Network Coding

Project presentation

Communication TheoryCommunication Theory16:332:54516:332:545

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 GoalGoal: To transfer data

at the maximum achievable throughput in a network.

IdeaIdea: Process incoming data at nodes in the network

IntroductionIntroduction

V1

V2V3

V4 V5

b1 b2

b1

V6

V7

b1 b2

b2

b2b1

b1 b2b1 b2

b1+b2

b1+b2b1+b2

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’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

IntroductionIntroduction

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

IntroductionIntroduction

Tnnee xxeY ]...][...[)( 1,1,

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]

Teeeeee mmyyeY ]...][...[*)(

1*

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eeieiei kkyyz ]...][...[

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IntroductionIntroduction

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

IntroductionIntroduction

Terminology and Notation

y

t2

s

x

t1

z

w

CutCut:: A partition of vertex set into 2 classes, S containing source and S’ containing the sink.

Value of the cutValue of the cut::

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

Min-Cut Max-Flow Lemma: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 l=1,2, the maxflowmaxflow from s to tl is the value of the min-cut between s and tl and is denoted by maxflow(s,tmaxflow(s,tll))

'

)(efromStoS

eC

IntroductionIntroduction

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

Algorithm for finding network codes

Given polynomial F(F(xx)), find aa such that

F(F(aa) ) ≠ 0≠ 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) ≠≠ 0 and F F(x)

If t = n then halt, else t t+1, goto previous step

‘a’ is the solution to the above problem

xt=at xt=at

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

Study and Implementation

)1(log2 NRm

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

b2

V1

V2V3

V4 V5

b1+b2 b2

b1+b2

b1 b2

b2b1+b2

b1 b2 b1 b2

b2

V1

V2V3

V4 V5

b1 b2

b1

b1 b2

b2b1

b1 b2 b1 b2

Study and Implementation

Simulation resultsAT=

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

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

)(log2 Nm

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

TheoremTheorem: 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