bolt/deps/llvm-18.1.8/mlir/lib/Analysis/Presburger/Barvinok.cpp

//===- 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.
MPInt mlir::presburger::detail::getIndex(ConeV cone) {
  if (cone.getNumRows() > cone.getNumColumns())
    return MPInt(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::unimodularConeGeneratingFunction(
    ParamPoint vertex, int sign, 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(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 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,
                   [&](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.push_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.size() != 0 &&
         "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, [&](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.push_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.push_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, ParamPoint v, std::vector<Point> ds,
                   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.push_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.push_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 (unsigned j = 0, e = dens.size(); j < e; ++j) {
    if (dens[j] < 0) {
      numNegExps += 1;
      sumNegExps += dens[j];
    }
  }

  // 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(QuasiPolynomial n,
                                                     unsigned r) {
  unsigned numParams = n.getNumInputs();
  std::vector<QuasiPolynomial> coefficients;
  coefficients.reserve(r + 1);
  coefficients.push_back(QuasiPolynomial(numParams, 1));
  for (unsigned j = 1; j <= r; ++j)
    // We use the recursive formula for binomial coefficients here and below.
    coefficients.push_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(Fraction n, Fraction r) {
  std::vector<Fraction> coefficients;
  coefficients.reserve((int64_t)floor(r));
  coefficients.push_back(1);
  for (unsigned j = 1; j <= r; ++j)
    coefficients.push_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 (unsigned j = 0, e = dens.size(); j < e; ++j)
      dens[j] = abs(dens[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.push_back(
          ArrayRef<Fraction>(singleTermDenCoefficients).slice(1));
    }

    // 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();
}