Description
Loop dependence analysis occur on a normalized loop of the form:
for i1 until U1 do for i2 until U2 do ... for in until Un do body done done
done
where body may contain:
S1 a := ... ...
S2 ... := a
Where a is an m-dimensional array and fn, hn, etc. are functions mapping from all iteration indexes (in) to a memory access in a particular dimension of the array.
For example, in C:
for (i = 0; i < U1; i++) for (j = 0; j < U2; j++) a = b + i*j;f1 would be i+4-j, controlling the write on the first dimension of a and h2 would be 2*i-j, controlling the read on the first dimension of b.
The scope of the problem is to find all possible dependencies between S1 and S2. To be conservative, any dependence which cannot be proven false must be assumed to be true.
Independence is shown by demonstrating that no two instances of S1 and S2 access or modify the same spot in array a. When a possible dependence is found, loop dependence analysis usually makes every attempt to characterize the relationship between dependent instances, as some optimizations may still be possible. It may also be possible to transform the loop to remove or modify the dependence.
In the course of (dis)proving such dependencies, a statement S may be decomposed according to which iteration it comes from. For instance, S refers to the iteration where i1 = 1, i2 = 3 and i3 = 5. Of course, references to abstract iterations, such as S, are both permitted and common.
Read more about this topic: Loop Dependence Analysis
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“The next Augustan age will dawn on the other side of the Atlantic. There will, perhaps, be a Thucydides at Boston, a Xenophon at New York, and, in time, a Virgil at Mexico, and a Newton at Peru. At last, some curious traveller from Lima will visit England and give a description of the ruins of St. Paul’s, like the editions of Balbec and Palmyra.”
—Horace Walpole (1717–1797)