1. ETH Zurich – Distributed Computing Group Stephan Holzer 1ETH Zurich – Distributed Computing – www.disco.ethz.ch Stephan Holzer Yvonne Anne Pignolet Jasmin Smula Roger Wattenhofer TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA A A AA AA Time-Optimal Information Exchange on Multiple Channels

2. ETH Zurich – Distributed Computing Group Stephan Holzer 2 Problem : Time-Optimal Information Exchange on Multiple Channels n:= # nodes

3. ETH Zurich – Distributed Computing Group Stephan Holzer 3 Problem : Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Have information Disseminate to all! ?

4. ETH Zurich – Distributed Computing Group Stephan Holzer 4 Problem : Time-Optimal Information Exchange on Multiple Channels Disseminate to all! ?

5. ETH Zurich – Distributed Computing Group Stephan Holzer 5 Problem : Time-Optimal Information Exchange on Multiple Channels Disseminate to all! ? Easy: O(n) Faster?

6. ETH Zurich – Distributed Computing Group Stephan Holzer 6 Problem : Time-Optimal Information Exchange on Multiple Channels n:= # nodes 1 2 3 4 5 n Unique IDs 1…n

7. ETH Zurich – Distributed Computing Group Stephan Holzer 7 I can: send / receive reach each node Time-Optimal Information Exchange on Multiple Channels

8. ETH Zurich – Distributed Computing Group Stephan Holzer 8 I can: send / receive ? reach each node Time-Optimal Information Exchange on Multiple Channels

9. ETH Zurich – Distributed Computing Group Stephan Holzer 9 Time-Optimal Information Exchange on Multiple Channels no collision detection I can: send / receive reach each node

10. ETH Zurich – Distributed Computing Group Stephan Holzer 10 switch channels no collision detection I can: send / receive reach each node 101 Mhz 117 Mhz 132 Mhz … Time-Optimal Information Exchange on Multiple Channels synchronus

11. ETH Zurich – Distributed Computing Group Stephan Holzer 11 switch channels no collision detection I can: send / receive reach each node complexity computation: free radio: time 1 Time-Optimal Information Exchange on Multiple Channels synchronus

12. ETH Zurich – Distributed Computing Group Stephan Holzer 12 One Information / log n bits per message O(1) Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information => Ω( k )

13. ETH Zurich – Distributed Computing Group Stephan Holzer 13 => Ω( k ) Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information + log n [Kushilevitz, Mansour SIAM JComp 1998] one channel ? Multi channel

14. ETH Zurich – Distributed Computing Group Stephan Holzer 14 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information TREE What can I do?

15. ETH Zurich – Distributed Computing Group Stephan Holzer 15 4 7 8 3 1 2 5 6 9 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information 10 11 12 TREE 13 14 15 16 Communicate in parallel on different channels

16. ETH Zurich – Distributed Computing Group Stephan Holzer 16 4 8 2 9 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information TREE Communicate in parallel on different channels

17. ETH Zurich – Distributed Computing Group Stephan Holzer 17 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information TREE

18. ETH Zurich – Distributed Computing Group Stephan Holzer 18 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information TREE

19. ETH Zurich – Distributed Computing Group Stephan Holzer 19 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information TREE

20. ETH Zurich – Distributed Computing Group Stephan Holzer 20 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information TREE

21. ETH Zurich – Distributed Computing Group Stephan Holzer 21 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information TREE

22. ETH Zurich – Distributed Computing Group Stephan Holzer 22 TREE Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information O( k + log n)

23. ETH Zurich – Distributed Computing Group Stephan Holzer 23 Time-Optimal Information Exchange on Multiple Channels What if k < log n? n:= # nodes k:= # information O( k ) if k > log n n channels TREE

24. ETH Zurich – Distributed Computing Group Stephan Holzer 24 Time-Optimal Information Exchange on Multiple Channels What if k < log n? n:= # nodes k:= # information

25. ETH Zurich – Distributed Computing Group Stephan Holzer 25 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known

26. ETH Zurich – Distributed Computing Group Stephan Holzer 26 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Balls into Bins

27. ETH Zurich – Distributed Computing Group Stephan Holzer 27 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Balls into Bins ID=1…2 k

28. ETH Zurich – Distributed Computing Group Stephan Holzer 28 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Balls into Bins ID=1…2 k

29. ETH Zurich – Distributed Computing Group Stephan Holzer 29 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Balls into Bins Listens on channel 1 Listens on channel 2 Listens on channel 3 Listens on channel 4 Send on random channel 1…2 ID=1…2 k k

30. ETH Zurich – Distributed Computing Group Stephan Holzer 30 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Balls into Bins Listens on channel 1 Listens on channel 2 Listens on channel 3 Listens on channel 4 ID=1…2 k Send on random channel 1…2 k

31. ETH Zurich – Distributed Computing Group Stephan Holzer 31 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Balls into Bins Pr no collision ID=1…2 k

32. ETH Zurich – Distributed Computing Group Stephan Holzer 32 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known ID=1…2 k Balls into Bins Pr no collision TREE

33. ETH Zurich – Distributed Computing Group Stephan Holzer 33 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known ID=1…2 k Balls into Bins Pr no collision TREE

34. ETH Zurich – Distributed Computing Group Stephan Holzer 34 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known ID=1…2 k Balls into Bins Pr no collision Repeat k times

35. ETH Zurich – Distributed Computing Group Stephan Holzer 35 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known ID=1…2 k Balls into Bins TREE Repeat k times k 2 channels

36. ETH Zurich – Distributed Computing Group Stephan Holzer 36 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset Time: O( k )

37. ETH Zurich – Distributed Computing Group Stephan Holzer 37 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset

38. ETH Zurich – Distributed Computing Group Stephan Holzer 38 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset

39. ETH Zurich – Distributed Computing Group Stephan Holzer 39 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset

40. ETH Zurich – Distributed Computing Group Stephan Holzer 40 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset

41. ETH Zurich – Distributed Computing Group Stephan Holzer 41 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset Not too big … Not too small … Just right!

42. ETH Zurich – Distributed Computing Group Stephan Holzer 42 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset Channel 3 Example: 3 channels Channel 1

43. ETH Zurich – Distributed Computing Group Stephan Holzer 43 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset Channel 3 Example: 3 channels Channel 1 Send k times

44. ETH Zurich – Distributed Computing Group Stephan Holzer 44 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset Channel 3 Example: 3 channels Channel 1 Send k times

45. ETH Zurich – Distributed Computing Group Stephan Holzer 45 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset Channel 3 Example: 3 channels Channel 1 Send k times

46. ETH Zurich – Distributed Computing Group Stephan Holzer 46 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset Channel 3 Example: 3 channels Channel 1 ? ? ? Send k times

47. ETH Zurich – Distributed Computing Group Stephan Holzer 47 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset Channel 3 Example: 3 channels Channel 1 Send k times

48. ETH Zurich – Distributed Computing Group Stephan Holzer 48 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset Channel 3 Example: 3 channels Channel 1 Send k times

49. ETH Zurich – Distributed Computing Group Stephan Holzer 49 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset Channel 3 Example: 3 channels Channel 1 Send k times

50. ETH Zurich – Distributed Computing Group Stephan Holzer 50 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset Example: 3 channels O( k )

51. ETH Zurich – Distributed Computing Group Stephan Holzer 51 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset Example: 3 channels O( k + k/2 + k/4 …

52. ETH Zurich – Distributed Computing Group Stephan Holzer 52 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset Example: 3 channels O( k )

53. ETH Zurich – Distributed Computing Group Stephan Holzer 53 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset Balls into Bins TREE

54. ETH Zurich – Distributed Computing Group Stephan Holzer 54 Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known unknown Unique Subset Balls into Bins TREE n channels 2 channels k n channels Future directions:deterministic less channels lower bounds

55. ETH Zurich – Distributed Computing Group Stephan Holzer 55 in Summary … Detect / Disseminate Information! Time-Optimal Information Exchange on Multiple Channels 101 Mhz 117 Mhz 132 Mhz …

56. ETH Zurich – Distributed Computing Group Stephan Holzer 56ETH Zurich – Distributed Computing – www.disco.ethz.ch Stephan Holzer Yvonne Anne Pignolet Jasmin Smula Roger Wattenhofer Thank You! Questions & Comments? TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA A A A