Post on 06-Jan-2016
description
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
)(2 z
0)(
z
0det2221
1211
hh
0
)(
)()(
)()(
det2221
1211
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