Optimal Distributed All Pairs Shortest Paths

1. Optimal Distributed • All Pairs Shortest Paths Stephan Holzer ETH Zürich Roger Wattenhofer ETH Zürich ETH Zurich – Distributed Computing Group TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAA

2. Distributed network Graph G of n nodes 2 1 5 3 4

3. Distributed network Graph G of n nodes Unique IDs 2 1 5 3 4

4. Distributed network Graph G of n nodes Unique IDs 2 Localinfor- mationonly 1 1 5 3 4

5. Distributed network Graph G of n nodes Unique IDs 2 Localinfor- mationonly 1 5 3 2 4 1 3 Slide inspiredbyDanuponNanongkai

6. Distributed network Graph G of n nodes Unique IDs 2 Localinfor- mationonly 1 5 3 2 ? 4 1 3

7. Distributed network Graph G of n nodes Unique IDs 2 log n Localinfor- mationonly log n log n 1 log n 5 log n 3 2 log n ? 4 1 3 Limited bandwidth

8. Distributed network Graph G of n nodes Unique IDs 2 Synchronized Internal computations negligible log n Localinfor- mationonly log n log n 1 log n 5 log n 3 2 log n 4 1 ? Time complexity: number of communication rounds 3 Limited bandwidth

9. Distributed algorithms:a simple example

10. Count thenodes!

11. Count thenodes! Compute BFS-Tree

12. Count thenodes! 2 Compute BFS-Tree 2. Count nodes in subtrees 1 2 1 2 2 1 0 2 1 Runtime: ?

13. Count thenodes! Runtime: ? 2 Compute BFS-Tree 2. Count nodes in subtrees 1 2 1 2 2 1 0 2 1 Diameter

14. Diameter of a network • Distancebetweentwonodes = Numberof hops ofshortestpath

15. Diameter of a network • Distancebetweentwonodes = Numberof hops ofshortestpath

16. Diameter of a network • Distance between two nodes = Number of hops of shortest path • Diameter of network = Maximum distance, between any two nodes

17. Diameter of a network Diameter of thisnetwork? Distancebetweentwonodes = Number of hops of shortestpath Diameter of network = Maximum distance, betweenanytwonodes

18. Fundamental problems • Spanning Tree – Broadcasting, Aggregation, etc • Minimum Spanning Tree – Efficient broadcasting, etc. • Shortest path – Routing, etc. • Steiner tree – Multicasting, etc. • Many other graph problems. • Diameter appears frequently in distributed computing

19. Fundamental problems • Spanning Tree – Broadcasting, Aggregation, etc • Minimum Spanning Tree – Efficient broadcasting, etc. • Shortest path – Routing, etc. • Steiner tree – Multicasting, etc. • Many other graph problems. • Diameter appears frequently in distributed computing • E.g. local vs. global

20. Fundamental problems 1. Formal definition? • Spanning Tree – Broadcasting, Aggregation, etc • Minimum Spanning Tree – Efficient broadcasting, etc. • Shortest path – Routing, etc. • Steiner tree – Multicasting, etc. • Many other graph problems. • Diameter appears frequently in distributed computing • E.g. local vs. global Complexity of computing D? . Known bounds: O(nD) Ω (D) [before 2012] ≈O(n2) ≈(1)

21. Fundamental problems 1. Formal definition? • Spanning Tree – Broadcasting, Aggregation, etc • Minimum Spanning Tree – Efficient broadcasting, etc. • Shortest path – Routing, etc. • Steiner tree – Multicasting, etc. • Many other graph problems. • Diameter appears frequently in distributed computing • E.g. local vs. global Complexity of computing D? . Known bounds: O(nD) Ω (D) New bounds: Ω (n) [before 2012] ≈O(n2) ≈(1) Even if D = 5 [2012] [FHW 2012]

22. Fundamental problems 1. Formal definition? • Spanning Tree – Broadcasting, Aggregation, etc • Minimum Spanning Tree – Efficient broadcasting, etc. • Shortest path – Routing, etc. • Steiner tree – Multicasting, etc. • Many other graph problems. • Diameter appears frequently in distributed computing • E.g. local vs. global Complexity of computing D? . Known bounds: O(nD) Ω (D) New bounds: O(n) Ω (n) [before 2012] ≈O(n2) ≈(1) Even if D = 5 3 [here] [FHW 2012]

23. Fundamental problems 1. Formal definition? Use APSP to Compute Diameter • Spanning Tree – Broadcasting, Aggregation, etc • Minimum Spanning Tree – Efficient broadcasting, etc. • Shortest path – Routing, etc. • Steiner tree – Multicasting, etc. • Many other graph problems. • Diameter appears frequently in distributed computing • E.g. local vs. global Independently: [Peleg, Roditty, Tal 2012] Complexity of computing D? . Known bounds: O(nD) Ω (D) New bounds: O(n) Ω (n) [before 2012] ≈O(n2) ≈(1) Even if D = 5 3 [here] [FHW 2012]

24. Extentions

25. Extentions Routing tables Socialnetworks Fightingspam

26. Extentions Combination with: [Peleg, Roditty, Tal 2012]

27. Extentions Combined with [Peleg, Roditty, Tal 2012]

28. APSP in O(n)

29. APSP in O(n) Compute All Pairs Shortest Paths 0

30. APSP in O(n) Knows its distance to all other nodes Compute All Pairs Shortest Paths 0

31. APSP in O(n) Compute All Pairs Shortest Paths 0

32. APSP in O(n) Compute All Pairs Shortest Paths For each node { compute distances to all other nodes; } 0

33. APSP in O(n) Compute All Pairs Shortest Paths For each node { compute distances to all other nodes; } 1 1 1 0 1

34. APSP in O(n) Compute All Pairs Shortest Paths For each node { compute distances to all other nodes; } 2 1 2 1 2 2 1 0 2 1

35. APSP in O(n) Compute All Pairs Shortest Paths For each node { compute distances to all other nodes; O(D) } 2 1 2 1 2 2 1 0 2 1

36. APSP in O(n) Compute All Pairs Shortest Paths For each node { compute distances to all other nodes; O(D) } 0

37. APSP in O(n) Compute All Pairs Shortest Paths For each node { compute distances to all other nodes; O(D) } 1 1 1 0 1

38. APSP in O(n) Compute All Pairs Shortest Paths For each node { compute distances to all other nodes; O(D) } 1 2 1 2 2 1 2 0 1

39. APSP in O(n) Compute All Pairs Shortest Paths For each node { compute distances to all other nodes; O(D) } 1 2 1 2 2 3 1 2 0 1

40. APSP in O(n) Compute All Pairs Shortest Paths For each node { O(n) compute distances to all other nodes; O(D) } 1 2 1 2 2 3 1 2 0 1

41. APSP in O(n) Compute All Pairs Shortest Paths For each node { O(n) compute distances to all other nodes; O(D) } O(nD) 1 2 1 2 2 3 1 2 0 1

42. APSP in O(n) Compute All Pairs Shortest Paths For each node { O(n) compute distances to all other nodes; O(D) } O(nD) 1 2 1 2 Limited parallelism: Only some nodes active. 2 3 1 2 0 1

43. APSP in O(n) Compute All Pairs Shortest Paths For each node { O(n) compute distances to all other nodes; O(D) } O(nD) 1 2 1 2 Limited parallelism: Only some nodes active. Wanted: All nodes active all the time! 2 3 1 2 0 1

44. APSP in O(n) Compute All Pairs Shortest Paths

45. APSP in O(n) Compute All Pairs Shortest Paths 1. Pick a root-node r r

46. APSP in O(n) Compute All Pairs Shortest Paths Pick a root-node r T := BFS-Tree(r) r

47. APSP in O(n) Compute All Pairs Shortest Paths Pick a root-node r; T := BFS-Tree(r); Pebble P traverses T in preorder; r

48. APSP in O(n) Compute All Pairs Shortest Paths • Pick a root-node r; • T := BFS-Tree(r); • Pebble P traverses T • in preorder; • If P visits node v first time{ • wait 1 timeslot; • start shortest paths(v); • } r

49. APSP in O(n) Compute All Pairs Shortest Paths • Pick a root-node r; • T := BFS-Tree(r); • Pebble P traverses T • in preorder; • If P visits node v first time{ • wait 1 timeslot; • start shortest paths(v); • } u w r v

50. APSP in O(n) Compute All Pairs Shortest Paths • Pick a root-node r; • T := BFS-Tree(r); • Pebble P traverses T • in preorder; • If P visits node v first time{ • wait 1 timeslot; • start shortest paths(v); • } Starts at t u w v