Loop dependence analysis

In compiler theory, loop dependence analysis is the task of determining whether statements within a loop body form a dependence, almost always with respect to array access and modification. For a normalized loop:

for i₁ from l₁ to u₁ do for i₂ from l₂ to u₂ do ... for i_n from l_n to u_n do "body" od od od

where "body" may contain:

S1 a [f1(i₁, ..., i_n), ..., fm(i₁, ..., i_n)] := ... ... S2 ... := a [h1(i₁, ..., i_n), ..., hm(i₁, ..., i_n)]

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.

In the course of (dis)proving such dependencies, a statement "S" may be decomposed according to which iteration it comes from. For instance, "S" [1,3,5] refers to the iteration where i1 = 1, i2 = 3 and i3 = 5. Of course, references to abstract iterations, such as "S" ["d1"+1,"d2","d3"] , are both permitted and common.

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 the dependence.

Iteration vectors

A specific iteration through a normalized loop is referenced through an iteration vector, which encodes the state of each iteration variable.

For a loop, an iteration vector is a member of the Cartesian product of the bounds for the loop variables. In the nomalized form given previously, this space is defined to be [l1..u1] × [l2..u2] × ... × [ln..un] . As noted, specific (or general) instances of statements may be parameterized by these iteration vectors, and they are also the domain of the array subscript functions found in the body of the loop. Of particular relevance, these vectors form a lexicographic order which corresponds with the chronological execution order.

Dependence

For the normalized form above, a dependence between "S1" and "S2" exists if and only if:

* a and b are two iteration vectors.
* f(a) = h(b).
* a < b
* Either a or b is a write/store

where vector function notation has been used. Without vector notation, the second constraint becomes: f1(a1,a2, ..., an) = h1(b1,b2, ..., bn) and ... and fm(a1,a2, ..., an) = hm(b1,b2, ..., bn).

If a is a write and b is a read, then this is a flow dependence. If a is a read and b is a write, then this is an antidependence.

If a is a write and b is a write, then this is an output dependence

Devices and techniques

Several established devices and techniques exist for tackling the loop dependence problem. For characterizing the nature of a dependence, there are distance vectors, direction vectors and dependence vectors. For determining whether a dependence exists, the GCD test and the Banerjee test are the most general tests in common use, while a variety of techniques exist for simpler cases.

Further reading

*cite book | author=Kennedy, Ken; & Allen, Randy. | title=Optimizing Compilers for Modern Architectures: A Dependence-based Approach | publisher=Morgan Kaufmann | year=2001 | id=ISBN 1-55860-286-0
*

See also

* Dependence analysis
* Loop transformation
* Loop splitting
* Loop fusion
* Loop interchange
* Loop skewing
* automatic parallelization