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Maximal Independent Set

Maximal Independent Set. Independent Set (IS):. In a graph, any set of nodes that are not adjacent. Maximal Independent Set (MIS):. An independent set that is no subset of any other independent set. Applications in Distributed Systems.

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Maximal Independent Set

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  1. Maximal Independent Set

  2. Independent Set (IS): In a graph, any set of nodes that are not adjacent

  3. Maximal Independent Set (MIS): An independent set that is no subset of any other independent set

  4. Applications in Distributed Systems • In a network graph consisting of nodes representing processors, a MIS defines a set of processors which can operate in parallel without interference • For instance, in wireless ad hoc networks, to avoid interference, a conflict graph is built, and a MIS on that defines a clustering of the nodes enabling efficient routing

  5. A Sequential Greedy algorithm Suppose that will hold the final MIS Initially

  6. Phase 1: Pick a node and add it to

  7. Remove and neighbors

  8. Remove and neighbors

  9. Phase 2: Pick a node and add it to

  10. Remove and neighbors

  11. Remove and neighbors

  12. Phases 3,4,5,…: Repeat until all nodes are removed

  13. Phases 3,4,5,…,x: Repeat until all nodes are removed No remaining nodes

  14. At the end, set will be an MIS of

  15. Running time of algorithm: Number of phases of the algorithm: Worst case graph (for number of phases): nodes

  16. A General Algorithm For Computing MIS Same as the sequential greedy algorithm, but at each phase we may select any independent set (instead of a single node)

  17. Example: Suppose that will hold the final MIS Initially

  18. Phase 1: Find any independent set And insert to :

  19. remove and neighbors

  20. remove and neighbors

  21. remove and neighbors

  22. Phase 2: On new graph Find any independent set And insert to :

  23. remove and neighbors

  24. remove and neighbors

  25. Phase 3: On new graph Find any independent set And insert to :

  26. remove and neighbors

  27. remove and neighbors No nodes are left

  28. Final MIS

  29. Observation: The number of phases depends on the choice of independent set in each phase: The larger the independent set at each phase the faster the algorithm

  30. Example: If is MIS, 1 phase is needed Example: If each contains one node, phases are needed (sequential greedy algorithm)

  31. A Randomized Sync. Distributed Algorithm Same with the general MIS algorithm At each phase the independent set is chosen randomly so that it includes many nodes of the remaining graph

  32. Let be the maximum node degree in the whole graph 2 1 Suppose that is known to all the nodes

  33. At each phase : Each node elects itself with probability 2 1 Elected nodes are candidates for independent set

  34. However, it is possible that neighbor nodes may be elected simultaneously Problematic nodes

  35. All the problematic nodes must be un-elected. The remaining elected nodes form independent set

  36. Analysis: Success for a node in phase : disappears at end of phase (enters or ) A good scenario that guarantees success No neighbor elects itself 2 1 elects itself

  37. Basics of Probability • E: finite universe of events; let A and B denote two events in E; then: • A  Bis the event that eitherA or B occurs; • A  Bis the event that both A and B occur.

  38. Probability of success in phase: At least No neighbor should elect itself 2 1 elects itself

  39. Fundamental inequalities

  40. Probability of success in phase: At least First ineq. with t =-1 For

  41. Therefore, node will enter and disappear at the end of phase with probability at least 2 1

  42. Expected number of phases until node disappears: phases at most

  43. Bad event for node : after phases node did not disappear Probability (First ineq. with t =-1 and n=2ed):

  44. Bad event for any node in : after phases at least one node did not disappear Probability:

  45. Good event for all nodes in : within phases all nodes disappear Probability: (high probability)

  46. Total number of phases: with high probability Time duration of each phase: Total time:

  47. Luby’s MIS Distributed Algorithm Runs in time in expected case with high probability this algorithm is asymptotically better than the previous

  48. Let be the degree of node 2 1

  49. At each phase : Each node elects itself with probability degree of in 2 1 Elected nodes are candidates for the independent set

  50. If two neighbors are elected simultaneously, then the higher degree node wins Example: if

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