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Hard-Potato Routing Costas Busch, Maurice Herlihy, and Roger - - PowerPoint PPT Presentation
Hard-Potato Routing Costas Busch, Maurice Herlihy, and Roger - - PowerPoint PPT Presentation
Hard-Potato Routing Costas Busch, Maurice Herlihy, and Roger Wattenhofer Brown University 1 Hard-Potato Routing 2 Hard-Potato Routing 3 Hard-Potato Routing 4 Hard-Potato Routing n x n Mesh synchronous one message / link one-shot
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Hard-Potato Routing
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Hard-Potato Routing
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Hard-Potato Routing
n x n Mesh synchronous
- ne message / link
- ne-shot problem
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Hot-Potato Routing
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Hot-Potato Routing
no buffers local decisions
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Hot-Potato Routing
no buffers local decisions simple hardware (optical networks)
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Hot-Potato Routing: Conflicts
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Hot-Potato Routing: Conflicts
- ne message / link
no buffers
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Hot-Potato Routing: Conflicts
- ne message / link
no buffers deflection!
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Hot-Potato Routing: Greedy
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Hot-Potato Routing: Greedy
Message prefers “good” link when there is no conflict. + simple + adaptive + very well in practice [Maxemchuk 89]
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Hot-Potato Routing: 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]
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Hard-Potato Routing: New? Hot Hard !
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…
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Lower Bound: Bandwidth
W = max # messages # links
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Lower Bound: Distance
D = max distance
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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.
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General Problem
Usual stubborn approach (often 1-bend path) does not work.
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Our Algorithm
Messages have priorities running high (excited) normal low ?
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Our Algorithm
Messages have priorities running high (excited) normal low deflection running with probability p!
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Our Algorithm: running
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Our Algorithm: running
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Time Analysis: Intuition
Running messages that want to take the
- pposite link always
have priority. Therefore a running message can only be interrupted when “starting” or “turning”. ? ? ? What’s the probability?
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Time Analysis: Intuition
Short answer: Depends on traffic. There are two extremes of traffic that can interfere: “local” and “global”. And there is the special case
- f “starters”.
?
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Time Analysis: Local Traffic
From the bandwidth lower bound we know that at most O(L) messages go into these two rectangles. ☺ ? Example: Interfering message is only one bend away…
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Time Analysis: Global Traffic
Possible destinations are much more! Up to O(n L)… But the interfering message also has made a lot of random choices! Traffic is the same. ☺ ? Example: Interfering message is still many bends away…
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Time Analysis: Start running
We make worst-case assumptions on the position
- f possible conflicting
messages. Conflicting messages only have one chance to start running. ? But before a message starts running it has to throw a coin. ☺
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Time Analysis: Probabilities
If we knew L we could set p so that a message manages to run home with constant probability. Since we do not know it, we let 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 p p(t) = c log t t
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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.
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Future Work
- More Dimensions
- Arbitrary Network Topologies
- Dynamic Analysis ()
- “Easy” Problems where L = o(n)
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