1. Page 1 Page 1 Network Coding Theory: Tutorial Presented by Avishek Nag Networks Research Lab UC Davis

2. Page 2 Page 2 11/26/2008 Outline Introduction Classifications Single-Source Network Coding –Global and Local Descriptions of a Network Code –Linear Multicast, Broadcast, and Dispersion –Static codes –Network Coding for Cyclic Networks

3. Page 3 Page 3 11/26/2008 Introduction DEFINITION: Network coding is a particular in-network data processing technique that exploits the characteristics of the broadcast communication channel in order to increase the capacity or the throughput of the network

4. Page 4 Page 4 11/26/2008 Communication networks TERMINOLOGY Communication network = finite directed graph Acyclic communication network = network without any directed cycle Source node = node without any incoming edges (square) Channel = noiseless communication link for the transmission of a data unit per unit time (edge) –WX has capacity equal to 2

5. Page 5 Page 5 11/26/2008 The canonical example (I) Without network coding –Simple store and forward –Multicast rate of 1.5 bits per time unit

6. Page 6 Page 6 11/26/2008 The canonical example (II) With network coding –X-OR  is one of the simplest form of data coding –Multicast rate of 2 bits per time unit

7. Page 7 Page 7 11/26/2008 NC and wireless communications b1b1 b2b2 b2b2 Problem: send b 1 from A to B and b 2 from B to A using node C as a relay A and B are not in communication range (r) Without network coding, 4 transmissions are required. With network coding, only 3 transmissions are needed A AB B C C b1b1 AB C r (a) (b)(c)

8. Page 8 Page 8 11/26/2008 Network Coding Classifications Based on Topology –Acyclic Network Coding –Cyclic Network Coding Based on number of nodes sourcing information –Single Source Network Coding: Simple Algebraic Notion –Multi Source Network Coding: Probabilistic Notion; the current understanding of multi-source network coding is quite far from being complete

9. Page 9 Page 9 11/26/2008 Single-Source Network Coding Network is acyclic. The message x, a  -dimensional row vector in a finite field F, is generated at the source node. A symbol in F can be sent on each channel.

10. Page 10 Page 10 11/26/2008 Definition of a Field A field is a set together with two operations, usually called addition (+) and multiplication (·), such that the following axioms hold: Closure of F under addition and multiplication –For all a, b in F, both a + b and a · b are in F (or more formally, + and · are binary operations on F). Associativity of addition and multiplication –For all a, b, and c in F, the following equalities hold: a + (b + c) = (a + b) + c and a · (b · c) = (a · b) · c. Commutativity of addition and multiplication –For all a and b in F, the following equalities hold: a + b = b + a and a · b = b · a.

11. Page 11 Page 11 11/26/2008 Definition of a Field Additive and multiplicative identity –There exists an element of F, called the additive identity element and denoted by 0, such that for all a in F, a + 0 = a. –Similarly, the multiplicative identity element denoted by 1, such that for all a in F, a · 1 = a. Additive and multiplicative inverses –For every a in F, there exists an element −a in F, such that a + (−a) = 0. –Similarly, for any a in F other than 0, there exists an element a−1 in F, such that a · a−1 = 1. Distributivity of multiplication over addition –For all a, b and c in F, the following equality holds: a · (b + c) = (a · b) + (a · c).

12. Page 12 Page 12 11/26/2008 Example: Binary Field A field with finite number of elements: finite field or Galois Field A binary field with elements 0 and 1 and operations XOR and AND is a GF(2) A message consisting of 1’s and 0’s and containing say, 3 bits is a 3-dimensional row vector in GF(2)

13. Page 13 Page 13 11/26/2008 Local Description of Network Code Let a pair of channels (d, e) be called an adjacent pair when there exists a node T with and Let F be a finite field and a positive integer. An - dimensional F-valued linear network code on an acyclic communication network consists of a scalar, called the local encoding kernel, for every adjacent pair (d, e) The local encoding kernel at the node T means the |In(T)| × |Out(T)| matrix

14. Page 14 Page 14 11/26/2008 Global Description of Network Code Let F be a finite field and a positive integer. An - dimensional F-valued linear network code on an acyclic communication network consists of a scalar for every adjacent pair (d, e) in the network as well as an -dimensional column vector for every channel e such that The vector is called the global encoding kernel for the channel e

15. Page 15 Page 15 11/26/2008 Local Description vs. Global Description Given the local encoding kernels for all channels in an acyclic network, the global encoding kernels can be calculated recursively in any upstream-to-downstream order by (1), while (2) provides the boundary conditions The global description and the local description are the two sides of a coin: –They are equivalent. –Both can describe the most general form of a (block) linear network code

16. Page 16 Page 16 11/26/2008 An Example

17. Page 17 Page 17 11/26/2008 T d e message x

18. Page 18 Page 18 11/26/2008 Desirable Properties of a Linear Network Code Law of information conservation: the content of information sent out from any group of non-source nodes must be derived from the accumulated information received by the group from outside maxflow(T): the maximum flow from S to a non- source node T maxflow( P ): the maximum flow from S to a collection P of non-source nodes Max-flow Min-cut Theorem: the information rate received by the node T cannot exceed maxflow(T)

19. Page 19 Page 19 11/26/2008 Desirable Properties of a Linear Network Code The network topology, the dimension, and the coding scheme determines achievability of the upper bound Three special classes of linear network codes are defined below by the achievement of this bound to three different extents –Linear Dispersion –Linear Broadcast –Linear Multicast Each notion is strictly weaker than the previous notion!

20. Page 20 Page 20 11/26/2008 Linear Multicast For each node v, if maxflow(v)  , then the message x can be recovered.

21. Page 21 Page 21 11/26/2008 Linear Broadcast For every node v, –If maxflow(v)  , the message x can be received. –If maxflow(v) < , maxflow(v) dimensions of the message x can be recovered. Linear Broadcast  Linear Multicast

22. Page 22 Page 22 11/26/2008 Linear Dispersion For every collection of nodes P, –If maxflow( P )  , the message x can be received. –If maxflow( P ) < , maxflow( P ) dimensions of the message x can be recovered. Linear Dispersion  Linear Broadcast  Linear Mulicast For a linear dispersion, a new comer who wants to receive the message x can do so by accessing a collection of nodes P such that maxflow( P )  , where each individual node u in P may have maxflow(u) < .

23. Page 23 Page 23 11/26/2008 Code Constructions Construction of multicast/broadcast/dispersion: consider a linear network code in which every collection of global encoding kernels that can possibly be linearly independent is linearly independent This motivates the following concept of a generic linear network code: A linear network code is said to be generic if: For every set of channels {e 1, e 2, …, e n }, where n   and e j  Out(v j ), the vectors f e1, f e2, …, f en are linearly independent provided that  {f d : d  In(v j )}    {f ek : k  j}  for 1  j  n

24. Page 24 Page 24 11/26/2008 Code Constructions A generic network code exists for all sufficiently large F and can be constructed by the Li-Yeung- Cai (LYC) algorithm. A linear dispersion, a linear broadcast, and a linear multicast can potentially be constructed with decreasing complexity since they satisfy a set of properties of decreasing strength. In particular, a polynomial time algorithm for constructing a linear multicast has been reported independently by Sanders et al. and Jaggi et al.

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27. Page 27 Page 27 11/26/2008 Static Network Codes Convention: A configuration of a network is a mapping from the set of channels in the network to the set {0,1} =0 for any link e signifies that the link e is absent due to link failure

28. Page 28 Page 28 11/26/2008 Static Network Codes Let F be a finite field and a positive integer. An - dimensional F-valued linear network code on an acyclic communication network consists of a scalar for every adjacent pair (d, e) in the network. The -global encoding kernel for the channel e, denoted by is - dimensional column vector calculated recursively in an upstream-to-downstream order by

29. Page 29 Page 29 11/26/2008 Static Codes The adjective “static” in the terms above stresses the fact that, while the configuration varies, the local encoding kernels remain unchanged The advantage of using a static network code in case of link failure is that the local operation at any node in the network is affected only at the minimum level

30. Page 30 Page 30 11/26/2008 Example

31. Page 31 Page 31 11/26/2008 Cyclic Networks Networks with at least one directed cycle Acyclic: the network coding problem independent of the propagation delay, operation at all nodes synchronized Cyclic: the global encoding kernels simultaneously implemented under the ideal assumption of delay- free communications (unrealistic) The time dimension is an essential part of the consideration in network coding Non-equivalence between local and global descriptions

32. Page 32 Page 32 11/26/2008 Non-Equivalence Example The local encoding kernels doesn’t give an unique solution for the global encoding kernels

33. Page 33 Page 33 11/26/2008 Convolutional Codes for Cyclic Networks Corresponding to a physical node X, there is a sequence of nodes X(0), X(1), X(2),... in the trellis network A channel in the trellis network represents a physical channel e only for a particular time slot t > 0, and is thereby identified by the pair (e, t) When e is from the node X to the node Y, the channel (e, t) is then from the node X(t) to the node Y(t+1)

34. Page 34 Page 34 11/26/2008 Convolutional Codes for Cyclic Networks

35. Page 35 Page 35 11/26/2008 References R. W. Yeung, S. Y. R. Li, N. Cai and Z. Zhang, “Network Coding Theory,” Now Publishers Inc., 2006. Elena Fasolo, “Wireless Systems Lecture: Network Coding Techniques,” March 2004

36. Page 36 Page 36 11/26/2008 Thank You!