Vectorization in the Polyhedral Model

Transcription

1 Vectorzaton n the Polyhedral Model Lous-Noël Pouchet pouchet@cse.oho-state.edu Dept. of Computer Scence and Engneerng, the Oho State Unversty October

2 Introducton: Overvew Vectorzaton: Detecton of parallel loops Vectorzaton n Pluto Vectorzaton n PoCC Algnment ssues OSU 2

3 Vectorzaton: Vectorzaton Pre-transformaton Exhbt nner-most parallel loops Ensure (f needed) strde- access Peel/shft for better algnment Code generaton Generate vector nstructon for vectorzable loops Hardware consderatons: Speed of dfferent nstructons Algnment constrants OSU 3

4 Vectorzaton: Vectorzaton n the Polyhedral Model Man consderaton: pre-transformaton Fnd a transformaton (schedulng) for nner parallelsm Complete the transformaton for algnment Detecton vs. transformatons Detect a loop s parallel, permutable, algned, etc. Transform: move parallel loops nwards, create parallel dmensons OSU 4

5 Affne Schedulng: Affne Schedulng Defnton (Affne schedule) Gven a statement S, a p-dmensonal affne schedule Θ R s an affne form on the outer loop terators x S and the global parameters n. It s wrtten: Θ S ( x S ) = T S x S n, T S K p dm( x S)+dm( n)+ A schedule assgns a tmestamp to each executed nstance of a statement If T s a vector, then Θ s a one-dmensonal schedule If T s a matrx, then Θ s a multdmensonal schedule OSU 5

6 Affne Schedulng: Program Transformatons Orgnal Schedule S: C[][j] = 0; S2: C[][j] += A[][k]* Θ S. x S = j n j Θ S2. x S2 = k n C[][j] = 0; C[][j] += A[][k]* Represent Statc Control Parts (control flow and dependences must be statcally computable) Use code generator (e.g. CLooG) to generate C code from polyhedral representaton (provded teraton domans + schedules) OSU 6

7 Affne Schedulng: Program Transformatons Orgnal Schedule S: C[][j] = 0; S2: C[][j] += A[][k]* Θ S. x S = j n j Θ S2. x S2 = k n C[][j] = 0; C[][j] += A[][k]* Represent Statc Control Parts (control flow and dependences must be statcally computable) Use code generator (e.g. CLooG) to generate C code from polyhedral representaton (provded teraton domans + schedules) OSU 7

8 Affne Schedulng: Program Transformatons Orgnal Schedule S: C[][j] = 0; S2: C[][j] += A[][k]* Θ S. x S = j n j Θ S2. x S2 = k n C[][j] = 0; C[][j] += A[][k]* Represent Statc Control Parts (control flow and dependences must be statcally computable) Use code generator (e.g. CLooG) to generate C code from polyhedral representaton (provded teraton domans + schedules) OSU 8

9 Affne Schedulng: Program Transformatons Dstrbute loops S: C[][j] = 0; S2: C[][j] += A[][k]* Θ S. x S = j n j Θ S2. x S2 = k n C[][j] = 0; for ( = n; < 2*n; ++) C[-n][j] += A[-n][k]* All nstances of S are executed before the frst S2 nstance OSU 9

10 Affne Schedulng: Program Transformatons Dstrbute loops + Interchange loops for S2 S: C[][j] = 0; S2: C[][j] += A[][k]* Θ S. x S = j n j Θ S2. x S2 = k n C[][j] = 0; for (k = n; k < 2*n; ++k) C[][j] += A[][k-n]* B[k-n][j]; The outer-most loop for S2 becomes k OSU 0

11 Affne Schedulng: Program Transformatons Illegal schedule S: C[][j] = 0; S2: C[][j] += A[][k]* Θ S. x S = 0 0. j n j Θ S2. x S2 = k n C[][j] += A[][k]* for ( = n; < 2*n; ++) C[-n][j] = 0; All nstances of S are executed after the last S2 nstance OSU

12 Affne Schedulng: Program Transformatons A legal schedule S: C[][j] = 0; S2: C[][j] += A[][k]* Θ S. x S = 0 0. j n 0 0 j Θ S2. x S2 = k n for ( = n; < 2*n; ++) C[][j] = 0; for (k= n+; k<= 2*n; ++k) C[][j] += A[][k-n-]* B[k-n-][j]; Delay the S2 nstances Constrants must be expressed between Θ S and Θ S2 OSU 2

13 Affne Schedulng: Program Transformatons Implct fne-gran parallelsm S: C[][j] = 0; S2: C[][j] += A[][k]* Θ S. x S = ( ). j n j Θ S2. x S2 = ( ). k n p C[][j] = 0; for (k = n; k < 2*n; ++k) p p C[][j] += A[][k-n]* B[k-n][j]; Number of rows of Θ number of outer-most sequental loops OSU 3

14 Affne Schedulng: Program Transformatons Representng a schedule S: C[][j] = 0; S2: C[][j] += A[][k]* Θ S. x S = 0 0. j n 0 0 j Θ S2. x S2 = k n for ( = n; < 2*n; ++) C[][j] = 0; for (k= n+; k<= 2*n; ++k) C[][j] += A[][k-n-]* B[k-n-][j]; Θ. x = ( j j k n n ) T OSU 4

15 Affne Schedulng: Program Transformatons Representng a schedule S: C[][j] = 0; S2: C[][j] += A[][k]* Θ S. x S = 0 0. j n 0 0 j Θ S2. x S2 = k n for ( = n; < 2*n; ++) C[][j] = 0; for (k= n+; k<= 2*n; ++k) C[][j] += A[][k-n-]* B[k-n-][j]; Θ. x = ( j j k n n ) T ı p c OSU 5

16 Affne Schedulng: Program Transformatons Representng a schedule S: C[][j] = 0; S2: C[][j] += A[][k]* Θ S. x S = 0 0. j n 0 0 j Θ S2. x S2 = k n for ( = n; < 2*n; ++) C[][j] = 0; for (k= n+; k<= 2*n; ++k) C[][j] += A[][k-n-]* B[k-n-][j]; ı p c Transformaton reversal skewng nterchange fuson dstrbuton peelng shftng Descrpton Changes the drecton n whch a loop traverses ts teraton range Makes the bounds of a gven loop depend on an outer loop counter Exchanges two loops n a perfectly nested loop, a.k.a. permutaton Fuses two loops, a.k.a. jammng Splts a sngle loop nest nto many, a.k.a. fsson or splttng Extracts one teraton of a gven loop Allows to reorder loops OSU 6

17 Affne Schedulng: Pctured Example Example of 2 extended dependence graphs OSU 7

18 Affne Schedulng: Checkng the Legalty of a Schedule Exercse: gven the dependence polyhedra, check f a schedule s legal eq eq 0 0 D : S S D 2 : 0 0 n 0. S S n Θ = 2 Θ = OSU 8

19 Affne Schedulng: Checkng the Legalty of a Schedule Exercse: gven the dependence polyhedra, check f a schedule s legal eq eq 0 0 D : S S D 2 : 0 0 n 0. S S n Θ = 2 Θ = Soluton: check for the emptness of the polyhedron [ ] S D P : S. S S n where: S S gets the consumer nstances scheduled after the producer ones For Θ =, t s S S, whch s non-empty OSU 9

20 Affne Schedulng: Detectng Parallel Dmensons Exercse: Wrte an algorthm whch detects f an nner-most loop s parallel OSU 20

21 Affne Schedulng: Lmtaton of Operatng on Dmensons As soon as there s one non-parallel teraton, the dmenson s not parallel Fuson/dstrbuton mpacts parallelsm After fuson/dstrbuton: On the generated code, some nner loop may be parallel The schedule for the program may not show the whole dmenson as parallel Exercse: Fnd a program where all schedule dmensons are sequental, but there are nner-most parallel loops OSU 2

22 Pre-Vectorzaton n the Polyhedral Model: Pluto s Approach for Pre-Vectorzaton Maxmze the number of outer-most parallel/permutable dmenson 2 An outer parallel dmenson can be moved nwards 3 Proceed from the nner-most dmenson, push nwards the "closest" parallel dmenson 4 Mssng consderatons: Algnment / strde- s not consdered Unable to model partally parallel dmensons (eg, those parallel only for some loop nests and not all) OSU 22

23 Pre-Vectorzaton n the Polyhedral Model: PoCC s Approach for Pre-Vectorzaton Very smple: decouple the problem Let Pluto transform the code for tlng, parallelsm, etc. Generate the transformed code Re-analyze the transformed code, to extract ts polyhedral representaton Operate on each loop nest ndvdually Not lmted to have a full dmenson as parallel (local to a loop nest now) Smple model to detect parallel loops wth good algnment Dfferent cost models can be used Possble pre-transformatons for vectorzaton: All of them! However, lmt to shft+peel+permute OSU 23

24 Pre-Vectorzaton n the Polyhedral Model: Strde- Access Defnton (Data Dstance Vector between two references) Consder two access functons fa and f A 2 to the same array A of dmenson n. Let ı and ı be two teratons of the nnermost loop. The data dstance vector s defned as an n-dmensonal vector δ(ı,ı ) f A,fA 2 = f A (ı) f A 2(ı ). Defnton (Strde-one memory access for an access functon) Consder an access functon f A surrounded by an nnermost loop. It has strde-one access f ı, δ(ı,ı + ) fa,f A = (0,...,0,). OSU 24

25 Pre-Vectorzaton n the Polyhedral Model: Detectng Strde- Access Exercse: Wrte an algorthm whch detects f an nner-most loop has strde- access for all memory references OSU 25