Presentation at IEEE AWSITC, June 4, 20101 Energy-Efficient Communications via Network Coding Jos Weber Delft University of Technology The Netherlands

Presentation at IEEE AWSITC, June 4, 2010 1

Energy-Efficient Communications via Network Coding

Jos WeberDelft University of Technology

The Netherlands

Visiting Professor at

Presentation at IEEE AWSITC, June 4, 2010

Based on joint work with Jasper Goseling


Outline

• Introduction on Network Coding

• Energy Benefit for Multiple Unicast in Wireless Networks

• Multi-Rate Network Coding for Minimum-Cost Multicasting

• Conclusions


Part 1




• Conclusions and Future Work


Network Coding Paradigm

Traditional routing solutions for communication networks keep independent data streams separate.

Network coding solutions allow nodes in the network to combine independent data streams.


Illustration: Traditional

Jos Jasper


Illustration: Network Coding

Jos Jasper


Illustration: Combining Messages

Since

Jos can decode

+ =

and Jasper can decode

-

-

=

=


Illustration: Result after Decoding

Jos Jasper


Network Coding Example

S1

R2

S2

R1

without network coding

m1

m2

m1

m1

m1

m1

S1S2

R1 R2

with network coding

m1

m1 m2

m2

m3m3

m3

m2=m3-m1

m3=m1+m2

m1=m3-m2

Possible benefits:

• throughput gain

• energy efficiency

• robustness

• adaptability

• security

• …

m2?

m2


“Bits are not cars!”(Ralf Koetter)

00001100

+

10101010 =

10100110

+

=


Wireless Example

1 2 3 21 3

Traditional Method

Network Coding

m1

m3

m1 m1

m3

m3

m1

m1+m3

m3

m1+m3

4 transmissions

3 transmissions

Information exchange between nodes 1 and 3 using node 2


Wireless Circular Network

Traditional Method

Network Coding1

46

8 2

3

5

7

46

8 2

5

7 3

1m1,m8,m2

m8,m7,m1 m2,m1,m3

m7,m6,m8 m3,m2,m4

m6,m5,m7

N(N-2)=8×6=48 transmissions

m4,m3,m

5

m5,m4,m

6

m2

m2

m2+m4

m2+m4

N(N-1)/2=8×7/2=28 transmissions

m6,m5,m7

m7,m6,m8

m8,m7,m1

m1,m8,m2

m2,m1,m3

m3,m2,m4

m4,m3,m

5

m5,m4,m

6


Random Network Coding

2

3

45

…

…

…

…

…

R

1

…

…

…

…

…

…

m1+m3 m1+m2+m3 m2+m3

m4

m3+m4+m5

m4

m3+m4+m5

y1= m3+m4+m5

m2

m3+m4

y2= m3+m4

y3=m1

y4=m1 +m3+m4+m5 y5=m1+m2

y6=m1 +m3 +m5


Encoding

Assume n original packets m1, m2, …, mn generated by one or several sources;

Each packet consists of K symbols from GF(2s): mi=(mi,1,mi,2,…,mi,K);

At a certain node, encoding vector g=(g1,g2,…,gn), with each giєGF(2s);

Information vector x=g1m1+g2m2+…+gnmn=(x1,x2,…,xK), where xk=g1m1,k+g2m2,k+…+gnmn,k;

Encoding can be performed recursively (to already encoded packets);

Encoding vector can be deterministic or random (in which case it is transmitted together with the information vector).


Decoding

Solving a linear system of equations with n unknowns (the original messages m1, m2, …, mn);

With random network coding, the probability of linearly dependent combinations becomes small if the field size 2s is sufficiently large;

Therefore, only (few more than) n information vectors need to be received in order to retrieve the original packets.


Max-Flow Min-Cut

Assume each link has unit capacity.Min-cut is two for both receiver nodes.

Max-flow is two for each receiver node. Not achievable simultaneously by traditional routing!Achievable simultaneously by network coding!

This works for all multicast networks:The upper bound on the obtainable data rate imposed

by the smallest maximum flow from the source to some receiver can be achieved simultaneously for all receivers using coding.

Source

R2R1


Network Coding in 2010

• Also other (theoretical) results on network coding have been derived since the start in 2000.

• Possible benefits with respect to throughput, energy efficiency, robustness, adaptability, security, …

• Potential for practical applications is under investigation, first results are available.

N.B. Work of North-West University, Potchefstroom


Part 2






Energy Benefit

Energy benefit of network coding for a wireless multiple unicast configuration:

minimum energy consumption of any routing solutionminimum energy consumption of any network coding solution


Energy Benefit: Wireless Example Revisited

1 2 3 21 3

Traditional Routing

Network Coding

m1

m3

m1 m1

m3

m3

m1

m1+m3

m3

m1+m3

4 transmissions

3 transmissions

Energy benefit of network coding in comparison to traditional routing is 4/3


Generalization of the Example

1 2 3 N-1… N

Energy Benefit: 2(N-1)/N → 2

Multiple Unicast: 1→N & N→1


Research Challenge

Line network example: ≥ 2

Effros et al.: ≥ 2.4

Our contribution: ≥ 3

Find the maximum energy benefit that network coding can offer


Network Used in Proof


Three Sets of Unicast Connections

Senders

Receivers

Senders

Receivers

Senders

Receivers


Number of Transmissions

Routing:

3K(K-1)/2 ≈

1.5K2

Network Coding:

3(K+1)K/2-

(K-2)(K-3) ≈

0.5K2

Hence, energy benefit of 1.5/0.5=3 for large K


Rx Energy

Energy benefit when taking also Rx energy into account:

Line network: 2E(Tx) + 2E(Rx) E(Tx) + 2E(Rx)

Triangle network:3E(Tx) + 3E(Rx)

E(Tx) + 6E(Rx)


Result for “Triangle Network”


Part 3






ExampleS

R2R1

Butterfly Network:• One source• Four relay nodes• Two receivers• Nine unit capacity edges of

cost 1


Throughput versus Cost

x y

x

x y

y

x+y

x+y

x+y

• Throughput 2• Cost/symbol 4.5

• Throughput 1• Cost/symbol 4

xx

xx


Goal

To construct a network code that enables the source to

control the throughput, achieving the minimum

possible cost at all throughputs.


Model and Definitions

Acyclic directed graph Capacity and cost on edges Multicast traffic Single network use

Throughput: number of symbols transmittedCost (per symbol) = (Σ costs of all edges

used)/throughputOperating point: throughput-cost pair


Network Coding at Minimum Cost

For a given throughput, find minimum-cost subgraph satisfying min-cut conditions: [Lun et al., IEEE IT, 2006]

Construct a code on the subgraph: [Jaggi et al., IEEE IT, 2005] [Ho et al., IEEE IT, 2006]

Multi-rate network coding: one subgraph for each operating point!

Challenge: Find a code that works on all subgraphs


Related Work

“Variable-Rate Linear Network Coding”, [Fong & Yeung, IEEE ITW, 2006]:Variable throughputSingle subgraph Changing set of receivers, i.e., those nodes in the

network that have the min-cut satisfied “Network Coding for Link Failures”,

[Koetter & Medard, IEEE/ACM TN, 2003], [Jaggi et al., IEEE IT, 2005]: Single throughputDifferent subgraphs


Outline of Code Construction

The source selects the throughput and encodes the data using one set of coding vectors. Take size of global coding vectors equal to maximum supported throughput.

At lower throughputs, fix unused symbols at zero. The chosen throughput is communicated to other nodes in the network, e.g., by including it in the header of a packet.

Intermediate nodes know the subgraphs used at each operating point and perform the same linear coding operation at all throughputs, i.e., there is only one set of local coding vectors.

Receivers know which symbols are used at each operating point and can decode accordingly.


Example Revisited

x x+y

x

x x+y

x+y

y

y y

Operating Point 1 • Throughput 2• Cost/symbol 4.5

Operating Point 2 • Throughput 1 (y=0)• Cost/symbol 4


Main Result

Theorem: For any network, a multi-rate code can be constructed achieving the minimum possible cost at all throughputs.

Proof (sketch): Consider transfer matrices for each receiver for

each operating point; Require all transfer matrices to have full rank; Consider product of all determinants; Follow [Koetter & Medard, IEEE/ACM TN, 2003]

algebraic framework.


Part 4






Conclusions

Network coding is a promising technique with possible benefits with respect to throughput, energy efficiency, robustness, adaptability, security, …

A better lower bound on the maximum possible energy benefit for multiple unicast on wireless networks has been derived

A multi-rate network code for minimum-cost multicasting has been proposed


Other/Future Research

Studying combined channel and network coding

Further exploring the possible energy benefit of network coding

Taking into consideration stochastic packet arrivals

Physical-layer network coding


Wireless Example Revisited Once More

1 2 3 21 3

Traditional Routing

Network Coding

m1

m3

m1m1

m3

m3

m1

m1+m3

m1+m3

4 transmissions

3 transmissions

PL Network Coding

31 2

2 transmissions

m3 m1+m3m1+

m3

Exploiting

Broadcast

Exploiting Broadcast &

MA

Documents

Presentation at IEEE AWSITC, June 4, 20101 Energy-Efficient Communications via Network Coding Jos Weber Delft University of Technology The Netherlands