access-optimal msr codes with optimal sub- packetization ... · facts: since the code is mds, it...
TRANSCRIPT
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
2
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
3
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
4
Among all MSR codes…
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
Our Tool – MDS Array Codes
5
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
6
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
7
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
8
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
9
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
10
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
11
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
12
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
13
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
14
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
15
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
16
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
17
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
18
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
Access Optimal Regenerating Codes
19
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
20
Explicit!
Explicit!
Non-explicit...
Access-optimal MSR codes with optimal sub-packetization over small fieldsNetanel Raviv
This work
Tamo et al. 2012
More Results
21
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
22
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