Transcript Network Coding - Texas A&M University

Network Coding and its
Applications in
Communication
Networks
Alex Sprintson
Computer Engineering Group
Department of Electrical and
Computer Engineering
Texas A&M University
1
Information
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Course webpage
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Office Hours
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http://cegroup.ece.tamu.edu/spalex/netcod
TBA, WERC 333D
Or by appointment [email protected]
Will use neo email
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Information vs. Commodity Flow
b1
b1
b1
Replication
b1
Replication
b2
b1+ b2
Encoding
Encoding
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Network coding
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New research area (Ahlswede et. al. 2000)
Benefits many areas
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Networking, Communication, Distributed
computing
Uses tools from Information Theory,
Algebra, Combinatorics
Has a potential for improving:
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throughput, robustness, reliability and
security of networking and distributed
systems.
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Communication Networks
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Directed graph
G=(V,E).
Source nodes S.
Terminal nodes T.
Requirement:
deliver data from
S to T.
s1
s3
G=(V,E)
s2
s4
t1
t4
t2
t3
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The multicast setting
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Multicast: One source transmits data to all
terminals.
s
t1
t4
t2
t3
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Standard approach
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Each node
forwards and
possibly duplicates
messages
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Paths for unicast
Trees for multicast
s2
s1
t1
t2
t4
t3
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Standard approach
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Steiner Tree
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Given a set T of
points (terminals),
interconnect them
by a tree of
shortest length.
s
Steiner nodes
t2
t3
t1
t4
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Standard tree packing
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Steiner Tree
Packing
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Given a source
node s and a set T
of terminals,
interconnect them
by a tree of
shortest length.
s
t2
t3
t1
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
F1(m1,m2,m3)
m2
m3
F2(m1,m2,m3)
encoding
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Network coding
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Linear Coding over finite fields
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m1
m2
m3
Examples
m1
m1+m2+m3
m2
m1+2m2+3m3
m3
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Traditional Approach
Steiner Tree Packing
s
s
0%
utilization
50%
utilization
t1
t2
Integer reservation 1 message per time unit
t1
t2
Fractional/time sharing solution
1.5 messages per time unit
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Network coding helps!
s
s
s
m1
m2
100%
utilization
0%
utilization
t1
t2
One message per
time unit
t1
m1Åm2
m1
50%
utilization
t2
1.5 messages per
time unit
t1
m2
t2
2 messages per
time unit
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Delay Minimization
s
s
b
a
b
a
a
aÅb
b
b
t1
t3
b
a
t3
t1
aÅb
a
t2
t2
Delay =3
Jain and Chou (2004)
Delay =2
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Energy Minimization
s1
s2
a
s2
a
b
s1
s2
b
s1
a
Without network coding - 4
messages
b
s1
s2
aÅb
aÅb
With network coding - 3
messages
Wu et al. (2003); Wu, Chou, Kung (2004)
Lun, Médard, Ho, Koetter (2004)
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Cost minimization
s
1
1
1
a
b
0
0
b
1
0
0
1
a,b
0
a
a
s
Edge Costs
aÅb
a,b
b
0
0
1
0
0
0
a,b
0
0
a,b
t1
t2
t1
t2
Cost without network coding 4
with network coding 3 - 33% reduction
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Research Issues
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How useful is network coding?
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What operations should be performed at each
node?
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Linear vs. non-linear operations
The minimum size of a packet
Under what conditions is a given network coding
problem solvable
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Multicast
Practical Implementation
Beyond Multicast
Networks with cycles
How to find suitable edge functions?
How many encoding nodes are needed?
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Syllabus
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In this class:
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Overview of the main results
Overview of the current research
Applications in communication networks
Directions for future research
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Syllabus
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Introduction
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Introduction to network coding
Introduction to network algorithms
Benefits and coding advantage
Diversity coding
Linear Network coding
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Course Outline
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Network optimization and Linear Programming
(LP)
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Basics of LP
LP duality
Primal-dual algorithms
Approximation of NP-hard problem
The probabilistic method
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Course Outline
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Network flows and disjoint path algorithms (3)
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Theory of network flows
Disjoint paths algorithms
Increasing network connectivity
Network reliability
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Course Outline
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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
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Practical network coding
Course Outline
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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
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Network Error Correction (4)
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Robust networks
Correcting adversarial errors
Secure network coding
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Course Outline
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Encoding Complexity
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Bounds on the number of encoding nodes
Hardness results
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Course Outline
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Coding for non-multicast networks
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Insufficiency of linear network codes
Non-linear network codes
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Course Outline
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Network coding applications
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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
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Web models and information retrieval
algorithms
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Modeling the web
Taxonomy of information retrieval models
Retrieval evaluation
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Course Outline
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Textbook:
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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
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Tentative Grading Policy:
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Assignments 40%
Project 40%
Student Presentations 20%
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Network Capacity
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In general: the
maximum number of
packets that can be
sent throughout the
network.
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For a given list of
source-destination
pairs
Each link has a
limited capacity
s2
s1
t1
t2
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Network Capacity
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Multicast connections:
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The maximum number of
packets that can be
sent from s to T.
Each link has a limited
capacity
t3
s1
t1
t2
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Cut
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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.
s
G
V1
c
V2
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Cut (cont.)
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We say that a cut (V1,V2) separates s
and t if sV1 and tV2.
s
t5
t1
t4
t2
t3
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Flow interpretation
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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
s
s
t5
t1
t5
t1
t4
t2
t3
t4
t2
t3
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Upper bound
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Let h be the
size of the mincut between s
and a terminal
tT
We cannot send
more than h
packets to this
terminal
t3
s1
t1
t2
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Challenge
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Let h be the size of the min-cut
between s and a terminal tT
Can send h packets to each terminal
tT separatly
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Problem:
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e.g., by using h disjoint paths
How to send them to all terminals
simultaneously ?
Solution:
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Network coding
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Challenge
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Path clashing is resolved by using NC
s
t1
s
s
t2
t1
t2
t1
t2
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