1 / 49

Maximal Independent Set

Maximal Independent Set. Independent Set (IS):. In a graph G=(V,E), |V|=n, |E|=m, any set of nodes that are not adjacent. Maximal Independent Set (MIS):. An independent set that is no subset of any other independent set. Maximum Independent Set:. A MIS of maximum size. A graph G….

stian
Download Presentation

Maximal Independent Set

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Maximal Independent Set

  2. Independent Set (IS): In a graph G=(V,E), |V|=n, |E|=m, 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. 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. A Sequential Greedy algorithm Suppose that will hold the final MIS Initially

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

  9. Remove and neighbors

  10. Remove and neighbors

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

  12. Remove and neighbors

  13. Remove and neighbors

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

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

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

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

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

  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. Example: Suppose that will hold the final MIS Initially

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

  22. remove and neighbors

  23. remove and neighbors

  24. remove and neighbors

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

  26. remove and neighbors

  27. remove and neighbors

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

  29. remove and neighbors

  30. remove and neighbors No nodes are left

  31. Final MIS

  32. Observation: 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

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

  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. 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)

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

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

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

  39. 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

  40. 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.

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

  42. Fundamental inequalities

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

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

  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) at most:

  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?

More Related