Title: Distributed zero-error network coding
1Distributed zero-errornetwork coding
Tracey Ho Michelle Effros
Sidharth Jaggi
2Decentralized zero-error
s
ACLY00
JCJ03, SET03 No error
Centralized design
HKMKE03, JCJ03 Prob(error) 2-n
Decentralized design
. . .
No error Decentralized
Design
t1
t2
Complexity
(Open CS problems)
3Convolutional network codes
F2(z)-linear network ACLY00,KM02,
b1
b2
Source- Generates a bit-stream
Every node- Perform linear combinations over
field F2(z)
degree-m polynomials
,JHEM04, Given an algebraic network code
over field of size q, exists a convolutional
network code with degree-log(q) polynomials
2mq
4Decentralized
zero-error
Information about network structure/code
percolates along network links
No error!
Guaranteeing linear independence of information
in each cut-set
is impossible! (Proof by example exercise for
audience)
Need some global information about network.
5Decentralized zero-error
Each node has unique ID i
Toy example (C2 (FS04))
- One linearly independent incoming vector
- Replicate on all outgoing links
2. Two linearly independent incoming
vectors Maintain MDS property
i
6Decentralized zero-error
Lexicographic ordering (Cantor-diagonal
assignment)
m polylog(i,j)
Really cool! ?
7Decentralized zero-error
General C, existence proof
Complete graph with V nodes and C links from vi
to vj, for all (i,j)
subgraphs with mincut C subgraphs
2CV(V-1)
KM02 Can design ONE network code which works
for MANY networks
Field size proportional to 2CV(V-1)
m proportional to CV(V-1) (exponential
increase, non-constructive -/)
8Decentralized zero-error
General C, construction
HMSEK04 Network code works
KM02 Line graph matrix
f(i,j) distinct for each (i,j) pair
(Cantor-diagonal assignment)
Each z term has distinct exponent
S(i,j) in term2f(i,j) distinct in each term
DOUBLE exponential field-size -(
Oh well next paper -)
9Summary/Where now?
- C2 toy example
- General C, exponentially larger fields required.
- High computational complexity OR
- High (double-exp) implementation complexity
- How much global info needed for tractable
- zero-error design?