Memory optimization by counting points in integer transformations of parametric polytopes

Seghir, Rachid; Loechner, Vincent

doi:10.1145/1176760.1176771

Cited by 6 publications

(5 citation statements)

References 26 publications

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“…Counting the number of integer points in such H-parametric polytopes can be performed very efficiently using Barvinok's algorithm [20]. Otherwise, i.e., if n ′ = 0, then additional techniques, e.g., [21], [22], need to be applied, which still work fairly well in practice.…”

Section: A An Upper Bound On the Number Of Elements In A Setmentioning

confidence: 99%

Symbolic Polynomial Maximization Over Convex Sets and Its Application to Memory Requirement Estimation

Clauss

Fernandez

Garbervetsky

et al. 2009

IEEE Trans. VLSI Syst.

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Memory requirement estimation is an important issue in the development of embedded systems, since memory directly influences performance, cost and power consumption. It is therefore crucial to have tools that automatically compute accurate estimates of the memory requirements of programs to better control the development process and avoid some catastrophic execution exceptions. Many important memory issues can be expressed as the problem of maximizing a parametric polynomial defined over a parametric convex domain. Bernstein expansion is a technique that has been used to compute upper bounds on polynomials defined over intervals and parametric "boxes". In this paper, we propose an extension of this theory to more general parametric convex domains and illustrate its applicability to the resolution of memory issues with several application examples.

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Section: A An Upper Bound On the Number Of Elements In A Setmentioning

confidence: 99%

Symbolic Polynomial Maximization Over Convex Sets and Its Application to Memory Requirement Estimation

Clauss

Fernandez

Garbervetsky

et al. 2009

IEEE Trans. VLSI Syst.

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“…The results of these systems are impressive, but are still slower than sampling based techniques for estimating the number of integer points in a polytope as used in the approximation thread of this work. Recent work on loop nest analysis [19,22] utilises the algorithmic results on point counting. However, abstract interpretation based analysis requires constraints between variables, not lattice point counts.…”

Section: Related Workmentioning

confidence: 99%

Integer Polyhedra for Program Analysis

Charles

Howe

King

2009

Algorithmic Aspects in Information and Management

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Abstract. Polyhedra are widely used in model checking and abstract interpretation. Polyhedral analysis is effective when the relationships between variables are linear, but suffers from imprecision when it is necessary to take into account the integrality of the represented space. Imprecision also arises when non-linear constraints occur. Moreover, in terms of tractability, even a space defined by linear constraints can become unmanageable owing to the excessive number of inequalities. Thus it is useful to identify those inequalities whose omission has least impact on the represented space. This paper shows how these issues can be addressed in a novel way by growing the integer hull of the space and approximating the number of integral points within a bounded polyhedron.

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“…For instance, for the MatVect kernel, dependence polyhedron D 1 R,S has a representative of 2: dim(D 1 R,S ) = 2, and the minimal loop depth of R and S is 1. Note that in the case of parametric loop bounds, an enumerator for the volume of a polyhedron can be computed as an Ehrhart quasi-polynomial in the form of the parameters [28,102,117]. For this case, using the normalization will in general allow removing the parameters from the picture: we simply keep the maximal degree of the polynomial and compare it with the loop depth.…”

Section: Dependencesmentioning

confidence: 99%

Iterative optimization in the polyhedral model

et al. 2008

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High-level loop optimizations are necessary to achieve good performance over a wide variety of processors. Their performance impact can be significant because they involve in-depth program transformations that aim to sustain a balanced workload over the computational, storage, and communication resources of the target architecture. Therefore, it is mandatory that the compiler accurately models the target architecture as well as the effects of complex code restructuring. However, because optimizing compilers (1) use simplistic performance models that abstract away many of the complexities of modern architectures, (2) rely on inaccurate dependence analysis, and (3) lack frameworks to express complex interactions of transformation sequences, they typically uncover only a fraction of the peak performance available on many applications. We propose a complete iterative framework to address these issues. We rely on the polyhedral model to construct and traverse a large and expressive search space. This space encompasses only legal, distinct versions resulting from the restructuring of any static control loop nest. We first propose a feedback-driven iterative heuristic tailored to the search space properties of the polyhedral model. Though, it quickly converges to good solutions for small kernels, larger benchmarks containing higher dimensional spaces are more challenging and our heuristic misses opportunities for significant performance improvement. Thus, we introduce the use of a genetic algorithm with specialized operators that leverage the polyhedral representation of program dependences. We provide experimental evidence that the genetic algorithm effectively traverses huge optimization spaces, achieving good performance improvements on large loop nests.

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Memory optimization by counting points in integer transformations of parametric polytopes

Cited by 6 publications

References 26 publications

Symbolic Polynomial Maximization Over Convex Sets and Its Application to Memory Requirement Estimation

Symbolic Polynomial Maximization Over Convex Sets and Its Application to Memory Requirement Estimation

Integer Polyhedra for Program Analysis

Iterative optimization in the polyhedral model

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