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TexPoint fonts used in EMF.

Time-Optimal Information Exchange on Multiple Channels. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box .: A A A A A A A A A A. Time-Optimal Information Exchange on Multiple Channels. n:= # nodes. Problem :.

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TexPoint fonts used in EMF.

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  1. Time-Optimal Information Exchange on Multiple Channels TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAA

  2. Time-Optimal Information Exchange on Multiple Channels n:= # nodes Problem:

  3. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Problem: Have information ? Disseminate to all!

  4. Time-Optimal Information Exchange on Multiple Channels Problem: ? Disseminate to all!

  5. Time-Optimal Information Exchange on Multiple Channels Problem: ? Disseminate to all! Easy: O(n) Faster?

  6. Time-Optimal Information Exchange on Multiple Channels n:= # nodes Problem: 1 5 2 3 Unique IDs 1…n 4 n

  7. Time-Optimal Information Exchange on Multiple Channels I can: send / receive reacheachnode

  8. Time-Optimal Information Exchange on Multiple Channels I can: ? send / receive reacheachnode

  9. Time-Optimal Information Exchange on Multiple Channels I can: send / receive reacheachnode nocollisiondetection

  10. Time-Optimal Information Exchange on Multiple Channels I can: send / receive reacheachnode nocollisiondetection 101 Mhz 117 Mhz 132 Mhz … switchchannels synchronus

  11. Time-Optimal Information Exchange on Multiple Channels I can: complexity computation: free radio: time 1 send / receive reacheachnode nocollisiondetection switchchannels synchronus

  12. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information One Information / log n bits per message O(1) => Ω( k )

  13. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Multi channel ? onechannel + log n => Ω( k ) [Kushilevitz, Mansour SIAM JComp 1998]

  14. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information What can I do? TREE

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

  16. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Communicate in parallel on different channels TREE 2 4 8 9

  17. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information TREE

  18. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information TREE

  19. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information TREE

  20. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information TREE

  21. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information TREE

  22. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information TREE O( k + log n)

  23. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information What if k < log n? n channels TREE O( k ) if k > log n

  24. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information What if k < log n?

  25. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known

  26. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Balls into Bins

  27. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known k ID=1…2 Balls into Bins

  28. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known k ID=1…2 Balls into Bins

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

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

  31. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known k ID=1…2 Pr no collision Balls into Bins

  32. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known k ID=1…2 Pr no collision TREE Balls into Bins

  33. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known k ID=1…2 Pr no collision TREE Balls into Bins

  34. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Repeat k times k ID=1…2 Pr no collision Balls into Bins

  35. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known What if ? Repeat k times k ID=1…2 Pr [no collision] > 1- If k Balls into Bins 2 channels TREE O( k ) if

  36. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known What if ? Unique Subset Time: O( k )

  37. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset

  38. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset

  39. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset Send on random channel }.

  40. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Unique Subset Send on random channel . Send on random channel }.

  41. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Pr[atmosthalf messages collide] 1- Unique Subset Not too big … Not too small … Just right! Send on random channel . Send on random channel }.

  42. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Example: 3 channels Unique Subset Channel 1 Channel 3

  43. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Example: 3 channels Unique Subset Channel 1 Channel 3 Send k times

  44. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Example: 3 channels Unique Subset Channel 1 Channel 3 Send k times

  45. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Example: 3 channels Unique Subset Channel 1 Channel 3 Send k times

  46. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Example: 3 channels ? ? Unique Subset ? Channel 1 Channel 3 Send k times

  47. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Example: 3 channels Unique Subset Channel 1 Channel 3 Send k times

  48. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Example: 3 channels Unique Subset Channel 1 Channel 3 Send k times

  49. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Example: 3 channels Unique Subset Channel 1 Channel 3 Send k times

  50. Time-Optimal Information Exchange on Multiple Channels n:= # nodes k:= # information Assume: k known Example: 3 channels Pr[atmosthalf messages collide]1- Unique Subset O( k )

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