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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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Outline

Motivation Acyclic networks Cyclic networks Conclusion

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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 Conditionsr sb

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Colored Cut Condition

Colors on nodes colors on edges

sr sb

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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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Therefore

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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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Step 1:

s1 s2

t1 t2

v1 v2

T1 T2

Easy

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Step 2:

s1 s2

t1 t2

v1 v2

T1 T2

Disjoint trees Disjoint paths

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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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s1 s2

c

u

t2 t1

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s1

t1

s2

t2

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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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A Simpler Example

1 2

3

4

5 6

87

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A Simpler Example

1 2

3

4

5 6

87

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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