Low Complexity Algebraic Multicast Network Codes

Sidharth “Sid” Jaggi

Philip Chou

Kamal Jain

The Multicast Problem Model

Network with directed edges, nodes.

Single source, rate R.|T| sinks, desired rate R, identical

information.

Examples –1. Online news broadcasts.2. Online gaming.

s

t1

t2

t|T|

Network

.

.

.

R

R

R

R

History-I AssumptionsAcyclic graph.Each link has unit capacity.Links have zero delay.Arithmetic operations allowed

at all nodes.

Upper bound for multicast capacity C,

C ≤ min{Ci}

s

t1

t2

t|T|

C|T|

C1

C2

Network

||,...,2,1,)(maxmin)( ||

TiCcutsize iflowtscut T

.

.

.

Multicast capacity C achievable! (Random coding argument, [ACLY 2000])

Example

s

t1 t2

b1 b2

b2

b2

b1

b1

(b1,b2)

b1+b2

b1+b2b1+b2

(b1,b2)Example due to [ACLY2000]

History-II

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

m Fbbb

2

k

b1b2 bm

1

kk ...2211

β1

β2

βk

F(2m)-linear network can achieve multicast capacity

C!

F(2m)-linear network[KM2001]

Source:- Group together `m’ bits,

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

Local Coding Vector: [β1 β2... βk]

2m>|T|C, Computational complexity high.

Results (Ours and Others’)

Importance Linear encoding/decoding. [SET2003] Capacity gap without coding arbitrarily large. Lower bound on field size 2m ..

[LL(preprint)] Lower bound on alphabet size ; finding smallest alphabet NP-hard. Multicast the only “easy and interesting” case.

Can be implemented as binary block-linear codes. Randomized construction (Also [SET2003],[HMKKE2003])

Faster design, More robust (Single code, zero error, small rate loss, arbitrary failure

pattern).

||T

Main Result (Also [SET2003]): For 2m ≥ |T|, exists an F(2m)-linear network which can be designed in polynomial time.

||T

Example: Local Coding Vectors

s

b1

b2b1

b1b1+b2

Local Coding Vector: [1]

Local Coding Vector: [1 1]

length = number of incoming edges(variable)

Idea: Global Coding Vectors

s

b1

b1+b2

Global Coding Vector: [1 0]Global Coding Vector: [0 1]

Information carried by edge↔(g.c.v.)edge.[s1s2...sC]T

Global Coding Vector: [1 1]

length = capacity(fixed)

b2

Idea: Linear IndependenceTASK: Find local coding vectors so that each receiver can decode.

METHOD: Find local coding vectors sequentially so that global coding vectors on every cut-set to every receiver are linearly independent.

s

t1 t2

PREPROCESSING:Find a set of C paths froms to each ti.

b2b1

b1+b2b1+b2 T2 = [1 1] [0 1]

T1 = [1 0] [1 1]

OUTPUT: Local coding vectors,Final global coding vectors.

Decoding: [x1...xk]=[Ti]-1[y1...yk]T

One edge at a time…

e1 e2

e4

e5

e3M1({1,4}) = [1 0]

[0 1]

b2b1

[1 0] [0 1]

[1 1]

M2({3,2}) = [1 0][0 1]

M2({2,5}) = [0 1][1 1]

M1({1,5}) = [1 0][1 1]

Design g.c.v. for edge 5

b1+b2

Choosing local coding vectors appropriately

Let v1,…,vk be global coding vectors feeding into e.Let Mi(n) be matrices of global coding vectorson nth cutset to ti. Inductive hypothesis – each Mi(n) has rank C

e

v1

v2

vkLet L = span {v1,…,vk},

L

Sj = span{rows(Mj(n))- vj}

Then we wish to find v in L such that

v not in Sj for all Sj (and therefore rank of each Mi is still C)

v1

vk

CCCC

jCjj

C

vvv

vvv

vvv

...

:

...

:

...

21

21

11211

V

Sk

S1

L

Cool Lemma

Lemma:

)(||1

j

k

jSLqT

)),((1

j

k

jSLL

,|| )(LrankqL

)(1)(

1

1)(

1)1(||||||

,||

LrankLrankj

k

j

Lrankj

qqTSLqT

qSL

0|))((|1

j

k

jSLL

VS1

L

Proof: Consider

V

L

1SLkSL

Hence Proved.(Quick deterministic algorithm,Faster randomized algorithms.)

Sk

Practical Optimal Network Codes.

Multicast the only “easy and interesting” case for network coding problems.

Capacity achieving codes. Linear encoding/decoding. Small field sizes.

At most quadratic gap. Smallest field-size determination hard.

Polynomial design complexity. (Random code design) Robustness. (Random code design)

Joint paper being prepared for submission to IT Trans.: Jaggi, Sanders, Chou, Effros, Egner, Jain, Tolhuizen