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Network Coding Theory: Tutorial. Presented by Avishek Nag Networks Research Lab UC Davis. Outline. Introduction Classifications Single-Source Network Coding Global and Local Descriptions of a Network Code Linear Multicast, Broadcast, and Dispersion Static codes

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## Network Coding Theory: Tutorial

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**Network Coding Theory: Tutorial**Presented by Avishek Nag Networks Research Lab UC Davis**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**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**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**The canonical example (I)**• Without network coding • Simple store and forward • Multicast rate of 1.5 bits per time unit**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**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**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**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.**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.**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).**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)**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**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**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**e**d T message x**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)**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!**Linear Multicast**• For each node v, if maxflow(v) , then the message x can be recovered.**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**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) < .**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 jn**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.**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**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**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**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**Non-Equivalence Example**• The local encoding kernels doesn’t give an unique solution for the global • encoding kernels**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)**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

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