On Linearity: A Taxonomy of Linear Network Codes

Sidharth Jaggi, Michelle Effros

Tracey Ho, Muriel Medard

Acknowledgement-Ralf Koetter

Network Codinghttp://tesla.csl.uiuc.edu/~koetter/NWC/Bibliography.html 72 papers… R. Ahlswede, N. Cai, S.-Y. R. Li and R. W. Yeung,

"Network information flow," IEEE Trans. on Information Theory, vol. 46, pp. 1204-1216, 2000.

S.-Y. R. Li, R. W. Yeung, and N. Cai. "Linear network coding". IEEE Transactions on Information Theory,  Feburary, 2003

*linear* Є 10 paper titles…, *algebraic* Є 5 paper titles…

# of papers in which linearity seems to play an integral part > 42

Linear Network Coding Li et al. - “Sufficiently large field” Koetter et al., … – “Algebraic”

)2(1,0)...( 110mm

m Fbbb

b0 b1 bm-1

2

k

1

kk ...2211

1

k

2

Linear Network Coding Li et al. - “Sufficiently large field” Koetter et al., … – “Algebraic” Jaggi et al., Ho et al., … - “Block” m1,0

b0 b1 bm-1

][ 11

2

k

][ 2

][ k

kk

][...][][ 2211

Linear Network Coding Li et al. - “Sufficiently large field” Koetter et al., … – “Algebraic” Jaggi et al., Ho et al., … - “Block” Erez et al., Fragouli et al, … - “Convolutional”

)(2 z

)(zk

)(1 z

)()(...)()( 11 zzzz kk

)()(1,0)...( 2110 zFzbbb m

b0 b1 )(z

)(1 z

)(2 z

)(zk

Linear Network Coding Li et al. - “Sufficiently large field” Koetter et al., … – “Algebraic” Jaggi et al., Ho et al., … - “Block” Erez et al., Fragouli et al, … - “Convolutional”

Dougherty et al., some networks need non-linear codes

A B C

Outline Inter-relationships

Global reduction FeasibilityA C

B?

?

?

A B?

Outline Inter-relationships

Global reduction Feasibility Local reduction Distributed single design

A C

B?

?

?

A B

A B

?

Outline Inter-relationships

Global reduction Feasibility Local reduction Distributed single design

I/O ≠

A C

B?

?

?

A B

A B

x y x y’

Outline Inter-relationships

Global reduction Feasibility Local reduction Distributed single design

I/O ≠ I/O = Different notions of linearity

co-exist in network

A C

B?

?

?

A B

A B

x y x y

Outline Inter-relationships

Global reduction Feasibility Local reduction Distributed single design

I/O ≠ I/O = Different notions of linearity

co-exist in network

A C

B?

?

?

A B

Multicast Vs. General

Outline Inter-relationships

Global reduction Feasibility Local reduction Distributed single design

I/O ≠ I/O = Different notions of linearity

co-exist in network

Complexity

A C

B?

?

?

A B

Multicast Vs. General

Outline Inter-relationships

Global reduction Feasibility Local reduction Distributed single design

I/O ≠ I/O = Different notions of linearity

co-exist in network

Complexity Unified Framework

A C

B?

?

?

Multicast Vs. General

B

CA

Inter-relationships

A C

B

?

?

?

Inter-relationships

A C

B

G General

Global

A AlgebraicB BlockC ConvolutionalG G? ?

BlockAlgebraic?

Convolutional?

Inter-relationships

A C

B

G General

Global

A AlgebraicB BlockC Convolutional

Does not exist

G G

Lehman et al/Ho et al example networks

“Switching” possibleonly for block codes

Inter-relationships

A C

B

M

M M

M Multicast G General

Global

A AlgebraicB BlockC Convolutional

Does not exist

G G

Block

Algebraic

Convolutional

Algebraic, Block and Convolutionalmulticast codes exist

Inter-relationships

A C

B

M

M M

M Multicast G General

Global

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

G Ga

a

a

Only true for acyclic networks

Block

Algebraic

Convolutional

Algebraic versus Block

A

B

MGa

M Multicast G General

Global

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

Algebraic versus Block

A

B

MGa

M Multicast G General

Global Local I/O ≠

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

?

B A

[β] β

?

Algebraic versus Block

.

.

.

.

.

.

S

R

n1 links n2 links

Algebraic versus Block

.

.

.

.

.

.

1 00 0[β1]=[ ]0 00 1[β2]=[ ]

S

R

[β1][β1]

[β1]

[β1]

[β1]

[β1]

[β2][β2]

[β2]

[β2]

[β2]

[β2]

n1 links n2 links ...

.

.

.

S

R

β1

β1

β1

β1

β1

β1

β2

β2

β2

β2

β2

β2

n1 links n2 links

(β1)n1= (β2)n2 =1

0

“Switching” possibleonly for block codes

“Destructive interference”

Algebraic versus Block

A

B

MGa

M Multicast G General

Global Local I/O ≠

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

Ma

?

Algebraic versus Block

A

B

MGa

M Multicast G General

Global Local I/O ≠ Local I/O =

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

G

Ma

Addition is identical in both.One can choose [β]s which mimic β multiplication

A B

β [β]

x y x y

Algebraic versus Block

A

B

MGa

M Multicast G General

Global Local I/O ≠ Local I/O =

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

G

Ma

Gives algorithm for block codes, if onehas already designed algebraic codes.

Algebraic versus Convolutional

A CMa

M Multicast G General

Global Local I/O ≠ Local I/O =

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

Algebraic versus Convolutional

A CMa

M Multicast G General

Global Local I/O ≠ Local I/O =

a Acyclic

A AlgebraicB BlockC Convolutional

Does not existMa

“Destructive interference”

Algebraic versus Block

A

M Multicast G General

Global Local I/O ≠ Local I/O =

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

CMa

Ma

Ma

Alternative to Erez et al.

Algebraic versus Convolutional

)(')( zz

)().( zz

Addition

Multiplication

Algebraic Convolutional

))(mod())().((. zpzz

mzp degree of polynomial eirreduciblan is )(

Addition

Multiplication

))((mod))(')((' zpzz

1

2

0

)(1 z

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0)(

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0det2221

1211

hh

0

)(

)()(

)()(

det2221

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z

zz

zz

hh

ij )(zij

Algebraic versus Convolutional

A

M Multicast G General

Global Local I/O ≠ Local I/O =

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

CMa

Ma

Ma

Algebraic versus Convolutional

A

M Multicast G General

Global Local I/O ≠ Local I/O =

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

CMa

Ma

Ma G

Algebraic/blocktransfer function

Convolutionaltransfer function

(FIR)

. . .

0

0

. . .

0

0

≠

Algebraic versus Convolutional

A

M Multicast G General

Global Local I/O ≠ Local I/O =

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

CMa

Ma

Ma

Algebraic/Blocktransfer function

. . .

0

0

Convolutionaltransfer function

(IIR)

. . .

0

G

≠

Algebraic versus Convolutional

A

M Multicast G General

Global Local I/O ≠ Local I/O =

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

CMa

Ma

Ma

G?

Ga

Block versus Convolutional

C

B

M

M Multicast G General

Global Local I/O ≠ Local I/O =

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

a

G

Block versus Convolutional

C

B

M

M Multicast G General

Global Local I/O ≠ Local I/O =

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

a

G

Blocktransfer function

Convolutionaltransfer function

G

≠

Block versus Convolutional

C

B

M

M Multicast G General

Global Local I/O ≠ Local I/O =

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

a

G

G

Fudge factor

Block versus Convolutional

C

B

M

M Multicast G General

Global Local I/O ≠ Local I/O =

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

Є epsilon rate loss

a

G

GЄ

G

Fudge factor

Block versus Convolutional

For all n>d (decoding delay),total throughput = C(n-d)

Convolutional

Sources

Sinks

n

n-d

n

n-d

n

n-d

Block versus Convolutional

For all n>d (decoding delay),total throughput = C(n-d)

0

0

n

n

d

Rate C(1-d/n)

Convolutional

Time-domain transfer matrix

Block versus Convolutional

C

B

M

M Multicast G General

Global Local I/O ≠ Local I/O =

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

Є epsilon rate loss

a

G

GЄ

G

Big picture

C

B

M

M Multicast G General

Global Local I/O ≠ Local I/O =

a Acyclic

A AlgebraicB BlockC Convolutional

Does not exist

Є epsilon rate loss

G

a

GЄ

A Ma

Ma

Ma

G?

M

G

a

G

Ma G

G

Complexity – Acyclic networks

. . .

. . .

1n

12 || nT

Minimum block length m ≈0.5(log(|T|))

intermediate nodes

receiversDitto block, evennon-linear codes(JCJ,TE,LL)

m>0.25(log(|T|))

m>0.125(log(|T|))(FIR)(IIR)

Unified Framework

AlgebraicBlockConvolutionalD???

What other features besides linearity?

101001101010…010101110101…001010110010…...

1. Causality

3. Finite number of memory elements

2. L-stationarity

Unified Framework

m

jiLLkLj

m

iiLLkLiLLk

m

jiLkj

m

iiLkiLk

m

jiLkj

m

iiLkiLk

xbyay

xbyay

xbyay

011,

111,1

011,

111,1

00,

10,

L-stationarity

Finite number of memory elements

Causality

Filter banks F(z) l l

Conclusions Reductions between different notions of linearity

Which notion is strongest? Single algorithm design. Co-existence of different types of linearity in a network.

Complexity Delay elements are necessary for some cyclic networks. Even for acyclic networks, convolutional codes can have

shorter block-lengths. Unified framework

Filter banks most general “reasonable” form of linearity.

Clip art acknowledgements

http://clear.msu.edu:16080/dennie/clipart/

http://particleadventure.org/particleadventure/frameless/weak.html

http://www.finalfantasy.8m.net/pics/ff3/the%20end.gif

All the following slides contain material from a previous draft, which might be useful, but will not show up in the Allerton presentation.

Unified Framework

Algebraic interpretation Implementational interpretation

Filter banks

F(z)

l l

Transfer function at any nodematrix of rational functions

Complexity

. . .

. . .

1n

12 || nT

Therefore q≥nTherefore minimum block length m = log(q) ≥log(n)≈0.5(log(|T|))

s

intermediate nodes

receivers

Distinct linear combinations

(1, 0)(1, 1)(1, 2)

… (1, q-1)

(0, 1)

q+1

Complexity

m>0.5(log(|T|))

Distinctlinear combinations

(1, 0)(1, 1)(1, 2)

… (1, q-1)

(0, 1)

2m+1

Ditto block, evennon-linear codes(JCJ,TE,LL)

(p(z),q(z))(p’(z),q’(z))

p(z)/q(z)≠p’(z)/q’(z)

# pairs of coprime polynomials (p(z),q(z))

≈22m [Morrison]

m>0.25(log(|T|))

In fact if IIR filters used, suspect m>0.125(log(|T|))suffices