Nick%20Harvey%20(MIT)

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Conservative Network Coding

• Nick Harvey (MIT)
• Kamal Jain (MSR)
• Lap Chi Lau (U. Toronto)
• Chandra Nair (MSR)
• Yunnan Wu (MSR)

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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
  Ti ? 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)

• 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.

IntegerRoutingRate
FractionalRoutingRate
NetworkCodingRate
Cut Bound



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Example
s
t4
t2
t1
t3

• 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,

IntegerRoutingRate
FractionalRoutingRate
NetworkCodingRate
?
?
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Outline

• Motivation
• Acyclic networks
• Cyclic networks
• Conclusion

IntegerRoutingRate
FractionalRoutingRate
NetworkCodingRate


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Colored Cut Condition
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Colored Cut Condition
sr
sb

• Colors on nodes ? colors on edges

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Colored Cut Condition
sr
sb

• Blue and Red need to cross the cut
• We have a red, blue edge, a red edge and a blue
  edge
• So okay!

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Colored Cut Condition
sr
sb

• Blue and Red need to cross the cut
• We have a red, blue edge and a blue edge
• So okay!

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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

tr
sr
sb
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 vk1
• 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

IntegerRoutingRate
FractionalRoutingRate
NetworkCodingRate


IntegerRoutingRate
FractionalRoutingRate
NetworkCodingRate
?
?
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Outline

• Motivation
• Acyclic networks
• Cyclic networks
• Conclusion

IntegerRoutingRate
FractionalRoutingRate
NetworkCodingRate


IntegerRoutingRate
FractionalRoutingRate
NetworkCodingRate
lt
lt
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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
Vertex Set V Sources s1,s2 Sinks t1,t2
G
t1
t2
T2
T1
Add vertices v1, v2 Add edges ti-vi Add edges
vi-u ? u ? V ti Set Ti V vi
s2
s1
G
v1
v2
t1
t2
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Step 1
T2
T1
s2
s1
v1
v2
t1
t2
Easy
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Step 2
T2
T1
s2
s1
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
s2
t1
t2
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Results for Cyclic Networks
IntegerRoutingRate
FractionalRoutingRate
NetworkCodingRate
lt
lt

• Buy one get one free Integer Routing Solution
  is NP-hard

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A Simpler Example
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2
3
4
5
6
8
7
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A Simpler Example
1
2
3
4
5
6
8
7
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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