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Reviews raid.c comments.
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Andrea Mazzoleni
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2456e88132
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e7804a5f8f
1 changed files with 32 additions and 31 deletions
63
raid.c
63
raid.c
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@ -16,17 +16,18 @@
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*/
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/*
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* The RAID5 and RAID6 support were originally derived from the libraid6 library
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* by H. Peter Anvin released with license "GPL2 or any later version"
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* used in the Linux Kernel 2.6.38.
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* The RAID5 and RAID6 support were originally derived from the libraid6
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* library by H. Peter Anvin released with license "GPL2 or any later
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* version", used in the Linux Kernel 2.6.38.
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* This support was later completely rewritten (many times), but the
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* H. Peter Anvin's Copyright may still apply.
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*
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* The RAIDTP (Triple Parity), RAIDQP (Quad parity), RAIDPP (Penta Parity) and
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* RAIDHP (Hexa parity) support is original work implemented from scratch.
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* The RAIDTP (Triple Parity), RAIDQP (Quad parity), RAIDPP (Penta Parity)
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* and RAIDHP (Hexa parity) support is original work implemented from
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* scratch.
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*
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* For RAID5 and RAID6 it works like the Linux Kernel RAID and it's based on
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* the Anvin's paper "The mathematics of RAID-6" [1].
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* For RAID5 and RAID6 it works like the Linux Kernel RAID and it's based
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* on the Anvin's paper "The mathematics of RAID-6" [1].
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*
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* We compute the parity in the Galois Field GF(2^8) with the primitive
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* polynomial x^8 + x^4 + x^3 + x^2 + 1 (285 decimal), starting from a set
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@ -35,16 +36,16 @@
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* P = sum(Di)
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* Q = sum(2^i * Di) with 0<=i<N
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*
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* To support RAIDTP (Triple Parity), it was first evaluated, and then dropped,
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* the use of an extension of the same approach, with additional parity
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* coefficients set as powers of 4, with equations:
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* To support RAIDTP (Triple Parity), it was first evaluated, and then
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* dropped, the use of an extension of the same approach, with additional
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* parity coefficients set as powers of 4, with equations:
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*
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* P = sum(Di)
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* Q = sum(2^i * Di)
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* R = sum(4^i * Di) with 0<=i<N
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*
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* This method is also the same used by ZFS to implement its RAIDZ3 support,
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* it works well, and it's very efficient.
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* This method is also the same used by ZFS to implement its RAIDZ3
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* support, it works well, and it's very efficient.
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*
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* Unfortunately, the same approach doesn't work for RAIDQP (Quad Parity).
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* Using additional parity coefficients set as power of 8 with equations:
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@ -72,7 +73,8 @@
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* this guarantee.
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*
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* Using the primitive polynomial 285, RAIDQP works for up to 21 data disks
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* with parity coefficients "1,2,4,8". Changing polynomial to one of 391/451/463/487,
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* with parity coefficients "1,2,4,8". Changing polynomial to one of
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* 391/451/463/487,
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* it works for up to 27 disks with the same parity coefficients.
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* Using different parity coefficients, like "5,13,27,35", it's possible to
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* make it working for up to 33 disks. But no more.
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@ -81,29 +83,30 @@
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* polynomial 100087 or 122563 that supports up to Hexa (6) Parity
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* with parity coefficients 1,2,4,8,16,32 for up to 89 disks.
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*
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* To overcome these limitations we instead use an Extended Cauchy Matrix [3][4]
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* to compute the parity.
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* To overcome these limitations we instead use an Extended Cauchy Matrix
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* [3][4] to compute the parity.
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* Such matrix has the mathematical property to have all the square
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* submatrices not singular, resulting in always solvable equations, for
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* any number of parities and for any number of disks.
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*
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* The problem of this approach is that the coefficients of the equations
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* are not powers of the same value, not allowing to use the fast implementation
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* described in the Anvin's paper [1].
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* are not powers of the same value, not allowing to use the fast
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* implementation described in the Anvin's paper [1].
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*
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* Hopefully there is a method to implement parallel multiplications
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* using SSSE3 instructions [1][5]. Method already competitive with the computation
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* of RAIDTP parity using power coefficients.
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* using SSSE3 instructions [1][5]. Method already competitive with the
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* computation of RAIDTP parity using power coefficients.
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*
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* Another important property of the Extended Cauchy matrix is that we can
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* setup the first two rows with power coeffients equal at the RAID5 and RAID6
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* approach described in Anvin's paper [1], resulting in a compatible extension,
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* and requiring SSSE3 instructions only if RAIDTP is used.
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* setup the first two rows with power coeffients equal at the RAID5 and
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* RAID6 approach described in Anvin's paper [1], resulting in a compatible
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* extension, and requiring SSSE3 instructions only if RAIDTP is used.
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*
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* We also "normalize" the matrix, multipling each row for a constant
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* factor to make the first column with all 1.
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*
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* This results in the "normalized" Extended Cauchy matrix A[row,col] defined as:
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* This results in the "normalized" Extended Cauchy matrix A[row,col]
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* defined as:
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*
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* 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 ...
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* 01 02 04 08 10 20 40 80 1d 3a 74 e8 cd 87 13 26 4c 98 2d 5a b4 75 ea c9 8f 03 06 ...
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* 01 2b 3f cf 73 2c d6 ed cb 74 15 78 8a c1 17 c9 89 68 21 ab 76 3b 4b 5a 6e 0e b9 ...
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* (see tables.h for the full matrix)
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*
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* with one row for each parity level, and one column for each data disk.
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*
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* This matrix supports 6 level of parity, one for each row, for up to 251
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* data disks, one for each column.
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* data disks, one for each column, with all the 377,342,351,231 square
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* submatrices not singular.
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*
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* This matrix can be extended to support any number of parities, just adding
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* additional rows, (but removing one column for each row added), without
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* the changing the already existing rows.
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* additional rows, and removing one column for each row added.
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* (see mktables.c for more details in how the matrix is generated)
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*
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* In details, parity is computed as:
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* Td = Ta + T
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* Ud = Ua + U
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*
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* we can sum these two set equations, obtaining:
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* we can sum these two sets of equations, obtaining:
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*
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* Pd = Dx + Dy + Dz + Dh + Dv + Dw
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* Qd = 2^x * Dx + 2^y * Dy + 2^z * Dz + 2^h * Dh + 2^v * Dv + 2^w * Dw
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* Td = A[4,x] * Dx + A[4,y] * Dy + A[4,z] * Dz + A[4,h] * Dh + A[4,v] * Dv + A[4,w] * Dw
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* Ud = A[5,x] * Dx + A[5,y] * Dy + A[5,z] * Dz + A[5,h] * Dh + A[5,v] * Dv + A[5,w] * Dw
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*
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* A linear system always solvable because the coefficients matrix is always
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* not singular due the properties of the matrix A[].
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* A linear system always solvable because the coefficients matrix is
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* always not singular due the properties of the matrix A[].
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*
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* Resulting speed in x64, with 24 data disks, using a stripe of 256 KiB,
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* for a Core i7-3740QM CPU @ 2.7GHz is:
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