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Hard-Potato Routing

Hard-Potato Routing. Costas Busch, Maurice Herlihy, and Roger Wattenhofer Brown University. Hard-Potato Routing. Hard-Potato Routing. Hard-Potato Routing. Hard-Potato Routing. n x n Mesh synchronous one message / link one-shot problem. Hot-Potato Routing. Hot-Potato Routing.

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Hard-Potato Routing

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  1. Hard-Potato Routing Costas Busch, Maurice Herlihy, and Roger Wattenhofer Brown University

  2. Hard-Potato Routing

  3. Hard-Potato Routing

  4. Hard-Potato Routing

  5. Hard-Potato Routing n x n Mesh synchronous one message / link one-shot problem

  6. Hot-PotatoRouting

  7. Hot-PotatoRouting no buffers local decisions

  8. Hot-PotatoRouting no buffers local decisions simple hardware (optical networks)

  9. Hot-PotatoRouting: Conflicts

  10. Hot-PotatoRouting: Conflicts one message / link no buffers

  11. Hot-PotatoRouting: Conflicts one message / link no buffers deflection!

  12. Hot-PotatoRouting: Greedy

  13. Hot-PotatoRouting: Greedy Message prefers“good” link when there is no conflict. + simple + adaptive + very well in practice [Maxemchuk 89]

  14. Hot-PotatoRouting: New? Hot-Potato Routing [Baran 64] Mesh-like Hot-Potato Routing [Feige and Raghavan 92] [Kaklamanis, Krizanc, Rao 93] [Kaufmann, Lauer, Schroder 94] [Newman and Schuster 95] [Spirakis and Triantafillou 97] [Ben-Dor, Halevi, Schuster 98] [BHW 00]

  15. Hard ! Hot Hard-Potato Routing: New? All papers tuned for permutation or random destinaton. This paper is about “hard” (“many-to-one”) routing. [Ben-Aroya, Eilam, Schuster 95] [Borodin, Rabani, Schieber 97] [Ben-Aroya, Newman, Schuster 97] [Ben-Dor, Halevi, Schuster 98] If n2 messages are injected, they need O(n2) time. We have the first algorithm that does better…

  16. Lower Bound: Bandwidth # messages W = max # links

  17. Lower Bound: Distance D = max distance

  18. Our Result: log3n-competitive Lower Bound L = W(D+W) Our Algorithm needs O(L log3n) to route all messages with high probability. Remarks: L = W(n) Distributed algorithm (local decisions only). L does not have to be known in advance. Greedy.

  19. Usual stubborn approach (often 1-bend path) does not work. General Problem

  20. Messages have priorities running high (excited) normal low ? Our Algorithm

  21. Our Algorithm deflection running with probability p! Messages have priorities running high (excited) normal low

  22. Our Algorithm: running

  23. Our Algorithm: running

  24. ? ? ? What’s the probability? Time Analysis: Intuition Running messagesthat want to take theopposite link alwayshave priority. Therefore a runningmessage can only beinterrupted when“starting” or “turning”.

  25. Time Analysis: Intuition Short answer: Depends on traffic. ? There are two extremes oftraffic that can interfere: “local” and “global”. And there is the special caseof “starters”.

  26. Time Analysis: Local Traffic Example: Interfering message is only one bend away… ? From the bandwidth lowerbound we know that at mostO(L) messages go into these two rectangles. 

  27. Time Analysis: Global Traffic Example: Interfering message is still many bends away… ? Possible destinations aremuch more! Up to O(n L)… But the interfering messagealso has made a lot ofrandom choices! Traffic isthe same. 

  28. Time Analysis: Start running Conflicting messages only have one chance to start running. We make worst-caseassumptions on the positionof possible conflictingmessages. But before a message starts running it has to throw a coin.  ?

  29. c log t p(t) = t Time Analysis: Probabilities If we knew L we could set p so that a message managesto run home with constant probability. p Since we do not know it, welet p change over time so that for a large enough window p is in the right order that we still have constant probability to run home. time

  30. Our Result: log3n-competitive Lower Bound L = W(D+W) Our Algorithm needs O(L log3n) to route all messages with high probability. Remarks: L = W(n) Distributed algorithm (local decisions only). L does not have to be known in advance. Greedy.

  31. Future Work • More Dimensions • Arbitrary Network Topologies • Dynamic Analysis () • “Easy” Problems where L = o(n)

  32. c log t p(t) = t Time Analysis: Details In interval [c’ L log3 n + 2n, 3 c’ L log3 n] a message is absorbed with high probability (1-1/n3). Therefore, all messages are absorbed in the sameinterval with probability 1-1/n.

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