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Maximal Independent Set

This article delves into the concept of Maximal Independent Sets (MIS) in graph theory, which consist of non-adjacent nodes. It discusses the characteristics of MIS, Maximum Independent Sets, and their applications in distributed systems, such as in wireless ad hoc networks where they enable parallel processor operations and efficient routing. It also explains a Sequential Greedy Algorithm for finding MIS and introduces a Randomized Distributed Algorithm that improves performance. Additionally, it covers the probability analysis and expected phases for successful execution of these algorithms.

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

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

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

  8. Remove and neighbors

  9. Remove and neighbors

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

  11. Remove and neighbors

  12. Remove and neighbors

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

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

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

  16. Running time of the algorithm: Θ(|E|) Number of phases of the algorithm: O(n) Worst case graph (for number of phases): nodes

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

  18. 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)

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

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

  21. remove and neighbors

  22. remove and neighbors

  23. remove and neighbors

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

  25. remove and neighbors

  26. remove and neighbors

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

  28. remove and neighbors

  29. remove and neighbors No nodes are left

  30. Final MIS

  31. 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

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

  33. 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

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

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

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

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

  38. 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

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

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

  41. Fundamental inequalities

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

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

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

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

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

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

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

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

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