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Transformations of Nested Loops with NonConvex Iteration Spaces
, 1996
"... When compiling for parallel machines, it is often necessary to generate a loop nest to scan a region of index points in lexicographic order. One wellknown application example is the use of loop transformations to restructure loop nests. Previous work shows how to generate code to scan a convex poly ..."
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Cited by 10 (2 self)
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polyhedron (possibly intersected with a lattice), a region derived from an application of a nonsingular transformation to a convex iteration space. The driving force behind this work is to investigate how to generate code for nonsingular transformations of nested loops with nonconvex iteration spaces
Nonunimodular Transformations of Nested Loops
 IN PROC. SUPERCOMPUTING 92
, 1992
"... This paper presents a linear algebraic approach to modeling loop transformations. The approach unifies apparently unrelated recent developments in supercompiler technology. Specifically we show the relationship between the dependence abstraction called dependence cones, and fully permutable loop nes ..."
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Cited by 46 (12 self)
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in order to "step aside from these holes" when traversing the transformed iteration space. For the class of nonunimodular loop transformations, we present algorithms for deriving the loop bounds, the array access expressions and step sizes of loops in the nest. The algorithms are based
Tiling Imperfectlynested Loop Nests
 In Proc. of SC 2000
, 2000
"... Tiling is one of the more important transformations for enhancing locality of reference in programs. Tiling of perfectlynested loop nests (which are loop nests in which all assignment statements are contained in the innermost loop) is well understood. In practice, most loop nests are imperfectlyne ..."
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Cited by 39 (0 self)
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, so existing compilers heuristically try to find a sequence of transformations that convert such loop nests into perfectlynested ones but not always succeed. In this paper, we propose a novel approach to tiling imperfectlynested loop nests. The key idea is to embed the iteration space of every
Synthesizing transformations for locality enhancement of imperfectlynested loop nests
 In Proceedings of the 2000 ACM International Conference on Supercomputing
, 2000
"... We present an approach for synthesizing transformations to enhance locality in imperfectlynested 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 perfectlynested loop ..."
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Cited by 64 (3 self)
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We present an approach for synthesizing transformations to enhance locality in imperfectlynested 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 perfectlynested loop
Beyond Convexity: Scanning ‘NonConvex Polyhedra’
"... The enumeration of points contained in an algebraically specified domain is one of the key algorithmic problems in the transformation of scientific programs. However, basic scanning algorithms accept only single convex polyhedra, requiring specialized techniques and causing runtime overhead if the ..."
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if the set of points to enumerate is not convex. In this paper, we review the existing approaches to the case of “regularly nonconvex” domains, and present an algorithm for scanning arbitrary unions of polyhedra. For this, we propose to use nested loop sequences instead of perfect loop nests, and present
Parametric Analysis of Polyhedral Iteration Spaces
 JOURNAL OF VLSI SIGNAL PROCESSING
, 1998
"... In the area of automatic parallelization of programs, analyzing and transforming loop nests with parametric affine loop bounds requires fundamental mathematical results. The most common geometrical model of iteration spaces, called the polytope model, is based on mathematics dealing with convex and ..."
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Cited by 85 (14 self)
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In the area of automatic parallelization of programs, analyzing and transforming loop nests with parametric affine loop bounds requires fundamental mathematical results. The most common geometrical model of iteration spaces, called the polytope model, is based on mathematics dealing with convex
Compile Time Partitioning of Nested Loop Iteration Spaces with Nonuniform Dependences
 Journal of Parallel Algorithms and Applications (special issue on Optimizing Compilers for Parallel Languages
, 1996
"... In this paper we address the problem of partitioning nested loops with nonuniform (irregular) dependence vectors. Parallelizing and partitioning of nested loops requires efficient interiteration dependence analysis. Although many methods exist for nested loop partitioning, most of these perform po ..."
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Cited by 8 (1 self)
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In this paper we address the problem of partitioning nested loops with nonuniform (irregular) dependence vectors. Parallelizing and partitioning of nested loops requires efficient interiteration dependence analysis. Although many methods exist for nested loop partitioning, most of these perform
Generation of Efficient Nested Loops from Polyhedra
 International Journal of Parallel Programming
, 2000
"... Automatic parallelization in the polyhedral model is based on affine transformations from an original computation domain (iteration space) to a target spacetime domain, often with a different transformation for each variable. Code generation is an often ignored step in this process that has a signi ..."
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Cited by 89 (5 self)
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Automatic parallelization in the polyhedral model is based on affine transformations from an original computation domain (iteration space) to a target spacetime domain, often with a different transformation for each variable. Code generation is an often ignored step in this process that has a
Multilevel Blocking in Complex Iteration Spaces
"... This paper presents a new unified method for simultaneously tiling the register and cache levels of the memory hierarchy. We will focus on the code transformation phase of tiling. Our algorithm uses stripmining and loop interchange on all memory hierarchy levels to determine the tiles as usual, and ..."
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, and, afterwards, and due to the special characteristics of the register level, we apply index set splitting, fully unrolling and unnecessary load/store elimination. We propose a technique to perform the loop interchange in nonconvex iteration spaces that computes the loop bounds exactly and we also
Unimodular Transformations of NonPerfectly Nested Loops
 Parallel Computing
, 1997
"... A framework is described in which a class of imperfectly nested loops can be restructured using unimodular transformations. In this framework, an imperfect loop nest is converted to a perfect loop nest using AbuSufah's NonBasictoBasicLoop transformation. Conditions for the legality of this ..."
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Cited by 21 (4 self)
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of this transformation and techniques for their verification are discussed. An iteration space, which extends the usual concept so as to represent explicitly the executions of individual statements, is proposed to model the converted loop nest. Since the converted loop nest is a perfect loop nest, data dependences can
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