1. 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

2. Outline Background Code-Length Extending Algorithm for Vertical RAID6 Codes Performance Analysis and Comparison Conclusion

3. Background RAID6 Codes – 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 RDP - Horizontal Code P-Code - Vertical Code

4. 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

5. Code-Length Extensions for Horizontal Codes 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 (1,5) (2,6) (3,7) (4,8) ( 1,6 ) (2,7) (3,8) (4) (1,7) (2,8) (3) (4,5) (1,8) (2) (3,5) (4,6) (1) (2,5) (3,6) (4,7) (5) (6) (7) (8) D1D2D3D4PQ Standard RDP Code with 6 columns Assume the column contain zeros, remove it from the code structure Extended RDP Code with 5 columns

6. 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 (2,6) (3,5) (1) D2 (3,6) (4,5) (2) D3 (1,2) (4,6) (3) D4 (1,3) (5,6) (4) D5 (1,4) (2,3) (5) D6 (1,5) (2,4) (6) (2,6)(3,6)(2,6) (4,6) (3,6)(2,6) (5,6)(4,6) (3,6)(2,6) Parity stripe P(6) loses the failure recovery ability

7. 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 (2,6) (3,5) (1) D2 (3,6) (4,5) (2) D3 (1,2) (4,6) (3) D4 (1,3) (5,6) (4) D5 (1,4) (2,3) (5) D6 (1,5) (2,4) (6) (2,6)(3,6)(2,6) (4,6) (3,6)(2,6) (5,6)(4,6) (3,6)(2,6) (4,6) (3,6) (5,6)(4,6) (3,6) (4,6)(5,6)(2,6)  =

8. 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 (3,5) (1) D2 (4,5) (2) D3 (1,2) (4,6) (3) D4 (1,3) (5,6) (4) D5 (1,4) (2,3) (5) D6 (1,5) (2,4) (6) (4,6)(5,6)(4,6)(3,6)(2,6)

9. 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: 2- 2/(n-2), (n-3)] – Impact CPU overhead of the RAID6 systems

10. 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

11. 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

12. 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

13. Summarize RAID-6 Codes Space Efficiency Computational Complexity Update Complexity Standard Lengths Extended Lengths Standard Lengths Extended Lengths Standard Lengths Extended Lengths RDP MDS (optimal) MDS (optimal) Optimal Of MDS Higher than optimal of MDS > 2 Higher than optimal > 2 Higher than optimal P-Code - first MDS (optimal) MDS (optimal) Optimal Of MDS Higher than optimal of MDS 2 (optimal) > 2 Higher than optimal P-Code - second Non-MDS (lower than optimal) Lower than optimal of MDS 2 (optimal)

14. 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 (2,6) (3,5) (1) D2 (3,6) (4,5) (2) D3 (1,2) (4,6) (3) D4 (1,3) (5,6) (4) D5 (1,4) (2,3) (5) D6 (1,5) (2,4) (6) (2,6)(3,6)(2,6) (4,6) (3,6)(2,6) (5,6)(4,6) (3,6)(2,6) Assume they are zero blocks and remove them Vertically shortened P-Code

15. 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.

16. Thanks! Email: chjinhust@gmail.com