A Comprehensive Study on RAID6 Codes:
Horizontal vs. Vertical
Chao Jin*, Dan Feng*, Hong Jiang†, Lei Tian*†
*Huazhong University of Science and Technology
†University of Nebraska-Lincoln
Outline
• Background
• Code-Length Extending Algorithm for Vertical
RAID6 Codes
• Performance Analysis and Comparison
• Conclusion
Background
• RAID6 Codes
RDP - Horizontal Code
– Can tolerate two concurrent disk
failures (column erasures)
• Horizontal RAID6 Codes
– Parity blocks held in dedicated
parity columns
– Data blocks held in data columns
• Vertical RAID6 Codes
– No dedicated parity column
– Data columns hold both data
and parity blocks
P-Code - Vertical Code
D1
D2
D3
D4
D5
D6
(1)
(2)
(3)
(4)
(5)
(6)
(2,6)
(3,6)
(1,2)
(1,3)
(1,4)
(1,5)
(3,5)
(4,5)
(4,6)
(5,6)
(2,3)
(2,4)
Parity Blocks
Data Blocks
Background
• RAID6 Code-Length Restrictions
– Lengths (number of columns in the code structure) can not
be an arbitrary number
– Usually related to a prime number
– RDP: prime+1; P-Code: prime-1, prime; …
• RAID6 Code-Length Extensions
– Horizontal Codes: easy to extend; just by removing data
columns directly
– Vertical codes: harder; can not be extended like horizontal
codes
Code-Length Extensions for Horizontal Codes
Standard RDP Code
with 6 columns
Extended RDP Code
with 5 columns
Assume the column
contain zeros, remove
it from the code
structure
•
•
•
•
D1
D2
(1,5)
(1,6)
D3
P
Q
(1)
(5)
(2,5)
(6)
(3,5) (3,6)
(7)
(4,5) (4,6) (4,7)
(8)
(1,7) (1,8)
(2,6) (2,7) (2,8)
(3,7) (3,8)
(4,8)
(4)
D4
(3)
(2)
Code-Length Extension: shortening from a standard code
Removing some columns directly from the standard code
Result in extended codes with shorter code lengths
Removed columns assumed to contain only zeros, thus do not
affect the fault-tolerating ability of the extended codes
Code-Length Extensions for Vertical Codes
• Vertical codes can not be shortened directly like horizontal
codes
• Each column contain not only data blocks but also parity blocks
• Removing a parity block may leave the parity stripe in an
inconsistent state
• Tow extending algorithms for vertical codes are proposed
D1
D2
D3
D4
D5
D6
(1)
(2,6)
(3,5)
(2)
(3,6)
(4,5)
(3)
(1,2)
(4,6)
(4)
(1,3)
(5,6)
(5)
(1,4)
(2,3)
(6)
(1,5)
(2,4)
Parity stripe P(6) loses the failure recovery ability
Code-Length Extensions for Vertical Codes
• First extending algorithm
– Select a new parity block for the parity stripe
– The parity block is originally a data block of the parity stripe
– The parity stripe remain consistent!
D1
D2
D3
D4
D5
D6
(1)
(2,6)
(3,5)
(2)
(3,6)
(4,5)
(3)
(1,2)
(4,6)
(4)
(1,3)
(5,6)
(5)
(1,4)
(2,3)
(6)
(1,5)
(2,4)

(5,6)
=
(3,6)

(4,6)
(2,6)
Code-Length Extensions for Vertical Codes
• Second extending algorithm
– Remove the entire parity stripe from the code structure
– Additional data blocks may be removed to keep an equal
number of blocks per column
– Removed blocks may be assumed to be zero blocks
D1
D2
D3
D4
D5
D6
(1)
(3,5)
(2,6)
(2)
(4,5)
(3,6)
(3)
(1,2)
(4,6)
(4)
(1,3)
(5,6)
(5)
(1,4)
(2,3)
(6)
(1,5)
(2,4)
RAID6 Code Performance Metrics
• Space Efficiency
– Ratio between data volume and whole volume (data and
parity volume). [Optimal (highest, MDS Codes): (n-2)/n]
– Related to the redundant rate of the RAID6 systems
• Update Complexity
– Average number of parity blocks need to be updated upon
each data block update. [Optimal (lowest): 2]
– Impact write overhead of the RAID6 systems
• Computational Complexity
– Average number of XOR per data block during
encoding/decoding. [Optimal (lowest) for MDS Codes: 22/(n-2) , (n-3)]
– Impact CPU overhead of the RAID6 systems
Space Efficiency
• Space efficiency comparison for RDP and P-Code
– Non-standard code lengths obtained by code-length extending
algorithms
– P-Code – first/second refers to extended P-Code by the first/second
extending algorithm for vertical codes
– RDP and P-Code – first are MDS codes with optimal space efficiency;
P-Code – second is non-MDS codes
Computational Complexity
• Computational complexity comparison for RDP and P-Code
– RDP and P-Code – first have non-optimal (higher) computational
complexity at extended code lengths
– P-Code – second has even lower computational complexity than the
optimal computational complexity for MDS codes
– Reveal that non-MDS codes has lower computational complexity than
MDS codes; proved trade offs between space efficiency and
computational complexity
Update Complexity
• Update complexity comparison between RDP and P-Code
– P-Code – second always has the optimal update complexity of 2
– P-Code – first has non-optimal update complexity at extended lengths
– RDP has non-optimal update complexity, with an asymptotic value of 3
Summarize
Space Efficiency
RAID-6 Codes
RDP
Standard
Lengths
MDS
(optimal)
MDS
(optimal)
P-Code - second
Update Complexity
Extended
Lengths
Standard
Lengths
Extended
Lengths
Standard
Lengths
Extended
Lengths
MDS
(optimal)
Optimal
Of
MDS
Higher
than
optimal of
MDS
>2
Higher
than
optimal
>2
Higher
than
optimal
MDS
(optimal)
P-Code - first
Computational
Complexity
Non-MDS
(lower than
optimal)
Optimal
Of
MDS
Higher
than
optimal of
MDS
Lower
than
optimal of
MDS
2
(optimal)
>2
Higher
than
optimal
2
(optimal)
Vertical shortening of vertical RAID6 codes
• Removing data rows instead of columns from the vertical code structure
• Data rows do not contain parity blocks, so vertical shortening do not
damage the parity stripe consistency
• Vertically shortened codes are non-MDS codes, but with lower
computational complexity
• Provide tradeoffs between space efficiency and computational complexity
D1
D2
D3
D4
D5
D6
(1)
(2)
(3)
(4)
(5)
(6)
(2,6) (3,6)
(1,2)
(1,3)
(1,4)
(1,5)
(3,5) (4,5)
(4,6)
(5,6)
(2,3)
(2,4)
Vertically shortened P-Code
Assume they are zero blocks
and remove them
Conclusions
• Horizontal codes are easy to be extended to eliminate the
code-length restrictions, but vertical codes are not easy to be
extended. We proposed two extending algorithms for vertical
codes. The first one selects a new parity block for the parity
stripe; the second one removes the entire parity stripe from
the code structure.
• We compared the performance of horizontal codes and vertical
codes at consecutive code lengths. We also studied the impact
of the code-length extending algorithms on the performance of
the RAID6 codes.
• We proposed the vertical shortening algorithm for vertical
codes. The algorithm can provide the tradeoff between space
efficiency and computational complexity for vertical codes.
Thanks!
Email:
[email protected]