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Decidability of distributed complexity of locally checkable problems on paths Alkida Balliu, Sebastian Brandt, Yi-Jun Chang, Dennis Olivetti, el Rabie , Jukka Suomela Mika ANR DESCARTES/ESTATE Tuesday, April 2 Mika el RABIE Decidability

  1. Decidability of distributed complexity of locally checkable problems on paths Alkida Balliu, Sebastian Brandt, Yi-Jun Chang, Dennis Olivetti, el Rabie , Jukka Suomela Mika¨ ANR DESCARTES/ESTATE Tuesday, April 2 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 1 / 19

  2. LCL Problems on Paths 1 28 37 52 8 32 46 47 73 5 3 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

  3. LCL Problems on Paths 1 28 37 52 8 32 46 47 73 5 3 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

  4. LCL Problems on Paths 1 28 37 52 8 32 46 47 73 5 3 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

  5. LCL Problems on Paths 1 28 37 52 8 32 46 47 73 5 3 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

  6. LCL Problems on Paths 1 28 37 52 8 32 46 47 73 5 3 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

  7. LCL Problems on Paths 1 28 37 52 8 32 46 47 73 5 3 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

  8. LCL Problems on Paths 1 28 37 52 8 32 46 47 73 5 3 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

  9. LCL Problems on Paths 1 28 37 52 8 32 46 47 73 5 3 1 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

  10. LCL Problems on Paths 1 28 37 52 8 32 46 47 73 5 3 1 2 3 2 3 1 2 1 3 2 1 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

  11. LCL Problems on Paths 1 28 37 52 8 32 46 47 73 5 3 1 2 3 2 3 1 2 1 3 2 1 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

  12. LCL Problems on Paths 1 28 37 52 8 32 46 47 73 5 3 1 2 3 2 3 1 2 1 3 2 1 3 Coloring. Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

  13. LCL Problems on Paths 1 28 37 52 8 32 46 47 73 5 3 1 2 1 2 1 2 1 2 1 2 1 3 Coloring. 2 Coloring. Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

  14. LCL Problems on Paths 1 28 37 52 8 32 46 47 73 5 3 1 0 0 1 0 1 0 1 0 0 1 3 Coloring. 2 Coloring. Maximal Independent Set. Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

  15. LCL Problems on Paths 1 28 37 52 8 32 46 47 73 5 3 0 0 0 1 0 0 0 0 1 0 0 3 Coloring. 2 Coloring. Maximal Independent Set. Independent Set. Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 2 / 19

  16. The log ∗ n Complexity 3-coloring a Path 3 Coloring a Path 1 28 37 52 8 32 46 47 73 5 3 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

  17. The log ∗ n Complexity 3-coloring a Path 3 Coloring a Path 1 28 37 52 8 32 46 47 73 5 3 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

  18. The log ∗ n Complexity 3-coloring a Path 3 Coloring a Path 1 28 37 52 8 32 46 47 73 5 3 1 1 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

  19. The log ∗ n Complexity 3-coloring a Path 3 Coloring a Path 1 28 37 52 8 32 46 47 73 5 3 1 1 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

  20. The log ∗ n Complexity 3-coloring a Path 3 Coloring a Path 1 28 37 52 8 32 46 47 73 5 3 2 1 2 2 1 2 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

  21. The log ∗ n Complexity 3-coloring a Path 3 Coloring a Path 1 28 37 52 8 32 46 47 73 5 3 2 1 2 2 1 2 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

  22. The log ∗ n Complexity 3-coloring a Path 3 Coloring a Path 1 28 37 52 8 32 46 47 73 5 3 1 2 1 2 1 1 2 1 2 1 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

  23. The log ∗ n Complexity 3-coloring a Path 3 Coloring a Path 1 28 37 52 8 32 46 47 73 5 3 1 2 1 2 1 1 2 1 2 1 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

  24. The log ∗ n Complexity 3-coloring a Path 3 Coloring a Path 1 28 37 52 8 32 46 47 73 5 3 2 1 2 1 2 3 1 2 1 2 1 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

  25. The log ∗ n Complexity 3-coloring a Path 3 Coloring a Path 21 22 23 24 25 26 27 28 29 30 31 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

  26. The log ∗ n Complexity 3-coloring a Path 3 Coloring a Path 21 22 23 24 25 26 27 28 29 30 31 Worst case communication complexity : Θ( n ). Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 3 / 19

  27. The log ∗ n Complexity 3-coloring a Path 3-coloring in O (log ∗ n ) Communications Cole, Vishkin (1986) There exists a LCL algorithm to 3-color a path in O (log ∗ n ) communications. Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 4 / 19

  28. The log ∗ n Complexity 3-coloring a Path From n colors to log n colors 42 102 36 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

  29. The log ∗ n Complexity 3-coloring a Path From n colors to log n colors 101010 1100110 100100 42 102 36 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

  30. The log ∗ n Complexity 3-coloring a Path From n colors to log n colors 101010 1100110 100100 42 102 36 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

  31. The log ∗ n Complexity 3-coloring a Path From n colors to log n colors 101010 1100110 100100 42 102 36 3#0 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

  32. The log ∗ n Complexity 3-coloring a Path From n colors to log n colors 101010 1100110 100100 42 102 36 3#0 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

  33. The log ∗ n Complexity 3-coloring a Path From n colors to log n colors 101010 1100110 100100 42 102 36 3#0 2#1 2#0 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

  34. The log ∗ n Complexity 3-coloring a Path From n colors to log n colors 110 101 100 42 102 36 3#0 2#1 2#0 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

  35. The log ∗ n Complexity 3-coloring a Path From n colors to log n colors 110 101 100 6 5 4 3#0 2#1 2#0 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

  36. The log ∗ n Complexity 3-coloring a Path From n colors to log n colors 110 101 100 6 5 4 3#0 2#1 2#0 n colors ⇒ log n bits ⇒ 2 log n new colors ⇒ log log n + 1 bits Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

  37. The log ∗ n Complexity 3-coloring a Path From n colors to log n colors 110 101 100 6 5 4 3#0 2#1 2#0 n colors ⇒ log n bits ⇒ 2 log n new colors ⇒ log log n + 1 bits After log ∗ n iterations, O (1) bits. Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 5 / 19

  38. log ∗ Lower Bound The log ∗ n Complexity Coloration Lower Bound Linial (1992) An algorithm which colors the n -cycle with three colors requires time at 2 (log ∗ n − 3). The same bound holds also for randomized algorithms. least 1 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 6 / 19

  39. log ∗ Lower Bound The log ∗ n Complexity Speed up Algorithm A : algorithm that k -colors nodes in T rounds. Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

  40. log ∗ Lower Bound The log ∗ n Complexity Speed up Algorithm A : algorithm that k -colors nodes in T rounds. T T Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

  41. log ∗ Lower Bound The log ∗ n Complexity Speed up Algorithm A : algorithm that k -colors nodes in T rounds. T c ∈ [1 , k ] T c Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

  42. log ∗ Lower Bound The log ∗ n Complexity Speed up Algorithm A : algorithm that k -colors nodes in T rounds. T c ∈ [1 , k ] T c T − 1 T − 1 Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

  43. log ∗ Lower Bound The log ∗ n Complexity Speed up Algorithm A : algorithm that k -colors nodes in T rounds. T c ∈ [1 , k ] T c T − 1 T − 1 ∀ id ≤ n Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

  44. log ∗ Lower Bound The log ∗ n Complexity Speed up Algorithm A : algorithm that k -colors nodes in T rounds. T c ∈ [1 , k ] T c T − 1 T − 1 S L ∈ 2 k ∀ id ≤ n Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

  45. log ∗ Lower Bound The log ∗ n Complexity Speed up Algorithm A : algorithm that k -colors nodes in T rounds. T c ∈ [1 , k ] T c T − 1 T − 1 S L ∀ id ≤ n Mika¨ el RABIE Decidability of LCL Problems on Paths Tuesday, April 2 7 / 19

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