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Near-Optimal Hot-Potato Routing on Trees

This research paper presents two near-optimal hot-potato routing algorithms for trees, aiming to minimize routing time in network systems. The algorithms are deterministic and randomized, with the latter resolving packet conflicts based on random packet priorities. The paper also discusses congestion, dilation, and deflection sequences in the context of routing on trees.

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Near-Optimal Hot-Potato Routing on Trees

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  1. Near-Optimal Hot-Potato Routing on Trees Costas BuschRensselaer Polytechnic Inst. Malik Magdon Ismail Rensselaer Polytechnic Inst. Marios Mavronicolas University of Cyprus Roger wattenhofer ETH Zurich

  2. Trees Trees are important in many networks (i.e. spanning trees)

  3. Routing on Trees • Every node generates at most one packet • Packets follow shortest paths

  4. Network Model • Synchronous network • Bi-directional links • One packet per time step

  5. Hot-Potato Routing Time 0 Buffer-less nodes

  6. conflict Time 1

  7. Time 2 deflected

  8. Time 3

  9. Time 4

  10. Hot-potato routing is interesting: • Optical networks • Simple hardware implementations • Works well in practice: Bartzis et al.: EUROPAR 2000 Maxemchuck: INFOCOM 1989

  11. Objective: Find hot-potato algorithm which minimizes routing time The time until the last packet is delivered to its destination

  12. Congestion: Maximum numbers of packets that share an edge

  13. Dilation: Maximum path length

  14. A lower bound on Routing Time: Congestion+Dilation We want to find an algorithm close to this lower bound

  15. Our contributions: • Deterministic Algorithm: node degree network size • Randomized Algorithm: degree independent

  16. Related Work for Trees • Matching Routing [ACG94] [PRS97] [Z97] • Direct Routing [AHLT98][BMMS04] • Hot-Potato routing [RSW00] Most results have routing time O(n) (worst case bound for O(C+D))

  17. Presentation Outiline • Deterministic Algorithm • Randomized Algorithm

  18. Deterministic Algorithm 1. Divide time into phases according to short nodes 2. At each phase send packets to their destinations greedily

  19. Short Node: …… every subtree has at most nodes

  20. Example short node

  21. Phase 1: Route packets that cross the short node ……

  22. Phase 2: In each subtree get the short nodes ……

  23. Phase 2: In each subtree get the short nodes ……

  24. Phase 2: Route packets that cross the short nodes …………

  25. There are at most phases

  26. Phase k: Route packets that cross the short node …… Bound on number of packets:

  27. Phase k: A packet follows its path greedily

  28. conflict deflected However, packets can conflict and get deflected

  29. Deflection Sequence [Borodin, Rabani, Schieber 1997] If a packet is deflected then some other packet reaches its destination

  30. Since there are at most packets, there are at most deflections Worst Routing Time for a packet: deflections Initial distance

  31. Total Routing Time Packet time In a phase Number of Phases

  32. Presentation Outiline • Deterministic Algorithm • Randomized Algorithm

  33. Randomized Algorithm Same with deterministic algorithm, with only difference: Packet conflicts are resolved according to random packet priorities

  34. Packet Priorities: Low:each packet starts with a low priority High: when a packet is deflected it increases its priority with probability

  35. A high priority packet can conflict with at most packets …… From those packets, are expected to be in high priority

  36. A high priority packet successfully reaches its destination with probability Thus attempts to become high priority are enough.

  37. Total Routing Time Packet time In a phase Number of Phases

  38. Discussion We presented two near-optimal hot-potato algorithms for trees (within logarithmic factors from optimal) Open problem: Remove the logarithmic factors

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