[Mlir-commits] [mlir] [MLIR][Presburger] Implement function to evaluate the number of terms in a generating function. (PR #78078)

[Arjun P]

================
@@ -245,3 +246,240 @@ QuasiPolynomial mlir::presburger::detail::getCoefficientInRationalFunction(
   }
   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.
+/// v represents the affine functions whose floors are multiplied by
+/// 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();
+  for (const Point &d : ds)
+    assert(d.size() == numDims &&
+           "μ has to have the same number of dimensions as the generators!");
+
+  // 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.
+/// Also, take the factors (-s) out of each term of the product.
+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];
+    }
+    // All exponents will be made positive; then reduce
+    // (1 - (s+1)^x)
+    // to
+    // (-s)*(Σ_{x-1} (s+1)^j) because x > 0
+    dens[j] = abs(dens[j]) - 1;
+  }
+
+  // If we have (1 - (s+1)^-c) in the denominator,
+  // multiply and divide by (s+1)^c.
+  // We convert all negative-exponent terms at once; therefore
+  // we multiply and divide by (s+1)^sumNegExps.
+  // Then we get
+  // -(1 - (s+1)^c) in the denominator,
+  // increase the numerator by c, and
+  // flip the sign.
+  if (numNegExps % 2 == 1)
+    sign = -sign;
+  num = num - QuasiPolynomial(num.getNumInputs(), sumNegExps);
+
+  // Take all the (-s) out.
+  unsigned r = dens.size();
+  if (r % 2 == 1)
+    sign = -sign;
+}
+
+/// 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 numerator and 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 by multiplying and dividing by (s+1)^(-dens[j]) if it is
+    // negative. We change the sign accordingly to keep the denominator in the
+    // same form.
+    // 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).
----------------
Superty wrote:

you can do that outside

https://github.com/llvm/llvm-project/pull/78078