Network Information Flow: Limits and their Achievability

1
Network Information Flow Limits and their
Achievability

• Shashibhushan P. Borade
• Dept. of Electrical Engineering,
• Indian Institute of Technology,
• Bombay
• ISIT2002
• Lausanne, Switzerland.

2
Point-to-point network on directed graph
G(V,E)
Cij
j
i
s1
t11
t12
3
The Network Properties

• Links - noisy, but mutually independent

Cij
j
i
s1
s2
t11
t12
t21
4
Network Code Procedure

• Source Node Encoder

Transmitted symbols f(W)X (from time 1 to n)
Message generated W
f
-Block length n (big enough for less
errors) -If rate of source R 2nR possible Ws
5

• Intermediate Node

Transmitted Symbols h(Y)X at time k
h
Received symbols Y (from time 1 to k-1)

• Sink Node Decoder

• Received symbols Y
• (from time 1 to n)

g
g(Y) W (message estimate)
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Main Result

• For every cutset S
• ? Rk ? Cij
• TS Sources inside cutset S with at least one
  sink outside S

S Sc
i j
S
Sc
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Examples

• 2 sources and
• 3 sources

S1
S2
RS2
RS1
RS1
RS3
S3
S2 and
S1 ,
RS1
RS2
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Examples

• 2 sources and

s1
s2
t2
s1
RS2

3
3
5
3
2
3
3
s2
3
2
RS1
t1
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Examples

• 3 sources

s3
s2 and
s1 ,
s2
t3

3
2
2
3
5
s1
t1
2
2
3
3
s3
t2
RS1
RS3
RS1
RS2
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Proof Sketch

• Result by Thomas Cover
• ? Rk I (XS,V YV,Sc XSc,V)

k
TS
Symbols transmitted from A to B
B
A
XA,B
YA,B
Symbols received at B from A
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Proof Sketch

• Using this
• H (Ya,Yb Xa,Xb,Xc) H (Ya Xa) H (Yb Xb)

• Independent Links

a
p(Ya,Y b,Y c Xa,Xb,Xc)
b
p(YaXa) p(Y bXb) p(Y cXc)
c
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Single Source case

• Ahlswede et al (2000) alpha-codes
• Yeung Li Linear code
• Extension to noisy links
• Song Yeung (2000) Asynchronous links

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Discussion

• Is routing is enough for monocast at least? Why?

s2
RS2
1
1
t3
2
t1
(2,2,2)
2
2
1
1
RS2
RS2
s1
s3
1
t2
1
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Acyclic Networks

• No critical comebacks

t3
t1
s2
s1
t2
s3
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Acyclic Networks

• No critical comebacks

s1
s3
s2
t11
t2
t3
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Routing suffices if..

• Multi-commodity flow achieves its bound
• Nagamochis characterization
• Planar, Acyclic graphs
• Graph is 2-connected
• All the sources or sinks on outer surface
• All nodes of type A and B on outer surface

Type A
Type B
(only outgoing edges)
(only incoming edges)
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Some musings..

• What if sources arent independent?
• What if links arent independent?
• Acyclic networks Routing always enough?
• When does memoryless coding suffice?

bit a
a

b
e.g. XOR
bit b

a

b
c
bit c
18

• Thank You

19
References

• T. M. Cover and J. A. Thomas. Elements of
  Information Theory. New York Wiley, 1991.
• R. Ahlswede et al, Network information flow,
  IEEE Trans. Inform. Theory, vol. 46, no. 4, pp.
  1204-1216, July 2000.
• R. Yeung and S.-Y.R Li, Linear network coding,
  IEEE Trans. Inform. Theory, Submitted for
  publication.
• L. Song and R. Yeung, On point-to-point
  communication networks, Proc. IEEE ISIT2000,
  pp. 21, June 2000.
• H. Nagamochi, Studies on Multicommodity Flows in
  Planar Networks, Ph.D. Thesis, Kyoto University,
  Japan, 1988.