24 - Advanced Topics Optimization for parallel machines and...

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Unformatted text preview: Advanced Topics Optimization for parallel machines and memory hierarchies Last Time • Dependence analysis Today • Loop transformations • An example - McKinley, Carr, Tseng loop transformations to improve cache performance CS 380C Lecture 24 1 Locality Analysis Which Loops are Parallel? review do I = 1, N do J = 1, N S 1 A(I,J) = A(I,J-1) + 1 do I = 1, N do J = 1, N S 2 A(I,J) = A(I-1,J-1) + 1 do I = 1, N do J = 1, N S 3 B(I,J) = B(I-1,J+1) + 1 J I • A dependence D = ( d 1 ,..., d k ) is carried at level i , if d i is the first nonzero element of the distance/direction vector. • A loop l i is parallel , if ∃ a dependence D j carried at level i . Either distance vector direction vector ∀ D j d 1 ,..., d i- 1 > d 1 ,..., d i- 1 = “ < OR d 1 ,..., d i = d 1 ,..., d i = “ = CS 380C Lecture 24 2 Locality Analysis Loop Transformations Taxonomy • Loop unrolling • Loop interchange • Loop fusion • Loop distribution (a.k.a. fission) • Loop skewing • Strip mine and interchange (a.k.a. tiling & blocking) • Unroll-and-jam (a variety of tiling) • Loop reversal CS 380C Lecture 24 3 Locality Analysis Loop Interchange do I = 1, N do J = 1, N S 1 A(I,J) = A(I-1,J) + 1 enddo enddo do I = 1, N do J = 1, N S 2 B(I,J) = B(I-1,J+1) + 1 enddo enddo I J I J Loop interchange is safe iff • it does not reverse the execution order of the source and sink of any dependence in the nest, i.e., if the distance vector would become negative. ◦ Enables parallelization of outer and/or inner loops ◦ Changes execution order of the statements ◦ Can improve reuse CS 380C Lecture 24 4 Locality Analysis Loop Fusion = ⇒ loop fusion = ⇒ do i = 2, n s 1 a(i) = b(i) do i = 2, n s 2 c(i) = b(i) * a(i-1) do i = 2, n s 1 a(i) = b(i) s 2 c(i) = b(i) * a(i-1) ⇐ = loop distribution ⇐ = Loop Fusion is safe iff • no forward dependence between nests becomes a backward loop carried dependence. ⇒ Would fusion be safe if s 2 referenced a ( i + 1 ) ? • Benefits ◦ Reuse ◦ Eliminates synchronization between parallel loops ◦ Reduced loop overhead CS 380C Lecture 24 5 Locality Analysis Loop Distribution = ⇒ loop distribution = ⇒ do i = 2, n s 1 a(i) = b(i) s 2 c(i) = b(i) * a(i+1) do i = 2, n s 2 c(i) = b(i) * a(i+1) do i = 2, n s 1 a(i) = b(i) Loop Distribution is safe iff • statements involved in a cycle of dependences ( recurrence ) remain in the same loop, & • if ∃ a dependence between two statements placed in different loops, it must be forward....
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24 - Advanced Topics Optimization for parallel machines and...

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