Locality Optimizations and Loop Transformations in Affine Transform Theory, Slides of Compilers

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Loop Transformations and

Locality

Agenda

• Introduction
• Loop Transformations
• Affine Transform Theory

Cache Locality

A[1,1] A[2,1] … A[1,2] A[2,2] … A[1,3] …

for i = 1, 100 for j = 1, 200 A[i, j] = A[i, j] + 3 end_for end_for

• Suppose array A has column-major layout
  • Loop nest has poor spatial cache locality.

Loop Interchange

A[1,1] A[2,1] … A[1,2] A[2,2] … A[1,3] …

for i = 1, 100 for j = 1, 200 A[i, j] = A[i, j] + 3 end_for end_for

• Suppose array A has column-major layout
• New loop nest has better spatial cache locality.

for j = 1, 200 for i = 1, 100 A[i, j] = A[i, j] + 3 end_for end_for

Dependence Vectors

i

j

 Distance vector (1,-1) = (4,2)-(3,3)  Direction vector (+, -) from the signs of distance vector  Loop interchange is not legal if there exists dependence (+, -)

Agenda

• Introduction
• Loop Transformations
• Affine Transform Theory

Loop Distribution

for i = 1, 1000 A[i] = A[i-1] + 3 end_for

for i = 1, 1000 C[i] = B[i] + 5 end_for

for i = 1, 1000 A[i] = A[i-1] + 3 C[i] = B[i] + 5 end_for

 2 nd^ loop is parallel

Register Blocking

for j = 1, 2m for i = 1, 2n A[i, j] = A[i-1, j] + A[i-1, j-1] end_for end_for

for j = 1, 2m, 2 for i = 1, 2n, 2 A[i, j] = A[i-1,j] + A[i-1,j-1] A[i, j+1] = A[i-1,j+1] + A[i-1,j] A[i+1, j] = A[i, j] + A[i, j-1] A[i+1, j+1] = A[i, j+1] + A[i, j] end_for end_for

 Better reuse between A[i,j] and A[i,j]

Scalar Replacement

for i = 2, N+ = A[i-1]+ A[i] = end_for

t1 = A[1] for i = 2, N+ = t1 + 1 t1 = A[i] = t end_for

 Eliminate loads and stores for array references

Unroll-and-Jam

for j = 1, 2*M for i = 1, N A[i, j] = A[i-1, j]

• A[i-1, j-1] end_for end_for

for j = 1, 2*M, 2 for i = 1, N A[i, j]=A[i-1,j]+A[i-1,j-1]

A[i, j+1]=A[i-1,j+1]+A[i-1,j] end_for end_for

 Expose more opportunity for scalar replacement

Loop Blocking

for v = 1, 1000, 20 for u = 1, 1000, 20 for j = v, v+ for i = u, u+ A[i, j] = A[i, j] + B[j, i] end_for end_for end_for end_for

 Access to small blocks of the arrays has good cache locality.

Loop Unrolling for ILP

for i = 1, 10 a[i] = b[i]; *p = ... end_for

for I = 1, 10, 2 a[i] = b[i]; *p = … a[i+1] = b[i+1]; *p = … end_for

 Large scheduling regions. Fewer dynamic branches

 Increased code size

Objective

• Unify a large class of program transformations.
• Example:

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float Z[100]; for i = 0, 9 Z[i+10] = Z[i]; end_for

Iteration Space

• A d-deep loop nest has d index variables, and is modeled by a d-dimensional space. The space of iterations is bounded by the lower and upper bounds of the loop indices.
• Iteration space i = 0,1, …

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for i = 0, 9 Z[i+10] = Z[i]; end_for