Universal and Robust Distributed Network Codes Lingxiao Xia

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

Distributed NCs [HKMKE 03] n Alice: Sends packets. “Small” rate-loss I X C packets n A Bob gets (Each column encoded with same transform T) T TX n Now Bob knows T and can decode. Prob(Error) < |E| |T| |F| / q Edges Sinks Network Configs Field size B 2

Example of limitations Infinite graph! [Wu 10] Prob(Error) < ∞ ∞ ∞/q NC distributed storage [DRWS 11]

Toy idea X α 1 X α 2 α 1 X α 3 α 2 α 1 X . . . αk…α 3α 2α 1 X αi from {0, 1, …, q} Pr(Y=0|X≠ 0)<Σk 1/q=k/q Large k… αi from {0, 1, …, 2 iq} Pr(Y=0|X≠ 0)<Σk 1/(2 iq)=1/q Large k… NOT Finite-field NC, but Integer NC, or Convolutional NC

(Generalized) SZ Lemma n SZ Lemma: If αi uniformly chosen from S in F Pr(P(α 1, α 2, …, αk)=0)≤d/|S| Polynomial n Degree of polynomial GSZ Lemma: If αi uniformly chosen from Si in F Pr(P(α 1, α 2, …, αk)=0)≤Σidi/|Si| Degree of variables

Universal codes X Code good |T 1||T 2|…|Tt| = P(α 1, α 2, …) nonzero αi Σidi/|Si| Geometric series? Y 1=T 1 X . . . 1. Which Si to use? # Nodes Yt=Tt. X 2. What’s di? # Terminals

Universal codes ~[LSB 05] “Similar” sized network Only 1. (i) Estimate depth D distributedly [BLS 07] Only ~ 2 D nodes at depth D (Roughly) (ii) Choose |Si| >> 2 D (say 23 D) # Σidi/|Si| 2. Bound number of “types of sinks”+ at each D Only ~ 2 D “types” at depth D (Roughly) * A couple slides later # 3 slides later need to code = ΣDΣi at Dd. D/23 D ≤ ΣD 2 Dd. D/23 D=ΣDd. D/22 D = ΣD 2 D/22 D=o(1)…! + Next slide

+ Types of sinks X D Actually… # types of “flow-sets” of depth at most D ~ 2 RD Y 1=T 1 X … Yt=Tt. X

* Robust universal codes

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

Zero-error codes Rate-2 codes, low complexity n General codes, high complexity n

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