Conservative
Network Coding
Nick Harvey (MIT)
Kamal Jain (MSR)
Lap Chi Lau (U. Toronto)
Chandra Nair (MSR)
Yunnan Wu (MSR)
1
Outline




Motivation
Acyclic networks
Cyclic networks
Conclusion
2
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…
3
“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}
4
Motivations

Practical motivation
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In peer-to-peer networks,
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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.
5
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 scut capacity.
Integer
Routing
Rate
=
Fractional
Routing
Rate
=
Network
Coding
Rate
=
Cut
Bound
6
Example
s
t4
t1

t3
t2
“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.”
7
Generalization?

For conservative networking,
Integer
Routing
Rate
?
Fractional
Routing
Rate
?
Network
Coding
Rate
8
Outline


Motivation
Acyclic networks
Integer
Routing
Rate


=
Fractional
Routing
Rate
=
Network
Coding
Rate
Cyclic networks
Conclusion
9
Colored Cut Condition
sr
sb
10
Colored Cut Condition
sr

sb
Colors on nodes  colors on edges
11
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!
12
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!
13
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
14
Proof that Colored Cut Bound
is Achievable by Routing

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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
15
Outline


Motivation
Acyclic networks
Integer
Routing
Rate

?
Fractional
Routing
Rate
=
Network
Coding
Rate
?
Network
Coding
Rate
Cyclic networks
Integer
Routing
Rate

=
Fractional
Routing
Rate
Conclusion
16
Outline


Motivation
Acyclic networks
Integer
Routing
Rate

=
Network
Coding
Rate
<
Network
Coding
Rate
Cyclic networks
Integer
Routing
Rate

=
Fractional
Routing
Rate
<
Fractional
Routing
Rate
Conclusion
17
Proof by Reduction


A k-pairs problem G  A conservative network problem G’
Find k-pairs problems such that
18
Therefore
19
Reduction
k-pairs  conservative networking
s1
t1
s2
G
Vertex Set V
Sources s1,s2
Sinks t1,t2
G’
Add vertices v1, v2
Add edges ti-vi
Add edges vi-u
∀ u ∈ V – ti
Set Ti = V + vi
t2
T2
T1
s1
s2
v2
v1
t1
t2
20
Step 1:
T2
T1
s1
s2
v2
v1
t1
t2
Easy
21
Step 2:
T2
T1
s1
s2
v2
v1
t1
t2
Disjoint trees Disjoint paths
22
Reduction does not preserve
rates for coding

A k-pairs problem G  A conservative network problem G’
“three butterflies flying together”
23
Proof by Reduction


A k-pairs problem G  A conservative network problem G’
Find k-pairs problems such that
24
s1
s2
c
u
t2
t1
25
s1
t2
s2
t1
26
Results for Cyclic Networks
Integer
Routing
Rate

<
Fractional
Routing
Rate
<
Network
Coding
Rate
“Buy one get one free”: Integer Routing Solution is NP-hard
27
A Simpler Example
1
2
3
4
5
6
7
8
28
A Simpler Example
1
2
3
4
5
6
7
8
29
Conclusion
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Conservative networking model, motivated by
practice and theory
Neat result for acyclic networks that generalize
Edmonds’ Theorem
Counter examples for cyclic networks

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Even if nodes are conservative, network coding can help
“Cycles are tricky!”
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Bound obtained by examining nodes in isolation is loose
Bound obtained by examining node-set cuts in isolation is
loose
Generally require entropy arguments
30