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Dependence Testing

Dependence Testing. Optimizing Compilers for Modern Architectures, Chapter 3 Allen and Kennedy Presented by Rachel Tzoref and Rotem Oshman. The General Problem. DO i 1 = L 1 , U 1 DO i 2 = L 2 , U 2 ... DO i n = L n , U n

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Dependence Testing

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  1. Dependence Testing Optimizing Compilers for Modern Architectures, Chapter 3Allen and Kennedy Presented by Rachel Tzoref and Rotem Oshman

  2. The General Problem DO i1 = L1, U1 DO i2 = L2, U2 ... DO in = Ln, Un S1A(f1(i1,...,in),...,fm(i1,...,in)) = ... S2... = A(g1(i1,...,in),...,gm(i1,...,in)) ENDDO ... ENDDO ENDDO Dependence exists , 9 iteration vectors , such that fj()=gj() for all j,1· j · m

  3. Basic Definitions • Index: A loop variable • Subscript: The pair of expressions that appear in a certain coordinate in a pair of array references A(i,j,m) = A(i,k,j) + C ‹i,i›is the first subscript ‹j,k›is the second subscript ‹m,j›is the third subscript

  4. Distance and Direction Vectors - Reminder DO I=1, N DO J=1, M DO K=1, L A(I+1,J,K-1) = A(I,J,K)+10 ENDDO ENDDO ENDDO Distance Vector: (1, 0, -1) Direction Vector: (<, =, >)

  5. What can we do? INPUT N; DO i = 1, 100 DO j = 1, 100 A(N+(i+j)3) = A(i(j+N)+(ji)+(j2+i2)) ENDDO ENDDO • The general case – the problem is undecidable during compile time

  6. What can we do? • Most subscripts are polynomials DO i = 1, 100 DO j = 1, 100 DO k = 1, 100 A(i2+j2,4¢k3) = A(k2,i4) ENDDO ENDDO ENDDO • Problem reduced to solving a system of Diophantine equations • Too complex: consider only linear subscript expressions

  7. What can we do? DO i = 1, 100 DO j = 1, 100 DO k = 1, 100 A(i+j,4¢k) = A(k,i) ENDDO ENDDO ENDDO • Finding integer solutions to a system of linearDiophantine Equations is NP-Complete • Testing may have to be imprecise

  8. Directions output by testing Directions where dependence exists Our Goal - Conservative Testing • Result = “no dependence” ) no dependence • Result = “dependence” + direction vector(s) ) dependence may or may not exist • Code is always correct, but may be less than optimal

  9. What About Nonlinear Subscripts? • Can’t analyze something ) assume dependence • Example: DO i = 1, 50 DO j = 51, 100 A(i3+j2,i) = A(j4,j) ENDDO ENDDO • May still prove independence or limit possible direction vectors

  10. Definition: Subscript Complexity A(5,i+1,j) = A(1,i,k) + C • ZIV Subscript: no indices • SIV Subscript: one index • MIV Subscript: more than one index

  11. Definition: Separability • A subscript is separable if its indices do not occur in other subscripts A(i+1,j) = A(k,j) + C Both subscripts are separable • If two different subscripts contain the same index they are coupled A(i,j,j) = A(i+1,j,k) + C Second and third subscripts are coupled

  12. Coupled Subscript Groups • Why are they important? DO i = 1, 100 S1A(i+1,i) = B(i) + C S2D(i) = A(i,i)¢E ENDDO • Coupling can cause imprecision in dependence testing

  13. Building Coupled Groups A(j-1,i+1,i+3,k) = A(j+2,j,i,k+1) • Test each group separately j j-1 j+2 < , > < , > < , > < , > i k+1 i+3 k i+1

  14. Subscript Tests • Input: • Separable subscript OR coupled group • Loop bounds • Output: • “No dependence”, OR • A set of direction vectors in which dependence may exist

  15. Merging Direction Vectors DO i = 1, 100 DO j = 1, 99 A(i+1,j) = B(i)+C D(j) = A(i,100-j) ENDDO ENDDO • Cartesian product: • Direction Vectors:

  16. General Algorithm Overview Input: a pair of array references and loop bounds. • Partition subscripts into separable and coupled groups. • Classify each subscript as ZIV, SIV or MIV. • For each separable subscript: apply single subscript test according to its complexity. If independence established: output “no dependence” and halt. • For each coupled group: apply multiple subscript test. If independence established: output “no dependence” and halt. • Merge all direction vectors computed in the previous steps into a single set of direction vectors.

  17. Roadmap: Part I • Single Subscript Tests: • ZIV • Strong SIV • Weak SIV • Weak-Zero SIV • Weak-Crossing SIV • SIV Tests in Complex Iteration Spaces

  18. Roadmap: Part II • MIV Tests: • GCD Test • Banerjee Inequality • Trapezoidal Banerjee Inequality • Coupled Subscripts: the Delta Test • Empirical Results

  19. Roadmap: Part I • Single Subscript Tests: • ZIV • Strong SIV • Weak SIV • Weak-Zero SIV • Weak-Crossing SIV • SIV Tests in Complex Iteration Spaces

  20. ZIV Test DO j = 1, 100 A(e1) = A(e2) + B(j) ENDDO • e1,e2 are loop invariant expressions • e1=5, e2=4 ) no dependence • e1=L, e2=L+1 ) no dependence • e1=L, e2=K ) dependence • If (e1-e2) is a non-zero constantthen no dependence exists

  21. ZIV Test & Breaking Conditions INPUT L,K DO i = 1, N S1 B(i+1) = A(L+K) + X(i) S2A(L) = B(i) + C ENDDO • The values of L and K are unknown • If K0 there is no dependence from S2 to S1 • K0 is called the Breaking Condition

  22. ZIV Test & Breaking Conditions IF (K0) THEN DO i = 1, N B(i+1) = A(L+K) + X(i) ENDDO DO i = 1, N A(L) = B(i) + C ENDDO ELSE DO i = 1, N B(i+1) = A(L+K) + X(i) A(L) = B(i) + C ENDDO ENDIF

  23. ZIV Test & Breaking Conditions IF (K0) THEN B(2:N+1) = A(L+K) + X(1:N) DO i = 1, N A(L) = B(i) + C ENDDO ELSE DO i = 1, N B(i+1) = A(L+K) + X(i) A(L) = B(i) + C ENDDO ENDIF

  24. Roadmap: Part I • Single Subscript Tests: • ZIV • Strong SIV • Weak SIV • Weak-Zero SIV • Weak-Crossing SIV • SIV Tests in Complex Iteration Spaces

  25. Strong SIV Examples:

  26. Strong SIV Test A(m1) Dependence exists if d is an integer and

  27. Strong SIV Test DO i = 1, N A(i+2¢N) = A(i+N) + C ENDDO N > N-1 ) No dependence

  28. Strong SIV & Breaking Conditions DO i = 1, L A(i+N) = A(i) + B ENDDO • N>L-1 is the breaking condition IF (N¸L) THEN A(N+1:N+L) = A(1:L) + B ELSE DOi = 1, L A(i+N) = A(i) + B ENDDO ENDIF

  29. Roadmap: Part I • Single Subscript Tests: • ZIV • Strong SIV • Weak SIV • Weak-Zero SIV • Weak-Crossing SIV • SIV Tests in Complex Iteration Spaces

  30. Weak SIV Examples: Dependence exists for all integer solutions to the following linear Diophantine equation:

  31. Roadmap: Part I • Single Subscript Tests: • ZIV • Strong SIV • Weak SIV • Weak-Zero SIV • Weak-Crossing SIV • SIV Tests in Complex Iteration Spaces

  32. Weak-Zero SIV Test Dependence equation (assume a2=0): Dependence exists if is an integer and is within the loop bounds

  33. Weak-Zero SIV Test Dependence always caused by a particular iteration

  34. Weak-Zero SIV & Loop Peeling DO i = 1, N A(i,N) = A(1,N) + A(N,N) ENDDO • Dependence caused by iterations 1 and N. • These iterations can be peeled from the loop: A(1,N) = A(1,N) + A(N,N) DO i = 2, N-1 A(i,N) = A(1,N) + A(N,N) ENDDO A(N,N) = A(1,N) + A(N,N)

  35. Weak-Zero SIV & Loop Peeling DO i = 1, N A(i,N) = A(1,N) + A(N,N) ENDDO • Dependence caused by iterations 1 and N. • These iterations can be peeled from the loop: A(1,N) = A(1,N) + A(N,N) A(N,N) = A(1,N) + A(N,N) A(2:N-1) = A(1,N) + A(N,N)

  36. Roadmap: Part I • Single Subscript Tests: • ZIV • Strong SIV • Weak SIV • Weak-Zero SIV • Weak-Crossing SIV • SIV Tests in Complex Iteration Spaces

  37. Weak-Crossing SIV Line of symmetry A(m3) A(m2) A(m1) When does dependence exist?

  38. Weak-Crossing SIV Test Line of symmetry A(m1) • Dependence exists if the line of symmetry is: • within the loop bounds ) L · (c2-c1)/2a · U

  39. Weak-Crossing SIV Test Line of symmetry  A(m1) • Dependence exists if the line of symmetry is: • within the loop bounds ) L · (c2-c1)/2a · U • an integer or has a non-integer part equal to ½ ) • The line of symmetry is halfway between two integers

  40. Weak-Crossing SIV & Loop Splitting DO i = 1, N A(i) = A(N-i+1) + C ENDDO • Line of symmetry: i=(N+1)/2 DO i = 1, (N+1)/2 A(i) = A(N-i+1) + C ENDDO DO i = (N+1)/2+1, N A(i) = A(N-i+1) + C ENDDO

  41. Weak-Crossing SIV & Loop Splitting DO i = 1, N A(i) = A(N-i+1) + C ENDDO • Line of symmetry: i=(N+1)/2 A(1:(N+1)/2) = A(N:(N+1)/2)+C A((N+1)/2+1:N) = A((N+1)/2-1:1)+C

  42. Roadmap: Part I • Single Subscript Tests: • ZIV • Strong SIV • Weak SIV • Weak-Zero SIV • Weak-Crossing SIV • SIV Tests in Complex Iteration Spaces

  43. Complex Iteration Spaces • Triangular: One of the loop bounds is a function of at least one other loop index DO i = 1, 5 DO j = 1, i 5 4 3 2 1 1 2 3 4 5

  44. Complex Iteration Spaces • Trapezoidal: Both loop bounds are functions of at least one other loop index 8 DO i = 1, 5 DO j = i-1, 2¢i-1 7 6 5 4 3 2 1 1 2 3 4 5

  45. Complex Iteration Spaces • SIV tests can be extended to such loops, with some loss of precision

  46. Strong SIV in Complex Iteration Spaces DO i = 1, N DO j = L0+L1¢i, U0+U1¢i S1 A(j+d) = … S2… = A(j) + B ENDDO ENDDO • No dependence if ) • Unless inequality holds for all values of i in the loop ) assume dependence

  47. Index Set Splitting DO i = 1, 100 DO j = 1, i A(j+20) = A(j) + B ENDDO ENDDO For there is no dependence

  48. Index Set Splitting • Use i<21 as a breaking condition DO i = 1, 20 DO j = 1, i A(j+20) = A(j) + B ENDDO ENDDO DO i = 21, 100 DO j = 1, i A(j+20) = A(j) + B ENDDO ENDDO

  49. Index Set Splitting • Use i<21 as a breaking condition DO i = 21, 100 DO j = 1, i A(j+20) = A(j) + B ENDDO ENDDO PARALLEL DO i = 1, 20 A(21:20+i) = A(1:i) + B ENDDO

  50. Weak-Zero SIV in Complex Iteration Spaces DO i = 1, N DO j = L0+L1¢i, U0+U1¢i S1 A(c) = … S2… = A(j) + B ENDDO ENDDO • No dependence if ) • Unless inequality holds for all values of i in the loop ) assume dependence

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