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Distributed minimum spanning tree problem Kustaa Kangas University of Helsinki, Finland November 8, 2012 K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 1 / 59 Minimum spanning tree A spanning tree of an

  1. Distributed minimum spanning tree problem Kustaa Kangas University of Helsinki, Finland November 8, 2012 K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 1 / 59

  2. Minimum spanning tree A spanning tree of an undirected graph is a subtree containing all vertices. K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 2 / 59

  3. Minimum spanning tree A spanning tree of an undirected graph is a subtree containing all vertices. K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 3 / 59

  4. Minimum spanning tree A spanning tree of an undirected graph is a subtree containing all vertices. K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 4 / 59

  5. Minimum spanning tree For a weighted graph, the weight of a spanning tree is the sum of the weights of its edges. A tree with a minimum weight is called a minimum spanning tree . For simplicity assume that all weights are disctinct so the MST is unique (proof in the report). K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 5 / 59

  6. Minimum spanning tree For a weighted graph, the weight of a spanning tree is the sum of the weights of its edges. A tree with a minimum weight is called a minimum spanning tree . For simplicity assume that all weights are disctinct so the MST is unique (proof in the report). K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 6 / 59

  7. Finding minimum spanning tree How to find the MST of a given graph? Fast and simple centralized algorithms: Kruskal: add edges greedily, avoid cycles. Prim: expand a single fragment greedily. Run in O ( | E | log | N | ). K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 7 / 59

  8. Kruskal’s algorithm K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 8 / 59

  9. Kruskal’s algorithm K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 9 / 59

  10. Kruskal’s algorithm K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 10 / 59

  11. Kruskal’s algorithm K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 11 / 59

  12. Kruskal’s algorithm K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 12 / 59

  13. Kruskal’s algorithm K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 13 / 59

  14. Kruskal’s algorithm K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 14 / 59

  15. Finding minimum spanning tree How to find the MST of a given graph? Fast and simple centralized algorithms: Kruskal: add edges greedily, avoid cycles. Prim: expand a single fragment greedily. Run in O ( | E | log | N | ). K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 15 / 59

  16. Prim’s algorithm K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 16 / 59

  17. Prim’s algorithm K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 17 / 59

  18. Prim’s algorithm K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 18 / 59

  19. Prim’s algorithm K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 19 / 59

  20. Prim’s algorithm K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 20 / 59

  21. Prim’s algorithm K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 21 / 59

  22. Finding minimum spanning tree Fast and common in the centralized setting. Slow in the distributed setting: Kruskal: requires global knowledge of the graph. Prim: requires communication within a large fragment. K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 22 / 59

  23. Finding minimum spanning tree Adding the minimum weight outgoing edge of any known fragment of the MST produces a new fragment of the MST. K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 23 / 59

  24. Finding minimum spanning tree Idea: Start with one fragment per node. Each fragment finds the minimum weight outgoing edge independently. K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 24 / 59

  25. Finding minimum spanning tree Fragments with a common minimum weight outgoing edge combine until there is one fragment containing the whole MST. K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 25 / 59

  26. Model of computation A classical algorithm by Gallager et al. [1983] does this in the asynchronous message passing model: Each node runs the same simple local algorithm. Each node initially knows only the weights of its incident edges. Adjacent nodes can exchange (short) messages. Asynchronous: messages arrive after an unknown but bounded delay. Performance measured in time and message complexity. K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 26 / 59

  27. Maintaining fragments on node level Each node keeps track of which incident edges are part of its fragment. How to find the minimum weight outgoing edge? Trivial for single nodes: just pick the minimum weight incident edge. All nodes know which incident edges are in the fragment. K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 27 / 59

  28. Maintaining fragments on node level How about larger fragments? When two fragments combine, the edge joining them becomes the core of the new fragment. It serves as 1 a unique identity of the fragment 2 a hub of computation within the fragment All nodes know the identity and the edge leading towards the core. K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 28 / 59

  29. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 29 / 59

  30. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 30 / 59

  31. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 31 / 59

  32. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 32 / 59

  33. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 33 / 59

  34. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 34 / 59

  35. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 35 / 59

  36. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 36 / 59

  37. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 37 / 59

  38. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 38 / 59

  39. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 39 / 59

  40. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 40 / 59

  41. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 41 / 59

  42. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 42 / 59

  43. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 43 / 59

  44. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 44 / 59

  45. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 45 / 59

  46. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 46 / 59

  47. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 47 / 59

  48. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 48 / 59

  49. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 49 / 59

  50. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 50 / 59

  51. Finding the minimum weight outgoing edge K. Kangas (U. Helsinki) Distributed minimum spanning tree problem November 8, 2012 51 / 59

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