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Network Coding: Theory andPractice

Apirath LimmaneeJacobs University

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Overview Theory Max-Flow Min-Cut Theorem Multicast Problem Network Coding

Practice

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Max-Flow Min-Cut Theorem Definition Graph Min-Cut and Max-Flow

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Definition (From Wiki) The max-flow min-cuttheorem is a statement in optimizationtheory about maximal flows in flow networks The maximal amount of flow is equal to thecapacity of a minimal cut.

In layman terms, the maximum flow in anetwork is dictated by its bottleneck.4

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Graph Graph G(V,E): consists ofa setV of verticesand a set E of edges. V consists of sources, sinks, and other nodes A member e(u,v) ofE has a capacity c(u,v) tosend information from u to v

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Min-Cuts and Max-Flows Cuts: Partition of vertices into two sets Size of a Cut = = Total Capacity Crossing the Cut Min-Cut: Minimum size of Cuts = = 5 Max-Flows from S to T Min-Cut = = Max-Flow

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Multicast Problem Butterfly Networks: Eachedge's capacity is 1. Max-Flow from A to D = = 2 Max-Flow from A to E = = 2 Multicast Max-Flow fromA to D and E = = 1.5 Max-Flow for eachindividual connection isnot achieved.

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Network Coding Introduction Linear Network Coding Transfer Matrix Network Coding Solution Connection between an Algebraic Quantityand A Graph Theoretic Tool Finding Network Coding Solution

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Introduction

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Ahlswede et ale (2000) With network coding,every sink obtains themaximum flow.

Li et ale (2003) Linear network codingis enough to achieve themaximum flow

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Linear Network Coding+ Random Processes in a Linear Network Source In ut: X v, I) ={ X o (v,I ~~___"_____-----I.--

Weighted Y (e )v WeightedCombination of { l i t Combination of I) ~ z 'ltiprocessesgenerated at v--

them WeightedCombination

from all+R

J l (V)Y e) =Lat,eX(v,l) +e' :head (e')=tai incoming edges= 1e comesout of v Z (v , i) = ~ e r .f(e)L . J e,j

e r : head (e') = v10

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Transfer Matrix

~,1 ~,2~,

t%,1 ~,2 ~,1 ~ ~7,1 7,2 7,

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z xM

Network Coding Solution We want z x

~A iL , t ~A f3~ Choose A to be ant iA f3eif6 tiA i LPJ B identity matrix.

~,t6 ~'BJ Choose B to be theinverse ofn. ~'4~40% ~ ' 4 4 , e ,

1 { , 3 1 { , 4 4 ' % 1 { , 4 4 ,e ,o 4,% n:12

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Connection between an AlgebraicQuantity and A Graph Theoretic Tool......____.I

Koetter and Medard (2003): Let a linearnetwork be given with source node v, sinknode v ; and a desired connection c=( v , v ' , x ( v , v ' ) )of rate R (c). The following three statementsare equivalent. 1. The connection c = = (v , v ',x (v , v ')) is possible. 2. The Min-Cut Max-Flow bound is satisfied 3. The determinant of the R(c)xR(c) transfermatrix M is non-zero over the Ring

F 2 ~ . . ,a l,e"" ,/3 e'{!,'&e',j'" J13

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Finding Network Coding Solution Koetter and Medard (2003): Greedy Algorithm Let a delay-free Communication Network G and aSolvable multicast problem be given with onesource and N receivers. Let R be the rate at whichthe source generates information. There exists asolution to the network coding problem in a finitefield F ;n with m

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Random Network Coding Jaggi, Sanders, et ale (2003): If the field sizeis at least E j /8, the encoding will beinvertible at any given receiver with probe atleast 1-8,while if the field size is at least E T /5then the encoding will be invertiblesimultaneously at all receivers with probe atleast 1-6.

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Practical Issues Network Delay Centralized Knowledge of Graph Topology Packet Loss Link Failures Change in Topology or Capacity

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Thank You

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You Are Welcome.

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