17

  21. Color Wave Algorithm Self-Stabilizing and Silent Complexities Arbitrary Initial Configuration. 3 Diam + O ( 1 ) Rounds to Achieve Legitimacy if Output = 1. 5 Diam + O ( 1 ) Rounds to Achieve Silence if Output = 1. IX Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 9 / 17

  22. Color Wave Algorithm Self-Stabilizing and Silent Complexities Arbitrary Initial Configuration. 3 Diam + O ( 1 ) Rounds to Achieve Legitimacy if Output = 1. 5 Diam + O ( 1 ) Rounds to Achieve Silence if Output = 1. O ( n ) Rounds if Output = 0. The Hard Case! IX Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 9 / 17

  23. Color Wave Algorithm Self-Stabilizing and Silent Complexities Arbitrary Initial Configuration. 3 Diam + O ( 1 ) Rounds to Achieve Legitimacy if Output = 1. 5 Diam + O ( 1 ) Rounds to Achieve Silence if Output = 1. O ( n ) Rounds if Output = 0. The Hard Case! Clean Initial Configuration. IX Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 9 / 17

  24. Color Wave Algorithm Self-Stabilizing and Silent Complexities Arbitrary Initial Configuration. 3 Diam + O ( 1 ) Rounds to Achieve Legitimacy if Output = 1. 5 Diam + O ( 1 ) Rounds to Achieve Silence if Output = 1. O ( n ) Rounds if Output = 0. The Hard Case! Clean Initial Configuration. If Output = 1: Same as Arbitrary. (No Help.) IX Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 9 / 17

  25. Color Wave Algorithm Self-Stabilizing and Silent Complexities Arbitrary Initial Configuration. 3 Diam + O ( 1 ) Rounds to Achieve Legitimacy if Output = 1. 5 Diam + O ( 1 ) Rounds to Achieve Silence if Output = 1. O ( n ) Rounds if Output = 0. The Hard Case! Clean Initial Configuration. If Output = 1: Same as Arbitrary. (No Help.) If Output = 0: Zero Rounds. (Already in Final Configuration) IX Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 9 / 17

  26. Color Wave Algorithm Self-Stabilizing and Silent Complexities Arbitrary Initial Configuration. 3 Diam + O ( 1 ) Rounds to Achieve Legitimacy if Output = 1. 5 Diam + O ( 1 ) Rounds to Achieve Silence if Output = 1. O ( n ) Rounds if Output = 0. The Hard Case! Clean Initial Configuration. If Output = 1: Same as Arbitrary. (No Help.) If Output = 0: Zero Rounds. (Already in Final Configuration) Space Complexity. IX Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 9 / 17

  27. Color Wave Algorithm Self-Stabilizing and Silent Complexities Arbitrary Initial Configuration. 3 Diam + O ( 1 ) Rounds to Achieve Legitimacy if Output = 1. 5 Diam + O ( 1 ) Rounds to Achieve Silence if Output = 1. O ( n ) Rounds if Output = 0. The Hard Case! Clean Initial Configuration. If Output = 1: Same as Arbitrary. (No Help.) If Output = 0: Zero Rounds. (Already in Final Configuration) Space Complexity. O ( log Diam + Degree ) Per Process. IX Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 9 / 17

  28. Purpose of Color Waves XIV Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 10 / 17

  29. Purpose of Color Waves Prevent Indefinite Growth of Fictitious Trees. XIV Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 10 / 17

  30. Purpose of Color Waves Prevent Indefinite Growth of Fictitious Trees. Side Effects of Color Waves XIV Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 10 / 17

  31. Purpose of Color Waves Prevent Indefinite Growth of Fictitious Trees. Side Effects of Color Waves Slow Down Algorithm by a Factor of Three XIV Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 10 / 17

  32. Purpose of Color Waves Prevent Indefinite Growth of Fictitious Trees. Side Effects of Color Waves Slow Down Algorithm by a Factor of Three After Legitimacy, Color Waves could Run Forever. XIV Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 10 / 17

  33. Purpose of Color Waves Prevent Indefinite Growth of Fictitious Trees. Side Effects of Color Waves Slow Down Algorithm by a Factor of Three After Legitimacy, Color Waves could Run Forever. Counteract Effect with Done Waves XIV Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 10 / 17

  34. Purpose of Color Waves Prevent Indefinite Growth of Fictitious Trees. Side Effects of Color Waves Slow Down Algorithm by a Factor of Three After Legitimacy, Color Waves could Run Forever. Counteract Effect with Done Waves Convergecast: Leaves Detect Algorithm Done XIV Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 10 / 17

  35. Purpose of Color Waves Prevent Indefinite Growth of Fictitious Trees. Side Effects of Color Waves Slow Down Algorithm by a Factor of Three After Legitimacy, Color Waves could Run Forever. Counteract Effect with Done Waves Convergecast: Leaves Detect Algorithm Done Root (Clusterhead) Color Freezes. XIV Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 10 / 17

  36. Purpose of Color Waves Prevent Indefinite Growth of Fictitious Trees. Side Effects of Color Waves Slow Down Algorithm by a Factor of Three After Legitimacy, Color Waves could Run Forever. Counteract Effect with Done Waves Convergecast: Leaves Detect Algorithm Done Root (Clusterhead) Color Freezes. Color Lock Results Within O ( Diam ) Rounds. XIV Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 10 / 17

  37. Purpose of Color Waves Prevent Indefinite Growth of Fictitious Trees. Side Effects of Color Waves Slow Down Algorithm by a Factor of Three After Legitimacy, Color Waves could Run Forever. Counteract Effect with Done Waves Convergecast: Leaves Detect Algorithm Done Root (Clusterhead) Color Freezes. Color Lock Results Within O ( Diam ) Rounds. Silence. XIV Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 10 / 17

  38. Color Wave Details: X Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 11 / 17

  39. Color Wave Details: 1 , 1 = 1 . Process with Output Bit = 1 has Color: 0 = X Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 11 / 17

  40. Color Wave Details: 1 , 1 = 1 . Process with Output Bit = 1 has Color: 0 = If Process X Executes Join , Attaching to Process Y : X Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 11 / 17

  41. Color Wave Details: 1 , 1 = 1 . Process with Output Bit = 1 has Color: 0 = If Process X Executes Join , Attaching to Process Y : 1 Y must have Color 1: X Y X Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 11 / 17

  42. Color Wave Details: 1 , 1 = 1 . Process with Output Bit = 1 has Color: 0 = If Process X Executes Join , Attaching to Process Y : 1 Y must have Color 1: X Y 1 1 Color of X Becomes 0: X Y X Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 11 / 17

  43. Color Wave Details: 1 , 1 = 1 . Process with Output Bit = 1 has Color: 0 = If Process X Executes Join, Attaching to Process Y : 1 Y must have Color 1: X Y 1 1 Color of X Becomes 0: X Y Process X can Change Color if the Following Conditions Hold: X Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 11 / 17

  44. Color Wave Details: 1 , 1 = 1 . Process with Output Bit = 1 has Color: 0 = If Process X Executes Join, Attaching to Process Y : 1 Y must have Color 1: X Y 1 1 Color of X Becomes 0: X Y Process X can Change Color if the Following Conditions Hold: X has Same Color as its Parent , or is Clusterhead: or or or 1 1 1 1 1 1 X X X X X Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 11 / 17

  45. Color Wave Details: 1 , 1 = 1 . Process with Output Bit = 1 has Color: 0 = If Process X Executes Join, Attaching to Process Y : 1 Y must have Color 1: X Y 1 1 Color of X Becomes 0: X Y Process X can Change Color if the Following Conditions Hold: X has Same Color as its Parent, or is Clusterhead: or or or 1 1 1 1 1 1 X X X X 1 1 1 1 or Children have Opposite Color : X X 1 1 X Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 11 / 17

  46. Color Wave Details: 1 , 1 = 1 . Process with Output Bit = 1 has Color: 0 = If Process X Executes Join, Attaching to Process Y : 1 Y must have Color 1: X Y 1 1 Color of X Becomes 0: X Y Process X can Change Color if the Following Conditions Hold: X has Same Color as its Parent, or is Clusterhead: or or or 1 1 1 1 1 1 X X X X 1 1 1 1 or Children have Opposite Color: X X 1 1 If Color = 1, X Cannot Change Color if Any Neighbor is Enabled to Attach to X: 1 X X Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 11 / 17

  47. Color Wave Details: 1 , 1 = 1 . Process with Output Bit = 1 has Color: 0 = If Process X Executes Join, Attaching to Process Y : 1 Y must have Color 1: X Y 1 1 Color of X Becomes 0: X Y Process X can Change Color if the Following Conditions Hold: X has Same Color as its Parent, or is Clusterhead: or or or 1 1 1 1 1 1 X X X X 1 1 1 1 or Children have Opposite Color: X X 1 1 If Color = 1: X Cannot Change Color if Any Neighbor is Enabled to Attach to X: 1 X When Clusterhead Changes Color, a Color Wave is Absorbed. X Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 11 / 17

  48. Color Wave Details: 1 , 1 = 1 . Process with Output Bit = 1 has Color: 0 = If Process X Executes Join, Attaching to Process Y : 1 Y must have Color 1: X Y 1 1 Color of X Becomes 0: X Y Process X can Change Color if the Following Conditions Hold: X has Same Color as its Parent, or is Clusterhead: or or or 1 1 1 1 1 1 X X X X 1 1 1 1 or Children have Opposite Color: X X 1 1 If Color = 1: X Cannot Change Color if Any Neighbor is Enabled to Attach to X: 1 X When Clusterhead Changes Color, a Color Wave is Absorbed. False Roots Cannot Absorb Color Waves . X Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 11 / 17

  49. Color Waves: XI Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 12 / 17

  50. Color Waves: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 Chain Example. 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 XI Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 12 / 17

  51. Color Waves: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 Chain Example. 0 0 0 0 0 0 0 1 1 One Process has Input = 1. 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 XI Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 12 / 17

  52. Color Waves: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 Chain Example. 0 0 0 0 0 0 0 1 1 One Process has Input = 1. 0 0 0 0 0 0 0 1 1 Arrow Shows Flow of Time. 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 XI Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 12 / 17

  53. Color Waves: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 Chain Example. 0 0 0 0 0 0 0 1 1 One Process has Input = 1. 0 0 0 0 0 0 0 1 1 Arrow Shows Flow of Time. 0 0 0 0 0 0 0 1 1 Color Waves Move 0 0 0 0 0 0 1 1 1 Toward Clusterhead 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 XI Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 12 / 17

  54. Color Waves: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 Chain Example. 0 0 0 0 0 0 0 1 1 One Process has Input = 1. 0 0 0 0 0 0 0 1 1 Arrow Shows Flow of Time. 0 0 0 0 0 0 0 1 1 Color Waves Move 0 0 0 0 0 0 1 1 1 Toward Clusterhead 0 0 0 0 0 0 1 1 1 Growth Rate = 1/3 . 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 XI Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 12 / 17

  55. Lazy Color Waves: 0 0 0 0 0 0 0 0 0 0 0 0 0 Chain Example. 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 One Process has Input = 1. 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 Arrow Shows Flow of Time. 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 Color Waves Move 0 0 0 0 0 0 1 1 1 1 1 0 0 Toward Clusterhead 0 0 0 0 0 0 0 0 1 1 1 1 1 Growth Rate = 1/3. 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 If Any Process is “Lazy": 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 0 XI 0 1 1 1 1 1 1 1 1 1 1 1 0 Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 12 / 17

  56. Lazy Color Waves: 0 0 0 0 0 0 0 0 0 0 0 0 0 Chain Example. 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 One Process has Input = 1. 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 Arrow Shows Flow of Time. 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 Color Waves Move 0 0 0 0 0 0 1 1 1 1 1 0 0 Toward Clusterhead 0 0 0 0 0 0 0 0 1 1 1 1 1 Growth Rate = 1/3. 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 If Any Process is “Lazy": 0 0 0 0 0 1 1 1 1 1 1 1 0 Eventual Color Lock : 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 0 XI 0 1 1 1 1 1 1 1 1 1 1 1 0 Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 12 / 17

  57. Lazy Color Waves: 0 0 0 0 0 0 0 0 0 0 0 0 0 Chain Example. 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 One Process has Input = 1. 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 Arrow Shows Flow of Time. 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 Color Waves Move 0 0 0 0 0 0 1 1 1 1 1 0 0 Toward Clusterhead 0 0 0 0 0 0 0 0 1 1 1 1 1 Growth Rate = 1/3. 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 If Any Process is “Lazy": 0 0 0 0 0 1 1 1 1 1 1 1 0 Eventual Color Lock : 0 0 0 0 1 1 1 1 1 1 1 1 0 That is, Lazy Process = 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 Only Enabled Process. 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 0 XI 0 1 1 1 1 1 1 1 1 1 1 1 0 Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 12 / 17

  58. Getting Rid of False Roots: XII Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 13 / 17

  59. Getting Rid of False Roots: If Any Process Stops Executing: XII Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 13 / 17

  60. Getting Rid of False Roots: If Any Process Stops Executing: Eventual Color Lock : XII Datta Devismes Larmore (UNLV/UJFG) Distributed Disjunction SSS 2012 13 / 17

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