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...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...

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...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...

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... 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...

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...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...

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...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...

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...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...

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... 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 ...

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...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...

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...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...

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...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...

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...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, ...,...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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... 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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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...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...

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... 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...

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.... 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...

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... 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...