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Loop Transformations

Loop Transformations. Motivation Loop level transformations catalogus Loop merging Loop interchange Loop unrolling Unroll-and-Jam Loop tiling Loop Transformation Theory and Dependence Analysis. Thanks for many slides go to the DTSE people from IMEC and

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Loop Transformations

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  1. Loop Transformations • Motivation • Loop level transformations catalogus • Loop merging • Loop interchange • Loop unrolling • Unroll-and-Jam • Loop tiling • Loop Transformation Theory and Dependence Analysis Thanks for many slides go to the DTSE people from IMEC and Dr. Peter Knijnenburg († 2007) from Leiden University

  2. Loop Transformations • Change the order in which the iteration space is traversed. • Can expose parallelism, increase available ILP, or improve memory behavior. • Dependence testing is required to check validity of transformation.

  3. j A i Why loop trafos Example 1: in-place mapping for (j=1; j<M; j++) for (i=0; i<N; i++) A[i][j] = f(A[i][j-1]); for (i=0; i<N; i++) OUT[i] = A[i][M-1]; for (i=1; i<N; i++){ for (j=0; j<M; j++) A[i][j] = f(A[i][j-1]); OUT[i] = A[i][M-1]; } OUT

  4. Foreground memory External memory interface CPU Background memory Why loop trafos Example 2: memory allocation for (i=0; i<N; i++) B[i] = f(A[i]); for (i=0; i<N; i++) C[i] = g(B[i]); for (i=0; i<N; i++){ B[i] = f(A[i]); C[i] = g(B[i]); } N cyc. 2N cyc. N cyc. 2 background ports 1 backgr. + 1 foregr. ports

  5. merge (fusion) improve locality bump extend/reduce body split reverse improve regularity interchange improve locality/regularity skew tiling index split unroll unroll-and-jam Loop transformation catalogue

  6. Loop Merge • Improve locality • Reduce loop overhead for Ia = exp1 to exp2 A(Ia) for Ib = exp1 to exp2 B(Ib) for I = exp1 to exp2 A(I) B(I) 

  7. Location Production/Consumption Consumption(s) Time Location Production/Consumption Consumption(s) Time Loop Merge:Example of locality improvement: Consumptions of second loop closer to productions and consumptions of first loop • Is this always the case after merging loops? for (i=0; i<N; i++) B[i] = f(A[i]); for (j=0; j<N; j++) C[j] = f(B[j],A[j]); for (i=0; i<N; i++) B[i] = f(A[i]); C[i] = f(B[i],A[i]);

  8. Loop Merge not always allowed ! • Data dependencies from first to second loopcan block Loop Merge • Merging is allowed if  I: cons(I) in loop 2  prod(I) in loop 1 • Enablers: Bump, Reverse, Skew for (i=0; i<N; i++) B[i] = f(A[i]); for (i=0; i<N; i++) C[i] = g(B[N-1]); for (i=0; i<N; i++) B[i] = f(A[i]); for (i=2; i<N; i++) C[i] = g(B[i-2]); N-1 >= i i-2 < i

  9. Loop Bump for I = exp1 to exp2 A(I)  for I = exp1+N to exp2+N A(I-N)

  10. Loop Bump: Example as enabler for (i=2; i<N; i++) B[i] = f(A[i]); for (i=0; i<N-2; i++) C[i] = g(B[i+2]); i+2 > i  merging not possible Loop Bump for (i=2; i<N; i++) B[i] = f(A[i]); for (i=2; i<N; i++) C[i-2] = g(B[i+2-2]);  i+2–2 = i  merging possible for (i=2; i<N; i++) B[i] = f(A[i]); C[i-2] = g(B[i]); Loop Merge 

  11. Loop Extend for I = exp1 to exp2 A(I) exp3 exp1 exp4 exp2  for I = exp3 to exp4 if Iexp1and Iexp2 A(I)

  12. Loop Extend: Example as enabler for (i=0; i<N; i++) B[i] = f(A[i]); for (i=2; i<N+2; i++) C[i-2] = g(B[i]); for (i=0; i<N+2; i++) if(i<N) B[i] = f(A[i]); for (i=0; i<N+2; i++) if(i>=2) C[i-2] = g(B[i); Loop Extend  for (i=0; i<N+2; i++) if(i<N) B[i] = f(A[i]); if(i>=2) C[i-2] = g(B[i); Loop Merge 

  13. Loop Reduce for I = exp1 to exp2 if Iexp3and Iexp4 A(I)  for I = max(exp1,exp3) to min(exp2,exp4) A(I)

  14. Loop Body Split A(I) must be single-assignment: its elements should be written once for I = exp1 to exp2 A(I) B(I) for Ia = exp1 toexp2 A(Ia) for Ib = exp1 to exp2 B(Ib) 

  15. Loop Body Split: Example as enabler for (i=0; i<N; i++) A[i] = f(A[i-1]); B[i] = g(in[i]); for (j=0; j<N; j++) C[i] = h(B[i],A[N]);  Loop Body Split for (i=0; i<N; i++) A[i] = f(A[i-1]); for (k=0; k<N; k++) B[k] = g(in[k]); for (j=0; j<N; j++) C[j] = h(B[j],A[N]); for (i=0; i<N; i++) A[i] = f(A[i-1]); for (j=0; j<N; j++) B[j] = g(in[j]); C[j] = h(B[j],A[N]);  Loop Merge

  16. Location Production Consumption Time Location Time Loop Reverse for I = exp1 to exp2 A(I) for I = exp2downto exp1 A(I)  OR for I = exp1 to exp2 A(exp2-(I-exp1))

  17. Loop Reverse: Satisfy dependencies No loop-carried dependencies allowed ! A[0] = … for (i=1; i<N; i++) A[i] = f(A[i-1]); A[0] = … for (i=1; i<=N; i++) A[i]= A[i-1]+ f(…); Enabler: data-flow transformations (+ is associative) A[N] = … for (i=N-1; i>=0; i--) A[i]= A[i+1]+ f(…);

  18. Loop Reverse: Example as enabler for (i=0; i<=N; i++) B[i] = f(A[i]); for (i=0; i<=N; i++) C[i] = g(B[N-i]); N–i > i  merging not possible Loop Reverse for (i=0; i<=N; i++) B[i] = f(A[i]); for (i=0; i<=N; i++) C[N-i] = g(B[N-(N-i)]);  N-(N-i) = i  merging possible Loop Merge  for (i=0; i<=N; i++) B[i] = f(A[i]); C[N-i] = g(B[i]);

  19. Loop Interchange for I1 = exp1 to exp2 for I2 = exp3 to exp4 A(I1, I2) for I2 = exp3 to exp4 for I1 = exp1 to exp2 A(I1, I2) 

  20. Loop Interchange: index traversal j j Loop Interchange i i for(i=0; i<W; i++) for(j=0; j<H; j++) A[i][j] = …; for(j=0; j<H; j++) for(i=0; i<W; i++) A[i][j] = …;

  21. Loop Interchange • Validity: check dependence direction vectors. • Mostly used to improve cache behavior. • The innermost loop (loop index changes fastest) should (only) index the right-most array index expression in case of row-major storage, like in C. • Can improve execution time by 1 or 2 orders of magnitude.

  22. Loop Interchange (cont’d) • Loop interchange can also expose parallelism. • If an inner-loop does not carry a dependence (entry in direction vector equals ‘=‘), this loop can be executed in parallel. • Moving this loop outwards increases the granularity of the parallel loop iterations.

  23. Loop Interchange:Example of locality improvement • In second loop: • A[j] reused N times (temporal locality) • However: loosing spatial locality in B (assuming row major ordering) • => Exploration required for (i=0; i<N; i++) for (j=0; j<M; j++) B[i][j] = f(A[j],B[i][j-1]); for (j=0; j<M; j++) for (i=0; i<N; i++) B[i][j] = f(A[j],B[i][j-1]);

  24. Loop Interchange: Satisfy dependencies • No loop-carried dependencies allowedunless  I2: prod(I2)  cons(I2) for (i=0; i<N; i++) for (j=0; j<M; j++) A[i][j] = f(A[i-1][j+1]); for (i=0; i<N; i++) for (j=0; j<M; j++) A[i][j] = f(A[i-1][j]); • Enablers: • Data-flow transformations • Loop Bump

  25. Loop Skew:Basic transformation for I1 = exp1 to exp2 for I2 = exp3 to exp4 A(I1, I2) for I1 = exp1 to exp2 for I2 = exp3+.I1+ to exp4+.I1+ A(I1, I2-.I1-) for I1 = exp1 to exp2 for I2 = exp3+.I1+ to exp4+.I1+ A(I1, I2-.I1-) 

  26. j j i i Loop Skew Loop Skewing for(j=0;j<H;j++) for(i=0;i<W;i++) A[i][j] = ...; for(j=0; j<H; j++) for(i=0+j; i<W+j;i++) A[I-j][j] = ...;

  27. Loop Extend & Interchange  Loop Skew: Example as enabler of regularity improvement for (i=0; i<N; i++) for (j=0; j<M; j++) … = f(A[i+j]); for (i=0; i<N; i++) for (j=i; j<i+M; j++) … = f(A[j]); Loop Skew  for (j=0; j<N+M; j++) for (i=0; i<N; i++) if (j>=i && j<i+M) … = f(A[j]);

  28. Loop Tiling for I = 0 to exp1 . exp2 A(I) Tile size exp1 Tile factor exp2 for I1 = 0 to exp2 for I2 = exp1.I1 to exp1.(I1 +1) A(I2) 

  29. Loop Tiling i Loop Tiling j i for(i=0;i<9;i++) A[i] = ...; for(j=0; j<3; j++) for(i=4*j; i<4*j+4; i++) if (i<9) A[i] = ...;

  30. Loop Tiling • Improve cache reuse by dividing the iteration space into tiles and iterating over these tiles. • Only useful when working set does not fit into cache or when there exists much interference. • Two adjacent loops can legally be tiled if they can legally be interchanged.

  31. for(i=0; i<N; i++) for(j=0; j<N; j++) A[i][j] = B[j][i]; 2-D Tiling Example for(TI=0; TI<N; TI+=16) for(TJ=0; TJ<N; TJ+=16) for(i=TI; i<min(TI+16,N); i++) for(j=TJ; j<min(TJ+16,N); j++) A[i][j] = B[j][i];

  32. 2-D Tiling • Show index space traversal

  33. What is the best Tile Size? • Current tile size selection algorithms use a cache model: • Generate collection of tile sizes; • Estimate resulting cache miss rate; • Select best one. • Only take into account L1 cache. • Mostly do not take into account n-way associativity.

  34. Loop Index Split for Ia = exp1 to exp2 A(Ia) for Ia = exp1 top A(Ia) for Ib = p+1 to exp2 A(Ib) 

  35. Loop Unrolling • Duplicate loop body and adjust loop header. • Increases available ILP, reduces loop overhead, and increases possibilities for common subexpression elimination. • Always valid !!

  36. (Partial) Loop Unrolling for I = exp1 to exp2 A(I) A(exp1) A(exp1+1) A(exp1+2) … A(exp2) for I = exp1/2 to exp2 /2 A(2l) A(2l+1) 

  37. Loop Unrolling: Downside • If unroll factor is not divisor of trip count, then need to add remainder loop. • If trip count not known at compile time, need to check at runtime. • Code size increases which may result in higher I-cache miss rate. • Global determination of optimal unroll factors is difficult.

  38. Loop Unroll: Example of removal of non-affine iterator for (L=1; L < 4; L++) for (i=0; i < (1<<L); i++) A(L,i); for (i=0; i < 2; i++) A(1,i); for (i=0; i < 4; i++) A(2,i); for (i=0; i < 8; i++) A(3,i);

  39. Unroll-and-Jam • Unroll outerloop and fuse new copies of the innerloop. • Increases size of loop body and hence available ILP. • Can also improve locality.

  40. Unroll-and-Jam Example • More ILP exposed • Spatial locality of B enhanced for (i=0;i<N;i++) for (j=0;j<N;j++) A[i][j] = B[j][i]; for (i=0;i<N;i++) for (j=0;j<N;j++) A[i][j] = B[j][i]; for (j=0;j<N;j++) A[i+1][j] = B[j][i+1]; for (i=0; i<N; i+=2) for (j=0; j<N; j++) { A[i][j] = B[j][i]; A[i+1][j] = B[j][i+1]; }

  41. Simplified loop transformation script • Give all loops same nesting depth • Use dummy 1-iteration loops if necessary • Improve regularity • Usually applying loop interchange or reverse • Improve locality • Usually applying loop merge • Break data dependencies with loop bump/skew • Sometimes loop index split or unrolling is easier

  42. Loop transformation theory • Iteration spaces • Polyhedral model • Dependency analysis

  43. Technical Preliminaries (1) do i = 2, N do j = i, N x[i,j]=0.5*(x[i-1,j]+ x[i,j-1]) enddo (1) j • Address expr (1) 4 • perfect loop nest • iteration space • dependence vector 3 2 i 2 3 4

  44. Technical Preliminaries (2) Switch loop indexes: do m = 2, N do n = 2, m x[n,m]=0.5*(x[n-1,m]+ x[n,m-1]) enddo (2) (2) affine transformation

  45. Polyhedral Model • Polyhedron is set x : Ax  cfor some matrix A and bounds vector c • Polyhedra (or Polytopes) are objects in a many-dimensional space without holes • Iteration spaces of loops (with unit stride) can be represented as polyhedra • Array accesses and loop transformations can be represented as matrices

  46. Iteration Space • A loop nest is represented as BI  b for iteration vector I • Example: for(i=0; i<10;i++) -1 0 0 for(j=i; j<10;j++) 1 0 i 9 1 -1 j 0 0 1 9 

  47. Array Accesses • Any array access A[e1][e2] for linear index expressions e1 and e2 can be represented as an access matrix and offset vector. A + a • This can be considered as a mapping from the iteration space into the storage space of the array (which is a trivial polyhedron)

  48. Unimodular Matrices • A unimodular matrix T is a matrix with integer entries and determinant 1. • This means that such a matrix maps an object onto another object with exactly the same number of integer points in it. • Its inverse T¹ always exist and is unimodular as well.

  49. Types of Unimodular Transformations • Loop interchange • Loop reversal • Loop skewing for arbitrary skew factor • Since unimodular transformations are closed under multiplication, any combination is a unimodular transformation again.

  50. Application • Transformed loop nest is given by AT¹ I’  a • Any array access matrix is transformed into AT¹. • Transformed loop nest needs to be normalized by means of Fourier-Motzkin elimination to ensure that loop bounds are affine expressions in more outer loop indices.

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