Three Steps of Proposed Approach

ent our integrated approach to enhance data locality in a single loop nest. This local l

atrix simu

3.4.1 Local Loop and Data Transformation Selection

oop transformation concept to explore tem

In this section, we pres

oop and data selection stage is used to produce necessary information to guild transformations in the next global data transformation stage, this part only focus on a single loop nest without considering other loop nests. Finally, we need to do a local loop transformation refinement to adapt global inconsistency.

For a single reference, determining both a loop and a data transformation m ltaneously is equivalent to solving a non-linear system with some additional constraints. As mentioned earlier, our approach is based on optimizing temporal locality using loop transformations and optimizing spatial locality via data transformations.

We divided this stage into two phases. The first phase uses l

poral locality; the second phase uses data transformation

concept to explore spatial locality.

(1) Temporal locality exploration

Loop transformation can improve both temporal and spatial locality, however, at this phase we only want to explore the potential temporal locality of loop transformations. Our framework first finds the temporal locality in the innermost loop nest for most number of references.

Let the reference matrices of array references in the loop nest be . Our approach first computes the spanning vectors for the kernel sets of these reference matrices. Consider all references, from among all spanning vectors, we choose the one which occurs most frequently. This approach tends to maximize the number of references for which temporal locality can be exploited.

Lk

L L₁, ₂,...,

(2) Spatial Locality Exploration

Data transformation can only improve spatial locality, at this phase, we want to explore the potential spatial locality of data transformations. A data transformation matrix we exploited for an array reference implies an associated array layout.

Previous research only exploit spatial locality for array references without temporal locality, however, our approach exploit spatial locality for all references no matter they have temporal locality or not. Nevertheless, references without temporal locality have higher priorities over the references with temporal locality.

We use hyperplane concept to represent array layouts, that is, for any given array layout, there is an associated hyperplane vector correspond to. To simplify our discussion, we consider in each loop nest there is a hyperplane vector corresponding to one optimal array layout. Our search for potential spatial locality starts at the last nonzero column denotes by , from inner loop to outer loop. Because a zero column simply implies the array reference corresponding to the loop nest exhibits temporal locality.

am

3.4.2 Global Data Transformation Decision

There are many loop nests and data arrays in the whole program, different loop nests maybe reference the same array; different arrays could be referenced in the same loop nest. In the previous related work, they do not consider the conflict situation when the array data layout determined by different loop nest is different.

In this section we discuss the main technique how we extend our approach to handle the multiple loop nest case. In the local stage, we have found all potential temporal and spatial locality. However, not all potential locality can exist at the same time. Our proposed method would use a global array layout solution to avoid conflict array layout situation, in other words, we do not consider changing array layouts at run time.

Conflict Array Layouts

The problem of determining of the optimal array data layouts has some factor which needs to consider separately. Because the effect of a data transformation is global in the sense that decisions regarding the memory layout of an array influence the locality characteristics of every part of the program that references the array. So we need to consider the following situation: layouts of some of the array references are constrained or fixed. If an array references is fixed, the changing of array data layout of that array is illegal because this transformation cannot guarantee the result is correct. If an array reference in a loop nest is constrained by other loop nest, then we need to decide which array layout will be used.

Resolution of Conflict Situation

If the references to the same array have more than one solution, then a conflict situation occurs. When there are conflict array layouts, we should make a decision to resolve the conflict situation. We need to decide which array layout is better. The resolution is based the following cost function. We use a cost function to analyze

different types of locality and relationship between different loop nests. Among all local layout possibilities of an array, we choose the one with minimal cost to be a best choice.

iterations ref

t =

× cos

3.4.3 Local Loop Transformation Refinement

After all array data layout is known, we can further change loop nests to adapt the data transformations. This phase is necessary because at the first stage we only adopt local consideration. But if a loop nest has conflict with other loop nests, there will be at least one loop nest need to be changed due to the conflict results. So we finalize the transformation task at this phase.

Loop transformations only consider exploiting the temporal locality; the spatial locality is unknown because the array layout is not decided yet. After we decide the global array data layout, then we know the spatial locality we can exploit. Finally, we have completed our approach by the resolution of conflict array layouts due to local consideration.