Download Network Coding

Optimal Multicast
Algorithms
Sidharth Jaggi
Michelle Effros
Philip A. Chou
Kamal Jain
Menger’s Theorem
Min-cut Max-flow Theorem
C  min max (cutsize)
cut ( S R )
flow
Ford-Fulkerson Algorithm
C
P1
P2
S
PC
R
Network Coding
S
b1
b2
b1
b2
b1+b2
b1
b2
b1+b2 b1+b2
R1
R2
(b1,b2)
(b1,b2)
Example due to Cai (2000)
Multicast algorithms
R1
C1
C2
S
Assumptions
Directed, acyclic graph.
Each link has unit capacity.
R2 Links have zero delay.
min max (cutsize)  Ci , i  1,2,..., r
cut ( S Ri ) flow
Network
Upper bound for multicast
capacity C,
C ≤ min{Ci}
Cr
Rr
Multicast algorithms
b1 b2

bm
(b1b2 ...bm )  0,1    F (2m )
m
1
2
k
F(2m)-linear network
(Koetter/Medard)
Source:- Group together `m’ bits,
Any node:- Perform linear combinations
over finite field F(2m)
β1
β2
βk
11   2 2  ...   k k
F(2m)-linear network can
achieve multicast capacity C!
Multicast algorithms

Caveats to Koetter/Medard algorithm

May “flood” the network unnecessarily

Field size may need to be “large” (2m > rC)
Design complexity may be “large” (related to flooding)


Our algorithm – you can have your cake and eat it too.



No “flooding”
Field size “small” (2m > r-1)
Design complexity smaller
Encoding/Decoding
v1
v2
vk
β1
β2
βk
Vc
Encoding:
Required β's provided by coefficients of linear
combinations of v's
Decoding:
If decoder Ri receives symbols [y1...yk], output
[x1...xk]=[Mi]-1[y1 ...yk]T
Minimum Field Size
q 1
 
q 1
2
...
...
This class of networks, for q(q+1)/2 receivers,
minimum field size = q
Minimum Field Size

Open Questions
 Either
q-1 or (q(q+1)-2)/2 tight?
 What, in general, is the smallest q for a
particular network?
Almost-optimal Random Binary
Linear Codes (ARBLCs)
b1 b2
bm
=

  (b1b2 ...bm )  0,1m
1
2
k
M (1 2 ... k )
Source:- Group together `m’ bits,
Any node:- Perform arbitrary linear
combinations over finite field F(2)
If m(C-R) > log(V.r),
ARBLCs can achieve multicast
rate R with zero error!
(V = |Vertex-set|)
Random, distributed, extremely
low complexity design. Can even
build in very strong robustness
properties...
Future work...


Only some nodes can encode
Practical implementation




Synchronicity/delays
Unknown topology
Packet losses
Issues related to next-generation network protocols
(FAST)
... Utility of WAN in Lab


Access to any subset of routers
Practical testing




Can introduce arbitrary delays patterns
Topology under our control
Have greater handle on packet loss statistics (needed to develop
theoretical models)
Examine behaviour of network codes with FAST