Barvinok.cpp

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//===- Barvinok.cpp - 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
//
//===----------------------------------------------------------------------===//
 
#include "mlir/Analysis/Presburger/Barvinok.h"
#include "mlir/Analysis/Presburger/Utils.h"
#include "llvm/ADT/Sequence.h"
#include <algorithm>
 
using namespace mlir;
using namespace presburger;
using namespace mlir::presburger::detail;
 
/// Assuming that the input cone is pointed at the origin,
/// converts it to its dual in V-representation.
/// Essentially we just remove the all-zeroes constant column.
ConeV mlir::presburger::detail::getDual(ConeH cone) {
  unsigned numIneq = cone.getNumInequalities();
  unsigned numVar = cone.getNumCols() - 1;
  ConeV dual(numIneq, numVar, 0, 0);
  // Assuming that an inequality of the form
  // a1*x1 + ... + an*xn + b ≥ 0
  // is represented as a row [a1, ..., an, b]
  // and that b = 0.
 
  for (auto i : llvm::seq<int>(0, numIneq)) {
    assert(cone.atIneq(i, numVar) == 0 &&
           "H-representation of cone is not centred at the origin!");
    for (unsigned j = 0; j < numVar; ++j) {
      dual.at(i, j) = cone.atIneq(i, j);
    }
  }
 
  // Now dual is of the form [ [a1, ..., an] , ... ]
  // which is the V-representation of the dual.
  return dual;
}
 
/// Converts a cone in V-representation to the H-representation
/// of its dual, pointed at the origin (not at the original vertex).
/// Essentially adds a column consisting only of zeroes to the end.
ConeH mlir::presburger::detail::getDual(ConeV cone) {
  unsigned rows = cone.getNumRows();
  unsigned columns = cone.getNumColumns();
  ConeH dual = defineHRep(columns);
  // Add a new column (for constants) at the end.
  // This will be initialized to zero.
  cone.insertColumn(columns);
 
  for (unsigned i = 0; i < rows; ++i)
    dual.addInequality(cone.getRow(i));
 
  // Now dual is of the form [ [a1, ..., an, 0] , ... ]
  // which is the H-representation of the dual.
  return dual;
}
 
/// Find the index of a cone in V-representation.
DynamicAPInt mlir::presburger::detail::getIndex(const ConeV &cone) {
  if (cone.getNumRows() > cone.getNumColumns())
    return DynamicAPInt(0);
 
  return cone.determinant();
}
 
/// Compute the generating function for a unimodular cone.
/// This consists of a single term of the form
/// sign * x^num / prod_j (1 - x^den_j)
///
/// sign is either +1 or -1.
/// den_j is defined as the set of generators of the cone.
/// num is computed by expressing the vertex as a weighted
/// sum of the generators, and then taking the floor of the
/// coefficients.
GeneratingFunction
mlir::presburger::detail::computeUnimodularConeGeneratingFunction(
    ParamPoint vertex, int sign, const ConeH &cone) {
  // Consider a cone with H-representation [0  -1].
  //                                       [-1 -2]
  // Let the vertex be given by the matrix [ 2  2   0], with 2 params.
  //                                       [-1 -1/2 1]
 
  // `cone` must be unimodular.
  assert(abs(getIndex(getDual(cone))) == 1 && "input cone is not unimodular!");
 
  unsigned numVar = cone.getNumVars();
  unsigned numIneq = cone.getNumInequalities();
 
  // Thus its ray matrix, U, is the inverse of the
  // transpose of its inequality matrix, `cone`.
  // The last column of the inequality matrix is null,
  // so we remove it to obtain a square matrix.
  FracMatrix transp = FracMatrix(cone.getInequalities()).transpose();
  transp.removeRow(numVar);
 
  FracMatrix generators(numVar, numIneq);
  transp.determinant(/*inverse=*/&generators); // This is the U-matrix.
  // Thus the generators are given by U = [2  -1].
  //                                      [-1  0]
 
  // The powers in the denominator of the generating
  // function are given by the generators of the cone,
  // i.e., the rows of the matrix U.
  std::vector<Point> denominator(numIneq);
  ArrayRef<Fraction> row;
  for (auto i : llvm::seq<int>(0, numVar)) {
    row = generators.getRow(i);
    denominator[i] = Point(row);
  }
 
  // The vertex is v \in Z^{d x (n+1)}
  // We need to find affine functions of parameters λ_i(p)
  // such that v = Σ λ_i(p)*u_i,
  // where u_i are the rows of U (generators)
  // The λ_i are given by the columns of Λ = v^T U^{-1}, and
  // we have transp = U^{-1}.
  // Then the exponent in the numerator will be
  // Σ -floor(-λ_i(p))*u_i.
  // Thus we store the (exponent of the) numerator as the affine function -Λ,
  // since the generators u_i are already stored as the exponent of the
  // denominator. Note that the outer -1 will have to be accounted for, as it is
  // not stored. See end for an example.
 
  unsigned numColumns = vertex.getNumColumns();
  unsigned numRows = vertex.getNumRows();
  ParamPoint numerator(numColumns, numRows);
  SmallVector<Fraction> ithCol(numRows);
  for (auto i : llvm::seq<int>(0, numColumns)) {
    for (auto j : llvm::seq<int>(0, numRows))
      ithCol[j] = vertex(j, i);
    numerator.setRow(i, transp.preMultiplyWithRow(ithCol));
    numerator.negateRow(i);
  }
  // Therefore Λ will be given by [ 1    0 ] and the negation of this will be
  //                              [ 1/2 -1 ]
  //                              [ -1  -2 ]
  // stored as the numerator.
  // Algebraically, the numerator exponent is
  // [ -2 ⌊ - N - M/2 + 1 ⌋ + 1 ⌊ 0 + M + 2 ⌋ ] -> first  COLUMN of U is [2, -1]
  // [  1 ⌊ - N - M/2 + 1 ⌋ + 0 ⌊ 0 + M + 2 ⌋ ] -> second COLUMN of U is [-1, 0]
 
  return GeneratingFunction(numColumns - 1, SmallVector<int>(1, sign),
                            std::vector({numerator}),
                            std::vector({denominator}));
}
 
/// We use Gaussian elimination to find the solution to a set of d equations
/// of the form
/// a_1 x_1 + ... + a_d x_d + b_1 m_1 + ... + b_p m_p + c = 0
/// where x_i are variables,
/// m_i are parameters and
/// a_i, b_i, c are rational coefficients.
///
/// The solution expresses each x_i as an affine function of the m_i, and is
/// therefore represented as a matrix of size d x (p+1).
/// If there is no solution, we return null.
std::optional<ParamPoint>
mlir::presburger::detail::solveParametricEquations(FracMatrix equations) {
  // equations is a d x (d + p + 1) matrix.
  // Each row represents an equation.
  unsigned d = equations.getNumRows();
  unsigned numCols = equations.getNumColumns();
 
  // If the determinant is zero, there is no unique solution.
  // Thus we return null.
  if (FracMatrix(equations.getSubMatrix(/*fromRow=*/0, /*toRow=*/d - 1,
                                        /*fromColumn=*/0,
                                        /*toColumn=*/d - 1))
          .determinant() == 0)
    return std::nullopt;
 
  // Perform row operations to make each column all zeros except for the
  // diagonal element, which is made to be one.
  for (unsigned i = 0; i < d; ++i) {
    // First ensure that the diagonal element is nonzero, by swapping
    // it with a row that is non-zero at column i.
    if (equations(i, i) != 0)
      continue;
    for (unsigned j = i + 1; j < d; ++j) {
      if (equations(j, i) == 0)
        continue;
      equations.swapRows(j, i);
      break;
    }
 
    Fraction diagElement = equations(i, i);
 
    // Apply row operations to make all elements except the diagonal to zero.
    for (unsigned j = 0; j < d; ++j) {
      if (i == j)
        continue;
      if (equations(j, i) == 0)
        continue;
      // Apply row operations to make element (j, i) zero by subtracting the
      // ith row, appropriately scaled.
      Fraction currentElement = equations(j, i);
      equations.addToRow(/*sourceRow=*/i, /*targetRow=*/j,
                         /*scale=*/-currentElement / diagElement);
    }
  }
 
  // Rescale diagonal elements to 1.
  for (unsigned i = 0; i < d; ++i)
    equations.scaleRow(i, 1 / equations(i, i));
 
  // Now we have reduced the equations to the form
  // x_i + b_1' m_1 + ... + b_p' m_p + c' = 0
  // i.e. each variable appears exactly once in the system, and has coefficient
  // one.
  //
  // Thus we have
  // x_i = - b_1' m_1 - ... - b_p' m_p - c
  // and so we return the negation of the last p + 1 columns of the matrix.
  //
  // We copy these columns and return them.
  ParamPoint vertex =
      equations.getSubMatrix(/*fromRow=*/0, /*toRow=*/d - 1,
                             /*fromColumn=*/d, /*toColumn=*/numCols - 1);
  vertex.negateMatrix();
  return vertex;
}
 
/// This is an implementation of the Clauss-Loechner algorithm for chamber
/// decomposition.
///
/// We maintain a list of pairwise disjoint chambers and the generating
/// functions corresponding to each one. We iterate over the list of regions,
/// each time adding the current region's generating function to the chambers
/// where it is active and separating the chambers where it is not.
///
/// Given the region each generating function is active in, for each subset of
/// generating functions the region that (the sum of) precisely this subset is
/// in, is the intersection of the regions that these are active in,
/// intersected with the complements of the remaining regions.
std::vector<std::pair<PresburgerSet, GeneratingFunction>>
mlir::presburger::detail::computeChamberDecomposition(
    unsigned numSymbols, ArrayRef<std::pair<PresburgerSet, GeneratingFunction>>
                             regionsAndGeneratingFunctions) {
  assert(!regionsAndGeneratingFunctions.empty() &&
         "there must be at least one chamber!");
  // We maintain a list of regions and their associated generating function
  // initialized with the universe and the empty generating function.
  std::vector<std::pair<PresburgerSet, GeneratingFunction>> chambers = {
      {PresburgerSet::getUniverse(PresburgerSpace::getSetSpace(numSymbols)),
       GeneratingFunction(numSymbols, {}, {}, {})}};
 
  // We iterate over the region list.
  //
  // For each activity region R_j (corresponding to the generating function
  // gf_j), we examine all the current chambers R_i.
  //
  // If R_j has a full-dimensional intersection with an existing chamber R_i,
  // then that chamber is replaced by two new ones:
  // 1. the intersection R_i \cap R_j, where the generating function is
  // gf_i + gf_j.
  // 2. the difference R_i - R_j, where the generating function is gf_i.
  //
  // At each step, we define a new chamber list after considering gf_j,
  // replacing and appending chambers as discussed above.
  //
  // The loop has the invariant that the union over all the chambers gives the
  // universe at every step.
  for (const auto &[region, generatingFunction] :
       regionsAndGeneratingFunctions) {
    std::vector<std::pair<PresburgerSet, GeneratingFunction>> newChambers;
 
    for (const auto &[currentRegion, currentGeneratingFunction] : chambers) {
      PresburgerSet intersection = currentRegion.intersect(region);
 
      // If the intersection is not full-dimensional, we do not modify
      // the chamber list.
      if (!intersection.isFullDim()) {
        newChambers.emplace_back(currentRegion, currentGeneratingFunction);
        continue;
      }
 
      // If it is, we add the intersection and the difference as chambers.
      newChambers.emplace_back(intersection,
                               currentGeneratingFunction + generatingFunction);
      newChambers.emplace_back(currentRegion.subtract(region),
                               currentGeneratingFunction);
    }
    chambers = std::move(newChambers);
  }
 
  return chambers;
}
 
/// For a polytope expressed as a set of n inequalities, compute the generating
/// function corresponding to the lattice points included in the polytope. This
/// algorithm has three main steps:
/// 1. Enumerate the vertices, by iterating over subsets of inequalities and
///    checking for satisfiability. For each d-subset of inequalities (where d
///    is the number of variables), we solve to obtain the vertex in terms of
///    the parameters, and then check for the region in parameter space where
///    this vertex satisfies the remaining (n - d) inequalities.
/// 2. For each vertex, identify the tangent cone and compute the generating
///    function corresponding to it. The generating function depends on the
///    parametric expression of the vertex and the (non-parametric) generators
///    of the tangent cone.
/// 3. [Clauss-Loechner decomposition] Identify the regions in parameter space
///    (chambers) where each vertex is active, and accordingly compute the
///    GF of the polytope in each chamber.
///
/// Verdoolaege, Sven, et al. "Counting integer points in parametric
/// polytopes using Barvinok's rational functions." Algorithmica 48 (2007):
/// 37-66.
std::vector<std::pair<PresburgerSet, GeneratingFunction>>
mlir::presburger::detail::computePolytopeGeneratingFunction(
    const PolyhedronH &poly) {
  unsigned numVars = poly.getNumRangeVars();
  unsigned numSymbols = poly.getNumSymbolVars();
  unsigned numIneqs = poly.getNumInequalities();
 
  // We store a list of the computed vertices.
  std::vector<ParamPoint> vertices;
  // For each vertex, we store the corresponding active region and the
  // generating functions of the tangent cone, in order.
  std::vector<std::pair<PresburgerSet, GeneratingFunction>>
      regionsAndGeneratingFunctions;
 
  // We iterate over all subsets of inequalities with cardinality numVars,
  // using permutations of numVars 1's and (numIneqs - numVars) 0's.
  //
  // For a given permutation, we consider a subset which contains
  // the i'th inequality if the i'th bit in the bitset is 1.
  //
  // We start with the permutation that takes the last numVars inequalities.
  SmallVector<int> indicator(numIneqs);
  for (unsigned i = numIneqs - numVars; i < numIneqs; ++i)
    indicator[i] = 1;
 
  do {
    // Collect the inequalities corresponding to the bits which are set
    // and the remaining ones.
    auto [subset, remainder] = poly.getInequalities().splitByBitset(indicator);
    // All other inequalities are stored in a2 and b2c2.
    //
    // These are column-wise splits of the inequalities;
    // a2 stores the coefficients of the variables, and
    // b2c2 stores the coefficients of the parameters and the constant term.
    FracMatrix a2(numIneqs - numVars, numVars);
    FracMatrix b2c2(numIneqs - numVars, numSymbols + 1);
    a2 = FracMatrix(
        remainder.getSubMatrix(0, numIneqs - numVars - 1, 0, numVars - 1));
    b2c2 = FracMatrix(remainder.getSubMatrix(0, numIneqs - numVars - 1, numVars,
                                             numVars + numSymbols));
 
    // Find the vertex, if any, corresponding to the current subset of
    // inequalities.
    std::optional<ParamPoint> vertex =
        solveParametricEquations(FracMatrix(subset)); // d x (p+1)
 
    if (!vertex)
      continue;
    if (llvm::is_contained(vertices, vertex))
      continue;
    // If this subset corresponds to a vertex that has not been considered,
    // store it.
    vertices.emplace_back(*vertex);
 
    // If a vertex is formed by the intersection of more than d facets, we
    // assume that any d-subset of these facets can be solved to obtain its
    // expression. This assumption is valid because, if the vertex has two
    // distinct parametric expressions, then a nontrivial equality among the
    // parameters holds, which is a contradiction as we know the parameter
    // space to be full-dimensional.
 
    // Let the current vertex be [X | y], where
    // X represents the coefficients of the parameters and
    // y represents the constant term.
    //
    // The region (in parameter space) where this vertex is active is given
    // by substituting the vertex into the *remaining* inequalities of the
    // polytope (those which were not collected into `subset`), i.e., into the
    // inequalities [A2 | B2 | c2].
    //
    // Thus, the coefficients of the parameters after substitution become
    // (A2 • X + B2)
    // and the constant terms become
    // (A2 • y + c2).
    //
    // The region is therefore given by
    // (A2 • X + B2) p + (A2 • y + c2) ≥ 0
    //
    // This is equivalent to A2 • [X | y] + [B2 | c2].
    //
    // Thus we premultiply [X | y] with each row of A2
    // and add each row of [B2 | c2].
    FracMatrix activeRegion(numIneqs - numVars, numSymbols + 1);
    for (unsigned i = 0; i < numIneqs - numVars; i++) {
      activeRegion.setRow(i, vertex->preMultiplyWithRow(a2.getRow(i)));
      activeRegion.addToRow(i, b2c2.getRow(i), 1);
    }
 
    // We convert the representation of the active region to an integers-only
    // form so as to store it as a PresburgerSet.
    IntegerPolyhedron activeRegionRel(
        PresburgerSpace::getRelationSpace(0, numSymbols, 0, 0), activeRegion);
 
    // Now, we compute the generating function at this vertex.
    // We collect the inequalities corresponding to each vertex to compute
    // the tangent cone at that vertex.
    //
    // We only need the coefficients of the variables (NOT the parameters)
    // as the generating function only depends on these.
    // We translate the cones to be pointed at the origin by making the
    // constant terms zero.
    ConeH tangentCone = defineHRep(numVars);
    for (unsigned j = 0, e = subset.getNumRows(); j < e; ++j) {
      SmallVector<DynamicAPInt> ineq(numVars + 1);
      for (unsigned k = 0; k < numVars; ++k)
        ineq[k] = subset(j, k);
      tangentCone.addInequality(ineq);
    }
    // We assume that the tangent cone is unimodular, so there is no need
    // to decompose it.
    //
    // In the general case, the unimodular decomposition may have several
    // cones.
    GeneratingFunction vertexGf(numSymbols, {}, {}, {});
    SmallVector<std::pair<int, ConeH>, 4> unimodCones = {{1, tangentCone}};
    for (const std::pair<int, ConeH> &signedCone : unimodCones) {
      auto [sign, cone] = signedCone;
      vertexGf = vertexGf +
                 computeUnimodularConeGeneratingFunction(*vertex, sign, cone);
    }
    // We store the vertex we computed with the generating function of its
    // tangent cone.
    regionsAndGeneratingFunctions.emplace_back(PresburgerSet(activeRegionRel),
                                               vertexGf);
  } while (std::next_permutation(indicator.begin(), indicator.end()));
 
  // Now, we use Clauss-Loechner decomposition to identify regions in parameter
  // space where each vertex is active. These regions (chambers) have the
  // property that no two of them have a full-dimensional intersection, i.e.,
  // they may share "facets" or "edges", but their intersection can only have
  // up to numVars - 1 dimensions.
  //
  // In each chamber, we sum up the generating functions of the active vertices
  // to find the generating function of the polytope.
  return computeChamberDecomposition(numSymbols, regionsAndGeneratingFunctions);
}
 
/// We use an iterative procedure to find a vector not orthogonal
/// to a given set, ignoring the null vectors.
/// Let the inputs be {x_1, ..., x_k}, all vectors of length n.
///
/// In the following,
/// vs[:i] means the elements of vs up to and including the i'th one,
/// <vs, us> means the dot product of vs and us,
/// vs ++ [v] means the vector vs with the new element v appended to it.
///
/// We proceed iteratively; for steps d = 0, ... n-1, we construct a vector
/// which is not orthogonal to any of {x_1[:d], ..., x_n[:d]}, ignoring
/// the null vectors.
/// At step d = 0, we let vs = [1]. Clearly this is not orthogonal to
/// any vector in the set {x_1[0], ..., x_n[0]}, except the null ones,
/// which we ignore.
/// At step d > 0 , we need a number v
/// s.t. <x_i[:d], vs++[v]> != 0 for all i.
/// => <x_i[:d-1], vs> + x_i[d]*v != 0
/// => v != - <x_i[:d-1], vs> / x_i[d]
/// We compute this value for all x_i, and then
/// set v to be the maximum element of this set plus one. Thus
/// v is outside the set as desired, and we append it to vs
/// to obtain the result of the d'th step.
Point mlir::presburger::detail::getNonOrthogonalVector(
    ArrayRef<Point> vectors) {
  unsigned dim = vectors[0].size();
  assert(llvm::all_of(
             vectors,
             [&dim](const Point &vector) { return vector.size() == dim; }) &&
         "all vectors need to be the same size!");
 
  SmallVector<Fraction> newPoint = {Fraction(1, 1)};
  Fraction maxDisallowedValue = -Fraction(1, 0),
           disallowedValue = Fraction(0, 1);
 
  for (unsigned d = 1; d < dim; ++d) {
    // Compute the disallowed values  - <x_i[:d-1], vs> / x_i[d] for each i.
    maxDisallowedValue = -Fraction(1, 0);
    for (const Point &vector : vectors) {
      if (vector[d] == 0)
        continue;
      disallowedValue =
          -dotProduct(ArrayRef(vector).slice(0, d), newPoint) / vector[d];
 
      // Find the biggest such value
      maxDisallowedValue = std::max(maxDisallowedValue, disallowedValue);
    }
    newPoint.emplace_back(maxDisallowedValue + 1);
  }
  return newPoint;
}
 
/// We use the following recursive formula to find the coefficient of
/// s^power in the rational function given by P(s)/Q(s).
///
/// Let P[i] denote the coefficient of s^i in the polynomial P(s).
/// (P/Q)[r] =
/// if (r == 0) then
///   P[0]/Q[0]
/// else
///   (P[r] - {Σ_{i=1}^r (P/Q)[r-i] * Q[i])}/(Q[0])
/// We therefore recursively call `getCoefficientInRationalFunction` on
/// all i \in [0, power).
///
/// https://math.ucdavis.edu/~deloera/researchsummary/
/// barvinokalgorithm-latte1.pdf, p. 1285
QuasiPolynomial mlir::presburger::detail::getCoefficientInRationalFunction(
    unsigned power, ArrayRef<QuasiPolynomial> num, ArrayRef<Fraction> den) {
  assert(!den.empty() && "division by empty denominator in rational function!");
 
  unsigned numParam = num[0].getNumInputs();
  // We use the `isEqual` method of PresburgerSpace, which QuasiPolynomial
  // inherits from.
  assert(llvm::all_of(num,
                      [&num](const QuasiPolynomial &qp) {
                        return num[0].isEqual(qp);
                      }) &&
         "the quasipolynomials should all belong to the same space!");
 
  std::vector<QuasiPolynomial> coefficients;
  coefficients.reserve(power + 1);
 
  coefficients.emplace_back(num[0] / den[0]);
  for (unsigned i = 1; i <= power; ++i) {
    // If the power is not there in the numerator, the coefficient is zero.
    coefficients.emplace_back(i < num.size() ? num[i]
                                             : QuasiPolynomial(numParam, 0));
 
    // After den.size(), the coefficients are zero, so we stop
    // subtracting at that point (if it is less than i).
    unsigned limit = std::min<unsigned long>(i, den.size() - 1);
    for (unsigned j = 1; j <= limit; ++j)
      coefficients[i] = coefficients[i] -
                        coefficients[i - j] * QuasiPolynomial(numParam, den[j]);
 
    coefficients[i] = coefficients[i] / den[0];
  }
  return coefficients[power].simplify();
}
 
/// Substitute x_i = t^μ_i in one term of a generating function, returning
/// a quasipolynomial which represents the exponent of the numerator
/// of the result, and a vector which represents the exponents of the
/// denominator of the result.
/// If the returned value is {num, dens}, it represents the function
/// t^num / \prod_j (1 - t^dens[j]).
/// v represents the affine functions whose floors are multiplied by the
/// generators, and ds represents the list of generators.
std::pair<QuasiPolynomial, std::vector<Fraction>>
substituteMuInTerm(unsigned numParams, const ParamPoint &v,
                   const std::vector<Point> &ds, const Point &mu) {
  unsigned numDims = mu.size();
#ifndef NDEBUG
  for (const Point &d : ds)
    assert(d.size() == numDims &&
           "μ has to have the same number of dimensions as the generators!");
#endif
 
  // First, the exponent in the numerator becomes
  // - (μ • u_1) * (floor(first col of v))
  // - (μ • u_2) * (floor(second col of v)) - ...
  // - (μ • u_d) * (floor(d'th col of v))
  // So we store the negation of the dot products.
 
  // We have d terms, each of whose coefficient is the negative dot product.
  SmallVector<Fraction> coefficients;
  coefficients.reserve(numDims);
  for (const Point &d : ds)
    coefficients.emplace_back(-dotProduct(mu, d));
 
  // Then, the affine function is a single floor expression, given by the
  // corresponding column of v.
  ParamPoint vTranspose = v.transpose();
  std::vector<std::vector<SmallVector<Fraction>>> affine;
  affine.reserve(numDims);
  for (unsigned j = 0; j < numDims; ++j)
    affine.push_back({SmallVector<Fraction>{vTranspose.getRow(j)}});
 
  QuasiPolynomial num(numParams, coefficients, affine);
  num = num.simplify();
 
  std::vector<Fraction> dens;
  dens.reserve(ds.size());
  // Similarly, each term in the denominator has exponent
  // given by the dot product of μ with u_i.
  for (const Point &d : ds) {
    // This term in the denominator is
    // (1 - t^dens.back())
    dens.emplace_back(dotProduct(d, mu));
  }
 
  return {num, dens};
}
 
/// Normalize all denominator exponents `dens` to their absolute values
/// by multiplying and dividing by the inverses, in a function of the form
/// sign * t^num / prod_j (1 - t^dens[j]).
/// Here, sign = ± 1,
/// num is a QuasiPolynomial, and
/// each dens[j] is a Fraction.
void normalizeDenominatorExponents(int &sign, QuasiPolynomial &num,
                                   std::vector<Fraction> &dens) {
  // We track the number of exponents that are negative in the
  // denominator, and convert them to their absolute values.
  unsigned numNegExps = 0;
  Fraction sumNegExps(0, 1);
  for (const auto &den : dens) {
    if (den < 0) {
      numNegExps += 1;
      sumNegExps += den;
    }
  }
 
  // If we have (1 - t^-c) in the denominator, for positive c,
  // multiply and divide by t^c.
  // We convert all negative-exponent terms at once; therefore
  // we multiply and divide by t^sumNegExps.
  // Then we get
  // -(1 - t^c) in the denominator,
  // increase the numerator by c, and
  // flip the sign of the function.
  if (numNegExps % 2 == 1)
    sign = -sign;
  num = num - QuasiPolynomial(num.getNumInputs(), sumNegExps);
}
 
/// Compute the binomial coefficients nCi for 0 ≤ i ≤ r,
/// where n is a QuasiPolynomial.
std::vector<QuasiPolynomial> getBinomialCoefficients(const QuasiPolynomial &n,
                                                     unsigned r) {
  unsigned numParams = n.getNumInputs();
  std::vector<QuasiPolynomial> coefficients;
  coefficients.reserve(r + 1);
  coefficients.emplace_back(numParams, 1);
  for (unsigned j = 1; j <= r; ++j)
    // We use the recursive formula for binomial coefficients here and below.
    coefficients.emplace_back(
        (coefficients[j - 1] * (n - QuasiPolynomial(numParams, j - 1)) /
         Fraction(j, 1))
            .simplify());
  return coefficients;
}
 
/// Compute the binomial coefficients nCi for 0 ≤ i ≤ r,
/// where n is a QuasiPolynomial.
std::vector<Fraction> getBinomialCoefficients(const Fraction &n,
                                              const Fraction &r) {
  std::vector<Fraction> coefficients;
  coefficients.reserve((int64_t)floor(r));
  coefficients.emplace_back(1);
  for (unsigned j = 1; j <= r; ++j)
    coefficients.emplace_back(coefficients[j - 1] * (n - (j - 1)) / (j));
  return coefficients;
}
 
/// We have a generating function of the form
/// f_p(x) = \sum_i sign_i * (x^n_i(p)) / (\prod_j (1 - x^d_{ij})
///
/// where sign_i is ±1,
/// n_i \in Q^p -> Q^d is the sum of the vectors d_{ij}, weighted by the
/// floors of d affine functions on p parameters.
/// d_{ij} \in Q^d are vectors.
///
/// We need to find the number of terms of the form x^t in the expansion of
/// this function.
/// However, direct substitution (x = (1, ..., 1)) causes the denominator
/// to become zero.
///
/// We therefore use the following procedure instead:
/// 1. Substitute x_i = (s+1)^μ_i for some vector μ. This makes the generating
/// function a function of a scalar s.
/// 2. Write each term in this function as P(s)/Q(s), where P and Q are
/// polynomials. P has coefficients as quasipolynomials in d parameters, while
/// Q has coefficients as scalars.
/// 3. Find the constant term in the expansion of each term P(s)/Q(s). This is
/// equivalent to substituting s = 0.
///
/// Verdoolaege, Sven, et al. "Counting integer points in parametric
/// polytopes using Barvinok's rational functions." Algorithmica 48 (2007):
/// 37-66.
QuasiPolynomial
mlir::presburger::detail::computeNumTerms(const GeneratingFunction &gf) {
  // Step (1) We need to find a μ such that we can substitute x_i =
  // (s+1)^μ_i. After this substitution, the exponent of (s+1) in the
  // denominator is (μ_i • d_{ij}) in each term. Clearly, this cannot become
  // zero. Hence we find a vector μ that is not orthogonal to any of the
  // d_{ij} and substitute x accordingly.
  std::vector<Point> allDenominators;
  for (ArrayRef<Point> den : gf.getDenominators())
    allDenominators.insert(allDenominators.end(), den.begin(), den.end());
  Point mu = getNonOrthogonalVector(allDenominators);
 
  unsigned numParams = gf.getNumParams();
  const std::vector<std::vector<Point>> &ds = gf.getDenominators();
  QuasiPolynomial totalTerm(numParams, 0);
  for (unsigned i = 0, e = ds.size(); i < e; ++i) {
    int sign = gf.getSigns()[i];
 
    // Compute the new exponents of (s+1) for the numerator and the
    // denominator after substituting μ.
    auto [numExp, dens] =
        substituteMuInTerm(numParams, gf.getNumerators()[i], ds[i], mu);
    // Now the numerator is (s+1)^numExp
    // and the denominator is \prod_j (1 - (s+1)^dens[j]).
 
    // Step (2) We need to express the terms in the function as quotients of
    // polynomials. Each term is now of the form
    // sign_i * (s+1)^numExp / (\prod_j (1 - (s+1)^dens[j]))
    // For the i'th term, we first normalize the denominator to have only
    // positive exponents. We convert all the dens[j] to their
    // absolute values and change the sign and exponent in the numerator.
    normalizeDenominatorExponents(sign, numExp, dens);
 
    // Then, using the formula for geometric series, we replace each (1 -
    // (s+1)^(dens[j])) with
    // (-s)(\sum_{0 ≤ k < dens[j]} (s+1)^k).
    for (auto &j : dens)
      j = abs(j) - 1;
    // Note that at this point, the semantics of `dens[j]` changes to mean
    // a term (\sum_{0 ≤ k ≤ dens[j]} (s+1)^k). The denominator is, as before,
    // a product of these terms.
 
    // Since the -s are taken out, the sign changes if there is an odd number
    // of such terms.
    unsigned r = dens.size();
    if (dens.size() % 2 == 1)
      sign = -sign;
 
    // Thus the term overall now has the form
    // sign'_i * (s+1)^numExp /
    // (s^r * \prod_j (\sum_{0 ≤ k < dens[j]} (s+1)^k)).
    // This means that
    // the numerator is a polynomial in s, with coefficients as
    // quasipolynomials (given by binomial coefficients), and the denominator
    // is a polynomial in s, with integral coefficients (given by taking the
    // convolution over all j).
 
    // Step (3) We need to find the constant term in the expansion of each
    // term. Since each term has s^r as a factor in the denominator, we avoid
    // substituting s = 0 directly; instead, we find the coefficient of s^r in
    // sign'_i * (s+1)^numExp / (\prod_j (\sum_k (s+1)^k)),
    // Letting P(s) = (s+1)^numExp and Q(s) = \prod_j (...),
    // we need to find the coefficient of s^r in P(s)/Q(s),
    // for which we use the `getCoefficientInRationalFunction()` function.
 
    // First, we compute the coefficients of P(s), which are binomial
    // coefficients.
    // We only need the first r+1 of these, as higher-order terms do not
    // contribute to the coefficient of s^r.
    std::vector<QuasiPolynomial> numeratorCoefficients =
        getBinomialCoefficients(numExp, r);
 
    // Then we compute the coefficients of each individual term in Q(s),
    // which are (dens[i]+1) C (k+1) for 0 ≤ k ≤ dens[i].
    std::vector<std::vector<Fraction>> eachTermDenCoefficients;
    std::vector<Fraction> singleTermDenCoefficients;
    eachTermDenCoefficients.reserve(r);
    for (const Fraction &den : dens) {
      singleTermDenCoefficients = getBinomialCoefficients(den + 1, den + 1);
      eachTermDenCoefficients.emplace_back(
          ArrayRef<Fraction>(singleTermDenCoefficients).drop_front());
    }
 
    // Now we find the coefficients in Q(s) itself
    // by taking the convolution of the coefficients
    // of all the terms.
    std::vector<Fraction> denominatorCoefficients;
    denominatorCoefficients = eachTermDenCoefficients[0];
    for (unsigned j = 1, e = eachTermDenCoefficients.size(); j < e; ++j)
      denominatorCoefficients = multiplyPolynomials(denominatorCoefficients,
                                                    eachTermDenCoefficients[j]);
 
    totalTerm =
        totalTerm + getCoefficientInRationalFunction(r, numeratorCoefficients,
                                                     denominatorCoefficients) *
                        QuasiPolynomial(numParams, sign);
  }
 
  return totalTerm.simplify();
}