Network Coding

Department of Computer Engineering Sharif University of Technology

Winter 2016

Arash Pourdamghani

A Short Introductionto

Outline

Background

Examples

Theories

Benefits & Challenges

2

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Background

3

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NetworkingSharing resources

Unify multiple devices

Packet switching

Through multiple layers

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RoutingPlanning trip for packets from source to destination

Model network by (weighted) graphs

0111

1001

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Routing drawbackTreat information as independent commodities!

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Examples

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Butterfly NetworkS

A B

C

𝑅1 𝑅2

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Butterfly Network(cont’d)

S

A B

C

𝑅1 𝑅2

D

S

A B

C

𝑅1 𝑅2

D

S

A B

C

𝑅1 𝑅2

D

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New Idea

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Butterfly Network(cont’d)

S

A B

C

𝑅1 𝑅2

D

m1 m2

m1 m2⊕

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Wireless Communication

A and B want to exchange 2 files by helping a

relay node R (e.g. a satellite link)

𝐴 𝑅 𝐵𝐴 𝑅 𝐵

𝐴 𝑅 𝐵𝐴 𝑅 𝐵

m1

m1

m2

m2

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Wireless Communication (Cont’d)

Energy efficient

Less delay

More wireless bandwidth

𝐴 𝑅 𝐵𝐴 𝑅 𝐵

m1 m2

m1 m2⊕ m1 m2⊕

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Content DistributionCombining collaborative content distribution &

network coding

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Content Distribution(cont’d)

Capacity increase with increasing the clients

number !

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Theories

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Other disciplines

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Overviewwant to transmit to simulatencly

𝑆1

𝑆2 𝑆𝑘

𝑅1

𝑅2

𝑅𝑛h

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Min-Cut Max-FlowAcyclic graph G = (V,E) with unit capacity edges,

a source vertex S, and a receiver vertex R.

If the min-cut between S and R equals h, then

the information can be send from S to R at a

maximum rate of h.

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Main TheoremThere exists a multicast transmission scheme over a

large enough finite field , in which intermediate network

nodes linearly combine their incoming information

symbols over , that delivers the information from the

sources simultaneously to each receiver at a rate equal to

h.

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Multicast Transmission One-to-many communication with specific

receiver addresses

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Finite FieldsAbelian Group -> Galois Field -> Extension Fields

Closure, Associativity, Commutativity

Identity & Inverse element

Closed on ‘+’ and ‘.’ + 0 10 0 11 1 0

. 0 10 0 01 0 1

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Finite Fields(cont’d)Prime Fields: where is prime number

Extension fields: where is prime number and

+ 0 1 A B0 0 1 A B1 1 0 B AA A B 0 1B B A 1 0

. 0 1 A B0 0 0 0 01 0 1 A BA 0 A B 1B 0 B 1 A

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Proof methods•Algebraic• There exist values in some large enough finite field for the

components of the local coding vectors, such that all matrices ,1 ≤ j ≤ N, defining the information that the receivers observe, are full rank .

•Information Theoretic• For each vertex select functions chosen uniformly at random• Each receiver could decode all source packets if get sufficient

input packets

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Benefits &

Challenges

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BenefitsThroughput increment

In place coding

Efficiency of wireless resources

Using current cables

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Coding AdvantageDIRECTED GRAPHS

Multicast : Multiple unicast:

UNDIRECTED GRAPHSUpper bound is 2Lower bound is

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ChallengesDynamic changes

Complexity of computations

Security of transmitted data

Integration with existing infrastructure

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ReferencesJ. Kurose, K. Ross, “Computer Networking: Top-

Down Approach”,6th edition, Addison Wesley, 2013

M. Jafari Siavoshani, ”A Very Short Introduction to

Network Coding”, Sharif University of Technology,

Fall 2014

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References(cont’d)C. Fragouli, E. Soljanin, “Network coding

fundamentals” Foundations and Trends in

Networking, 2007

A. Sprintson, Theory and application of network

coding, Texas A&M University, 2016

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Thank You