Assignment 8 Solution

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Problem 1 – Dependencies

Suppose we have the following program (a sequence of instructions), with each instruction labeled as

S<num>:

  • 1 : a := 4;

  • 2 : b := 2;

  • 3 : c := 5;

  • 4 : read ( d );

  • 5 : a := b + 3;

  • 6 : b := a – 3;

  • 7 : c := d * b ;

  • 8 : e := a + 6;

  • 9 : print ( c );

  • 10 : print ( e );

    1. Give the statement-level dependence graph for the above program. A node in the statement-level dependence graph represents a statement, an edge represents dependence between the statements (i.e. the nodes). Label each edge as a true data dependence, an anti data dependence, or an output data dependence.

    1. Assume that each statement takes 1 cycle to execute. What is the execution time of the sequen-tial code? What is the fastest parallel execution time of the program (i.e. the critical path)? You may assume that I/O operations (read and print) can be done in parallel.

Problem 2 – Dependence Analysis

Give the source and sink references, the type (whether a dependence is true, anti, or output), and the distance vectors for all dependences in the following loops.

  1. do i = 3, 100

a(i) = a(i – 1) + a(i + 1) – a(i – 2)

enddo

  1. do i = 2, 100

a(3 * i) = a(3 * i – 3) + a(3 * i + 3)

enddo

  1. do i = 1, 10

a(i) = a(5) + a(i)

enddo

Use aW (i) and aR(i) to annotate the write access to a(i) and the read access to a(i) respectively.

Problem 3 – Loop Parallelization

Given the following nested loop:

1

do i =

2

,

100

do

j

=

2 ,

100

S1 :

a (i , j ) = b ( i – 1 , j – 1) + 2

  • 2 :b (i , j ) = i + j – 1 enddo

enddo

  1. Give the statement-level dependence graph. Show the dependence graph for statement instances in a part of the iteration space: i = 2 … 5, j = 2 … 5.

  1. In its current form, can any loop be parallelized? If so, which loop(s)? If not, justify your answer.

  1. Provide one valid a ne schedule for statements S1 and S2 such that p(S1)= C11*i + C12*j + d1 and p(S2)= C21*i + C22*j + d2 in order to achieve synchronization-free parallelism. There could be many possible solutions for fC11; C12; C21; C22; d1; d2g. You only need to provide one feasible solution. (Hint: You can let d1 = d2 = 0.)

  1. Generate two-level loop code for the a ne schedule you provided. Please use p as outermost loop and i as innermost loop in the transformed loop. Calculate the loop bounds for p and i using Fourier-Motzkin elimination. You might need to calculate the overlapping polyhedron for S1 and S2 in order to eliminate the j loop. Please refer to the techniques for code generation in lecture 20 and lecture 21.

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