[Mlir-commits] [mlir] [MLIR][Presburger] Implement vertex enumeration and chamber decomposition for polytope generating function computation. (PR #78987)
Arjun P
llvmlistbot at llvm.org
Tue Jan 30 08:41:38 PST 2024
================
@@ -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
+/// generating function to the chambers where the corresponding vertex is
+/// active and separating the chambers where it is not.
+///
+/// Given the region each vertex is active in, for each subset of vertices,
+/// the region that precisely this subset is in, is the intersection of the
+/// regions that these are active in, intersected with the complements of the
+/// remaining regions. The generating function for this region is the sum of
+/// generating functions for the vertices' tangent cones.
+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).
+ 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 as a chamber where this vertex is
+ // active.
+ newChambers.emplace_back(intersection,
+ currentGeneratingFunction + generatingFunction);
+
+ // We also add the difference if it is full-dimensional.
+ PresburgerSet difference = currentRegion.subtract(region);
+ if (difference.isFullDim())
+ newChambers.emplace_back(difference, 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.
+ std::vector<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 (std::find(vertices.begin(), vertices.end(), vertex) != vertices.end())
+ continue;
+ // If this subset corresponds to a vertex that has not been considered,
+ // store it.
+ vertices.push_back(*vertex);
+
+ // 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.,
+ // [A2 | B2 | c2].
----------------
Superty wrote:
ok then `i.e., into the matrix` would be clearer. I couldn't understand what is [A2 | B2 | c2] in the `i.e. [A2 | B2 | c2]`
https://github.com/llvm/llvm-project/pull/78987
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