Parallel Reed/Solomon Coding on Multicore Processors

Transcription

1 / 8 Parallel Reed/Solomon Coding on Multicore Processors Peter Sobe Institute of Computer Engineering University of Luebeck, Germany sobe@iti.uni-luebeck.de currently at Institute of Computer Science University of Potsdam, Germany SNAPI Workshop, May 3rd,

2 / 8 Contents Basics of Erasure-tolerant Coding Cauchy Reed/Solomon 3 Equation-based Coding Parallel Coding 5 Evaluation

3 3 / 8 Erasure-tolerant Codes k data storage resources, e.g. disks m redundant resources regular data striping across k resources encoding: calculation of m independent redundant blocks k : original data m : redundant data a code tolerates f failed storage resources: f m Criteria: Number tolerated faults: f = m as the optimum Flexibility, when choosing k, m Computational cost for en- and decoding

4 / 8 Cauchy Reed/Solomon Encoding: Multiplication of original data word o with a generator matrix G Example Reed/Solomon, 5+: operations +, within GF( 3 ) h o i» I a = = G o = c j 7 3 c = G sub ff o o as a Cauchy-Reed/Solomon code: projection to GF( ), binary logic operations XOR, AND o 5 storage resources with 3 partitions each ressources units u d u u u3 d u u5 u6 d u 7 u8 u9 d3 u u u d u 3 u a G sub a sub matrix of G XOR XOR XOR c redundant resources u5 d u 6 5 u7 u8 d 6 u 9 u

5 5 / 8 Cauchy Reed/Solomon Decoding: system used for data recalculation when nd and 3rd resource fail: operations +, within GF( 3 ) 8 >< o = >: o = G a >= >< >; >: d d5 d6 d3 d 9 >= >; by Cauchy-Reed/Solomon: projection to GF( ), binary logic operations XOR, AND a o d d d d3 d u u u u3 u u5 u6 u 7 u8 u9 u u u u 3 u c G u5 d u 6 5 u7 u8 d u 9 6 u * d d 5 d6 d 3 d = d d d d3 d

6 6 / 8 Cauchy Reed/Solomon: Equation-based definition Equations refer different bits within storage resources different bits units on resources partitions on disks Unit assignment for k=5, m= data parities resource r r r r 3 r r 5 r 6 units number of units per resource (ω) ω = 3, generally ω > k + m data 6 redundancy

7 7 / 8 Cauchy Reed/Solomon: Equation-based definition Coding algorithm is an execution of several : either: instant application on data or: store and transform, apply them on a sequence of code words Example of a 5+ Reed/Solomon code: direct encoding (5 XOR op.) 5 = XOR(, 3,, 5, 7, 9,, ) 6 = XOR(,, 3, 7, 8, 9,,, 3) 7 = XOR(, 3,, 6, 8,,, ) 8 = XOR(,,, 6, 7, 8,,, 3) 9 = XOR(,,,, 5, 6, 9,, ) = XOR(,, 3, 5, 6, 7,, ) encoding (33 XOR op.) 5 = XOR(B, C, D) 6 = XOR(D, E, F) 7 = XOR(3,, 8, E, H) 8 = XOR(,, 6, 7, C, F) 9 = XOR(,, 9,, B, H) = XOR(5, 7,,, A, G) A = XOR(, 3) E = XOR(, ) B = XOR(, 5) F = XOR(, 8, 3) C = XOR(, ) G = XOR(, 6) D = XOR(7, 9, A) H = XOR(, G) direct decoding ( XOR op.) 6 = XOR(, 5, 7, 8,, 3, ) 7 = XOR(, 3, 5, 5, 7, 8, 9,,, 3) 8 = XOR(,, 6, 9,,, 3, ) 9 = XOR(, 3,, 5, 6, 7, 9,, 3) = XOR(,, 5, 5, 9,, ) = XOR(, 3, 5, 6, 8,, ) decoding (9 XOR op.) 6 = XOR(B, C) 7 = XOR(5, C, D, F) 8 = XOR(9,, A, G) 9 = XOR(3, 7, 3, D, G) = XOR(,, B, D) = XOR(5, 6, 8,, F) A = XOR(, 3) D = XOR(5, 9) B = XOR(, 5, ) F = XOR(, 3, ) C = XOR(7, 8, A) G = XOR(,,, 6)

8 8 / 8 Equations for coding Separation: equation preparation equation interpretation for coding equation generator written data encoding encoder failure description storage system decoding decoder read data

9 8 / 8 Equations for coding Separation: equation preparation equation interpretation for coding one time, a few milliseconds, kbytes written data encoding encoder equation generator failure description storage system decoding decoder read data

10 8 / 8 Equations for coding Separation: equation preparation equation interpretation for coding equation generator written data encoding encoder failure description storage system decoding decoder read data many times hours... days, MBytes... TBytes

11 8 / 8 Equations for coding Separation: equation preparation equation interpretation for coding equation generator written data encoding encoder failure description storage system decoding decoder read data many times hours... days MByte... TByte

12 Equations for coding Separation: equation preparation equation interpretation for coding equation generator written data encoding encoder failure description storage system triggered by fault detection decoding decoder read data 8 / 8

13 8 / 8 Equations for coding Separation: equation preparation equation interpretation for coding written data encoding encoder equation generator failure description storage system seldom a few millisec., kbytes decoding decoder read data

14 8 / 8 Equations for coding Separation: equation preparation equation interpretation for coding equation generator written data encoding encoder failure description storage system decoding decoder read data many times hours... days MByte... TByte

15 A 3 B 5 C A 3 7 D B 9 5 A E C 7 F D A G A E 6 3 H B F G C G B C D D H D E F 9 A G 7 E E H B C D F 6 7 C F D E F G B H 6 H 5 7 A G 6 G 9 B C D B H D E F 5 7 A G E H C F 9 9 B H 5 7 A G 9 / 8 Parallel Coding Obvious parallelism: block parallel coding same coding function on different data blocks a core interprets all a core streams only a part of the input data tile tile 6 computation on core 6 computation on core tile tile 6 original data storage resources redundancy storage resources

16 / 8 Parallel Coding - Equation oriented Equation-oriented coding a core interprets dedicated a core streams data which is refered by the dedicated on core (7),(),() on core (8),(),() on core 3 (3),(),(3) on core (9),() on core 5 (),(5) redundancy storage resources original data storage resources on core 6 equation (6)

17 / 8 Parallel Coding Schedules encoding and decoding extended to schedules schedule: assigned to cores XOR ops assigned to time steps data dependencies resolved cores steps B C D 7 9 A 3 F E D E H G C F 5 9 B H A G Schedule preparation: stacking of

18 / 8 Parallel Coding Schedules Stacking of Encoding, 33 XOR operations Terminal 5 = XOR(B,C,D) 6 = XOR(D,E,F) 7 = XOR(3,,8,E,H) 8 = XOR(,,6,7,C,F) 9 = XOR(,,9,,B,H) = XOR(5,7,,,A,G) Temporary A = XOR(,3) B = XOR(,5) C = XOR(,) D = XOR(7,9,A) E = XOR(,) F = XOR(,8,3) G = XOR(,6) H = XOR(,G) cores steps B C D 7 9 A 3 F E D E H G C F 5 9 B H A G temporary units required G A B,A, D,G, H F,C E temporary units available A,B, H D,F C,G,E

19 3 / 8 Evaluation Question: Is equation-oriented parallel coding beneficial? Criteria: accessed data per core number of referenced storage resources (input, output) number of referenced units per resource and per core multiplicity of references number of temporary results taken from other cores number of time steps of a schedule (under absence of access delays, and synchrony of XOR operations)

20 / 8 Evaluation.8.7 ratio: accessed data per core 7 6 # accessed input resources block parallel direct eqn. oriented.66 improved block parallel direct eqn. oriented improved Lower values are better

21 5 / 8 Evaluation 3.5 # accessed output resources 3.5 # refered units per resource, per core block parallel direct eqn. oriented improved block parallel direct eqn. oriented improved Lower values are better

22 6 / 8 Evaluation.5 # multiple accesses (local on core) 3.5 # acc. distant temporary units block parallel. direct.3 eqn. oriented.3 improved block parallel. direct.66 eqn. oriented.6 improved Lower values are better

23 7 / 8 Evaluation Equation-oriented parallel coding: only! 33 schedule length improve: selecting good Cauchy matrices Performance benefits: minimal schedule length multiple accesses reduced block parallel direct eqn. oriented improved locality of accesses (resources, units) Performance obstacles: access to temporary results from other cores

24 8 / 8 Summary Cauchy-Reed/Solomon code: XOR based Decomposition of coding into several parts, described by Equations: parameterize the encoding and decoding function Schedules: pre-calculated placement of on cores Iterative schedules: concentration of data accesses of a core on local regions Advantage for software-based coding performance on multicore processors