A short introduction to Network coding for math students of Sharif University of Technology

1. Network Coding
Department of Computer Engineering
Sharif University of Technology
Winter 2016
Arash Pourdamghani
A Short Introduction
to

2. Outline
Background
Examples
Theories
Benefits & Challenges
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3. Arash PourdamghaniNetwork Coding
Background
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4. Arash PourdamghaniNetwork Coding
Networking
Sharing resources
Unify multiple devices
Packet switching
Through multiple layers
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5. Arash PourdamghaniNetwork Coding
Routing
Planning trip for packets from source to destination
Model network by (weighted) graphs
0111
1001
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6. Arash PourdamghaniNetwork Coding
Routing drawback
Treat information as independent commodities!
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7. Arash PourdamghaniNetwork Coding
Examples
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8. Arash PourdamghaniNetwork Coding
Butterfly Network
S
A B
C
𝑅1 𝑅2
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9. Arash PourdamghaniNetwork Coding
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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10. Arash PourdamghaniNetwork Coding
New Idea
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11. Arash PourdamghaniNetwork Coding
Butterfly Network(cont’d)
S
A B
C
𝑅1 𝑅2
D
m1 m2
m1 ⊕ m2
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12. Arash PourdamghaniNetwork Coding
Wireless Communication
A and B want to exchange 2 files by helping a
relay node R (e.g. a satellite link)
𝐴 𝑅 𝐵
𝐴 𝑅 𝐵
𝐴 𝑅 𝐵
𝐴 𝑅 𝐵
m1
m1
m2
m2
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13. Arash PourdamghaniNetwork Coding
Wireless Communication (Cont’d)
Energy efficient
 Less delay
 More wireless bandwidth
𝐴 𝑅 𝐵
𝐴 𝑅 𝐵
m1 m2
m1 ⊕ m2 m1 ⊕ m2
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14. Arash PourdamghaniNetwork Coding
Content Distribution
Combining collaborative content distribution &
network coding
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15. Arash PourdamghaniNetwork Coding
Content Distribution(cont’d)
Capacity increase with increasing the clients
number !
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16. Arash PourdamghaniNetwork Coding
Theories
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17. Arash PourdamghaniNetwork Coding
Other disciplines
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18. Arash PourdamghaniNetwork Coding
Overview
𝑆1, 𝑆2, … 𝑆 𝑘 want to transmit to 𝑅1, 𝑅2, … 𝑅 𝑛
simulatencly
𝑆1
𝑆2
𝑆 𝑘
𝑅1
𝑅2
𝑅 𝑛
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h

19. Arash PourdamghaniNetwork Coding
Min-Cut Max-Flow
Acyclic 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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20. Arash PourdamghaniNetwork Coding
Main Theorem
There exists a multicast transmission scheme over a
large enough finite field 𝑭 𝒒, in which intermediate
network nodes linearly combine their incoming
information symbols over 𝐹𝑞, that delivers the
information from the sources simultaneously to each
receiver at a rate equal to h.
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21. Arash PourdamghaniNetwork Coding
Multicast Transmission
One-to-many communication with specific
receiver addresses
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22. Arash PourdamghaniNetwork Coding
Finite Fields
Abelian Group -> Galois Field -> Extension Fields
 Closure, Associativity, Commutativity
Identity & Inverse element
Closed on ‘+’ and ‘.’
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+ 0 1
0 0 1
1 1 0
. 0 1
0 0 0
1 0 1

23. Arash PourdamghaniNetwork Coding
Finite Fields(cont’d)
Prime Fields: 𝐺𝐹(𝑝) where 𝑝 is prime number
Extension fields: 𝐺𝐹(𝑝 𝑚) where 𝑝 is prime
number and 𝑚 > 1
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+ 0 1 A B
0 0 1 A B
1 1 0 B A
A A B 0 1
B B A 1 0
. 0 1 A B
0 0 0 0 0
1 0 1 A B
A 0 A B 1
B 0 B 1 A

24. Arash PourdamghaniNetwork Coding
Proof methods
•Algebraic
• There exist values in some large enough finite field 𝐹𝑞 for the
components {𝛼 𝑘} of the local coding vectors, such that all
matrices 𝑨𝒋,1 ≤ j ≤ N, defining the information that the
receivers observe, are full rank .
•Information Theoretic
• For each vertex 𝑣 select |𝑂𝑢𝑡(𝑣)| functions 𝑓𝑖
𝑣
: 2ℎ 𝐼𝑛 𝑣 → 2ℎ
chosen uniformly at random
• Each receiver could decode all source packets if get sufficient input
packets
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25. Arash PourdamghaniNetwork Coding
Benefits
&
Challenges
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26. Arash PourdamghaniNetwork Coding
Benefits
Throughput increment
In place coding
Efficiency of wireless resources
Using current cables
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27. Arash PourdamghaniNetwork Coding 27
Coding Advantage
DIRECTED GRAPHS
Multicast : Ω( 𝑛)
Multiple unicast: Ω(𝑛)
UNDIRECTED GRAPHS
Upper bound is 2
Lower bound is
8
7
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28. Arash PourdamghaniNetwork Coding
Challenges
Dynamic changes
Complexity of computations
Security of transmitted data
Integration with existing infrastructure
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29. Arash PourdamghaniNetwork Coding
References
J. 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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30. Arash PourdamghaniNetwork Coding
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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31. Arash PourdamghaniNetwork Coding
Thank You
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