Orderly Broadcasting in Multidimensional Tori

1. Orderly Broadcasting in Multidimensional Tori Presentation by Perouz Taslakian May 6, 2004

2. Outline • Introduction and Motivation • Types of Broadcasting • Orderly Broadcasting • Lower Bound on 2-dimensional Tori • Upper Bound for 2-dimensional Tori • Upper Bound for d-dimensional Tori • Conclusion and Future Work

3. Introduction Broadcasting : process of sending a message from one node of a communication network to the rest of the nodes. Originator: the node of the network which initiates broadcasting by sending its message to the rest of the nodes. Broadcast problem : the problem of determining the amount of time needed to transmit a message to every node in an interconnection network

4. Motivation • Computer networks • Parallel processing • Cache coherence

5. 1 u v 2 w Constraints Broadcasting is subject to the following constraints: • Each message transmission takes 1 time unit • A node can transmit only to an adjacent node • A node can transmit a message to 1 node in 1 time unit

6. Types of Broadcasting Broadcast models can be divided into two major categories: • Classical Broadcasting (Slater in 1977) • Messy Broadcasting (Alshwede, Khatchadrian & Harutyunyan in 1994)

7. Notations A network is modeled as a connected graph G=(V, E) b(u) : min number of time units required to complete broadcasting when u is the originator (classical) bm(u): max number of time units required to complete broadcasting when u is the originator (messy) b(G) = max {b(u) | u  V} bm(G) = max {bm(u) | u  V}

8. Classical Broadcasting Find a scheme so as information dissemination takes the least amount of time. Assumption: each vertex knows the • graph topology • originator of the message • time the message was sent. When a vertex is informed, it transmits the message to its neighbor in the most cleverway.

9. Example : Classical Broadcasting u v1 v4 v2 v3

10. Example : Classical Broadcasting 1 u v1 v4 v2 v3

11. Example : Classical Broadcasting 1 u v1 2 2 v4 v2 v3

12. Example : Classical Broadcasting 1 u v1 2 2 v4 3 v2 v3 b(u) = 3

13. Messy Broadcasting Analyzes broadcast schemes that take the mostamount of time Assumption: each vertex knows nothing about the graph topology and the originator. When a vertex is informed, it transmits the message to a randomly chosen neighbor in each time unit.

14. Example : Messy Broadcasting u v1 v4 v2 v3

15. Example : Messy Broadcasting 1 u v1 v4 v2 v3

16. Example : Messy Broadcasting 1 u v1 2 2 v4 v2 v3

17. Example : Messy Broadcasting 1 u v1 2 3 2 v4 3 v2 v3

18. Example : Messy Broadcasting 1 u v1 2 3 2 v4 3 4 4 v2 v3

19. Example : Messy Broadcasting 1 u v1 2 3 2 v4 3 4 4 v2 v3 5

20. Example : Messy Broadcasting 1 u v1 2 3 2 v4 3 4 4 6 v2 v3 5 bm(u) = 6

21. How hard is it? Finding b(u) for an arbitrary graph isNP-Complete

22. Practicality Storing information about the graph topology and the originator is not efficient in practice Sometimes, network nodes have primitive structures with small memories that cannot store such information. Building networks where the vertices have no decision making responsibility is easier.

23. Orderly Broadcasting The neighbors of each vertex are assigned a unique number. Assumption: each vertex knows nothing about the graph topology and the originator. When a vertex is informed, it transmits the message first to vertex numbered 1, then to vertex numbered 2, …etc.

24. Orderly Broadcasting Problem: Find an orderingof the neighbors of all the vertices of a given graph that will minimize the broadcast time.

25. 1 2 3 Example : Orderly Broadcasting u v1 v4 v2 v3

26. 1 2 3 Example : Orderly Broadcasting 1 u v1 v4 v2 v3

27. 1 2 3 Example : Orderly Broadcasting 1 u v1 2 2 v4 v2 v3

28. 1 2 3 Example : Orderly Broadcasting 1 u v1 3 2 3 2 v4 v2 v3

29. 1 2 3 Example : Orderly Broadcasting 1 u v1 3 2 3 2 v4 4 v2 v3 b(u) = 4

30. Previous Results

31. 2-dimensional Tori Torus Tmn: wrap-around grid with m rows & n columns Each vertex in a 2-dimensional torus has degree 4 Diameter of Tmn is : Every vertex in Tmn has : 1 vertex at distance D(Tmn) if m and n are even 2 vertices at distance D(Tmn) if one of m or n is odd 4 vertices at distance D(Tmn) if m and n are odd

32. Lower Bound

33. A 2-dim Torus Torus T69

34. 1 2 3 4 Ordering  on 2D Tori 0 1 ℓ 2 …………… n …………… 0 1 2 . . . . . . . . . . . . . . m

35. 1 2 3 4 Ordering  on 2D Tori 0 1 ℓ 2 …………… n …………… 0 1 2 . . . . . . . . . . . . . . m

36. 1 2 3 4 Ordering  on 2D Tori 0 1 ℓ 2 …………… n …………… 0 1 2 . . . . . . . . . . . . . . m

37. 1 2 3 4 Ordering  on 2D Tori 0 1 ℓ 2 …………… n …………… 0 1 2 . . . . . . . . . . . . . . m

38. 1 2 3 4 Ordering  on 2D Tori 0 1 ℓ 2 …………… n …………… 0 1 2 . . . . . . . . . . . . . . m

39. The Variable ℓ

40. Upper Bound on 2-dim Tori

41. How it works: an example T813 0 1 3 4 6 7 10 11 2 5 8 9 12 0 0 1 2 1 2 3 4 3 4 5 6 7

42. How it works: an example T813 0 1 3 4 6 7 10 11 2 5 8 9 12 0 0 1 3 2 1 2 3 4 3 4 5 6 7

43. How it works: an example T813 0 1 3 4 6 7 10 11 2 5 8 9 12 0 0 1 6 5 4 3 2 1 2 3 4 3 4 5 6 7

44. How it works: an example T813 0 1 3 4 6 7 10 11 2 5 8 9 12 0 0 1 6 5 4 3 2 7 1 2 3 4 3 8 4 5 6 7

45. How it works: an example T813 0 1 3 4 6 7 10 11 2 5 8 9 12 0 0 1 6 5 4 3 2 7 9 10 11 1 2 3 4 3 8 4 5 6 7

46. How it works: an example T813 0 1 3 4 6 7 10 11 2 5 8 9 12 0 0 P1 1 6 5 4 3 2 7 9 10 11 1 2 3 4 3 8 12 12 14 4 5 6 7

47. How it works: an example T813 0 1 3 4 6 7 10 11 2 5 8 9 12 0 0 1 2 3 4 P1 1 6 5 4 3 2 7 9 10 11 1 2 3 4 3 8 12 12 14 4 5 6 7

48. How it works: an example T813 0 1 3 4 6 7 10 11 2 5 8 9 12 0 0 1 2 3 4 P1 1 6 5 4 3 2 7 9 10 11 1 2 3 4 3 8 12 12 14 9 4 8 5 7 6 6 7 5

49. How it works: an example T813 0 1 3 4 6 7 10 11 2 5 8 9 12 0 0 1 2 3 4 P1 1 6 5 4 3 2 7 9 10 11 P2 1 2 3 4 3 8 12 12 14 13 12 11 9 4 8 5 7 6 6 7 5

50. How it works: an example T813 0 1 3 4 6 7 10 11 2 5 8 9 12 0 0 1 2 3 4 P1 1 6 5 4 3 2 7 9 10 11 P2 P3 1 2 3 4 3 8 12 12 14 13 12 11 9 12 11 10 4 8 5 7 6 6 7 5