1. 1 Maximal Independent Set

2. 2 Independent Set (IS): In a graph G=(V,E), |V|=n, |E|=m, any set of nodes that are not adjacent

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

4. 4 Maximum Independent Set: A MIS of maximum size A graph G……a MIS of G… …a MIS of max size

5. 5 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

6. 6 Applications in Distributed Systems (2) A MIS is always a Dominating Set (DS) of the graph (the converse in not true), namely every node in G must be at distance at most 1 from at least one node in the MIS  In a network graph G consisting of nodes representing processors, a MIS defines a set of processors which can monitor the correct functioning of all the nodes in G

7. 7 A Sequential Greedy algorithm Suppose that will hold the final MIS Initially

8. 8 Pick a node and add it to Phase 1:

9. 9 Remove and neighbors

10. 10 Remove and neighbors

11. 11 Pick a node and add it to Phase 2:

12. 12 Remove and neighbors

13. 13 Remove and neighbors

14. 14 Repeat until all nodes are removed Phases 3,4,5,…:

15. 15 Repeat until all nodes are removed No remaining nodes Phases 3,4,5,…,x:

16. 16 At the end, set will be an MIS of

17. 17 Worst case graph (for number of phases): n nodes, n-1 phases Running time of the algorithm: Θ(m) Number of phases of the algorithm: O(n)

18. 18 Homework Can you see a distributed version of the algorithm just given?

19. 19 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)

20. 20 Suppose that will hold the final MIS Initially Example:

21. 21 Find any independent set Phase 1: And insert to :

22. 22 remove and neighbors

23. 23 remove and neighbors

24. 24 remove and neighbors

25. 25 Phase 2: Find any independent set And insert to : On new graph

26. 26 remove and neighbors

27. 27 remove and neighbors

28. 28 Phase 3: Find any independent set And insert to : On new graph

29. 29 remove and neighbors

30. 30 remove and neighbors No nodes are left

31. 31 Final MIS

32. 32 The number of phases depends on the choice of independent set in each phase: The larger the subgraph removed at the end of a phase, the smaller the residual graph, and then the faster the algorithm Observation:

33. 33 Example: If is MIS, 1 phase is needed Example: If each contains one node, phases are needed (sequential greedy algorithm)

34. 34 A Randomized Sync. Distributed Algorithm Follows the general MIS algorithm paradigm, by choosing randomly at each phase the independent set, in such a way that it is expected to include many nodes of the remaining graph

35. 35 Let be the maximum node degree in the whole graph 1 2 Suppose that d is known to all the nodes (this may require a pre-processing)

36. 36 Elected nodes are candidates for independent set Each node elects itself with probability At each phase : 1 2

37. 37 However, it is possible that neighbor nodes may be elected simultaneously Problematic nodes

38. 38 All the problematic nodes must be un-elected. The remaining elected nodes form independent set

39. 39 Success for a node in phase : disappears at end of phase (enters or ) Analysis: 1 2 No neighbor elects itself A good scenario that guarantees success elects itself

40. 40 Basics of Probability E: finite universe of events; let A and B denote two events in E; then: 1.A  B is the event that A or (non-exclusive) B occurs; 2.A  B is the event that both A and B occur.

41. 41 Probability of success in a phase: is at least the probability that a node elects itself and no neighbor elects itself, i.e.: 1 2 No neighbor should elect itself elects itself

42. 42 Fundamental inequalities

43. 43 Probability of success in phase: At least For First (left) ineq. with t =-1

44. 44 Therefore, node disappears at the end of phase with probability at least 1 2

45. 45 after phases Definition: Bad event for node : node did not disappear This happens with probability (first (right) ineq. with t =-1 and n =2ed) at most:

46. 46 after phases Bad event for G: at least one node did not disappear This happens with probability: P(OR x  G (bad event for x)) ≤

47. 47 within phases Good event for G: all nodes disappear This happens with probability: (high probability)

48. 48 Total number of phases: # rounds for each phase: 3 1.In round 1, each node tries to elect itself and notifies neighbors; 2.In round 2, each node receives notifications from neighbors, decide whether is in I k, and notifies neighbors; 3.In round 3, each node receiving notifications from elected neighbors, realizes to be in N(I k ).  total # of rounds: (with high probability)

49. 49 Homework Can you provide a good bound on the number of messages?