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Distributed Vertex Coloring

This article presents a distributed algorithm for vertex coloring in a graph using a randomized approach. The running time of the algorithm is analyzed with high probability.

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Distributed Vertex Coloring

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  1. Distributed Vertex Coloring Part II

  2. A Randomized -Coloring Algorithm • Distributed Algorithm • Randomized Algorithm Running time: with high probability (similar with the -coloring algorithm, but now the color palette size is )

  3. Each node has a palette with colors Palette of node Initially all colors in palette are available (Recall: is the node’s degree)

  4. At the beginning of a phase: : uncolored neighbors of : uncolored degree of Example:

  5. In conflicts, the node with the highest uncolored degree wins Example 1 Beginning of phase

  6. In conflicts, the node with the highest uncolored degree wins Random color choices

  7. In conflicts, the node with the highest uncolored degree wins End of phase

  8. In conflicts, the node with the highest uncolored degree wins Example 2 Beginning of phase

  9. In conflicts, the node with the highest uncolored degree wins Random color choices

  10. In conflicts, the node with the highest uncolored degree wins End of phase

  11. If both nodes have same degree, both reject their color Example 3 Beginning of phase

  12. If both nodes have same degree, both reject their color Random color choices

  13. If both nodes have same degree, both reject their color End of phase

  14. Algorithm for node Repeat (iteration = phase) Pick a color uniformly at random from available palette colors; Send color to neighbors; If (some neighbor with chose same color ) Then Reject color ; Else Accept color ; Inform neighbors about color ; (so that they mark color as unavailable) Until color is accepted;

  15. Example execution

  16. Phase 1: (iteration 1) Nodes pick random colors

  17. Conflicts For this phase, uncolored degree = degree The nodes of higher uncolored degree win

  18. Successful colors

  19. Phase 2: (iteration 2) Nodes pick random colors

  20. Conflicts The nodes of higher uncolored degree win

  21. Successful colors

  22. Phase 3: (iteration 3) Nodes pick random colors

  23. Successful colors End of execution

  24. Consider phase (iteration ) Analysis Palette of uncolored node available color unavailable color Set of available colors: Example:

  25. Number of available colors: Palette size Maximum unavailable colors

  26. If then every color choice is a success So suppose that

  27. We want to compute the probability of: Event : node successfully accepts a color in current phase We will prove that is at least constant

  28. Event : node successfully picks and then accepts color (available color)

  29. Note that the events and are mutually exclusive for any pair of colors Any two such events cannot occur simultaneously

  30. since, success for node in current phase is to successfully accept some color in

  31. since are mutually exclusive

  32. :nodes in with same or higher uncolored degree than Example:

  33. Note that for any

  34. Event that node picks randomly color Event that no node in picks randomly color

  35. Probability that node picks color Since node picks randomly and uniformly a color from the available colors

  36. Event that no uncolored neighbor of picks randomly color Event that uncolored neighbor does not pick color

  37. This holds because the events are independent

  38. Consider some

  39. Fundamental inequalities

  40. This holds since and are independent events (the nodes pick randomly their colors independent from one another)

  41. Probability of success

  42. Probability that node succeeds in a phase: at least Probability that node fails in a phase: at most Note that is a constant

  43. is the number of nodes Probability that node fails for phases: at most

  44. Probability that node fails for phases: at most Probability that some node fails for phases: at most Probability that every node succeeds in the first phases: at least

  45. Duration of each phase: time steps The algorithm terminates in phases with probability at least Total time steps: (with high probability) END OF ANALYSIS

  46. Alternative Randomized -Coloring Algorithm • Distributed Algorithm • Randomized Algorithm Running time: with high probability

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