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Parallel Computing
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Loop Transformation Using Nonunimodular Matrices
IEEE Transactions on Parallel and Distributed Systems
Extension Of The Alpha Language To Recurrences On Sparse Periodic Domains
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Code Generation in the Polyhedral Model Is Easier Than You Think
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Conference record of the 33rd ACM SIGPLAN-SIGACT symposium on Principles of programming languages
Memory optimization by counting points in integer transformations of parametric polytopes
CASES '06 Proceedings of the 2006 international conference on Compilers, architecture and synthesis for embedded systems
Finding Synchronization-Free Parallelism Represented with Trees of Dependent Operations
ICA3PP '08 Proceedings of the 8th international conference on Algorithms and Architectures for Parallel Processing
Integer affine transformations of parametric ℤ-polytopes and applications to loop nest optimization
ACM Transactions on Architecture and Code Optimization (TACO)
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The polyhedral model is a well developed formalism and has been extensively used in a variety of contexts viz. the automatic parallelization of loop programs, program verification, locality, hardware generationand more recently, in the automatic reduction of asymptotic program complexity. Such analyses and transformations rely on certain closure properties. However, the model is limited in expressivity and the need for a more general class of programs is widely known. We provide the extension to ⁰-polyhedra which are the intersection of polyhedra and lattices. We prove the required closure properties using a novel representation and interpretation of ⁰-polyhedra. In addition, we also prove closure in the ⁰-polyhedral model under images by dependence functions---thereby proving that unions of LBLs, widely assumedto be a richer class of sets, is equal to unions of ⁰-polyhedra. Another corollary of this result is the equivalence of the unions of ⁰-polyhedraand Presburger sets. Our representation and closure properties constitute the foundations of the ⁰-polyhedral model. As an example, we presentthe transformation for automatic reduction of complexity in the ⁰-polyhedral model.