On uniformization of affine dependence algorithms

Journal of Systems and Software, 2002

In process of automatic parallelizing/vectorizing constant-bound loops with multi-dimensional arrays under specific dependence direction, the Lambda test is claimed to be an efficient and precise data dependence analysis method that can check whether there exist generally inexact 'real-valued' solutions to the derived dependence equations. In this paper, we propose a precise data dependence analysis method -the multi-dimensional direction vector I test. The multi-dimensional direction vector I test can be applied towards testing whether there exist generally accurate 'integer-valued' solutions to the dependence equations derived from multi-dimensional arrays under specific dependence direction in constant-bound loops. Experiments with benchmark showed that the accuracy rate and the improvement rate for the proposed method are approximately 33.3% and 21.6%, respectively.

The extent of parallelization of a loop is largely determined by the dependences between its statements. While dependence free loops are fully parallelizable, those with loop carried dependences are not. Dependence distance is a measure of absolute difference between a pair of dependent iterations. Loops with constant distance data dependence, because of uniform distance between the dependent iterations, lead to easy partitioning of the iteration space and hence they have been successfully dealt with. Parallelization of loops with variable distance data dependences is a considerably difficult problem. It is our belief that partitioning the iteration space in such loops cannot be done without examining solutions of the corresponding Linear Diophantine Equations. Focus of this work is to study variable distance data dependences and examine the relation between dependent iterations. Our analysis based on parametric solution leads to a mathematical formulation capturing dependence betwe...

Optimizations and Machine Code Generation, 2002

ACM SIGPLAN Notices, 1991

Data dependence testing is the basic step in detecting loop level parallelism in numerieal programs. The problem is equivalent to integer linear programming and thus in general cannot be solved efficiently. Current methods in use employ inexact methods that sacrifice potential parallelism in order to improve compiler efficiency. This paper shows that in practice, data dependence can be computed exactly and efficiently. There are three major ideas that lead to this result. First, we have developed and assembled a small set of efficient algorithms, each one exact for special case inputs. Combined with a moderately expensive backup test, they are exact for all the cases we have seen in practice. Second, we introduce a memorization technique to save results of previous tests, thus avoiding calling the data dependence routines multiple times on the same input. Third, we show that this approach can both be extended to compute distance and direction vectors and to use unknowns ymbolic terms without any loss of accuracy or efficiency, We have implemented our algorithm in the SUIF system, a general purpose compiler system developwl at Stanford. We ran the algorithm on the PERFECT Club Benchmarks and our data dependence analyzer gave an exact solution in all cases efficiently, Proceedings of the ACM SIGPLAN '91 Conference on Programming Language Design and Implementation.

A long running program often spends most of its time in nested loops. The polyhedral model provides powerful abstractions to optimize loop nests with regular accesses for parallel execution. Affine transformations in this model capture a complex sequence of execution-reordering loop transformations that improve performance by parallelization as well as better locality. Although a significant amount of research has addressed affine scheduling and partitioning, the problem of automatically finding good affine transforms for communication-optimized coarse-grained parallelization along with locality optimization for the general case of arbitrarily-nested loop sequences remains a challenging problem -most frameworks do not treat parallelization and locality optimization in an integrated manner, and/or do not optimize across a sequence of producer-consumer loops.

Journal of Combinatorial Optimization, 2014

In this paper, we present a modified Floyd-Warshall algorithm, where the most time-consuming part-calculating transitive closure describing self-dependences for each loop statement-is computed applying basis dependence distance vectors derived from all vectors describing self-dependences. We demonstrate that the presented approach reduces the transitive closure calculation time for parameterized graphs representing all dependences in the loop in comparison with that yielded by means of techniques implemented in the Omega and ISL libraries. This increases the applicability scope of techniques based on transitive closure of dependence graphs and being aimed at building optimizing compilers. Experimental results for NASA Parallel Benchmarks are discussed.

… Mathematics and its …, 2003

It is therefore of interest to try to parallelize nested loops in order to reduce the overall computation time.

… Mathematics and its …, 2003

The Computer Journal, 1997

Although many methods exist for nested loop partitioning, most of them perform poorly when parallelizing loops with non-uniform dependences. This paper addresses the issue of automatic parallelization of loops with non-uniform dependences. Such loops are normally not parallelized by existing parallelizing compilers and transformations. Even when parallelized in rare instances, the performance is very poor. Our approach is based on the`convex hull' theory which has adequate information to handle non-uniform dependences. We introduce the concept of`complete dependence convex hull',`unique head and tail sets' and abstract the dependence information into these sets. These sets form the basis of the iteration space partitions. The properties of the unique head and tail sets are derived. Depending on the relative placement of these unique sets, partitioning schemes are suggested for implementation of our technique. Implementation results of our scheme on the Cray J916 and comparison with other schemes show the superiority of our technique.

A long running program often spends most of its time in nested loops. The polyhedral model pro-vides powerful abstractions to optimize loop nests with regular accesses for parallel execution. Affine transformations in this model capture a complex sequence of execution-reordering loop transforma-tions that improve performance by parallelization as well as better locality. Although a significant amount of research has addressed affine scheduling and partitioning, the problem of automatically finding good affine transforms for communication-optimized coarse-grained parallelization along with locality optimization for the general case of arbitrarily-nested loop sequences remains a challenging problem -most frameworks do not treat parallelization and locality optimization in an integrated manner, and/or do not optimize across a sequence of producer-consumer loops. In this paper, we develop an approach to communication minimization and locality optimization in tiling of arbitrarily nested...