Low Complexity Encoding for Network Codes

Yuval Cassuto

Michelle EffrosSidharth Jaggi

Obligatory Example/Historys

t1 t2

b1 b2

b2

b2

b1

b1 b1

b1 b1

b1 (b1,b2)

b1+b2

b1+b2b1+b2

(b1,b2)

[ACLY00] [ACLY00] Characterization Non-constructive

[LYC03], [KM02] Constructive (linear) Exp-time design

[JCJ03], [SET03] Poly-time design Centralized design

[HKMKE03], [JCJ03] Decentralized design

SIMPLER

.

.

.

C=2

[This work] All the above, plus optimal implementation complexity.

Complexity

)2(1,0)...( 21mm

m Fbbb

2

k

b1b2 bm

1

kk ...2211

β1

β2

βk

F(2m)-linear network[KM02],[HKMKE03],…

Source:- Group together `m’ bits,

Every node:- Perform linear combinations over finite field F(2m)

?

Complexity

1011

0111

0101

1010

1

0

1

1

“Thm”: For any algebraic code, at least half the β matrices in F(2m) have at least m2/2 non-zero elements

Randomly chosen algebraic encoders require O(m2) bit operations

[HKMKE03],…

1

0

1

0

=

…,[JEHM04],…

[KKHRM05]

Simplicity – Permute-and-add

1000

0001

0100

0010

1

0

1

1

“Thm”: With “high” probability, permute-and-add codes have “almost” the same performance as algebraic codes

Permute-and-add encoders require O(m) bit operations

Tight! “Thm”: To achieve capacity, need O(m) bit operations

1

1

0

1

=

Permutation matrix (sparse)

Simplicity – Permute-and-add

0100

0010

1

0

1

1

“Thm”: With “high” probability, permute-and-add codes have “almost” the same performance as algebraic codes

Permute-and-add encoders require O(m) bit operations

Tight! “Thm”: To achieve capacity, need O(m) bit operations

0000

0000

0100

0010

0

0

0

1

=

Loss of information Loss of information

Permute-and-add Codesm “sufficiently” large

b1b2 bm

b’1b’2 b’m

b’’1b’’2 b’’m

’

’’

Uniformly at random

b2b1 bm

b’1b’m b’2

b’’mb’’1 b’’2

Permute-and-add Codes

’

’’

Uniformly at random

b2 b1 bm

b’1b’m b’2

b’’mb’’1 b’’2

]''[...]'[][

Transfer matrix

Permute-and-add Codes

Percolate transfer matrices acrosssuccessive cutsets (in header)

If each transfer matrix full rank,

Final transfer matrices full rank

Decode by inverting final transfer matrix, QED

Not true, with probability c > 0

Permute-and-add Codesm “sufficiently” large

Each transfer matrix “almost” full rank,

Final transfer matrices “almost” full rank

Decode by inverting final transfer matrix, QED

Prπ [Row rank > (1-εm) fraction] > 1-2-O(mεm)Prπ [Row rank > (1-|E|εm) fraction] > 1-|T||E|2-O(mεm)

R=C- |E|εm - εm

Prπ [Final transform invertible] > 1-(|T||E|+1)2-O(mεm)

(1-|E|εm) fraction

(1-εm) fraction

Thm: Permute-and-add codes achieve R=C-(|E|+1)εm , Pr > 1-(|T||E|+1)2-O(mεm)

Proof of Lemma

][

0

21 LL

ITransform

I

L1,L2

][0

0

2 L

IGaussian

Elimination

“Almost” full rank, w.h.p.

mV

mVL

2

1]0])[[(Pr 2][,

The End/Where now?

Low-complexity decoding? Fewer packets encoded together at nodes? Same permutation at each node? Zero-error/Deterministic?

Permute-and-add Vs. Algebraic [HKMKE03]

• Rate Almost Same (ε loss)• Probability of error Almost same (smaller exp)• Block-length Almost same (1/ε increase)• Simple Distributed Design Ditto• Implementation Complexity Quadratically better• Randomness required Quadratically better