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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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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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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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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