Prof. Tuvi Etzion Netanel Raviv D.S.S. Based on Equidistant - - PowerPoint PPT Presentation

D.S.S. Based on Equidistant Subspace Codes Netanel Raviv

Distributed Storage Systems Based on Equidistant Subspace Codes

Netanel Raviv

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Joint work with:

• Prof. Tuvi Etzion

Coding Seminar, Technion, December 2014

D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

Distributed Storage

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D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

Since we consider linear codes, node stores inner products of with Node can compute the inner product of and any linear combination of We may say that node “knows” on Properties of the subspace code

Distributed Storage – Subspace Interpretation

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• Shah, Rashmi, Kumar, Ramchandran, “Explicit Codes Minimizing Repair Bandwidth for Distributed Storage”, ITW ‘10.
• Hollmann, “Storage Codes – Coding Rate and Repair Locality”, ICNC ‘13.

D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

Send to node By the definition of

Subspace Interpretation – Storage

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Spanning matrix

D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

Subspace Interpretation – Repair

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• Repair node 3 from nodes 1,2:
• We need –
• And then –
• Possible setting –

Spanning matrix

D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

Reconstruct from nodes 1,2,3:

We need –

Subspace Interpretation – Reconstruction

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The data collector restores the data by solving a non- singular system of equations.

D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

The code should satisfy:

Constant dimension Constant intersection (Every set / There exists a set) of subspaces span . For a given , (Every set / There exists a set) of subspaces satisfies:

Subspace Interpretation – Properties

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D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

Let Identify the coordinates of as For define In every entry, is

Skew symmetric: Alternating: Bilinear:

Construction of Subspace Code

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Julius Plücker 1801-1868

D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

Hence, is

Skew-symmetric, Alternating, Bilinear. E.g.,

For , let

Construction of Subspace Code

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D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

If then If , then According to the bilinearity, (and vice versa) If then There are such subspaces.

Subspace code - Properties

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D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

Choose Define Send to node

DSS from Subspace Code - Storage

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Explicit choice later…

D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

Lemma: Let If is a basis for and then for any such that Proof: Similar to

DSS from Subspace Code - Repair

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D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

Corollary: To repair node , let be a set of vectors such that

DSS from Subspace Code - Repair

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The new node can restore

D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

Observation: Lemma: If , then Corollary: To reconstruct , download For instance: Download all data from nodes associated with s.t. Better:

DSS from Subspace Code – Reconstruction

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D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

If the reconstructing nodes know the identity of each

• ther, they may only send

Reduces the reconstruction bandwidth to (optimal).

If not, all nodes but the last give their entire data.

Reduces communication to

DSS from Subspace Code – Reconstruction (2)

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D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

Lemma: if then Proof: If then for all Corollary: Repair node by downloading all data from nodes

DSS from Subspace Code - Locality

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D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

All properties of the DSS depend on the choice of Note:

Repair – requires active nodes whose vectors form a basis for Reconstruction – same. Local repair of - requires a small set of nodes whose span contains

Solution: (Repair, Reconstruction)

Set to be columns of a generator matrix of a linear code Lemma: By removing any columns, the remaining ones span Resilient to node failures. If the linear code is MDS, every nodes suffice for Repair/Reconstruction. If not, there always exists a set of nodes suffices for Repair/Reconstruction.

DSS from Subspace Code – Choice of Vectors

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D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

Solution: (Local Repair)

Take a basis for , divide it to subsets and add the linear span of each subset. It is possible to repair any node by contacting at most active nodes. Resilient to failures.

DSS from Subspace Code – Choice of Vectors

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D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

Product Matrix Codes: Let be columns of a generator matrix of an ECC. Node stores Repair: Download from active nodes which span and solve a non-singular system of equations. Reconstruction: Download from active nodes which span and multiply by a proper matrix. Locality: Download from such that and restore

Comparison to Product-Matrix Codes

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• Rashmi, Shah, Kumar, “Optimal Exact-Regenerating Codes for DS at the MSR and MBR points via a Product-Matrix Construction”, ISIT ‘11.

D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

Comparison to Product-Matrix Codes

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Equidistant Subspace Codes Product-Matrix Codes

D.S.S. Based on Equidistant Subspace Codes Netanel Raviv December, 2014

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