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Universal Bufferless Routing

Study on buffer-less routing in arbitrary networks with focus on minimizing congestion and dilation, presenting a new algorithm for efficient packet routing within logarithmic factors from optimal solutions.

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Universal Bufferless Routing

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  1. Universal Bufferless Routing Marios MavronicolasUniversity of Cyprus Costas BuschRensselaer Polytechnic Inst. Malik Magdon Ismail Rensselaer Polytechnic Inst.

  2. Routing Problem • Arbitrary network • Arbitrary set of packets

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

  4. Network Model Buffer-less nodes Time 0 Packets are always moving

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

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

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

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

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

  10. the time until the last Packet is absorbed Routing Time: Objective: Minimize Routing Time

  11. Congestion Maximum number of packets that cross an edge

  12. Dilation Maximum path length

  13. Lower bound on Routing Time: Congestion Dilation We want to find algorithms with routing time close to lower bound

  14. Our contribution: A buffer-less algorithm with routing time nodes packets

  15. Related Work For routing with buffersthere exist optimal algorithms: [Leighton - Maggs - Rao, Combinatorica 94] [BS99, LMR99, MV99, OR97, RT96] What about buffer-less routing?

  16. Related Work Buffer-less Routing has been studied for specific topologies Mesh [BRST93, BES97, BHS98, BU96, BHW00] Hypercube [BH85, BC95, FR92, H91] Trees [BMMW04, RSW00] Leveled [BBPRRS96, B02] Vertex-symmetric[MS95] No previous work for time on arbitrary networks

  17. Presentation Outiline • Algorithm • Analysis • Conclusion

  18. Partition the Network into Regions • Each edge belongs to only one region • Regions share nodes

  19. Construct Region Graph Region Graph Regions are connected if they share a node in the original graph

  20. Translate Packet Paths Region Graph

  21. Translate Packet Paths Region Graph

  22. Execute a Buffered Algorithm on Region Graph Packet Buffer

  23. Simulate Buffered Algorithm Buffered Algorithm Buffers No Buffers

  24. Region Graph Original Graph Buffered Algorithm Action for a Packet Bufferless Algorithm Simulation Action 1 time step 1 phase • Circulare in region • Buffer in node • Move from • region to • Move from • node to • Inject/Absorb • at source/destination • in region • Inject/Absorb • at node

  25. Packet Buffer Simulation- Circulation Follow repeatetly an Euler tour

  26. Packet Buffer Simulation- Circulation Euler tour (abstract)

  27. Simulation- Circulation Packet Buffer Multiple packets Pipeline packets on Euler Tour

  28. Packet Move Simulation of move Packet moves to new region when it reaches common node

  29. Packet Move Simulation of move Packet moves to new region when it reaches common node

  30. Packet Move Simulation of move Packet moves to new region when it reaches common node

  31. Packet Move Simulation of move Insert packet when there is an empty spot in multiple packets in

  32. Oscillate here until empty slot is found Packet Move Slow Circulation Sending Mode Fast Circulation Receiving Mode

  33. Color the Region Graph 0 1 0 1 2 Let be the maximum color

  34. Change Colors 12 9 12 9 6 Replace each color with bits original color complement

  35. Each phase consists of rounds At round a node is in sending mode if its color bit is equal to 1

  36. Round 0 Round 1 sending 12 9 12 9 12 12 9 6 9 6 Round 2 Round 3 12 9 12 9 12 12 9 6 9 6

  37. There is a round that is receiving is sending And vice-versa

  38. Presentation Outiline • Algorithm • Analysis • Conclusion

  39. We give a buffered algorithm with Buffer size: Routing time: “Time Steps” : #nodes, node congestion, dilation in region graph

  40. We partition original graph so that each region has edges edges Buffer size

  41. It holds: Edges in region Edge congestion

  42. Each simulation phase has duration duration of round number of rounds

  43. Total Routing Time Number of phases Phase Duration

  44. Presentation Outiline • Algorithm • Analysis • Conclusion

  45. We presented the first bufferless algorithm for arbitrary networks within logarithmic factors from optimal • Future research: • Improve logarithmic factors • Maintain original paths

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