### Citations

215 | Scanning polyhedra with do loops
- Ancourt, Irigoin
- 1991
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Citation Context ...arametric polyhedron. To our knowledge, so far it has been used in: • compilation of computer programs, where parts of programs (a class of nested for / do / while loops) are represented by polyhedra =-=[2]-=-. Their computational load [27], power consumption, execution time [13, 20, 6, 31, 32], amount of parallelism [32, 10, 5, 23, 16, 15], and memory behavior [30, 9, 18, 22, 14, 17, 1, 7, 36] are then de... |

179 |
A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed
- Barvinok
- 1994
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Citation Context ...led explicit form. Its data structure has only a polynomial complexity in function of the number of bits used to code the problem. Morevover, the algorithm proposed by Barvinok to compute such a form =-=[3]-=-, extended and implemented by Verdoolaege et al [34], has also polynomial computational complexity for a fixed number of dimensions. Note that the computational complecity of all the existing algorith... |

128 | An algorithmic theory of lattice points in polyhedra
- BARVINOK, POMMERSHEIM
- 1999
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Citation Context ... parameters and variables [26]. Trying to evaluate its computation complexity leads to compare it with Barvinok’s algorithm for computing Ehrhart polynomials, which also uses simplicial decomposition =-=[4, 35]-=-, but on each of P ’s cones (whose vertices are parametric). Decomposing a polytope into simplices has bigger computation complexity than separately decomposing its cones. Due to the (at least) expone... |

106 |
On estimating and enhancing cache effectiveness.
- Ferrante, Sarkar, et al.
- 1991
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Citation Context ...while loops) are represented by polyhedra [2]. Their computational load [27], power consumption, execution time [13, 20, 6, 31, 32], amount of parallelism [32, 10, 5, 23, 16, 15], and memory behavior =-=[30, 9, 18, 22, 14, 17, 1, 7, 36]-=- are then derived from the number of integer points in these polyhedra. These metrics are then used to drive optimizations and parallelization of the compiled program. More examples of Ehrhart polynom... |

105 | Counting solutions to linear and nonlinear constraints through ehrhart polynomials: applications to analyze ans transform scientific programs.
- Clauss
- 1996
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Citation Context ...while loops) are represented by polyhedra [2]. Their computational load [27], power consumption, execution time [13, 20, 6, 31, 32], amount of parallelism [32, 10, 5, 23, 16, 15], and memory behavior =-=[30, 9, 18, 22, 14, 17, 1, 7, 36]-=- are then derived from the number of integer points in these polyhedra. These metrics are then used to drive optimizations and parallelization of the compiled program. More examples of Ehrhart polynom... |

100 |
Decompositions of rational convex polytopes,
- Stanley
- 1980
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Citation Context ...iodic, let us review two contributions about mathematical relations between the faces of P and the period of EP (N). The first one is a corollary of Ehrhart’s conjecture [12], proved later by Stanley =-=[29]-=- and McMullen [24] and extended by Clauss [8] to the case of several parameters: Theorem 1. If all the vertices of P are integer for any integer values of its parameters N, then EP (N) is a (non-perio... |

94 |
Polynomes arithmetiques et Methode des Polyedres en Combinatoire,
- Ehrhart
- 1977
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Citation Context ... Ehrhart polynomial is not periodic, let us review two contributions about mathematical relations between the faces of P and the period of EP (N). The first one is a corollary of Ehrhart’s conjecture =-=[12]-=-, proved later by Stanley [29] and McMullen [24] and extended by Clauss [8] to the case of several parameters: Theorem 1. If all the vertices of P are integer for any integer values of its parameters ... |

85 | Parametric analysis of polyhedral iteration spaces
- Clauss, Loechner
- 1998
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Citation Context ...ts of programs (a class of nested for / do / while loops) are represented by polyhedra [2]. Their computational load [27], power consumption, execution time [13, 20, 6, 31, 32], amount of parallelism =-=[32, 10, 5, 23, 16, 15]-=-, and memory behavior [30, 9, 18, 22, 14, 17, 1, 7, 36] are then derived from the number of integer points in these polyhedra. These metrics are then used to drive optimizations and parallelization of... |

80 |
Footprints in the cache,”
- Thiebaut, Stone
- 1987
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Citation Context ...while loops) are represented by polyhedra [2]. Their computational load [27], power consumption, execution time [13, 20, 6, 31, 32], amount of parallelism [32, 10, 5, 23, 16, 15], and memory behavior =-=[30, 9, 18, 22, 14, 17, 1, 7, 36]-=- are then derived from the number of integer points in these polyhedra. These metrics are then used to drive optimizations and parallelization of the compiled program. More examples of Ehrhart polynom... |

52 | Fully automatic, parametric worst-case execution time analysis.
- Lisper
- 2003
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Citation Context ...compilation of computer programs, where parts of programs (a class of nested for / do / while loops) are represented by polyhedra [2]. Their computational load [27], power consumption, execution time =-=[13, 20, 6, 31, 32]-=-, amount of parallelism [32, 10, 5, 23, 16, 15], and memory behavior [30, 9, 18, 22, 14, 17, 1, 7, 36] are then derived from the number of integer points in these polyhedra. These metrics are then use... |

32 | Polylib: A library for manipulating parameterized polyhedra
- Loechner
- 1999
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Citation Context ...an Barvinok’s exact method. 30s7 Implementation and performance The roughest approximation method, i.e. the orthogonal expansion independent of the validity domains, has been implemented in PolyLib 5 =-=[21]-=-. The algorithm for computing Ehrhart polynomials in PolyLib uses interpolation, counting the integer points by scanning them, for some instances of the parameters. As we reduce the period to (1, ...,... |

32 |
Lattice invariant valuations on rational polytopes, Archiv der Mathematik 31
- McMullen
- 1978
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Citation Context ...ew two contributions about mathematical relations between the faces of P and the period of EP (N). The first one is a corollary of Ehrhart’s conjecture [12], proved later by Stanley [29] and McMullen =-=[24]-=- and extended by Clauss [8] to the case of several parameters: Theorem 1. If all the vertices of P are integer for any integer values of its parameters N, then EP (N) is a (non-periodic) polynomial. T... |

30 | Analytical computation of ehrhart polynomials: Enabling more compiler analyses and optimizations
- Verdoolaege, Seghir, et al.
- 2004
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22 |
Estimation of nested loop execution time by integer arithmetics in convex polyhedra
- Tawbi
- 1994
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Citation Context ...compilation of computer programs, where parts of programs (a class of nested for / do / while loops) are represented by polyhedra [2]. Their computational load [27], power consumption, execution time =-=[13, 20, 6, 31, 32]-=-, amount of parallelism [32, 10, 5, 23, 16, 15], and memory behavior [30, 9, 18, 22, 14, 17, 1, 7, 36] are then derived from the number of integer points in these polyhedra. These metrics are then use... |

21 | On the Quest for Perfect Load Balance in Loop-Based Parallel Computations
- Sakellariou
- 1996
(Show Context)
Citation Context ...nowledge, so far it has been used in: • compilation of computer programs, where parts of programs (a class of nested for / do / while loops) are represented by polyhedra [2]. Their computational load =-=[27]-=-, power consumption, execution time [13, 20, 6, 31, 32], amount of parallelism [32, 10, 5, 23, 16, 15], and memory behavior [30, 9, 18, 22, 14, 17, 1, 7, 36] are then derived from the number of intege... |

17 | Design Space Exploration for Massively Parallel Processor Arrays
- Hannig, Teich
- 2001
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Citation Context ...ts of programs (a class of nested for / do / while loops) are represented by polyhedra [2]. Their computational load [27], power consumption, execution time [13, 20, 6, 31, 32], amount of parallelism =-=[32, 10, 5, 23, 16, 15]-=-, and memory behavior [30, 9, 18, 22, 14, 17, 1, 7, 36] are then derived from the number of integer points in these polyhedra. These metrics are then used to drive optimizations and parallelization of... |

17 | An experimental evaluation of tiling and shackling for memory hierarchy
- Kodukula, Pingali, et al.
- 1999
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14 | On synthesizing parametric specifications of dynamic memory utilization
- Braberman, Garbervetsky, et al.
- 2004
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14 | Probabilistic memory disambiguation and its application to data speculation.
- Ju, Collard, et al.
- 1998
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14 |
On Ehrhart polynomials and probability calculations in voting theory, Social Choice and Welfare
- Lepelley, Louichi, et al.
- 2008
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Citation Context ...ilers by Clauss, Loechner and Wilde [8, 11], are now acknowledged as the most relevant way to do it. • high-level embedded systems hardware synthesis [33]. • probability calculations in voting theory =-=[19]-=-, where election paradigms as well as cases of interest can be phrased as parametric linear constraints. Unfortunately, the use of Ehrhart polynomials in production toolchains is hindered by their com... |

14 |
Processor allocation and loop scheduling on multiprocessor computers
- Tawbi, Feautrier
- 1992
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Citation Context ...compilation of computer programs, where parts of programs (a class of nested for / do / while loops) are represented by polyhedra [2]. Their computational load [27], power consumption, execution time =-=[13, 20, 6, 31, 32]-=-, amount of parallelism [32, 10, 5, 23, 16, 15], and memory behavior [30, 9, 18, 22, 14, 17, 1, 7, 36] are then derived from the number of integer points in these polyhedra. These metrics are then use... |

13 | Deriving formulae to count solutions to parameterized linear systems using Ehrhart polynomials: Applications to the analysis of nested-loop programs
- Clauss, Wilde
- 1997
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Citation Context ...hart polynomials use in the context of embedded systems can be found in [34]. Ehrhart polynomials, initially introduced in the field of program compilation for compilers by Clauss, Loechner and Wilde =-=[8, 11]-=-, are now acknowledged as the most relevant way to do it. • high-level embedded systems hardware synthesis [33]. • probability calculations in voting theory [19], where election paradigms as well as c... |

12 | A compile time based approach for solving out-of-order communication in kahn process networks
- Turjan, Kienhuis, et al.
- 2002
(Show Context)
Citation Context ...duced in the field of program compilation for compilers by Clauss, Loechner and Wilde [8, 11], are now acknowledged as the most relevant way to do it. • high-level embedded systems hardware synthesis =-=[33]-=-. • probability calculations in voting theory [19], where election paradigms as well as cases of interest can be phrased as parametric linear constraints. Unfortunately, the use of Ehrhart polynomials... |

9 | Near-optimal loop tiling by mean of cache miss equations and genetic algorithms
- Abella, Gonzalez, et al.
- 2002
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9 | Tight timing estimation with the newton-gregory formulae
- Engelen, Gallivan, et al.
- 2003
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9 | Analytical computation of ehrhart polynomials and its applications for embedded systems
- Verdoolaege, Beyls, et al.
- 2004
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Citation Context ... polyhedra. These metrics are then used to drive optimizations and parallelization of the compiled program. More examples of Ehrhart polynomials use in the context of embedded systems can be found in =-=[34]-=-. Ehrhart polynomials, initially introduced in the field of program compilation for compilers by Clauss, Loechner and Wilde [8, 11], are now acknowledged as the most relevant way to do it. • high-leve... |

8 | Volume driven data distribution for numamachines
- Heine, Slowik
- 2000
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Citation Context ...ts of programs (a class of nested for / do / while loops) are represented by polyhedra [2]. Their computational load [27], power consumption, execution time [13, 20, 6, 31, 32], amount of parallelism =-=[32, 10, 5, 23, 16, 15]-=-, and memory behavior [30, 9, 18, 22, 14, 17, 1, 7, 36] are then derived from the number of integer points in these polyhedra. These metrics are then used to drive optimizations and parallelization of... |

7 | Stating and Manipulating Periodicity in the Polytope Model. Applications to Program Analysis and Optimization
- Meister
- 2004
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Citation Context ...ts in P can be defined by a linear recurrence relation on N over a validity domain, which implies that the period of P is independent of the constant part K in (1). This property is also retrieved in =-=[25]-=- (chapter 2): the period of the integer hull of P 3 a point I0(N) is said to saturate a constraint aI + bN + c = ≥ 0 iff aI0 + bN + c = 0 10sis independent of K, and the Ehrhart polynomial of P equals... |

6 | The volume of a lattice polyhedron to enumerate processors and parallelism
- Clauss
- 1995
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Citation Context ...hart polynomials use in the context of embedded systems can be found in [34]. Ehrhart polynomials, initially introduced in the field of program compilation for compilers by Clauss, Loechner and Wilde =-=[8, 11]-=-, are now acknowledged as the most relevant way to do it. • high-level embedded systems hardware synthesis [33]. • probability calculations in voting theory [19], where election paradigms as well as c... |

6 |
Precise data locality optimization of nested loops
- Clauss
- 2002
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5 | Performance prediction of distributed applications running on a network of workstations
- Bourgeois, Spies, et al.
- 1999
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5 | Parametric timing estimation with the Newton-Gregory formulae
- Engelen, Gallivan, et al.
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4 | Communication optimization for affine recurrence equations using broadcast and locality
- Loechner, Mongenet
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1 |
Communication pre-evaluation in hpf. Research Report 93-36, LIP, Ecole Normale Supérieure de
- Boulet, Redon
- 1998
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1 |
Volume calculation and volume estimation of parameterized integer polytopes
- Rabl
- 2006
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Citation Context ... are the n-dimensional coordinates of ∆’s vertices. This technique has been extended to the parametric case and even further to cases where inequalities involving products of parameters and variables =-=[26]-=-. Trying to evaluate its computation complexity leads to compare it with Barvinok’s algorithm for computing Ehrhart polynomials, which also uses simplicial decomposition [4, 35], but on each of P ’s c... |

1 |
Counting points in integer affine transformations of parametric z-polytopes
- Seghir, Loechner, et al.
- 2007
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Citation Context .... The variables, I ∈ Z n and the parameters, N ∈ Z p , can then be written as: ⎛ ⎞ ⎛ ⎜ I ⎟ ⎜ I ⎜ ⎟ ⎜ ⎜ ⎜N ⎟ = G. ⎜ ⎝ ⎠ ⎝ 1 ′ N ′ ⎞ ⎟ ⎠ 1 , I′ ∈ Z n , N ′ ∈ Z p 7sLat(G) is called the validity lattice =-=[28]-=- of the polyhedron. A upper-triangular G can always be found, and its (p × p) right bottom sub-matrix of G defines a lattice on the parameters: ⎛ ⎞ ⎛ ⎞ ⎜ ⎝ N⎟ ⎠ = G 1 ′ ′ ⎜N ⎟ . ⎝ ⎠ 1 The polyhedron c... |

1 |
Computation and maniupulation of enumerators of integer projections of parametric polytopes
- Verdoolaege, Woods, et al.
- 2005
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Citation Context ... parameters and variables [26]. Trying to evaluate its computation complexity leads to compare it with Barvinok’s algorithm for computing Ehrhart polynomials, which also uses simplicial decomposition =-=[4, 35]-=-, but on each of P ’s cones (whose vertices are parametric). Decomposing a polytope into simplices has bigger computation complexity than separately decomposing its cones. Due to the (at least) expone... |