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Network Information Flow: Limits and their Achievability

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Title: 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)
6
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
7
Examples
  • 2 sources and
  • 3 sources

S1
S2
RS2
RS1
RS1
RS3
S3
S2 and
S1 ,
RS1
RS2
8
Examples
  • 2 sources and

s1
s2
t2
s1
RS2

3
3
5
3
2
3
3
s2
3
2
RS1
t1
9
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
10
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
11
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
12
Single Source case
  • Ahlswede et al (2000) alpha-codes
  • Yeung Li Linear code
  • Extension to noisy links
  • Song Yeung (2000) Asynchronous links

13
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
14
Acyclic Networks
  • No critical comebacks

t3
t1
s2
s1
t2
s3
15
Acyclic Networks
  • No critical comebacks

s1
s3
s2
t11
t2
t3
16
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)
17
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.
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