On the Complexity of Distributed Network Decomposition

1. Eliran Natan On the Complexity of Distributed Network Decomposition Alessandro Panconesi, Aravind Srinivasan Open Problems in Distributed Computing (236825) Winter 2013

2. Definitions Let be an undirected graph.

3. Definitions Let be an undirected graph. Diameter The length of the longest shortest path between any two graph vertices.

4. Definitions Let be an undirected graph. Diameter The length of the longest shortest path between any two graph vertices.

5. Definitions Let be an undirected graph. Diameter The length of the longest shortest path between any two graph vertices.

6. Definitions Let be an undirected graph. Induced Subgraph Given , the Induced Subgraph has exactly the edges that appear in  over the vertex set .

7. Definitions Let be an undirected graph. Induced Subgraph Given , the Induced Subgraph has exactly the edges that appear in  over the vertex set . c a b d e

8. Definitions Let be an undirected graph. Induced Subgraph Given , the Induced Subgraph has exactly the edges that appear in  over the vertex set . c a b d e

9. Definitions Let be an undirected graph. Cluster Graph Given a partition: the cluster graph is , where: ,

10. Definitions Let be an undirected graph. Cluster Graph Given a partition: the cluster graph is , where: c a b d , e

11. Definitions Let be an undirected graph. Cluster Graph Given a partition: the cluster graph is , where: c a b d , e

12. Definitions Let be an undirected graph. Cluster Graph Given a partition: the cluster graph is , where: c a b d , e 1 2 3

13. Definitions Let be an undirected graph. Vertex Coloring Assignment of colors to each vertex such that neighbors do not share the same color.

14. Definitions Let be an undirected graph. Vertex Coloring Assignment of colors to each vertex such that neighbors do not share the same color. c a b d e

15. Definitions Let be an undirected graph. Vertex Coloring Assignment of colors to each vertex such that neighbors do not share the same color. c a b d e

16. Definitions Let be an undirected graph. - Graph Decomposition A partition of such that: • Every is of diameter. • The cluster graph is vertex colored with colors. -coloring is a -decomposition.

17. Main Idea Given Prove that it is possible to compute a -Network Decomposition in a rounds, where . -Network Decomposition -Network Decomposition

18. Ruling Forest A forest of rooted trees , where each tree is a subgraph of , such that: • For every , the root of , called leader of and donated by , is in . • Every vertex in belongs to a unique tree. • Trees are vertex-disjoint. • for every • Trees depth is at most

19. Ruling Forest A forest of rooted trees , where each tree is a subgraph of , such that: • For every , the root of , called leader of and donated by , is in . • Every vertex in belongs to a unique tree. • Trees are vertex-disjoint. • for every • Trees depth is at most

20. Ruling Forest A forest of rooted trees , where each tree is a subgraph of , such that: • For every , the root of , called leader of and donated by , is in . • Every vertex in belongs to a unique tree. • Trees are vertex-disjoint. • for every • Trees depth is at most

21. Ruling Forest A forest of rooted trees , where each tree is a subgraph of , such that: • For every , the root of , called leader of and donated by , is in . • Every vertex in belongs to a unique tree. • Trees are vertex-disjoint. • for every • Trees depth is at most

22. Merging Decompositions Let be an undirected graph, where is partitioned to such that: : Given: -decomposition of -coloring of We can compute -decomposition of in time.

23. Algorithm

24. Algorithm Input An undirected graph such that Output A -network decomposition of .

25. Algorithm Compute .

26. Algorithm Compute .

27. Algorithm Compute a -ruling forest of with respect to .

28. Algorithm Compute a -ruling forest of with respect to . Lbe the graph induced by the resulting forest.

29. Algorithm Lbe the graph induced by the resulting forest. .

30. Algorithm Let be the rest of the graph.

31. Algorithm Let be the rest of the graph. Compute a -colouring to the rest of the graph.

32. Algorithm Merge and to a single decomposition of . time.

33. Algorithm Merge and to a single decomposition of . time.

34. Algorithm – Complexity Minimumis attained when: For this choice of we get: = Where .

35. The Trivial Distributed Algorithm

36. The Trivial Distributed Algorithm • Given a graph and its decomposition: • Iterate through the color classes.

37. The Trivial Distributed Algorithm • Given a graph and its decomposition: • Iterate through the color classes. • All clusters from the same color computes a local solution in parallel.

38. The Trivial Distributed Algorithm • Given a graph and its decomposition: • Iterate through the color classes. • All clusters from the same color computes a local solution in parallel. • Inside each cluster: the leader computes a solution.

39. The Trivial Distributed Algorithm • Given a graph and its decomposition: • Iterate through the color classes. • All clusters from the same color computes a local solution in parallel. • Inside each cluster: the leader computes a solution. Efficient when the clusters are crowded, so massage passing is fast.

40. Shortcut Graph

41. Shortcut Graph ,

42. Shortcut Graph ,

43. Shortcut Graph ,

44. Shortcut Graph ,

45. Algorithm Input An undirected graph Output A -network decomposition of .

46. Algorithm Use to compute a -decomposition of the -shortcut graph of

47. Algorithm For each :

48. Algorithm For each : Iterate over the color classes of so that all clusters of color are being processed in parallel.

49. Algorithm For each : Iterate over the color classes of so that all clusters of color are being processed in parallel. For each cluster keep a copy of , and execute the following routine:

50. Algorithm For each : Iterate over the color classes of so that all clusters of color are being processed in parallel. For each cluster keep a copy of , and execute the following routine: Choose an unvisited vertex.