This document provides an introduction to network coding, including: - Examples of how network coding can increase throughput in butterfly networks and wireless communications - Theories showing how intermediate nodes can linearly combine information to deliver data at the maximum possible rate - Benefits of network coding like increased throughput and efficiency, but also challenges like integrating it into existing infrastructure

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
2

3. Arash PourdamghaniNetwork Coding
Background
3

4. Arash PourdamghaniNetwork Coding
Networking
Sharing resources
Unify multiple devices
Packet switching
Through multiple layers
4

5. Arash PourdamghaniNetwork Coding
Routing
Planning trip for packets from source to destination
Model network by (weighted) graphs
0111
1001
5

6. Arash PourdamghaniNetwork Coding
Routing drawback
Treat information as independent commodities!
6

7. Arash PourdamghaniNetwork Coding
Examples
7

8. Arash PourdamghaniNetwork Coding
Butterfly Network
S
A B
C
𝑅1 𝑅2
8

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
9

10. Arash PourdamghaniNetwork Coding
New Idea
10

11. Arash PourdamghaniNetwork Coding
Butterfly Network(cont’d)
S
A B
C
𝑅1 𝑅2
D
m1 m2
m1 ⊕ m2
11

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
12

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
13

14. Arash PourdamghaniNetwork Coding
Content Distribution
Combining collaborative content distribution &
network coding
14

15. Arash PourdamghaniNetwork Coding
Content Distribution(cont’d)
Capacity increase with increasing the clients
number !
15

16. Arash PourdamghaniNetwork Coding
Theories
16

17. Arash PourdamghaniNetwork Coding
Other disciplines
17

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
18
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.
19

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.
20

21. Arash PourdamghaniNetwork Coding
Multicast Transmission
One-to-many communication with specific
receiver addresses
21

22. Arash PourdamghaniNetwork Coding
Finite Fields
Abelian Group -> Galois Field -> Extension Fields
 Closure, Associativity, Commutativity
Identity & Inverse element
Closed on ‘+’ and ‘.’
22
+ 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
23
+ 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
24

25. Arash PourdamghaniNetwork Coding
Benefits
&
Challenges
25

26. Arash PourdamghaniNetwork Coding
Benefits
Throughput increment
In place coding
Efficiency of wireless resources
Using current cables
26

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
27

28. Arash PourdamghaniNetwork Coding
Challenges
Dynamic changes
Complexity of computations
Security of transmitted data
Integration with existing infrastructure
28

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
29

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
30

31. Arash PourdamghaniNetwork Coding
Thank You
31