Network Coding Theory: Tutorial

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

2. 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. 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. 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. The canonical example (I) • Without network coding • Simple store and forward • Multicast rate of 1.5 bits per time unit

6. 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. b1 C A r B NC and wireless communications • Problem: send b1 from A to B and b2 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) (b) (c) b2 b2 b1 C C B A B A

8. 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. Single-Source Network Coding • Network is acyclic. • The message x, a -dimensional row vector ina finite field F, is generated at the source node. • A symbol in F can be sent on each channel.

10. 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. 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. 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. 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. 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. 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. An Example

17. e d T message x

18. 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. 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. Linear Multicast • For each node v, if maxflow(v)  , then the message x can be recovered.

21. 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. 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 thatmaxflow(P)  , where each individual node u in P may have maxflow(u) < .

23. 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 {e1, e2, … , en}, where n and ej Out(vj), the vectors fe1, fe2, … , fen are linearly independent provided that {fd: d In(vj)}{fek: k  j} for 1 jn

24. 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.

25. 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

26. 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 byis -dimensional column vector calculated recursively in an upstream-to-downstream order by

27. 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

28. Example

29. 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

30. Non-Equivalence Example • The local encoding kernels doesn’t give an unique solution for the global • encoding kernels

31. 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)

32. Convolutional Codes for Cyclic Networks

33. 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

34. Thank You!