Code generation in the polyhedral model is easier than you think

Many advances in automatic parallelization and optimization have been achieved through the polyhedral model. It has been extensively shown that this computational model provides convenient abstractions to reason about and apply program transformations. Nevertheless, the complexity of code generation has long been a deterrent for using polyhedral representation in optimizing compilers. First, code generators have a hard time coping with generated code size and control overhead that may spoil theoretical benefits achieved by the transformations. Second, this step is usually time consuming, hampering the integration of the polyhedral framework in production compilers or feedback-directed, iterative optimization schemes. Moreover, current code generation algorithms only cover a restrictive set of possible transformation functions. This paper discusses a general transformation framework able to deal with nonunimodular, noninvertible, nonintegral or even nonuniform functions. It presents several improvements to a state-of-the-art code generation algorithm. Two directions are explored: generated code size and code generator efficiency. Experimental evidence proves the ability of the improved method to handle real-life problems.

[1]  P. Feautrier.  Parametric integer programming , 1988 .

[2]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[3]  Yves Robert,et al.  Mapping Uniform Loop Nests Onto Distributed Memory Architectures , 1993, Parallel Comput..

[4]  W. Kelly,et al.  Code generation for multiple mappings , 1995, Proceedings Frontiers '95. The Fifth Symposium on the Frontiers of Massively Parallel Computation.

[5]  J. Ramanujam,et al.  Beyond unimodular transformations , 1995, The Journal of Supercomputing.

[6]  H. Le Verge,et al.  A Note on Chernikova's algorithm , 1992 .

[7]  Keshav Pingali,et al.  A Singular Loop Transformation Framework Based on Non-Singular Matrices , 1992, LCPC.

[8]  Martin Griebl,et al.  Code generation in the polytope model , 1998, Proceedings. 1998 International Conference on Parallel Architectures and Compilation Techniques (Cat. No.98EX192).

[9]  Jingling Xue.  Automating Non-Unimodular Loop Transformations for Massive Parallelism , 1994, Parallel Comput..

[10]  Corinne Ancourt,et al.  Scanning polyhedra with DO loops , 1991, PPOPP '91.

[11]  Doran Wilde,et al.  Loop nest synthesis using the polyhedral library , 1994 .

[12]  Doran Wilde,et al.  A LIBRARY FOR DOING POLYHEDRAL OPERATIONS , 2000 .

[13]  Philippe Clauss.  Counting Solutions to Linear and Nonlinear Constraints Through Ehrhart Polynomials: Applications to Analyze and Transform Scientific Programs , 1996, International Conference on Supercomputing.

[14]  Paul Feautrier,et al.  Some efficient solutions to the affine scheduling problem. Part II. Multidimensional time , 1992, International Journal of Parallel Programming.

[15]  Jingling Xue.  Transformations of Nested Loops with Non-Convex Iteration Spaces , 1996, Parallel Comput..

[16]  Youcef Bouchebaba.  Optimisation des transferts de données pour le traitement du signal : pavage, fusion et réallocation des tableaux , 2002 .

[17]  QuilleréFabien,et al.  Generation of Efficient Nested Loops from Polyhedra , 2000 .

[18]  Cédric Bastoul,et al.  Efficient code generation for automatic parallelization and optimization , 2003, Second International Symposium on Parallel and Distributed Computing, 2003. Proceedings..

[19]  Nicolas Halbwachs,et al.  Automatic discovery of linear restraints among variables of a program , 1978, POPL.

[20]  Paul Feautrier,et al.  Dataflow analysis of array and scalar references , 1991, International Journal of Parallel Programming.

[21]  Albert Cohen,et al.  Putting Polyhedral Loop Transformations to Work , 2003, LCPC.

[22]  Marc Le Fur.  Parcours de polyèdre paramétré avec l'élimination de Fourier-Motzkin , 1994 .

[23]  Philippe Clauss,et al.  Counting solutions to linear and nonlinear constraints through Ehrhart polynomials: applications to analyze and transform scientific programs , 1996 .

[24]  Paul Feautrier,et al.  Construction of Do Loops from Systems of Affine Constraints , 1995, Parallel Process. Lett..

[25]  David L. Kuck,et al.  The Structure of Computers and Computations , 1978 .

[26]  Pierre Boulet,et al.  Loop Parallelization Algorithms: From Parallelism Extraction to Code Generation , 1998, Parallel Comput..

[27]  Martin Griebl,et al.  Index Set Splitting , 2000, International Journal of Parallel Programming.

[28]  Paul Feautrier,et al.  Some efficient solutions to the affine scheduling problem. I. One-dimensional time , 1992, International Journal of Parallel Programming.