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Network Coding and its Applications in Communication

Networks

Alex Sprintson

Computer Engineering GroupDepartment of Electrical and

Computer EngineeringTexas A&M University

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Information

Course webpage http://cegroup.ece.tamu.edu/spalex/netco

d Office Hours

TBA, WERC 333D Or by appointment spalex@ece.tamu.edu

Will use neo email

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Information vs. Commodity Flow

b1

b1 b1

Replication

b1b2

b1+ b2

Encoding

Replication

Encoding

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Network coding New research area (Ahlswede et. al. 2000) Benefits many areas

Networking, Communication, Distributed computing

Uses tools from Information Theory, Algebra, Combinatorics

Has a potential for improving: throughput, robustness, reliability and security

of networking and distributed systems.

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Communication Networks

Directed graph G=(V,E).

Source nodes S. Terminal nodes T. Requirement:

deliver data from S to T.

t1

t2t3

t4

s1

s2s3

s4

G=(V,E)

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The multicast setting

Multicast: One source transmits data to all terminals.

t1

t2t3

t4

s

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Standard approach

Each node forwards and possibly duplicates messages Paths for unicast Trees for

multicast

t1

t2t3

t4

s1

s2

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Standard approach

Steiner Tree Given a set T of

points (terminals), interconnect them by a tree of shortest length.

s

t1

t2

t3

t4

Steiner nodes

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Standard tree packing

Steiner Tree Packing Given a source node

s and a set T of terminals, interconnect them by a tree of shortest length.

s

t1

t2

t3

t4

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Network coding

Introduce nodes that do more than forwarding + duplication.

Each outgoing packets is a function of incoming packets.

m1

m2

m3

F1(m1,m2,m3)

F2(m1,m2,m3)

encoding

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Network coding

Linear Coding over finite fields Examples

m2

m1

m3

m1+m2+m3

m1

m2

m3

m1+2m2+3m3

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Traditional Approach

Steiner Tree Packing

Integer reservation -1 message per time unit

Fractional/time sharing solution1.5 messages per time unit

t1 t2

s

50% utilization

t1 t2

s

0% utilization

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Network coding helps!

One message per time unit

1.5 messages per time unit

t1 t2

s

50% utilization

t1 t2

s

0% utilization

s

t1 t2

100% utilization

m1 m2

m1m2m1Åm2

2 messages per time unit

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Delay Minimization

Jain and Chou (2004)

t1 t3

t2

a

a

a

b

b

b

s

Delay =3 Delay =2

t1 t3

t2

a

aÅb

b

b

a

s

aÅb

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Energy Minimization

Wu et al. (2003); Wu, Chou, Kung (2004)Lun, Médard, Ho, Koetter (2004)

s1 s2

a bs1 s2

b a

Without network coding - 4 messages

s1 s2

a bs1 s2

aÅbaÅb

With network coding - 3 messages

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Cost minimization

Cost without network coding 4with network coding 3 - 33%

reduction

t1 t2

s

0 0

00

1 1

0 0

1

Edge Costs

t1 t2

s

0 0

00

1 1

0 0

1

a

aa

b

bb

a,b

a,ba,b

a,b

aÅb

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Research Issues

How useful is network coding? Multicast Practical Implementation Beyond Multicast Networks with cycles

What operations should be performed at each node?

Linear vs. non-linear operations The minimum size of a packet

Under what conditions is a given network coding problem solvable

How to find suitable edge functions? How many encoding nodes are needed?

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Syllabus

In this class: Overview of the main results Overview of the current research Applications in communication networks Directions for future research

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Syllabus

Introduction Introduction to network coding Introduction to network algorithms Benefits and coding advantage Diversity coding Linear Network coding

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Course Outline

Network optimization and Linear Programming (LP) Basics of LP LP duality Primal-dual algorithms Approximation of NP-hard problem The probabilistic method

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Course Outline

Network flows and disjoint path algorithms (3)

Theory of network flows Disjoint paths algorithms Increasing network connectivity Network reliability

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Course Outline

Network coding (12) Algebraic framework Polynomial – time algorithms for network code desing Randomized algorithms Distributed network coding Information-flow decomposition Network coding for cyclic networks Encoding complexity Practical network coding

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Course Outline

Design of robust communication networks

Bandwidth allocation Network coding-based methods Connection to convolution codes Knots and special cases Information-Theoretic approach for

network management

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Course Outline

Network Error Correction (4) Robust networks Correcting adversarial errors Secure network coding

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Course Outline

Encoding Complexity Bounds on the number of encoding nodes Hardness results

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Course Outline

Coding for non-multicast networks   Insufficiency of linear network codes Non-linear network codes

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Course Outline

Network coding applications Applications in distributed computing Applications in sensor networks Network planning in wireless and ad-hoc

networks Applications for network storage 

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Course Outline

Web models and information retrieval algorithms Modeling the web Taxonomy of information retrieval models Retrieval evaluation

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Course Outline

Textbook: No textbook will be used. Notes and research papers will be

available on the course website. A good reference site:

www.networkcoding.info

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Course Outline

Tentative Grading Policy: Assignments 40%  Project 40% Student Presentations 20%

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Network Capacity

In general: the maximum number of packets that can be sent throughout the network. For a given list of

source-destination pairs

Each link has a limited capacity

s1

t1

s2

t2

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Network Capacity

Multicast connections: The maximum number

of packets that can be sent from s to T.

Each link has a limited capacity

s1

t1

t3

t2

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Cut

Definition: A cut (V1,V2) is a partition of the nodes of the graph into two subsets V1 and V2=V\V1

Size of the cut: The total capacity of links between nodes in V1 and V2.

G V1

V2

s

c

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Cut (cont.)

We say that a cut (V1,V2) separates s and t if sV1 and tV2.

t3

s

t1

t2t4

t5

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Flow interpretation

Menger’s theorem: The minimum size of a cut between

s and t is h There are h disjoint paths between s and each t

A special case of min-cut – max-flow theorem

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Flow interpretation

t3

s

t1

t2t4

t5

t3

s

t1

t2t4

t5

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Upper bound

Let h be the size of the min-cut between s and a terminal tT

We cannot send more than h packets to this terminal

s1

t1

t3

t2

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Challenge

Let h be the size of the min-cut between s and a terminal tT

Can send h packets to each terminal tT separatly e.g., by using h disjoint paths

Problem: How to send them to all terminals

simultaneously ? Solution:

Network coding

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Challenge

Path clashing is resolved by using NC

s

t1 t2

s

t1 t2

s

t1 t2