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

Coding Seminar, Technion, December 2014 Distributed Storage Systems Based on Equidistant Subspace Codes Netanel Raviv Joint work with: Prof. Tuvi Etzion Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 1

Distributed Storage December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 2

Distributed Storage – Subspace Interpretation 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 • Shah, Rashmi, Kumar, Ramchandran, “ Explicit Codes Minimizing Repair Bandwidth for Distributed Storage ” , ITW ‘ 10. • Hollmann, “ Storage Codes – Coding Rate and Repair Locality ” , ICNC ‘ 13. December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 3

Subspace Interpretation – Storage Spanning matrix Send to node By the definition of December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 4

Subspace Interpretation – Repair • Repair node 3 from nodes 1,2: • We need – • • • • And then – Spanning matrix • Possible setting – December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 5

Subspace Interpretation – Reconstruction Reconstruct from nodes 1,2,3: We need – The data collector restores the data by solving a non- singular system of equations. December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 6

Subspace Interpretation – Properties 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: December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 7

Construction of Subspace Code Let Julius Plücker 1801-1868 Identify the coordinates of as For define In every entry, is Skew symmetric: Alternating: Bilinear: December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 8

Construction of Subspace Code Hence, is Skew-symmetric, Alternating, Bilinear. E.g., For , let December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 9

Subspace code - Properties If then If , then According to the bilinearity, (and vice versa) If then There are such subspaces. December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 10

DSS from Subspace Code - Storage Choose Explicit choice later … Define Send to node December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 11

DSS from Subspace Code - Repair Lemma: Let If is a basis for and then for any such that Proof: Similar to December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 12

DSS from Subspace Code - Repair Corollary: To repair node , let be a set of vectors such that The new node can restore December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 13

DSS from Subspace Code – Reconstruction Observation: Lemma: If , then Corollary: To reconstruct , download For instance: Download all data from nodes associated with s.t. Better: December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 14

DSS from Subspace Code – Reconstruction (2) If the reconstructing nodes know the identity of each other, they may only send Reduces the reconstruction bandwidth to (optimal). If not, all nodes but the last give their entire data. Reduces communication to December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 15

DSS from Subspace Code - Locality Lemma: if then Proof: If then for all Corollary: Repair node by downloading all data from nodes December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 16

DSS from Subspace Code – Choice of Vectors 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. December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 17

DSS from Subspace Code – Choice of Vectors 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. December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 18

Comparison to Product-Matrix Codes 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 • Rashmi, Shah, Kumar, “ Optimal Exact-Regenerating Codes for DS at the MSR and MBR points via a Product-Matrix Construction ” , ISIT ‘ 11. December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 19

Comparison to Product-Matrix Codes Product-Matrix Codes Equidistant Subspace Codes December, 2014 Netanel Raviv D.S.S. Based on Equidistant Subspace Codes 20

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