Many compiler optimization techniques depend on the ability to calculate the number of elements that satisfy certain conditions. If these conditions can be represented by linear constraints, then such problems are equivalent to counting the number of integer points in (possibly) parametric polytopes.It is well known that the enumerator of such a set can be represented by an explicit function consisting of a set of quasi-polynomials, each associated with a chamber in the parameter space. Previously, interpolation was used to obtain these quasi-polynomials, but this technique has several disadvantages. Its worst-case computation time for a single quasi-polynomial is exponential in the input size, even for fixed dimensions. The worst-case size of such a quasi-polynomial (measured in bits needed to represent the quasi-polynomial) is also exponential in the input size. Under certain conditions this technique even fails to produce a solution.Our main contribution is a novel method for calculating the required quasi-polynomials analytically. It extends an existing method, based on Barvinok's decomposition, for counting the number of integer points in a non-parametric polytope. Our technique always produces a solution and computes polynomially-sized enumerators in polynomial time (for fixed dimensions).

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Abstract. Many compiler optimization techniques depend on the ability to calculate the number of integer values that satisfy a given set of linear constraints. This count (the enumerator of a parametric polytope) is a function of the symbolic parameters that may appear in the constraints. In an extended problem (the "integer projection" of a parametric polytope), some of the variables that appear in the constraints may be existentially quantified and then the enumerated set corresponds to the projection of the integer points in a parametric polytope. This paper shows how to reduce the enumeration of the integer projection of parametric polytopes to the enumeration of parametric polytopes. Two approaches are described and experimentally compared. Both can solve problems that were considered very difficult to solve analytically.

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Modern instruction sets extend their load/store-instructions with cache hints, as an additional means to bridge the processor-memory speed gap. Cache hints are used to specify the cache level at which the data is likely to be found, as well as the cache level where the data is stored after accessing it. In order to improve a program's cache behavior, the cache hint is selected based on the data locality of the instruction. We represent the data locality of an instruction by its reuse distance distribution. The reuse distance is the amount of data addressed between two accesses to the same memory location. The distribution allows to efficiently estimate the cache level where the data will be found, and to determine the level where the data should be stored to improve the hit rate. The Open64 EPIC-compiler was extended with cache hint selection. Execution on an Itanium multiprocessor shows speedups of up to 36% in numerical and 23% in non-numerical programs.