Network Coding Project presentation

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Network Coding Project presentation

• Communication Theory
• 16332545

Amith Vikram Atin Kumar Jasvinder
Singh Vinoo Ganesan
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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

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Network Coding Concept

• Goal To transfer data at the maximum achievable
  throughput in a network.
• Idea Process incoming data at nodes in the
  network

Introduction
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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
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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

Introduction
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Terminology and Notation

• Information on outgoing link e on intermediate
  node
• 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

Introduction
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Terminology and Notation

• Output vector z is z x M
• where M is the system transfer matrix
• M A G B
• where A is ai,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 ei,j is a k n matrix

Introduction
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Terminology and Notation
Cut A partition of vertex set into 2 classes, S
containing source and S containing the
sink. Value of the cut where C(e) is the
rate constraint of each link
Min-Cut Max-Flow
Lemma Let G be a graph with source node s
and sink nodes t1 and t2, and rate
constraints R .Then for l1,2, the maxflow from
s to tl is the value of the min-cut between s and
tl and is denoted by maxflow(s,tl)

y
t2
Introduction
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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
• Multiple source multiple sink det (Mii) ? 0
• 
  det (Mii) 0

i
Study and Implementation
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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 gt ?

Study and Implementation
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Algorithm for finding network codes

• Find an element at in F2i such that
• F(x) ? 0 and F
  F(x)
• If t n then halt, else t t1, goto previous
  step
• a is the solution to the above problem

xtat
xtat
Study and Implementation
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Bound on Field size

• There exists a solution to the single source
  multicast network coding problem in a finite
  field 2m with

Study and Implementation
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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
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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
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Simulation results
b2
b2
Study and Implementation
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Simulation results
AT
Study and Implementation
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Simulation results
Study and Implementation
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Low Complexity Network Codes

• Gives a solution to the single source multicast
  network coding problem in a finite field 2m with
• Uses only union of edge-disjoint paths to each
  receiver thus avoiding flooding

Study and Implementation
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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
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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
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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
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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

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Future direction of research

• Joint source-channel-network coding