1 nick harvey (mit) kamal jain (msr) lap chi lau (u. toronto) chandra nair (msr) yunnan wu (msr)...
Post on 22-Dec-2015
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Nick Harvey (MIT) Kamal Jain (MSR)
Lap Chi Lau (U. Toronto)Chandra Nair (MSR)
Yunnan Wu (MSR)
Conservative Network Coding
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The General Multi-Session Network Coding Problem Given a network coding problem:
Directed graph G = (V,E) k “commodities” (streams of information) Sources: s1, …, sk
Receiver set for each source T1, …, Tk
At what rate can the sources transmit? This is very general and very hard…
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“Conservative Network Coding” Given a network coding problem:
Directed graph G = (V,E) k “commodities” (streams of information) Sources: s1, …, sk
Receiver set for each source T1, …, Tk
At what rate can the sources transmit? Consider solutions where intermediate nodes are
conservative i.e., a node rejects anything it does not want. i.e., commodity i is not allowed to leave the set T i {si}
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Motivations
Practical motivation In peer-to-peer networks,
a node may not have incentive to relay traffic for others
a node may be concerned about security troubles
Theoretical motivation In the special case when there is a single
commodity, there are elegant results.
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Single Session Conservative Networking (Broadcasting)
IntegerRouting
Rate
FractionalRouting
Rate
NetworkCodingRate
= = =Cut
Bound
Edmonds’ Theorem (1972): Given a directed graph and a source node s, the maximum number of edge disjoint spanning trees rooted at s is equal to the minimum s-cut capacity.
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Example
t3t1 t2
s
t4
“As long as we can route information to each node individually at rate C, we can route information simultaneously to all destinations at rate C.”
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Generalization?
For conservative networking,
IntegerRouting
Rate
FractionalRouting
Rate
NetworkCodingRate
? ?
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Outline
Motivation Acyclic networks
Cyclic networks Conclusion
IntegerRouting
Rate
FractionalRouting
Rate
NetworkCodingRate
= =
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Colored Cut Condition
Blue and Red need to cross the cut We have a {red, blue} edge, a red edge and a blue edge So okay!
sr sb
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Colored Cut Condition
Blue and Red need to cross the cut We have a {red, blue} edge and a blue edge So okay!
sr sb
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Colored Cut Condition
Generally, for each node-set cut, the set of edges across the cut must enable that the colors that need to cross the cut indeed can cross. A bi-partite matching condition
sr
sb
tr
tb
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Proof that Colored Cut Bound is Achievable by Routing Visit the nodes in the topological order, v1,…,vn
By inductive hypothesis, the previous nodes v1,…,vk can indeed recover the messages they want.
Consider node vk+1
Colored cut condition must hold; Conversely, if it holds, there exists an integer routing solution.
tr,g tg,b
tr,b
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Outline Motivation Acyclic networks
Cyclic networks
Conclusion
IntegerRouting
Rate
FractionalRouting
Rate
NetworkCodingRate
= =
IntegerRouting
Rate
FractionalRouting
Rate
NetworkCodingRate
? ?
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Outline Motivation Acyclic networks
Cyclic networks
Conclusion
IntegerRouting
Rate
FractionalRouting
Rate
NetworkCodingRate
= =
IntegerRouting
Rate
FractionalRouting
Rate
NetworkCodingRate
< <
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Proof by Reduction
A k-pairs problem G A conservative network problem G’
Find k-pairs problems such that
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Reductionk-pairs conservative networking
s1 s2
t1 t2
s1 s2
t1 t2
v1 v2
T1 T2
Vertex Set VSources s1,s2
Sinks t1,t2
Add vertices v1, v2
Add edges ti-vi
Add edges vi-u ∀ u ∈ V – ti
Set Ti = V + vi
G
G’
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Reduction does not preserve rates for coding
A k-pairs problem G A conservative network problem G’
“three butterflies flying together”
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Proof by Reduction
A k-pairs problem G A conservative network problem G’
Find k-pairs problems such that
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Results for Cyclic Networks
IntegerRouting
Rate
FractionalRouting
Rate
NetworkCodingRate
< <
“Buy one get one free”: Integer Routing Solution is NP-hard
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Conclusion
Conservative networking model, motivated by practice and theory
Neat result for acyclic networks that generalize Edmonds’ Theorem
Counter examples for cyclic networks Even if nodes are conservative, network coding can help
“Cycles are tricky!” Bound obtained by examining nodes in isolation is loose Bound obtained by examining node-set cuts in isolation is
loose Generally require entropy arguments