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Diffstat (limited to 'c_src/raid/raid.c')
-rw-r--r-- | c_src/raid/raid.c | 586 |
1 files changed, 0 insertions, 586 deletions
diff --git a/c_src/raid/raid.c b/c_src/raid/raid.c deleted file mode 100644 index 3052675f..00000000 --- a/c_src/raid/raid.c +++ /dev/null @@ -1,586 +0,0 @@ -/* - * Copyright (C) 2013 Andrea Mazzoleni - * - * This program is free software: you can redistribute it and/or modify - * it under the terms of the GNU General Public License as published by - * the Free Software Foundation, either version 2 of the License, or - * (at your option) any later version. - * - * This program is distributed in the hope that it will be useful, - * but WITHOUT ANY WARRANTY; without even the implied warranty of - * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the - * GNU General Public License for more details. - */ - -#include "internal.h" -#include "gf.h" - -/* - * This is a RAID implementation working in the Galois Field GF(2^8) with - * the primitive polynomial x^8 + x^4 + x^3 + x^2 + 1 (285 decimal), and - * supporting up to six parity levels. - * - * For RAID5 and RAID6 it works as as described in the H. Peter Anvin's - * paper "The mathematics of RAID-6" [1]. Please refer to this paper for a - * complete explanation. - * - * To support triple parity, it was first evaluated and then dropped, an - * extension of the same approach, with additional parity coefficients set - * as powers of 2^-1, with equations: - * - * P = sum(Di) - * Q = sum(2^i * Di) - * R = sum(2^-i * Di) with 0<=i<N - * - * This approach works well for triple parity and it's very efficient, - * because we can implement very fast parallel multiplications and - * divisions by 2 in GF(2^8). - * - * It's also similar at the approach used by ZFS RAIDZ3, with the - * difference that ZFS uses powers of 4 instead of 2^-1. - * - * Unfortunately it doesn't work beyond triple parity, because whatever - * value we choose to generate the power coefficients to compute other - * parities, the resulting equations are not solvable for some - * combinations of missing disks. - * - * This is expected, because the Vandermonde matrix used to compute the - * parity has no guarantee to have all submatrices not singular - * [2, Chap 11, Problem 7] and this is a requirement to have - * a MDS (Maximum Distance Separable) code [2, Chap 11, Theorem 8]. - * - * To overcome this limitation, we use a Cauchy matrix [3][4] to compute - * the parity. A Cauchy matrix has the property to have all the square - * submatrices not singular, resulting in always solvable equations, - * for any combination of missing disks. - * - * The problem of this approach is that it requires the use of - * generic multiplications, and not only by 2 or 2^-1, potentially - * affecting badly the performance. - * - * Hopefully there is a method to implement parallel multiplications - * using SSSE3 or AVX2 instructions [1][5]. Method competitive with the - * computation of triple parity using power coefficients. - * - * Another important property of the Cauchy matrix is that we can setup - * the first two rows with coeffients equal at the RAID5 and RAID6 approach - * decribed, resulting in a compatible extension, and requiring SSSE3 - * or AVX2 instructions only if triple parity or beyond is used. - * - * The matrix is also adjusted, multipling each row by a constant factor - * to make the first column of all 1, to optimize the computation for - * the first disk. - * - * This results in the matrix A[row,col] defined as: - * - * 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01... - * 01 02 04 08 10 20 40 80 1d 3a 74 e8 cd 87 13 26 4c 98 2d 5a b4 75... - * 01 f5 d2 c4 9a 71 f1 7f fc 87 c1 c6 19 2f 40 55 3d ba 53 04 9c 61... - * 01 bb a6 d7 c7 07 ce 82 4a 2f a5 9b b6 60 f1 ad e7 f4 06 d2 df 2e... - * 01 97 7f 9c 7c 18 bd a2 58 1a da 74 70 a3 e5 47 29 07 f5 80 23 e9... - * 01 2b 3f cf 73 2c d6 ed cb 74 15 78 8a c1 17 c9 89 68 21 ab 76 3b... - * - * This matrix supports 6 level of parity, one for each row, for up to 251 - * data disks, one for each column, with all the 377,342,351,231 square - * submatrices not singular, verified also with brute-force. - * - * This matrix can be extended to support any number of parities, just - * adding additional rows, and removing one column for each new row. - * (see mktables.c for more details in how the matrix is generated) - * - * In details, parity is computed as: - * - * P = sum(Di) - * Q = sum(2^i * Di) - * R = sum(A[2,i] * Di) - * S = sum(A[3,i] * Di) - * T = sum(A[4,i] * Di) - * U = sum(A[5,i] * Di) with 0<=i<N - * - * To recover from a failure of six disks at indexes x,y,z,h,v,w, - * with 0<=x<y<z<h<v<w<N, we compute the parity of the available N-6 - * disks as: - * - * Pa = sum(Di) - * Qa = sum(2^i * Di) - * Ra = sum(A[2,i] * Di) - * Sa = sum(A[3,i] * Di) - * Ta = sum(A[4,i] * Di) - * Ua = sum(A[5,i] * Di) with 0<=i<N,i!=x,i!=y,i!=z,i!=h,i!=v,i!=w. - * - * And if we define: - * - * Pd = Pa + P - * Qd = Qa + Q - * Rd = Ra + R - * Sd = Sa + S - * Td = Ta + T - * Ud = Ua + U - * - * we can sum these two sets of equations, obtaining: - * - * Pd = Dx + Dy + Dz + Dh + Dv + Dw - * Qd = 2^x * Dx + 2^y * Dy + 2^z * Dz + 2^h * Dh + 2^v * Dv + 2^w * Dw - * Rd = A[2,x] * Dx + A[2,y] * Dy + A[2,z] * Dz + A[2,h] * Dh + A[2,v] * Dv + A[2,w] * Dw - * Sd = A[3,x] * Dx + A[3,y] * Dy + A[3,z] * Dz + A[3,h] * Dh + A[3,v] * Dv + A[3,w] * Dw - * 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 - * 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 - * - * A linear system always solvable because the coefficients matrix is - * always not singular due the properties of the matrix A[]. - * - * Resulting speed in x64, with 8 data disks, using a stripe of 256 KiB, - * for a Core i5-4670K Haswell Quad-Core 3.4GHz is: - * - * int8 int32 int64 sse2 ssse3 avx2 - * gen1 13339 25438 45438 50588 - * gen2 4115 6514 21840 32201 - * gen3 814 10154 18613 - * gen4 620 7569 14229 - * gen5 496 5149 10051 - * gen6 413 4239 8190 - * - * Values are in MiB/s of data processed by a single thread, not counting - * generated parity. - * - * You can replicate these results in your machine using the - * "raid/test/speedtest.c" program. - * - * For comparison, the triple parity computation using the power - * coeffients "1,2,2^-1" is only a little faster than the one based on - * the Cauchy matrix if SSSE3 or AVX2 is present. - * - * int8 int32 int64 sse2 ssse3 avx2 - * genz 2337 2874 10920 18944 - * - * In conclusion, the use of power coefficients, and specifically powers - * of 1,2,2^-1, is the best option to implement triple parity in CPUs - * without SSSE3 and AVX2. - * But if a modern CPU with SSSE3 or AVX2 is available, the Cauchy - * matrix is the best option because it provides a fast and general - * approach working for any number of parities. - * - * References: - * [1] Anvin, "The mathematics of RAID-6", 2004 - * [2] MacWilliams, Sloane, "The Theory of Error-Correcting Codes", 1977 - * [3] Blomer, "An XOR-Based Erasure-Resilient Coding Scheme", 1995 - * [4] Roth, "Introduction to Coding Theory", 2006 - * [5] Plank, "Screaming Fast Galois Field Arithmetic Using Intel SIMD Instructions", 2013 - */ - -/** - * Generator matrix currently used. - */ -const uint8_t (*raid_gfgen)[256]; - -void raid_mode(int mode) -{ - if (mode == RAID_MODE_VANDERMONDE) { - raid_gen_ptr[2] = raid_genz_ptr; - raid_gfgen = gfvandermonde; - } else { - raid_gen_ptr[2] = raid_gen3_ptr; - raid_gfgen = gfcauchy; - } -} - -/** - * Buffer filled with 0 used in recovering. - */ -static void *raid_zero_block; - -void raid_zero(void *zero) -{ - raid_zero_block = zero; -} - -/* - * Forwarders for parity computation. - * - * These functions compute the parity blocks from the provided data. - * - * The number of parities to compute is implicit in the position in the - * forwarder vector. Position at index #i, computes (#i+1) parities. - * - * All these functions give the guarantee that parities are written - * in order. First parity P, then parity Q, and so on. - * This allows to specify the same memory buffer for multiple parities - * knowning that you'll get the latest written one. - * This characteristic is used by the raid_delta_gen() function to - * avoid to damage unused parities in recovering. - * - * @nd Number of data blocks - * @size Size of the blocks pointed by @v. It must be a multipler of 64. - * @v Vector of pointers to the blocks of data and parity. - * It has (@nd + #parities) elements. The starting elements are the blocks - * for data, following with the parity blocks. - * Each block has @size bytes. - */ -void (*raid_gen_ptr[RAID_PARITY_MAX])(int nd, size_t size, void **vv); -void (*raid_gen3_ptr)(int nd, size_t size, void **vv); -void (*raid_genz_ptr)(int nd, size_t size, void **vv); - -void raid_gen(int nd, int np, size_t size, void **v) -{ - /* enforce limit on size */ - BUG_ON(size % 64 != 0); - - /* enforce limit on number of failures */ - BUG_ON(np < 1); - BUG_ON(np > RAID_PARITY_MAX); - - raid_gen_ptr[np - 1](nd, size, v); -} - -/** - * Inverts the square matrix M of size nxn into V. - * - * This is not a general matrix inversion because we assume the matrix M - * to have all the square submatrix not singular. - * We use Gauss elimination to invert. - * - * @M Matrix to invert with @n rows and @n columns. - * @V Destination matrix where the result is put. - * @n Number of rows and columns of the matrix. - */ -void raid_invert(uint8_t *M, uint8_t *V, int n) -{ - int i, j, k; - - /* set the identity matrix in V */ - for (i = 0; i < n; ++i) - for (j = 0; j < n; ++j) - V[i * n + j] = i == j; - - /* for each element in the diagonal */ - for (k = 0; k < n; ++k) { - uint8_t f; - - /* the diagonal element cannot be 0 because */ - /* we are inverting matrices with all the square */ - /* submatrices not singular */ - BUG_ON(M[k * n + k] == 0); - - /* make the diagonal element to be 1 */ - f = inv(M[k * n + k]); - for (j = 0; j < n; ++j) { - M[k * n + j] = mul(f, M[k * n + j]); - V[k * n + j] = mul(f, V[k * n + j]); - } - - /* make all the elements over and under the diagonal */ - /* to be zero */ - for (i = 0; i < n; ++i) { - if (i == k) - continue; - f = M[i * n + k]; - for (j = 0; j < n; ++j) { - M[i * n + j] ^= mul(f, M[k * n + j]); - V[i * n + j] ^= mul(f, V[k * n + j]); - } - } - } -} - -/** - * Computes the parity without the missing data blocks - * and store it in the buffers of such data blocks. - * - * This is the parity expressed as Pa,Qa,Ra,Sa,Ta,Ua in the equations. - */ -void raid_delta_gen(int nr, int *id, int *ip, int nd, size_t size, void **v) -{ - void *p[RAID_PARITY_MAX]; - void *pa[RAID_PARITY_MAX]; - int i, j; - int np; - void *latest; - - /* total number of parities we are going to process */ - /* they are both the used and the unused ones */ - np = ip[nr - 1] + 1; - - /* latest missing data block */ - latest = v[id[nr - 1]]; - - /* setup pointers for delta computation */ - for (i = 0, j = 0; i < np; ++i) { - /* keep a copy of the original parity vector */ - p[i] = v[nd + i]; - - if (ip[j] == i) { - /* - * Set used parities to point to the missing - * data blocks. - * - * The related data blocks are instead set - * to point to the "zero" buffer. - */ - - /* the latest parity to use ends the for loop and */ - /* then it cannot happen to process more of them */ - BUG_ON(j >= nr); - - /* buffer for missing data blocks */ - pa[j] = v[id[j]]; - - /* set at zero the missing data blocks */ - v[id[j]] = raid_zero_block; - - /* compute the parity over the missing data blocks */ - v[nd + i] = pa[j]; - - /* check for the next used entry */ - ++j; - } else { - /* - * Unused parities are going to be rewritten with - * not significative data, becase we don't have - * functions able to compute only a subset of - * parities. - * - * To avoid this, we reuse parity buffers, - * assuming that all the parity functions write - * parities in order. - * - * We assign the unused parity block to the same - * block of the latest used parity that we know it - * will be written. - * - * This means that this block will be written - * multiple times and only the latest write will - * contain the correct data. - */ - v[nd + i] = latest; - } - } - - /* all the parities have to be processed */ - BUG_ON(j != nr); - - /* recompute the parity, note that np may be smaller than the */ - /* total number of parities available */ - raid_gen(nd, np, size, v); - - /* restore data buffers as before */ - for (j = 0; j < nr; ++j) - v[id[j]] = pa[j]; - - /* restore parity buffers as before */ - for (i = 0; i < np; ++i) - v[nd + i] = p[i]; -} - -/** - * Recover failure of one data block for PAR1. - * - * Starting from the equation: - * - * Pd = Dx - * - * and solving we get: - * - * Dx = Pd - */ -void raid_rec1of1(int *id, int nd, size_t size, void **v) -{ - void *p; - void *pa; - - /* for PAR1 we can directly compute the missing block */ - /* and we don't need to use the zero buffer */ - p = v[nd]; - pa = v[id[0]]; - - /* use the parity as missing data block */ - v[id[0]] = p; - - /* compute the parity over the missing data block */ - v[nd] = pa; - - /* compute */ - raid_gen(nd, 1, size, v); - - /* restore as before */ - v[id[0]] = pa; - v[nd] = p; -} - -/** - * Recover failure of two data blocks for PAR2. - * - * Starting from the equations: - * - * Pd = Dx + Dy - * Qd = 2^id[0] * Dx + 2^id[1] * Dy - * - * and solving we get: - * - * 1 2^(-id[0]) - * Dy = ------------------- * Pd + ------------------- * Qd - * 2^(id[1]-id[0]) + 1 2^(id[1]-id[0]) + 1 - * - * Dx = Dy + Pd - * - * with conditions: - * - * 2^id[0] != 0 - * 2^(id[1]-id[0]) + 1 != 0 - * - * That are always satisfied for any 0<=id[0]<id[1]<255. - */ -void raid_rec2of2_int8(int *id, int *ip, int nd, size_t size, void **vv) -{ - uint8_t **v = (uint8_t **)vv; - size_t i; - uint8_t *p; - uint8_t *pa; - uint8_t *q; - uint8_t *qa; - const uint8_t *T[2]; - - /* get multiplication tables */ - T[0] = table(inv(pow2(id[1] - id[0]) ^ 1)); - T[1] = table(inv(pow2(id[0]) ^ pow2(id[1]))); - - /* compute delta parity */ - raid_delta_gen(2, id, ip, nd, size, vv); - - p = v[nd]; - q = v[nd + 1]; - pa = v[id[0]]; - qa = v[id[1]]; - - for (i = 0; i < size; ++i) { - /* delta */ - uint8_t Pd = p[i] ^ pa[i]; - uint8_t Qd = q[i] ^ qa[i]; - - /* reconstruct */ - uint8_t Dy = T[0][Pd] ^ T[1][Qd]; - uint8_t Dx = Pd ^ Dy; - - /* set */ - pa[i] = Dx; - qa[i] = Dy; - } -} - -/* - * Forwarders for data recovery. - * - * These functions recover data blocks using the specified parity - * to recompute the missing data. - * - * Note that the format of vectors @id/@ip is different than raid_rec(). - * For example, in the vector @ip the first parity is represented with the - * value 0 and not @nd. - * - * @nr Number of failed data blocks to recover. - * @id[] Vector of @nr indexes of the data blocks to recover. - * The indexes start from 0. They must be in order. - * @ip[] Vector of @nr indexes of the parity blocks to use in the recovering. - * The indexes start from 0. They must be in order. - * @nd Number of data blocks. - * @np Number of parity blocks. - * @size Size of the blocks pointed by @v. It must be a multipler of 64. - * @v Vector of pointers to the blocks of data and parity. - * It has (@nd + @np) elements. The starting elements are the blocks - * for data, following with the parity blocks. - * Each block has @size bytes. - */ -void (*raid_rec_ptr[RAID_PARITY_MAX])( - int nr, int *id, int *ip, int nd, size_t size, void **vv); - -void raid_rec(int nr, int *ir, int nd, int np, size_t size, void **v) -{ - int nrd; /* number of data blocks to recover */ - int nrp; /* number of parity blocks to recover */ - - /* enforce limit on size */ - BUG_ON(size % 64 != 0); - - /* enforce limit on number of failures */ - BUG_ON(nr > np); - BUG_ON(np > RAID_PARITY_MAX); - - /* enforce order in index vector */ - BUG_ON(nr >= 2 && ir[0] >= ir[1]); - BUG_ON(nr >= 3 && ir[1] >= ir[2]); - BUG_ON(nr >= 4 && ir[2] >= ir[3]); - BUG_ON(nr >= 5 && ir[3] >= ir[4]); - BUG_ON(nr >= 6 && ir[4] >= ir[5]); - - /* enforce limit on index vector */ - BUG_ON(nr > 0 && ir[nr-1] >= nd + np); - - /* count the number of data blocks to recover */ - nrd = 0; - while (nrd < nr && ir[nrd] < nd) - ++nrd; - - /* all the remaining are parity */ - nrp = nr - nrd; - - /* enforce limit on number of failures */ - BUG_ON(nrd > nd); - BUG_ON(nrp > np); - - /* if failed data is present */ - if (nrd != 0) { - int ip[RAID_PARITY_MAX]; - int i, j, k; - - /* setup the vector of parities to use */ - for (i = 0, j = 0, k = 0; i < np; ++i) { - if (j < nrp && ir[nrd + j] == nd + i) { - /* this parity has to be recovered */ - ++j; - } else { - /* this parity is used for recovering */ - ip[k] = i; - ++k; - } - } - - /* recover the nrd data blocks specified in ir[], */ - /* using the first nrd parity in ip[] for recovering */ - raid_rec_ptr[nrd - 1](nrd, ir, ip, nd, size, v); - } - - /* recompute all the parities up to the last bad one */ - if (nrp != 0) - raid_gen(nd, ir[nr - 1] - nd + 1, size, v); -} - -void raid_data(int nr, int *id, int *ip, int nd, size_t size, void **v) -{ - /* enforce limit on size */ - BUG_ON(size % 64 != 0); - - /* enforce limit on number of failures */ - BUG_ON(nr > nd); - BUG_ON(nr > RAID_PARITY_MAX); - - /* enforce order in index vector for data */ - BUG_ON(nr >= 2 && id[0] >= id[1]); - BUG_ON(nr >= 3 && id[1] >= id[2]); - BUG_ON(nr >= 4 && id[2] >= id[3]); - BUG_ON(nr >= 5 && id[3] >= id[4]); - BUG_ON(nr >= 6 && id[4] >= id[5]); - - /* enforce limit on index vector for data */ - BUG_ON(nr > 0 && id[nr-1] >= nd); - - /* enforce order in index vector for parity */ - BUG_ON(nr >= 2 && ip[0] >= ip[1]); - BUG_ON(nr >= 3 && ip[1] >= ip[2]); - BUG_ON(nr >= 4 && ip[2] >= ip[3]); - BUG_ON(nr >= 5 && ip[3] >= ip[4]); - BUG_ON(nr >= 6 && ip[4] >= ip[5]); - - /* if failed data is present */ - if (nr != 0) - raid_rec_ptr[nr - 1](nr, id, ip, nd, size, v); -} - |