[Mlir-commits] [mlir] [MLIR][Presburger] Generating functions and quasi-polynomials for Barvinok's algorithm (PR #75702)

[llvmlistbot at llvm.org]

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+//===- Barvinok.h - Barvinok's Algorithm ------------------------*- C++ -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+//
+// Functions and classes for Barvinok's algorithm in MLIR.
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef MLIR_ANALYSIS_PRESBURGER_BARVINOK_H
+#define MLIR_ANALYSIS_PRESBURGER_BARVINOK_H
+
+#include "mlir/Analysis/Presburger/Fraction.h"
+#include "mlir/Analysis/Presburger/IntegerRelation.h"
+#include "mlir/Analysis/Presburger/Matrix.h"
+#include "mlir/Analysis/Presburger/PresburgerSpace.h"
+#include "mlir/Analysis/Presburger/Utils.h"
+#include "mlir/Support/LogicalResult.h"
+#include <optional>
+
+namespace mlir {
+namespace presburger {
+
+// The H (inequality) representation of both general
+// polyhedra and cones specifically is an integer relation.
+using PolyhedronH = IntegerRelation;
+using ConeH = PolyhedronH;
+
+// The V (generator) representation of both general
+// polyhedra and cones specifically is simply a matrix
+// whose rows are the generators.
+using PolyhedronV = Matrix<MPInt>;
+using ConeV = PolyhedronV;
+
+// A parametric point is a vector, each of whose elements
+// is an affine function of n parameters. Each row
+// in the matrix represents the affine function and
+// has n+1 elements.
+using ParamPoint = Matrix<Fraction>;
+
+// A point is simply a vector.
+using Point = SmallVector<Fraction>;
+
+// A class to describe the type of generating function
+// used to enumerate the integer points in a polytope.
+// Consists of a set of terms, where the ith term has
+// * a sign, ±1, stored in `signs[i]`
+// * a numerator, of the form x^{n},
+//      where n, stored in `numerators[i]`,
+//      is a parametric point (a vertex).
+// * a denominator, of the form (1 - x^{d1})...(1 - x^{dn}),
+//      where each dj, stored in `denominators[i][j]`,
+//      is a vector (a generator).
+//
+// Represents functions f : Q^n -> Q of the form
+//
+// f(x) = \sum_i s_i * (x^n_i(p)) / (\prod_j (1 - x^d_{ij})
+//
+// where s_i is ±1,
+// n_i \in (Q^d -> Q)^n is an n-vector of affine functions on d parameters, and
+// g_{ij} \in Q^n are vectors.
+class GeneratingFunction {
+public:
+  GeneratingFunction(unsigned numParam, SmallVector<int, 8> signs,
+                     std::vector<ParamPoint> nums,
+                     std::vector<std::vector<Point>> dens)
+      : numParam(numParam), signs(signs), numerators(nums), denominators(dens) {
+    for (const ParamPoint &term : numerators)
+      assert(term.getNumColumns() - 1 == numParam &&
+             "dimensionality of numerator exponents does not match number of "
+             "parameters!");
+  }
+
+  // Find the number of parameters involved in the function
+  // from the dimensionality of the affine functions.
+  unsigned getNumParams() { return numParam; }
+
+  GeneratingFunction operator+(const GeneratingFunction &gf) {
+    assert(numParam == gf.getNumParams() &&
+           "two generating functions with different numbers of parameters "
+           "cannot be added!");
+    signs.append(gf.signs);
+    numerators.insert(numerators.end(), gf.numerators.begin(),
+                      gf.numerators.end());
+    denominators.insert(denominators.end(), gf.denominators.begin(),
+                        gf.denominators.end());
+    return *this;
+  }
+
+  llvm::raw_ostream &print(llvm::raw_ostream &os) const {
+    for (unsigned i = 0, e = signs.size(); i < e; i++) {
+      if (signs[i] == 1)
+        os << " + ";
+      else
+        os << " - ";
+
+      os << "x^[";
+      for (unsigned j = 0, e = numerators[i].size(); j < e - 1; j++)
+        os << numerators[i][j] << ",";
+      os << numerators[i].back() << "]/";
+
+      for (Point den : denominators[i]) {
+        os << "(x^[";
+        for (unsigned j = 0, e = den.size(); j < e - 1; j++)
+          os << den[j] << ",";
+        os << den[den.size() - 1] << "])";
+      }
+    }
+    return os;
+  }
+
+  SmallVector<int, 8> signs;
+  std::vector<ParamPoint> numerators;
+  std::vector<std::vector<Point>> denominators;
+
+private:
+  unsigned numParam;
+};
+
+// A class to describe the quasi-polynomials obtained by
+// substituting the unit vector in the type of generating
+// function described above.
+// Consists of a set of terms.
+// The ith term is a constant `coefficients[i]`, multiplied
+// by the product of a set of affine functions on n parameters.
+// Represents functions f : Q^n -> Q of the form
+//
+// f(x) = \sum_i c_i * \prod_j ⌊g_{ij}(x)⌋
+//
+// where c_i \in Q and
+// g_{ij} : Q^n -> Q are affine functionals.
+class QuasiPolynomial {
+public:
+  QuasiPolynomial(unsigned numParam, SmallVector<Fraction> coeffs = {},
+                  std::vector<std::vector<SmallVector<Fraction>>> aff = {})
+      : numParam(numParam), coefficients(coeffs), affine(aff) {
+    // Find the first term which involves some affine function.
+    for (const std::vector<SmallVector<Fraction>> &term : affine) {
+      if (term.size() == 0)
+        continue;
+      // The number of elements in the affine function is
+      // one more than the number of parameters.
+      assert(term[0].size() - 1 == numParam &&
+             "dimensionality of affine functions does not match number of "
+             "parameters!")
+    }
+  }
+
+  // Find the number of parameters involved in the polynomial.
+  unsigned getNumParams() { return numParam; }
+
+  QuasiPolynomial operator+(const QuasiPolynomial &x) {
+    assert(numParam == x.getNumParams() &&
+           "two quasi-polynomials with different numbers of parameters cannot "
+           "be added!");
+    coefficients.append(x.coefficients);
+    affine.insert(affine.end(), x.affine.begin(), x.affine.end());
+    return *this;
+  }
+
+  QuasiPolynomial operator-(const QuasiPolynomial &x) {
+    bool sameNumParams = (getNumParams() == -1) || (x.getNumParams() == -1) ||
+                         (getNumParams() == x.getNumParams());
+    assert(numParam == x.getNumParams() &&
+           "two quasi-polynomials with different numbers of parameters cannot "
+           "be subtracted!");
+    QuasiPolynomial qp(x.coefficients, x.affine);
+    for (unsigned i = 0, e = x.coefficients.size(); i < e; i++)
+      qp.coefficients[i] = -qp.coefficients[i];
+    return (*this + qp);
+  }
+
+  QuasiPolynomial operator*(const QuasiPolynomial &x) {
+    assert(numParam = x.getNumParams() &&
+                      "two quasi-polynomials with different numbers of "
+                      "parameters cannot be multiplied!");
+    QuasiPolynomial qp();
+    std::vector<SmallVector<Fraction>> product;
+    for (unsigned i = 0, e = coefficients.size(); i < e; i++) {
+      for (unsigned j = 0, e = x.coefficients.size(); j < e; j++) {
+        qp.coefficients.append({coefficients[i] * x.coefficients[j]});
+
+        product.clear();
+        product.insert(product.end(), affine[i].begin(), affine[i].end());
+        product.insert(product.end(), x.affine[j].begin(), x.affine[j].end());
+
+        qp.affine.push_back(product);
+      }
+    }
+
+    return qp;
+  }
+
+  QuasiPolynomial operator/(Fraction x) {
+    assert(x != 0 && "division by zero!");
+    for (Fraction &coeff : coefficients)
+      coeff /= x;
+    return *this;
+  };
+
+  // Removes terms which evaluate to zero from the expression.
+  QuasiPolynomial simplify() {
+    SmallVector<Fraction> newCoeffs({});
+    std::vector<std::vector<SmallVector<Fraction>>> newAffine({});
+    bool prodIsNonz, sumIsNonz;
+    for (unsigned i = 0, e = coefficients.size(); i < e; i++) {
+      prodIsNonz = true;
+      for (unsigned j = 0, e = affine[i].size(); j < e; j++) {
+        sumIsNonz = false;
+        for (unsigned k = 0, e = affine[i][j].size(); k < e; k++)
+          if (affine[i][j][k] != Fraction(0, 1))
+            sumIsNonz = true;
+        if (sumIsNonz == false)
+          prodIsNonz = false;
+      }
+      if (prodIsNonz && coefficients[i] != Fraction(0, 1)) {
+        newCoeffs.append({coefficients[i]});
+        newAffine.push_back({affine[i]});
+      }
+    }
+    return QuasiPolynomial(newCoeffs, newAffine);
+  }
+
+  SmallVector<Fraction> coefficients;
+  std::vector<std::vector<SmallVector<Fraction>>> affine;
+
----------------
Abhinav271828 wrote:

They need to be set during the constant term computation

https://github.com/llvm/llvm-project/pull/75702