Idiom Recognition in the Polaris Parallelizing Compiler Bill Pottenger and Rudolf Eigenmann Presented by Vincent Yau

Induction Variable Substitution Most of the compilers do not able to transform some general forms of induction variable triangular nested loops multiplicative expressions.

General Induction Variable Algorithm Step 1: Recognize the induction variable pattern iv = iv + inc_expression inc_expression - outerloop index, other iv or loop-invariant Step 2: Compute 3 Definitions (next slide) Compute closed forms Step 3: Direct substitution of the closed forms

iv = 0 do i = 1, n do j = 1, i a(iv) = … iv = iv + 1 enddo Example:

iv = 0 do i = 1, n do j = 1, i a(iv) = … iv = iv + 1 enddo Example: = j

iv = 0 do i = 1, n do j = 1, i a(iv) = … iv = iv + 1 enddo Example: = j = i*(i - 1)/2

iv = 0 do i = 1, n do j = 1, i a(iv) = … iv = iv + 1 enddo Example: do i = 1, n do j = 1, i a(j+(i 2 - i) / 2 - 1) = … enddo = j = i*(i - 1)/2

Symbolic sum function base on Bernoulli numbers Bernoulli numbers are defined by special case of Bernoulli polynomials

Wrap-Around Variables Definition: The variable that takes on the value of an induction variable after one iteration of a loop.

Example: m = 0 do i = 1, n do j = 1, i lb = j ub = i do k = i, n do l = lb, ub m = m + 1 a(m) = … enddo lb = 1 ub = k + 1 enddo m = m + i enddo

Example: m = 0 do i = 1, n do j = 1, i lb = j ub = i do k = i, n do l = lb, ub m = m + 1 a(m) = … enddo lb = 1 ub = k + 1 enddo m = m + i enddo Step 1: recognized the wrap-around variable Step 2: remove the wrap-around variable (lb, ub) by peeling the first iteration of the k loop. (next slide) Step 3: apply induction variable substitution Powerful symbolic manipulation is needed

Example: m = 0 do i = 1, n do j = 1, i lb = j ub = i do k = i, n do l = lb, ub m = m + 1 a(m) = … enddo lb = 1 ub = k + 1 enddo m = m + i enddo m = 0 do i = 1, n do j = 1, i do l = j, i m = m + 1 a(m) = … enddo do k = 1 + i, n do l = 1, k m = m + 1 a(m) = … enddo m = m + i enddo step 2

m = 0 do i = 1, n do j = 1, i do l = j, i m = l + (i + (-9i 2 - 3i 4 + 6i + 6i 3 - 6in - 6in 2 + 6i 2 n 2 )/4 - 3n - 3j 2 -n 2 + 2i 3 + 3j + 3i 2 - 3ij - 3ji 2 + 3jn + 3jn 2 )/6 - 2i + 2ij a(m) = … enddo do k = 1 + i, n do l = 1, k m = l + ((-9i 2 - 3i 4 + 6i + 6i 3 - 6in - 6in 2 + 6i 2 n 2 )/4 - 3k - 3j 2 - 3n 2 - 2i + 3j 3 + 3j + 3k 2 - 3ij - 3ji 2 +3jn +3jn 2 )/6 - 2i + 2ij a(m) = … enddo Example: m = 0 do i = 1, n do j = 1, i do l = j, i m = m + 1 a(m) = … enddo do k = 1 + i, n do l = 1, k m = m + 1 a(m) = … enddo m = m + i enddo step 3

Reduction Recognition Step 1: Recognition Pass A(x 1, x 2, x 3, …) = A(x 1, x 2, x 3, …) + B set the reduction variable flag for A( ) Step 2: Data Dependence Pass analyzes candidate reduction variables. removes the reduction flag, if it can be proven to be independence Step 3: Transformation Pass 3 different types of parallel reduction transformation:  blocked  privatized  expanded

Transformation Pass insert synchronization primitives around each reduction statement synchronization overhead was high.

Performance Result Overall Program Speedups (running on 8-processor set of an SGI Challenge R440)

Performance Results