page 1 page 1 network coding theory: tutorial presented by avishek nag networks research lab uc...

Network Coding Theory: Tutorial

Presented by

Avishek Nag

Networks Research LabUC Davis

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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

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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

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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

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The canonical example (I)

• Without network coding– Simple store and forward

– Multicast rate of 1.5 bits per time unit

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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

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NC and wireless communications

b1 b2b2

• 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 A BB CC

b1

A BC

r

(a)

(b) (c)

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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

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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.

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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.

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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).

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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)

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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

( )d In T ( )e Out T

,d ek

, ( ), ( )T d e d In T e Out TK k

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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

,d ek

ef

,( )

(1) , where ( )

(2)The vectors for the imaginary channels ( ) form the

natural basis of the vector space

e d e dd In T

e

f k f e Out T

f e In S

F

ef

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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

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An Example

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Td e

message x

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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)

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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!

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Linear Multicast

• For each node v, if maxflow(v) , then the message x can be recovered.

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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

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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) < .

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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

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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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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

( )e

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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

,d ek

,ef

, ,( )

(1) ( ) , where ( )

(2)The -global encoding kernals for the imaginary channels are

independent of and form the natural basis of the space

e d e dd In T

f e k f e Out T

F

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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

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Example

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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

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Non-Equivalence Example

• The local encoding kernels doesn’t give an unique solution for the global encoding kernels

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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)

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Convolutional Codes for Cyclic Networks

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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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Thank You!

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