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Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
Access-optimal MSR codes with optimal sub-packetization over small fields
Netanel Raviv
Joint work with:
Dr. Natalia Silberstein
Technion Coding Theory Seminar, June 2015
Prof. Tuvi Etzion
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
Distributed Storage
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Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
Lemma [Dimakis et al.]:
Provides a tradeoff between parameters.
Two extremal points achieve equality:
Regenerating Codes
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Graph-theoretic proof.
Minimum Storage Regenerating (MSR) Codes
Minimal storage that allows the reconstruction from any set of
nodes.
Minimum Bandwidth Regenerating (MBR) Codes
Minimal data transmission during node repair.
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
Goal –
Devise MSR codes that achieve minimal bandwidth repair.
Sub-Goal –
Devise MSR codes that achieve minimal bandwidth repair of systematic node failures.
Sub-Sub-Goal –
Devise MSR codes that achieve minimal bandwidth repair of a single systematic node failure.
MSR Codes with Minimum Bandwidth Repair
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Among all MSR codes…
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
Our Tool – MDS Array Codes
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A file
Encode the file using an MDS array code:
Systematic nodes Parity nodes
Invertible coding matrices
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
Facts:
Since the code is MDS, it can tolerate up to failures (erasures).
If failures occur, fraction of the information must be downloaded.
This talk:
Optimal repair of a single failure of a systematic node.
Download fraction of the information from nodes.
MDS Array Codes for Distributed Storage
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Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
Tamo et al.: Optimal repair of a systematic node is possible if there exist subspaces of dimension of such that
Optimal Repair of a Systematic Node
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independent
Independence property:
Invariance property:
Nonsingular property:Every square block submatrix of is invertible.
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
Our Goal
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Goal: Construct a set of pairs as large as possible. as small as possible.
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
Assume For the repair, each node project his data on
Systematic nodes send Parity nodes send
Goal: Cancel out the redundant termsfrom the data received from the parity nodes, and find
The Subspace Condition – Repair
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Symbols from each node.
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
By the Independence Property:
and thus the system is solvable.
Systematic: Parity:
The Subspace Condition – Repair
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By the Invariance Property , thus, given we may cancel in the sums received from parity nodes.
We remain with:
To recover , we must solve the system:
Parity:
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
Consider the matrix –
Notice:
Thus:
The subspace is an independent subspace.
The eigenvalues are the roots of unity of order
The eigenvectors are
Our Construction – Underlying Principles
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and
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
Corollary: For ,
The subspace is an independentsubspace.
Subspaces of the form are eigenspaces, and thus also invariant subspaces.
Problems:
Choose the change-of-basis matrices such that the invariance and the independence property are satisfied.
Multiply each by a field constant such that the nonsingular property is satisfied.
Our Construction – Matrices and Subspaces
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Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
Identify the unit vectors as nodes in the complete uniform hypergraph.
Let be a perfect colored matching.
contributes pairs , such that
Change Basis Matrices from Perfect Colored Matchings
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a function of
a function of
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
Eigenspacefor
Independent subspace
Eigenspacefor
Change Basis Matrices from Perfect Colored Matchings
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a function of
a function of
Ensure that:
e.g.,
a function of
For
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
One matching pairs.
More than one matching?
Construct codes from different matchings which satisfy a mutual relation.
Each edge in one matching is monochromatic in any other matching.
The independence property obviously holds.
What about the invariance property?
Observe:
From One Matching to Many Matchings
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a function of
a function of
a function of
a function of
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
From One Matching to Many Matchings
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a function of
a function of
a function of
a function of
Take two matchings such that -Each edge in one matching is monochromatic in the other.
We get:
a function of
a function of
a function of
a function of
A colored subspace from
anothermatching
A colored subspace from
anothermatching
The invariance property holds. Easy to construct such matchings.
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
So far, the construction works for any number of parities.
What about the nonsingular property?
Every square block sub-matrix of must be nonsingular.
For two parities, requires full rank-distance between matrices.
Lemma: if then
For three parities, requires:
Full rank-distance between matrices.
Full rank-distance between squares of matrices.
Non singularity of
The Nonsingular Property
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Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
For three parities, requires
Full rank-distance between matrices.
Full rank-distance between squares of matrices.
Non singularity of
Lemma: if
Lemma: Matrices from different matchings are simultaneously diagonalizable, and hence they commute.
Corollary:is a Vandermonde matrix!
The Nonsingular Property – Three Parities
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Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
Access Optimal Regenerating Codes
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All repair subspaces are spanned by unit vectors.
During repair, to project the nodes’ data on the subspace we may choose a spanning matrix whose rows are unit vectors.
Node sends:
Corollary: A node participating in the repair sends parts of his data as-is.
Theorem [Tamo et al. 2012]: If the code is access optimal then
Corollary: Our codes are optimal access-optimal codes.
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
This work
Tamo et al. 2012
Li et al. 2015
This work
Tamo et al. 2012
Two parities:
Three parities:
Results
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Explicit!
Explicit!
Non-explicit...
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
This work
Tamo et al. 2012
More Results
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Each matching can contribute one more subspace.
Three parities
Easy to construct such matchings.
Non-explicit…
Explicit!
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
Future Research
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Our work:
Use non-diagonalizable matrices to further reduce the field size.
Nonsingularity for more than three parities.
Other constructions of matchings.
In general:
Non-systematic node failure.
More than one simultaneous failure?
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv23
