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					         Network Coding
                Project presentation
                Communication Theory
                            16:332:545

Amith Vikram   Atin Kumar     Jasvinder Singh   Vinoo Ganesan
Outline
   Introduction
       Network coding concept
       Literature Survey
       Terminology and Notation
   Study and Implementation
       Solvability in Multicast Networks
       Algorithm and Pseudo-code
       Low Complexity Network Codes
       Network Recovery and Management
   Scope for future work
Network Coding Concept
                                                             b1 b2
   Goal: To transfer data                     V1

    at the maximum
    achievable throughput                 b1                         b2


    in a network.               V2
                                                                               V3

                                                b1             b2


                                     b1
                                               V6
   Idea: Process incoming                                  b1+b2
                                                                          b2


    data at nodes in the
                              V4                       V7                       V5
    network                                    b1+b2         b1+b2

                                                                               b1 b2
                             b1 b2




                      Introduction
Literature Survey
   Network Information Flow - Ahlswede, Cai, Li, Yeung, 2000
      Characterized the admissible coding rate region for multicast networks
      Proved that maximum throughput in a network can be achieved using ‘coding’
   Linear network Coding – Li, Yeung, Cai, 2003
      Coding at nodes treated as linear transformation of incoming data
      Showed that individual maxflow bounds of each receiver can be achieved but
        over a time period of the LCM of the maxflow bounds
   Algebraic Approach – Koetter and Medard, 2002
      Proposed algebraic framework to study networks and capacity
      Necessary and sufficient conditions for coding to be acheivable
      Necessary and sufficient conditions for robustness to link failures
   Network Management – Ho, Koetter and Medard,2002
      Quantify Network Management information required to affect link failure
        recovery
   Low complexity Network Codes – Jaggi, Kamal Jain, Philip Chou,2003
      Field size and thus arithmetic complexity is small; link usage is lower




                                 Introduction
Terminology and Notation
   Network denoted as a graph G=(V,E)
      V ----- Set of vertices (nodes)

      E ----- Set of Edges (line joining
        pairs of vertices)
   Input vector at source ’s’ x =
    [x1,x2,…,xn]
   Information on each outgoing link ‘e’
    of source

       Y (e)  [ e,1... e,n ][ x1...xn ]T


                                     Introduction
Terminology and Notation
   Information on outgoing link e* on intermediate
    node
                    Y (e*)  [  e* ,e ... e* ,e ][ ye1 ...yem ]
                                    T
                                      1         m


    where ‘m’ is the number of incoming edges on the node e*
          ‘ye’ is the incoming information on the incoming link e
   Output vector at the destination (sink) node
                   z = [z1,…,zn]
                     zi  [ e1 ,i ... ek ,i ][ ye1 ...yek ]T



                                  Introduction
Terminology and Notation
   Output vector ‘z’ is z = x * M
    where ‘M’ is the system transfer matrix
   M=A*G*B
    where A is [αi,j] is a n * k matrix where ‘k’ is total
    number of edges in the network.
          G = (I-F)-1 is the k * k adjacency matrix
          B is [εi,j] is a k * n matrix




                           Introduction
  Terminology and Notation
Cut: A partition of vertex set into 2 classes, S                s

containing source and S’ containing the sink.

Value of the cut:             C (e)
                         efro m S to S'
                                                        x           y

where ‘C(e)’ is the rate constraint of each link

Min-Cut Max-Flow Lemma:                                     z


         Let ‘G’ be a graph with source node ‘s’
and sink nodes ‘t1’ and ‘t2’, and rate                      w

constraints ‘R’ .Then for l=1,2, the maxflow       t1
                                                                        t2

from s to tl is the value of the min-cut between
s and tl and is denoted by maxflow(s,tl)



                               Introduction
Study and Implementation
   Finding a network code for a given multicast
    problem

   Solvability conditions
       Single source single sink :    det (M) ≠ 0
       Single source multiple sink : ∏ det (Mi) ≠ 0
                                       i

       Multiple source multiple sink : det (Mii) ≠ 0
                                        det (Mii) = 0


                       Study and Implementation
Algorithm for finding network codes
   Given polynomial F(x), find a such that
                      F(a) ≠ 0

    Find maximal degree ‘∂’ of F in any
    variable xi and choose smallest ‘i’ such that
                        2i > ∂



                 Study and Implementation
Algorithm for finding network codes
   Find an element ‘at’ in F2i such that
                  F(x) xt=at ≠ 0 and F   F(x) xt=at



   If t = n then halt, else t t+1, goto previous
    step
   ‘a’ is the solution to the above problem



                 Study and Implementation
Bound on Field size
   There exists a solution to the single source
    multicast network coding problem in a finite
    field 2m with

                m  log 2 ( NR  1)




                   Study and Implementation
Simulation Steps
   Generate a random network (single source
    multicast)
   Find the network capacity using maxflow
    algorithm
   Generate matrices A,G, B from the network
    topology
   Solve for the network parameters


                Study and Implementation
Coding vs Routing
   Is coding really required?
   How to check if routing achieves capacity?
   Routing is a special case of coding with
    constraints on codes
   Put constraints on codes and solve to see if
    routing is feasible



                 Study and Implementation
Simulation results
                               b1 b2
                                                                                            b1 b2


                          V1
                                                                                       V1
                b1                     b2
                                                                       b1+b2                        b2




           V2
                                                    V3            V2
                                                                                                         V3
                     b1            b2
                                                                               b1+b2            b2
                                                          b1+b2

     b1
                                                     b2
                                                                                                              b2




          V4                                        V5
                                                              V4                                         V5


                                                          b1 b2                                                b1 b2
  b1 b2                                     b1 b2




                                   Study and Implementation
Simulation results
     AT=




           Study and Implementation
Simulation results




           Study and Implementation
Low Complexity Network Codes
   Gives a solution to the single source multicast
    network coding problem in a finite field 2m with


                   m  log 2 ( N )

   Uses only union of edge-disjoint paths to each
    receiver thus avoiding ‘flooding’



                   Study and Implementation
Network Recovery and Management
   Nodes need to change their ‘behavior’ for
    recovery from link failures
   Network management involves switching
    between appropriate codes for recovery
    from link failures
   Management requirement can be quantified
    by the number of different codes needed


                Study and Implementation
Network Recovery and Management
   Two formulations of quantification
       Centralized formulation
            Network behavior described by an overall code
            Network management requirement quantified by
             logarithm of the number of codes needed
       Node based formulation
            Network behavior described by the number of
             nodes which change behavior
            Quantified by the sum of the logarithm of the
             number of different behaviors of each node

                       Study and Implementation
 Network Recovery and Management
Theorem: For a single receiver network with r processes and a
minimum capacity of C, tight bounds on the number of codes needed for
the no-failure scenario and all single link failures, assuming they are
recoverable are :
To be included in the final report
   Faster implementation of the code-
    generating algorithm

   Comparison of Routing vs Coding on large
    number of random networks
Future direction of research
   Joint source-channel-network coding

				
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posted:8/16/2011
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