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Introduction Centralized Algorithm Distributed Algorithm Conclusion

Efficient Bufferless Routing on Leveled Networks Costas Busch Shailesh Kelkar Malik Magdon-Ismail Rensselaer Polytechnic Institute. Talk Outline. Introduction Centralized Algorithm Distributed Algorithm Conclusion. Leveled Networks. Level:. 0. 1. 2. 3. L-1. L.

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Introduction Centralized Algorithm Distributed Algorithm Conclusion

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  1. Efficient Bufferless Routing on Leveled NetworksCostas BuschShailesh KelkarMalik Magdon-IsmailRensselaer Polytechnic Institute

  2. Talk Outline • Introduction • Centralized Algorithm • Distributed Algorithm • Conclusion

  3. Leveled Networks Level: 0 1 2 3 L-1 L

  4. Examples of Leveled Networks 0 1 2 3 3 4 5 6 2 1 0 Butterfly Mesh

  5. Network Model • Synchronous network (time steps) • Bi-directional links One packet per direction, per time step

  6. Buffer-less nodes Time 0 Packets are always moving

  7. Buffer-less nodes Time 1 Packets are always moving

  8. Buffer-less nodes Time 2 Packets are always moving

  9. Buffer-less nodes Time 3 Packets are always moving

  10. Buffer-less nodes Time 4 Packets are always moving

  11. Bufferless routing is interesting: • Optical networks • Simple hardware implementations • Works well in practice: Bartzis et al.: EUROPAR 2000 Maxemchuck: INFOCOM 1989

  12. the time until the last packet is absorbed Routing Time: Objective: Minimize Routing Time

  13. Each packet has a pre-selected path source destination Packet path is from left to right

  14. The packet follows the pre-selected path source destination

  15. The packet follows the pre-selected path source destination

  16. The packet follows the pre-selected path source destination

  17. There are packets Each packet has its own path

  18. Dilation D: The maximum length of any path Routing time:

  19. CongestionC: The maximum number of packets traversing any edge Routing time:

  20. Lower bound on Routing Time: Congestion Dilation We want algorithms with Routing Time close to:

  21. Our Contributions • Centralized Algorithm: • Distributed Algorithm: Both algorithms are randomized Results hold with high probability : number of packets

  22. Related Work Networks with buffers Leveled networks: Leighton, Maggs, Ranade, Rao: J. Algorithms 1992 Arbitrary networks: [Leighton - Maggs - Rao, Combinatorica 94] [BS99, LMR99, MV99, OR97, RT96]

  23. Bufferless networks Mesh [BRST93, BES97, BHS98, BU96, BHW00] Hypercube [BH85, BC95, FR92, H91] Trees [BMMW04, RSW00, BMMW] Leveled [BBPRRS96, B02] Vertex-symmetric[MS95] Arbitrary networks [BMM04]

  24. Most related work Arbitrary networks: Busch, Magdon-Ismail, Mavronicolas WAOA’04 Leveled Networks: Busch TCS’04 Leveled networks in different routing model: Bhatt, Bilardi, Pucci, Ranade, Rosenberg, Scwabe TC’96

  25. Talk Outline • Introduction • Centralized Algorithm • Distributed Algorithm • Conclusion

  26. Centralized Algorithm A central node knows all the parameters of the problem and computes a packet schedule

  27. Packet Grouping A Group 1 Group 2 Group 3 Packet Grouping B Group 4 Group 5

  28. Send packets of Grouping A • Send packets of Grouping B Group 1 Increases routing time by only a factor of 2 Packet Grouping A Group 1 Group 2 Group 3 Packets in different groups can be sent simultaneously We focus only in one group

  29. Group 1 set of packets Congestion Dilation

  30. Partition the packets randomly and uniformly into sets packets #of packets

  31. Benefit: congestion drops Congestion packets New Congestion w.h.p.

  32. Before partitioning Edge Congestion

  33. After partitioning Expected one packet from each packet set

  34. Expected Congestion 1 (Congestion w.h.p.)

  35. We partition the levels into frames # number of frames:

  36. We send packets from frame to frame Wave

  37. We send packets from frame to frame Wave Duration: Wave

  38. We send packets from frame to frame Wave

  39. We send packets from frame to frame Wave

  40. We send packets from frame to frame Wave

  41. Injection wave A packet follows its path from source to Destination along the wave

  42. A packet follows its path from source to Destination along the wave

  43. A packet follows its path from source to Destination along the wave

  44. Absorption wave A packet follows its path from source to Destination along the wave

  45. Absorption wave A packet follows its path from source to Destination along the wave

  46. Sending packets of different packet sets simultaneously Wave 1

  47. Sending packets of different packet sets simultaneously Wave 1

  48. Sending packets of different packet sets simultaneously Wave 2 Wave 1

  49. Sending packets of different packet sets simultaneously Wave 2 Wave 1

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