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

[Arjun P]

================
@@ -147,6 +149,310 @@ 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 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 a vertex v_j, whose
+  // generating function is 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 (if it is full-dimensional), where the
+  // generating function is gf_i + gf_j.
+  // 2. the difference R_i - R_j (if it is full-dimensional), where v_j is
+  // inactive and the generating function is gf_i.
+  //
+  // At each step, we define a new chamber list after considering vertex v_j,
+  // and its generating function, 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 (modulo lower-dimensional spaces).
----------------
Superty wrote:

this modulo lower dimensional spaces is not correct. it has to actually give the union at every step.

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