Network Coding Theory: Tutorial

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Network Coding Theory Tutorial

• Presented by
• Avishek Nag
• Networks Research Lab
• UC 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
(a)

• 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

(b)
(c)
b2
b2
b1
C
C
B
A
B
A
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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 1s and 0s 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

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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

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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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e
d
T
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) lt ?, 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) lt ?, 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) lt ?.

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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

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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

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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 doesnt 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 gt 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(t1)

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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!