introduction to network coding

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Presented by:

Mohamed Y. SelimVisiting Scholar at Iowa State University

Lecturer Assistant at Suez Canal University

[email protected]

I. Introduction to Network Coding

Network Network CodingCodingWhyWhy??

DefinitionsDefinitions

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CodingCodingWhyWhy??

What?What?


� What is Network Coding

�Network Coding is a field of information theoryand coding theory and is a method of attainingmaximum information flow in a network.

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maximum information flow in a network.

�Network Coding Theory points out that it isnecessary to consider encoding/decoding data onnodes in network in order to achieve optimalthroughput.



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Information Information theorytheory

Wireless Wireless networksnetworks

ChannelChannelcodingcoding

Quantum Quantum information information

theorytheory

GraphGraphtheorytheory

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Computer Computer networksnetworks

SwitchingSwitchingtheorytheory

ComputerComputersciencescience

CryptographyCryptography

OptimizationOptimizationtheorytheory

Game Game theorytheory

Matroid Matroid theorytheory Data Data

storagestorage


� Example: Two-way Relay Comm.

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� Network Coding: The Butterfly

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� Kirchhoff Versus Network Coding

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� Linear Network Coding

• Random Processes in a Linear Network

� Source Input:� Info. Along Edges:

{ }),...,(),,(),( 10 lvxlvxlvX =

The index is a { }),...(),()( 10 eyeyeY =Weighted Combination

of processes

Weighted Combination of processes from

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� Info. Along Edges:� Sink Output:

• Relationship among them

The index is a time index

{ }),...(),()( 10 eyeyeY ={ }),...,(),,(),( 10 lvzlvzlvZ =

∑∑==

+=)(´)(´:

´,

)(

1, ´)(),()(

etaileheadeee

v

lel eYlvXeY βα

µ

∑=

=veheade

je eYjvZ´)(´:

´, ´)(),( εe comes out of v

of processes generated at v

of processes from adjacent edges of e

Weighted Combination from all

incoming edges


� Linear Network Coding (The Transfer Matrix)

v2e1

e2

e5

)1,( 1vX )1,(vZ

)3,()2,()1,()(111 ,3,2,11 vXvXvXeY eee ααα ++=

)3,()2,()1,()(222 ,3,2,12 vXvXvXeY eee ααα ++=

)3,()2,()1,()(333 ,3,2,13 vXvXvXeY eee ααα ++=

)()()( eYeYeY ββ +=

))3,(),2,(),1,(( 111 vXvXvXxLet =))3,(),2,(),1,(( 444 vZvZvZz=

Mxz ⋅=

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e4v1

v3

v4

e2

e3

e6

e7

)1,( 1vX

)2,( 1vX

)3,( 1vX

)1,( 4vZ)2,( 4vZ

)3,( 4vZ

)()()( 2,1,4 4241eYeYeY eeee ββ +=

)()()( 2,1,5 5251eYeYeY eeee ββ +=

)()()( 4,3,6 6463eYeYeY eeee ββ +=

)()()( 4,3,7 7473eYeYeY eeee ββ +=

)()()()1,( 71,61,51,4 765eYeYeYvZ eee εεε ++=

)()()()2,( 72,62,52,4 765eYeYeYvZ eee εεε ++=

)()()()3,( 73,63,53,4 765eYeYeYvZ eee εεε ++=

BAM

eeee

eeeeeeeeee

eeeeeeeeee

⋅

⋅=

7363

7442644252

7441644151

,,

,,,,,

,,,,,

0 ββββββββββββ

=

321

321

321

,3,3,3

,2,2,2

,1,1,1

eee

eee

eee

A

ααααααααα

=

3,2,1,

3,2,1,

3,2,1,

777

666

555

eee

eee

eee

B

εεεεεεεεε


� Linear Network Coding (Solution)

• We want

• Choose to be an identity matrix.

BAM eeeeeeeeee

eeeeeeeeee

⋅

⋅=7442644252

7441644151

,,,,,

,,,,,

ββββββββββ

Mxz ⋅=xz =

A

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an identity matrix.

• Choose B to be the inverse of

=

3,2,1,

3,2,1,

3,2,1,

777

666

555

eee

eee

eee

B

εεεεεεεεε

,

321

321

321

,3,3,3

,2,2,2

,1,1,1

=

eee

eee

eee

A

ααααααααα

BAM

eeee

eeeeeeeeee ⋅

⋅=

7363

7442644252

,,

,,,,,

0 βββββββ

7363

7442644252

7441644151

,,

,,,,,

,,,,,

0 eeee

eeeeeeeeee

eeeeeeeeee

ββββββββββββ

NETWORK CODING SOLUTION EXISTS IF DETERMINANT OF M

IS NON-ZERO


� Random Linear Network Coding

� Ho, Koetter, Medard, Karger, Effros (2003/06)

� Random coefficients for linear network coding

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� Can decode with probability ≈ 1

� Enables network coding in unknown network topologies


� Random Linear Network Coding

• How to select the coefficients ξ ?

� Randomly Select

� Coefficients are chosen uniformly at random from a finite field F

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� Coefficients are chosen uniformly at random from a finite field Fq(Fq is the set of integers from 0 to q-1, where q=2g )

� If q is large, then the probability of that two coefficient vectors are dependent is small.


� Signal-Level Network Coding (PNC)

�Allows wireless signals to add up physically

�Can further improve the efficiency of wireless network coding

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�Physical-Layer NC: Zhang, Liew, and Lam (2006)

�Analog NC: Katti, Gollakota, and Katabi (2007)



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For more details:

http://arxiv.org/ftp/arxiv/papers/1105/1105.4261.pdf


� Network Coding Applications (VANET Application)

� Computer Networks

� Wireless/Satellite Communications

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� Distributed information storage/ dissemination

(e.g., Bit-Torrent)

� Robust Network Management

� Network Error Correction



C1 Sends Block 1

B1

C3 Sends Block 2

B2

C2 Sends Block 2

B1 B2B2 B2

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C3C2C1

C6C5C4

B1 B2B2

C5 Sends Block 2

B2 B2

B1 is STILL missing!!Without Network

Coding


� Network Coding Applications

C1 Sends Block 1

B1

C3 Sends Block 2

B2

C2 Sends a Coded Block: B1+B2

B1 B2B2 B1+B2

B1+B2B1

B1+B2

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C3C2C1

C6C5C4

B1 B2B1+B2

C5 Sends a Coded Block: B1+B2

B1+B2 B1+B2B2 B1

C4 and C6 successfully recovered both blocksWith Network Coding



�CarTorrent is a BitTorrent-like file protocol VANET.

�A car passing by an access point, pulls as many blocks as

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A car passing by an access point, pulls as many blocks as

possible. Once it’s out-of-range, it’ll start talking to others to

pull blocks.

�Each peer sends the availability of blocks to others.



Internet

RY

YY

RRR

Without Network Coding

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Downloading Blocks from AP

Exchange Blocks via multi-hop pulling

G

Y2

Gossiping Availability of Blocks

Y



InternetBuffer

BufferBuffer

With Network Coding

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Downloading Coded Blocks from AP

Outside Range of AP

Buffer

Re-Encoding: Random Linear Comb.of Encoded Blocks in the Buffer

Exchange Re-Encoded Blocks

Meeting Other Vehicles with Coded Blocks

“coded” block

B1

File

: kbl

ocks B2

B3

Bk

+

*a1

*a2*a3

*ak

Random Linear Combination



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� Satellite/Wireless Application


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A+B



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� Network Coding Practical Problems

• Network Delay

• Centralized Knowledge of Graph Topology

• Packet Loss

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• Packet Loss

• Link Failures

• Change in Topology or Capacity


� References

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� References

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ا��د � رب ا��ن

Thank You

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ا��د � رب ا��ن

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