Synthesizing transformations for locality enhancement of imperfectly-nested loop nests

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Synthesizing transformations for locality enhancement of imperfectly-nested loop nests (2000)

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by Nawaaz Ahmed , Nikolay Mateev , Keshav Pingali

Venue:	In Proceedings of the 2000 ACM International Conference on Supercomputing
Citations:	64 - 3 self

BibTeX

@INPROCEEDINGS{Ahmed00synthesizingtransformations,
    author = {Nawaaz Ahmed and Nikolay Mateev and Keshav Pingali},
    title = {Synthesizing transformations for locality enhancement of imperfectly-nested loop nests},
    booktitle = {In Proceedings of the 2000 ACM International Conference on Supercomputing},
    year = {2000},
    pages = {141--152}
}

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Abstract

We present an approach for synthesizing transformations to enhance locality in imperfectly-nested loops. The key idea is to embed the iteration space of every statement in a loop nest into a special iteration space called the product space. The product space can be viewed as a perfectly-nested loop nest, so embedding generalizes techniques like code sinking and loop fusion that are used in ad hoc ways in current compilers to produce perfectly-nested loops from imperfectly-nested ones. In contrast to these ad hoc techniques however, our embeddings are chosen carefully to enhance locality. The product space is then transformed further to enhance locality, after which fully permutable loops are tiled, and code is generated. We evaluate the effectiveness of this approach for dense numerical linear algebra benchmarks, relaxation codes, and the tomcatv code from the SPEC benchmarks. 1. BACKGROUND AND PREVIOUSWORK Sophisticated algorithms based on polyhedral algebra have been developed for determining good sequences of linear loop transformations (permutation, skewing, reversal and scaling) for enhancing locality in perfectly-nested loops 1. Highlights of this technology are the following. The iterations of the loop nest are modeled as points in an integer lattice, and linear loop transformations are modeled as nonsingular matrices mapping one lattice to another. A sequence of loop transformations is modeled by the product of matrices representing the individual transformations; since the set of nonsingular matrices is closed under matrix product, this means that a sequence of linear loop transformations can be represented by a nonsingular matrix. The problem of finding an optimal sequence of linear loop transformations is thus reduced to the problem of finding an integer matrix that satisfies some desired property, permitting the full machinery of matrix methods and lattice theory to ¢ This work was supported by NSF grants CCR-9720211, EIA-9726388, ACI-9870687,EIA-9972853. £ A perfectly-nested loop is a set of loops in which all assignment statements are contained in the innermost loop.

Keyphrases

linear loop transformation perfectly-nested loop imperfectly-nested loop nest locality enhancement product space nonsingular matrix loop transformation code sinking background previouswork sophisticated algorithm permutable loop ad hoc technique loop fusion individual transformation imperfectly-nested one innermost loop spec benchmark generalizes technique integer matrix imperfectly-nested loop relaxation code special iteration space integer lattice matrix method key idea iteration space desired property current compiler dense numerical linear algebra benchmark lattice theory full machinery tomcatv code ad hoc way assignment statement nsf grant ccr-9720211 optimal sequence matrix product polyhedral algebra good sequence

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