Description
Problem 1 – Dependencies
Suppose we have the following program (a sequence of instructions), with each instruction labeled as
S<num>:
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1 : a := 4;
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2 : b := 2;
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3 : c := 5;
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4 : read ( d );
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5 : a := b + 3;
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6 : b := a – 3;
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7 : c := d * b ;
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8 : e := a + 6;
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9 : print ( c );
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10 : print ( e );
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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.
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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.
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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.
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do i = 3, 100
a(i) = a(i – 1) + a(i + 1) – a(i – 2)
enddo
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do i = 2, 100
a(3 * i) = a(3 * i – 3) + a(3 * i + 3)
enddo
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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
2 |
, |
100 |
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do |
j |
= |
2 , |
100 |
S1 : |
a (i , j ) = b ( i – 1 , j – 1) + 2 |
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2 :b (i , j ) = i + j – 1 enddo
enddo
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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.
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In its current form, can any loop be parallelized? If so, which loop(s)? If not, justify your answer.
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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.)
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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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