1. Distributed Network Codes
Universal and
Robust
Lingxiao Xia
Sidharth Jaggi
Svitlana Vyetrenko
Tracey Ho

2. Prob(Error) < |E| |T| |F| / q
Distributed NCs
[HKMKE03]
Alice: Sends packets.
Bob gets (Each column encoded with same transform T)
Now Bob knows T and can decode. A
“Small” rate-loss
C packets X I TX T B2
Prob(Error) < |E| |T| |F| / q
Network
Configs
Edges
Sinks
Field
size

3. Example of limitations
Infinite graph! [Wu10]
Prob(Error) < ∞ ∞ ∞/q
NC distributed storage [DRWS11]

4. Toy idea . . . X α1X α2α1X α3α2α1X αk…α3α2α1X αi from {0,1,…,q}
Pr(Y=0|X≠0)<Σk1/q=k/q
Large k… 
αi from {0,1,…,2iq}
Pr(Y=0|X≠0)<Σk1/(2iq)=1/q
Large k… 
NOT Finite-field NC, but Integer NC, or Convolutional NC

5. (Generalized) SZ Lemma
SZ Lemma: If αi uniformly chosen from S in F
Pr(P(α1,α2,…,αk)=0)≤d/|S|
Polynomial
Degree of polynomial
GSZ Lemma: If αi uniformly chosen from Si in F
Pr(P(α1,α2,…,αk)=0)≤Σidi/|Si|
Degree of variables

6. |T1||T2|…|Tt| = P(α1,α2,…) nonzero
Universal codes X
Code good
|T1||T2|…|Tt| = P(α1,α2,…) nonzero αi
Σidi/|Si|
Geometric series?
1. Which Si to use?  # Nodes
Y1=T1X
. . .
Yt=TtX
2. What’s di?  # Terminals

7. “Similar” sized network
Universal codes
~[LSB05]
“Similar” sized network
Only
need to code
1. (i) Estimate depth D distributedly [BLS07]
Only ~ 2D nodes at depth D (Roughly)
(ii) Choose |Si| >> 2D (say 23D) #
Σidi/|Si|
= ΣDΣi at DdD/23D
≤ ΣD2DdD/23D=ΣDdD/22D
2. Bound number of “types of sinks”+ at each D
= ΣD2D/22D=o(1)…!
Only ~ 2D “types” at depth D (Roughly)
* A couple slides later
# 3 slides later
+ Next slide

8. # types of “flow-sets” of
+ Types of sinks X
Actually… D
# types of “flow-sets” of
depth at most D
~ 2RD
Y1=T1X …
Yt=TtX
Y1=T1X

9. * Robust universal codes

10. # Complexity bounds
~ 2D 
1. Only ~ D bits…
2. Provable lower bound…

11. Zero-error codes Rate-2 codes, low complexity
General codes, high complexity

12. Summary Randomized Distributed Robust Rate-optimal Poly-time
Matches complexity lower bounds
Deterministic
Rate 2 – low complexity
General – high complexity