Maximal Independent Set

1. Maximal Independent Set

2. Independent Set (IS): 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. A Sequential Greedy algorithm Suppose that will hold the final MIS Initially

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

6. Remove and neighbors

7. Remove and neighbors

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

9. Remove and neighbors

10. Remove and neighbors

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

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

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

14. Running time of algorithm: Worst case graph: nodes

15. 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)

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

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

18. remove and neighbors

19. remove and neighbors

20. remove and neighbors

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

22. remove and neighbors

23. remove and neighbors

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

25. remove and neighbors

26. remove and neighbors No nodes are left

27. Final MIS

28. 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

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

30. A Simple 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

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

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

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

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

35. 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

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

37. Fundamental inequalities

38. Probability of success in phase: At least For

39. Therefore, node will enter and disappear in phase with probability at least 2 1

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

41. Bad event for node : after phases node did not disappear Probability:

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

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

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

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

46. Let be the degree of node 2 1

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

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

49. If both have the same degree, ties are broken arbitrarily Example: if

50. Using previous rules, problematic nodes are removed Problematic nodes