An Upper Bound on Locally Recoverable Codes Viveck R. Cadambe (MIT) Arya Mazumdar (University of Minnesota)

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)

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Failure Tolerance versus Storage versus Access:

Erasure Codes: Classical Trade-off

codeword-symbol (storage node)

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Failure Tolerance versus Storage versus Access:

Erasure Codes: Recently studied trade-off

codeword-symbol (storage node)

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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