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ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 1 Stephan Holzer Silvio Frischknecht Roger Wattenhofer Networks Cannot Compute Their Diameter.

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Presentation on theme: "ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 1 Stephan Holzer Silvio Frischknecht Roger Wattenhofer Networks Cannot Compute Their Diameter."— Presentation transcript:

1 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 1 Stephan Holzer Silvio Frischknecht Roger Wattenhofer Networks Cannot Compute Their Diameter in Sublinear Time ETH Zurich – Distributed Computing Group

2 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Distributed network Graph G of n nodes 2 4 5 1 3

3 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Distributed network Graph G of n nodes 2 4 5 1 3

4 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Distributed network Graph G of n nodes 2 4 5 1 3 1 Local infor- mation only

5 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Distributed network Graph G of n nodes 2 4 5 1 3 2 1 3 Local infor- mation only Slide inspired by Danupon Nanongkai

6 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Distributed network Graph G of n nodes 2 4 5 1 3 2 1 3 Local infor- mation only ?

7 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Distributed network Graph G of n nodes 2 4 5 1 3 Limited bandwidth log n 2 1 3 Local infor- mation only ?

8 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Distributed network Graph G of n nodes 2 4 5 1 3 Limited bandwidth log n 2 1 3 Local infor- mation only ? Time complexity: number of communication rounds Synchronized Internal computations negligible

9 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Distributed algorithms: a simple example

10 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Count the nodes!

11 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Count the nodes! 1.Compute BFS-Tree

12 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 0 Count the nodes! 1.Compute BFS-Tree

13 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 1 1 0 1 1 Count the nodes! 1.Compute BFS-Tree

14 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Count the nodes! 1 1 0 2 1 2 1 2 2 2 Runtime: ? Diameter 1.Compute BFS-Tree 2. Count nodes in subtrees

15 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Diameter of a network Distance between two nodes = Number of hops of shortest path

16 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Diameter of a network Distance between two nodes = Number of hops of shortest path

17 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Diameter of a network Distance between two nodes = Number of hops of shortest path Diameter of network = Maximum distance, between any two nodes

18 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Diameter of a network Distance between two nodes = Number of hops of shortest path Diameter of network = Maximum distance, between any two nodes Diameter of this network?

19 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Fundamental graph-problems Spanning Tree – Broadcasting, Aggregation, etc Minimum Spanning Tree – Efficient broadcasting, etc. Shortest path – Routing, etc. Steiner tree – Multicasting, etc. Many other graph problems. Thanks for slide to Danupon Nanongkai

20 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Fundamental graph-problems Spanning Tree – Broadcasting, Aggregation, etc Minimum Spanning Tree – Efficient broadcasting, etc. Shortest path – Routing, etc. Steiner tree – Multicasting, etc. Many other graph problems. Global problems: Ω( D ) 2 1 3 Local infor- mation only ?

21 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Fundamental graph-problems Maximal Independent Set Coloring Matching

22 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Fundamental graph-problems Maximal Independent Set Coloring Matching Local problems: runtime independent of / smaller than D e.g. O(log n)

23 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Fundamental problems Spanning Tree Spanning Tree – Broadcasting, Aggregation, etc Minimum Spanning Tree Minimum Spanning Tree – Efficient broadcasting, etc. Shortest path Shortest path – Routing, etc. Steiner tree Steiner tree – Multicasting, etc. Many other graph problems. Diameter appears frequently in distributed computing

24 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Spanning Tree Spanning Tree – Broadcasting, Aggregation, etc Minimum Spanning Tree Minimum Spanning Tree – Efficient broadcasting, etc. Shortest path Shortest path – Routing, etc. Steiner tree Steiner tree – Multicasting, etc. Many other graph problems. Diameter appears frequently in distributed computing Fundamental problems 1. Formal definition? Complexity of computing D ?. Known: Ω (D) Ω(n) ≈Ω(1) This talk: Even if D = 3

25 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Networks cannot compute their diameter in sublinear time!

26 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Diameter of a network Diameter of this network? Distance between two nodes = Number of hops of shortest path Diameter of network = Maximum distance, between any two nodes

27 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Networks cannot compute their diameter in sublinear time!

28 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Networks cannot compute their diameter in sublinear time!

29 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Networks cannot compute their diameter in sublinear time!

30 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Networks cannot compute their diameter in sublinear time!

31 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Networks cannot compute their diameter in sublinear time!

32 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Networks cannot compute their diameter in sublinear time! Base graph has diameter 3

33 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Networks cannot compute their diameter in sublinear time! has diameter 3 2?

34 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Networks cannot compute their diameter in sublinear time! Pair of nodes not connected on both sides? has diameter 3 2?

35 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Networks cannot compute their diameter in sublinear time! Pair of nodes not connected on both sides? has diameter 3 2?

36 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Pair of nodes not connected on both sides? Now: slightly more formal Networks cannot compute their diameter in sublinear time!

37 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Pair of nodes not connected on both sides? Label potential edges Networks cannot compute their diameter in sublinear time!

38 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Pair of nodes not connected on both sides? Label potential edges 11 Networks cannot compute their diameter in sublinear time!

39 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Pair of nodes not connected on both sides? Label potential edges 11 2 2 Networks cannot compute their diameter in sublinear time!

40 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Pair of nodes not connected on both sides? Label potential edges 11 2 2 33 Networks cannot compute their diameter in sublinear time!

41 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Pair of nodes not connected on both sides? Label potential edges 11 2 2 33 4 4 Networks cannot compute their diameter in sublinear time!

42 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Pair of nodes not connected on both sides? 11 2 2 33 4 4 Networks cannot compute their diameter in sublinear time!

43 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Pair of nodes not connected on both sides? 2 4 2 3 11 3 4 Networks cannot compute their diameter in sublinear time! Given graph

44 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Pair of nodes not connected on both sides? Θ(n) edges 2 4 2 3 11 3 4 4 2 3 4 A B Networks cannot compute their diameter in sublinear time!

45 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Pair of nodes not connected on both sides? Θ(n) edges 2 4 2 3 11 3 4 2 3 4 Same as “A and B not disjoint?” Networks cannot compute their diameter in sublinear time! A B 4 2 3 4

46 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Pair of nodes not connected on both sides? Θ(n) edges 2 4 2 3 11 3 4 2 3 4 Same as “A and B not disjoint?” Networks cannot compute their diameter in sublinear time! A ⊆ [n 2 ] B ⊆ [n 2 ] 4 2 3 4

47 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 A ⊆ [n 2 ] B ⊆ [n 2 ] “A and B not disjoint?” 4 2 3 4 Networks cannot compute their diameter in sublinear time!

48 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Pair of nodes not connected on both sides? Θ(n) edges A ⊆ [n 2 ] B ⊆ [n 2 ] “A and B not disjoint?” Communication Complexity randomized: Ω(n 2 ) bits 4 2 3 4 Ω(n) time Networks cannot compute their diameter in sublinear time! Same as

49 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Diameter Approximation

50 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 3/2-ε approximating the diameter takes  (n 1/2 ) Extend 2 vs. 3 D = 9 base graph

51 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 3/2-ε approximating the diameter takes  (n 1/2 ) Extend 2 vs. 3 D = 9 base graph

52 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 3/2-ε approximating the diameter takes  (n 1/2 ) Extend 2 vs. 3 9 D = 9 base graph

53 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 3/2-ε approximating the diameter takes  (n 1/2 ) Extend 2 vs. 3 D = 9 base graph

54 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 3/2-ε approximating the diameter takes  (n 1/2 ) Extend 2 vs. 3 D = 9 base graph 5

55 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 3/2-ε approximating the diameter takes  (n 1/2 ) Extend 2 vs. 3 D = 9 base graph D = 7 good graph ------------- Factor: 3/2 7 5

56 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 3/2-ε approximating the diameter takes  (n 1/2 ) Extend 2 vs. 3 D = 9 base graph D = 7 good graph ------------- Factor: 3/2 -ε 7

57 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Technique is general

58 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 G Technique is general

59 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 G Technique is general

60 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 G cut Technique is general

61 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 cut Technique is general

62 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 cut Technique is general

63 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 cut f’(G a,G b )

64 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Alice Bob cut f’(G a,G b )

65 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Alice Bob cut f’(G a,G b )

66 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Alice Bob cut f’(G a,G b )

67 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Alice Bob cut Time(g) Time(f) ≥ ----------- |cut| f’(G a,G b )

68 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Summary Diameter Ω(n) 7 3/2-eps approximation takes Ω(n 1/2 ) cut general technique

69 ETH Zurich – Distributed Computing Group Stephan HolzerSODA 2012 Thanks!

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