Network coding theory consolidation and extensions
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Network Coding Theory: Consolidation and Extensions. Raymond Yeung Joint work with Bob Li, Ning Cai and Zhen Zhan. Outline. Single-Source Network Coding Global and Local Descriptions of a Network Code Linear Multicast, Broadcast, and Dispersion Static codes Multi-Source Network Coding

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Network coding theory consolidation and extensions

Network Coding Theory:Consolidation and Extensions

Raymond Yeung

Joint work with

Bob Li, Ning Cai and Zhen Zhan


Outline

Outline

  • Single-Source Network Coding

    • Global and Local Descriptions of a Network Code

    • Linear Multicast, Broadcast, and Dispersion

    • Static codes

  • Multi-Source Network Coding

    • Fundamental Limits of Linear Codes

      Based on an upcoming paper to appear in Foundation and Trends in Communications and Information Theory (Editor: Sergio Verdu).


Single source network coding

Single-Source Network Coding

  • Network is acyclic.

  • The message x, a -dimensional row vector in F, is generated at the source node.

  • A symbol in F can be sent on each channel.


Global description

Global Description

  • The symbol sent on channel e is a function of the message, called the global encoding mapping for channel e.

  • For any node v, the global encoding mappings have to satisfy the local constraints, i.e., the local encoding mapping for every node v is well defined.


A globally linear network code

A Globally Linear Network Code

  • A code is globally linear if all the global encoding mappings are linear (and all the local constraints satisfied).

  • A globally linear code is the most general linear code that can possibly be defined.

  • The global encoding mapping for channel e is characterized by a column vector fe, s.t. the symbol sent on e is x fe.

  • It can be proved that if a code is globally linear, then it is also locally linearly, i.e., all local encoding mappings are linear.


Global description vs local description

Global Description vs Local Description

  • Since the local encoding mapping at a node v is linear, it follows that

    for any eOut(v),fe is a linear combination of fe’, e’  In(v).

     Global description (Li-Yeung-Cai).

  • These linear combination forms the local encoding kernel.

     Local description (Koetter-Medard)


Global description local description

Global Description = Local Description

  • 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!


Generic network code

Generic Network Code

  • Definition (LYC)

    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.

  • The idea: Whenever a collection of vectors can possibly be linear independent, they are.


Special cases of a generic network code

Special Cases of a Generic Network Code

Generic network code

Linear dispersion

Linear Broadcast

Linear Multicast

Each notion is strictly weaker than the previous notion!


Linear multicast

Linear Multicast

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


Linear broadcast

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

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 (Generic network code implies all)

  • 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

Code Constructions

  • A generic network code exists for all sufficiently large F and can be constructed by the 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 codes

Static Codes

  • Static linear multicast was introduced by KM which finds applications in robust network multicast.

  • Static versions of linear broadcast and linear dispersion can be defined accordingly.

  • The LYC algorithm can be modified for constructing a static generic network code.

  • This means that the static versions of a linear dispersion, a linear broadcast, and a linear multicast can all be constructed.


Multi source network coding

Multi-Source Network Coding

  • A network is given.

  • Independent information sources of rates

     = (1, 2, …, S)

    are generated at possibly different nodes, and each source is to be multicast to a specific sets of nodes.

  • The set of all achievable rates is called the achievable information rate regionR.

  • If all the sources are multicast to the same set of nodes, then it reduces to a single-source network coding problem, otherwise it does not.


Network coding theory consolidation and extensions

  • A multi-source network coding problem cannot be decomposed into single-source network coding problems even when all the information sources are generated at the same node (Yeung 95).

  • Special multi-source network coding problems have been shown to be decomposable (Roche, Hau, Yeung, Zhang 95-99).


An example of indecomposability with wireless application

An Example of Indecomposability(with Wireless Application)

Independent sources need to be coded jointly

b1

b2

b1

b2

b1+b2

b1

b2


Characterization of the information rate region r

Characterization of the Information Rate region R

  • Inner and outer bounds on R acyclic networks can be expressed in term of the region of all entropy functions of random variables (Yeung 97, Yeung-Zhang 99, Song et al. 03).

  • A computable outer bound on R, called RLP, has also been obtained.

  • Only existence proofs by random coding are available  no code construction.


The region

The region Γ*

  • Let Γ* be the set of all entropy functions of a collection of random variables labeled by the information sources and the channels.


Outer bound r out

Outer Bound Rout

If an information rate tuple  is achievable, then there exists h  closure(Γ*) which satisfies a set of constraints denoted by Cwhich specifies

  • the independence of the information sources

  • the rate tuple

  • local constraints of the code

  • the channel capacity constraints

  • the multicast requirements.

    C is a collection of hyperplanes in the Eucledian space.


Linear codes for multiple sources

Linear Codes for Multiple Sources

  • The global description for a linear network code can be generalized to multiple sources.

  • Each channel is characterized by a column vector of an appropriate dimension.

  • The existence of a linear code is nothing but the existence of a collections of vectors satisfying the set of constraints C.


The region1

The Region *

  • Let * be the set of all rank functions for a collection of -dimensional column vectors labeled by the information sources and the channels over some finite field F, where   1.


Linear codes vs nonlinear codes

Linear Codes vs Nonlinear Codes

Linear codes  Rlinear

An information rate tuple  is linearly achievable iff there exists h  closure(*) which satisfies the set of constraints C.

Note: Rlinear includes all rate tuples that are inferior to some rate tuples achievable by mixing linear codes.

Nonlinear codes outer bound Rout

If an information rate tuple  is achievable, then there exists h  closure(Γ*) which satisfies the set of constraints C.


Similarity between rank and entropy

Similarity between Rank and Entropy

  • The rank function satisfies

    • 0  rank(A).

    • rank(A)  rank(B) if A  B.

    • rank(A) + rank(B)  rank(AB) + rank(AB).

    • rank(A)  |A|.

  • The entropy function in general satisfies

    • 0  H(A).

    • H(A)  H (B) if A  B.

    • H(A) + H (B)  H (AB) + H (AB).

      1 - 3 are called the polymatroidal axioms.


The bridge from rank to entropy

The Bridge from Rank to Entropy

Theorem 1:

Let F be a finite field, Y be an -dimensional random row vector that distributes uniformly on F, and A be an   l matrix. Let Z = Y·A. Then H(Z) = rank(A) log |F|.

Using this theorem, it can be shown that

*  Γ*.


A gap between and

A Gap between * and Γ*

  • In addition to the polymatroidal axioms, the rank function also satisfies the Ingleton inequality:

    r(A13)+ r(A14)+ r(A23)+ r(A24)+ r(A34)

     r(A3)+ r(A4)+ r(A12)+ r(A134)+ r(A234)

  • The Ingleton inequality is satisfied by algebraic structures as general as Abelian groups.

  • The corresponding inequality is not satisfied by the entropy function (Zhang-Yeung 99), so there is a gap between * and Γ*.

  • This gap between * and Γ* suggests that nonlinear codes may actually perform better for some multi-source problems.


Vector linear codes

Vector Linear Codes

  • Vector Linear Codes (Riis, Lehman2, Medard, Effros, Ho, Karger, Koetter)

    • It can be regarded as a linear code over a network obtained by expanding all the capacities by an integer factor.

    • It has been shown that some multi-source problems do not have linear solutions but have vector linear solutions.

  • Question 1: Are these vector linear solutions better than all mixtures of linear solutions?

    Question 2: Do these vector linear solutions exceed the Ingleton inequality? (If so, the answer to Q1 is yes.)


Codes beyond fields

Codes Beyond Fields

  • Dougherty, Frieling and Zeger have recently shown that there exist a multi-source problem that has no linear solution even in the more general algebraic context of modules, which includes all finite rings and Abelian groups.

  • Question 1: Is the nonlinear solution given by DFZ better than all mixtures of linear solutions?

    Question 2: Does the nonlinear solution given by DFZ exceed the Ingleton inequality? (If so, the answer to Q1 is yes.)


Ingleton inequality classification

Ingleton Inequality Classification

  • Codes abide by the Ingleton inequality

    • Linear codes, module codes

  • Codes not necessarily abide by the Ingleton inequality

    • Vector linear codes (abide by the Ingleton inequality in an extended space)

  • Codes not abide by the Ingleton inequality

    • Non-Abelian group codes are asymptotically as good as all nonlinear codes (Chan, submitted to ISIT 2005).


Network coding theory consolidation and extensions

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