[Mlir-commits] [mlir] [MLIR][Presburger] Implement vertex enumeration and chamber decomposition for polytope generating function computation. (PR #78987)

[Arjun P]

================
@@ -147,6 +149,289 @@ GeneratingFunction mlir::presburger::detail::unimodularConeGeneratingFunction(
                             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 their generating
+/// functions. We iterate over the list of regions, each time adding the
----------------
Superty wrote:

It seems unnecessary to bring in vertices here. when you update the header comment according to my suggestion, you can also drop mentions of vertices here and directly talk about active GFs.

https://github.com/llvm/llvm-project/pull/78987