An Upper Bound on Locally Recoverable Codes
Viveck R. Cadambe (MIT)Arya Mazumdar (University of Minnesota)
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Failure Tolerance versus Storage versus Access:
Erasure Codes: Classical Trade-off
codeword-symbol (storage node)
3
Failure Tolerance versus Storage versus Access:
Erasure Codes: Classical Trade-off
codeword-symbol (storage node)
4
Failure Tolerance versus Storage versus Access:
Erasure Codes: Recently studied trade-off
codeword-symbol (storage node)
5
Failure Tolerance versus Storage versus Access*:
* Locality important in practice [Huang et. al. 2012, Sathiamoorthy et. al. 2013]* Repair bandwidth is another measure [See a survey by Datta and Oggier 2013]
codeword-symbol (storage node)
Erasure Codes: Recently studied trade-off
Trade-off between distance and rate
Singleton Bound
Trade-off between distance and rate
Singleton Bound
Singleton Bound
Trade-off between distance and rate
Singleton Bound
Singleton Bound
Trade-off between distance and rate and locality?
Singleton Bound
[Gopalan et. al. 11, Papailiopoulous et. al. 12]
Singleton Bound
Singleton Bound[Gopalan et. al.]
Trade-off between distance and rate and locality?
MRRW Bounds are best known locality-unaware bounds
[Gopalan et. al.]
MRRW bound
Singleton Bound
Trade-off between distance and rate and locality?
[Gopalan et. al. 11, Papailiopoulous et. al. 12]
Main Result: A New Upper bound on the price of locality
This talk!
[Gopalan et. al.]
MRRW bound
Our Bound
• At least as strong as previously derived bounds.- Information theoretic (also applicable for non-linear codes )
• At least as strong as previously derived bounds.- Information theoretic (also applicable for non-linear codes )
• Analytical insights from Plotkin Bound:
Distance-expansion
• At least as strong as previously derived bounds.- Information theoretic (also applicable for non-linear codes )
• Analytical insights from Plotkin Bound:
• A bound on the capacity of a particular multicast network for a fixed alphabet (field) size.
• Because of achievability of [Papailiopoulous et. al. 12]
Distance-expansion
Open Question What is the largest distance achievable by a locally recoverable code, for a fixed alphabet and locality?
Our Bound
A naïve code
A naïve code:
Gallager’s LDPC ensemble seems to do better
Thank you.
Proof Sketch
In the code, t(r+1) nodes that contain tr “q-its of information”, for a certain range of t
Remove Locality-induced Redundancy
Measure Locality-induced Redundancy
