[Mlir-commits] [mlir] [MLIR][Presburger][WIP] Implement vertex enumeration and chamber decomposition for polytope generating function computation. (PR #78987)
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Mon Jan 22 06:57:34 PST 2024
llvmbot wrote:
<!--LLVM PR SUMMARY COMMENT-->
@llvm/pr-subscribers-mlir
Author: None (Abhinav271828)
<details>
<summary>Changes</summary>
We implement a function to compute the generating function corresponding to a full-dimensional parametric polytope whose tangent cones are all unimodular.
---
Full diff: https://github.com/llvm/llvm-project/pull/78987.diff
5 Files Affected:
- (modified) mlir/include/mlir/Analysis/Presburger/Barvinok.h (+12)
- (modified) mlir/include/mlir/Analysis/Presburger/GeneratingFunction.h (+1-1)
- (modified) mlir/include/mlir/Analysis/Presburger/IntegerRelation.h (+7)
- (modified) mlir/lib/Analysis/Presburger/Barvinok.cpp (+312)
- (modified) mlir/lib/Analysis/Presburger/IntegerRelation.cpp (+42)
``````````diff
diff --git a/mlir/include/mlir/Analysis/Presburger/Barvinok.h b/mlir/include/mlir/Analysis/Presburger/Barvinok.h
index b70ec33b693235..a5bc49d79408c6 100644
--- a/mlir/include/mlir/Analysis/Presburger/Barvinok.h
+++ b/mlir/include/mlir/Analysis/Presburger/Barvinok.h
@@ -27,6 +27,7 @@
#include "mlir/Analysis/Presburger/GeneratingFunction.h"
#include "mlir/Analysis/Presburger/IntegerRelation.h"
#include "mlir/Analysis/Presburger/Matrix.h"
+#include "mlir/Analysis/Presburger/PresburgerRelation.h"
#include "mlir/Analysis/Presburger/QuasiPolynomial.h"
#include <optional>
@@ -84,6 +85,17 @@ ConeH getDual(ConeV cone);
GeneratingFunction unimodularConeGeneratingFunction(ParamPoint vertex, int sign,
ConeH cone);
+/// Find the solution of a set of equations that express affine constraints
+/// between a set of variables and a set of parameters. The solution expresses
+/// each variable as an affine function of the parameters.
+/// If there is no solution, return std::null.
+std::optional<ParamPoint> findVertex(IntMatrix equations);
+
+/// Compute the generating function corresponding to a polytope.
+/// All tangent cones of the polytope must be unimodular.
+std::vector<std::pair<PresburgerRelation, GeneratingFunction>>
+polytopeGeneratingFunction(PolyhedronH poly);
+
/// Find a vector that is not orthogonal to any of the given vectors,
/// i.e., has nonzero dot product with those of the given vectors
/// that are not null.
diff --git a/mlir/include/mlir/Analysis/Presburger/GeneratingFunction.h b/mlir/include/mlir/Analysis/Presburger/GeneratingFunction.h
index c38eab6efd0fc1..35f15319b4568e 100644
--- a/mlir/include/mlir/Analysis/Presburger/GeneratingFunction.h
+++ b/mlir/include/mlir/Analysis/Presburger/GeneratingFunction.h
@@ -72,7 +72,7 @@ class GeneratingFunction {
return denominators;
}
- GeneratingFunction operator+(GeneratingFunction &gf) const {
+ GeneratingFunction operator+(GeneratingFunction gf) const {
assert(numParam == gf.getNumParams() &&
"two generating functions with different numbers of parameters "
"cannot be added!");
diff --git a/mlir/include/mlir/Analysis/Presburger/IntegerRelation.h b/mlir/include/mlir/Analysis/Presburger/IntegerRelation.h
index c476a022a48272..141dc1f8fa97ed 100644
--- a/mlir/include/mlir/Analysis/Presburger/IntegerRelation.h
+++ b/mlir/include/mlir/Analysis/Presburger/IntegerRelation.h
@@ -711,6 +711,13 @@ class IntegerRelation {
/// return `this \ set`.
PresburgerRelation subtract(const PresburgerRelation &set) const;
+ // Remove equalities which have only zero coefficients.
+ void removeTrivialEqualities();
+
+ // Verify whether the relation is full-dimensional, i.e.,
+ // has the same number of dimensions as the number of variables.
+ bool isFullDim();
+
void print(raw_ostream &os) const;
void dump() const;
diff --git a/mlir/lib/Analysis/Presburger/Barvinok.cpp b/mlir/lib/Analysis/Presburger/Barvinok.cpp
index e0fd0dd8caa4d3..3afc9ad8638ccd 100644
--- a/mlir/lib/Analysis/Presburger/Barvinok.cpp
+++ b/mlir/lib/Analysis/Presburger/Barvinok.cpp
@@ -10,6 +10,7 @@
#include "mlir/Analysis/Presburger/Utils.h"
#include "llvm/ADT/Sequence.h"
#include <algorithm>
+#include <bitset>
using namespace mlir;
using namespace presburger;
@@ -147,6 +148,317 @@ 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::findVertex(IntMatrix equations) {
+ // equations is a d x (d + p + 1) matrix.
+ // Each row represents an equation.
+
+ unsigned numEqs = equations.getNumRows();
+ unsigned numCols = equations.getNumColumns();
+
+ // First, we check that the system has a solution, and return
+ // null if not.
+ IntMatrix coeffs(numEqs, numEqs);
+ for (unsigned i = 0; i < numEqs; i++)
+ for (unsigned j = 0; j < numEqs; j++)
+ coeffs(i, j) = equations(i, j);
+
+ if (coeffs.determinant() == 0)
+ return std::nullopt;
+
+ // We work with rational numbers.
+ FracMatrix equationsF(equations);
+
+ for (unsigned i = 0; i < numEqs; ++i) {
+ // First ensure that the diagonal element is nonzero, by swapping
+ // it with a nonzero row.
+ if (equationsF(i, i) == 0) {
+ for (unsigned j = i + 1; j < numEqs; ++j) {
+ if (equationsF(j, i) != 0) {
+ equationsF.swapRows(j, i);
+ break;
+ }
+ }
+ }
+
+ Fraction b = equationsF(i, i);
+
+ // Set all elements except the diagonal to zero.
+ for (unsigned j = 0; j < numEqs; ++j) {
+ if (equationsF(j, i) == 0 || j == i)
+ continue;
+ // Set element (j, i) to zero
+ // by subtracting the ith row,
+ // appropriately scaled.
+ Fraction a = equationsF(j, i);
+ equationsF.addToRow(j, equationsF.getRow(i), -a / b);
+ }
+ }
+
+ // Rescale diagonal elements to 1.
+ for (unsigned i = 0; i < numEqs; ++i) {
+ Fraction a = equationsF(i, i);
+ for (unsigned j = 0; j < numCols; ++j)
+ equationsF(i, j) = equationsF(i, j) / a;
+ }
+
+ // We copy the last p+1 columns of the matrix as the values of x_i.
+ // We shift the parameter terms to the RHS, and so flip their sign.
+ ParamPoint vertex(numEqs, numCols - numEqs);
+ for (unsigned i = 0; i < numEqs; ++i)
+ for (unsigned j = 0; j < numCols - numEqs; ++j)
+ vertex(i, j) = -equationsF(i, numEqs + j);
+
+ return vertex;
+}
+
+/// For a polytope expressed as a set of inequalities, compute the generating
+/// function corresponding to the number of lattice points present. This
+/// algorithm has three main steps:
+/// 1. Enumerate the vertices, by iterating over subsets of inequalities and
+/// checking for solubility.
+/// 2. For each vertex, identify the tangent cone and compute the generating
+/// function corresponding to it. The sum of these GFs is the GF of the
+/// polytope.
+/// 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<PresburgerRelation, GeneratingFunction>>
+mlir::presburger::detail::polytopeGeneratingFunction(PolyhedronH poly) {
+ unsigned numVars = poly.getNumRangeVars();
+ unsigned numParams = poly.getNumSymbolVars();
+ unsigned numIneqs = poly.getNumInequalities();
+
+ // The generating function of the polytope is computed as a set of generating
+ // functions, each one associated with a region in parameter space (chamber).
+ std::vector<std::pair<PresburgerRelation, GeneratingFunction>> gf({});
+
+ // The active region will be defined as activeRegionCoeffs @ p +
+ // activeRegionConstant ≥ 0. The active region is a polyhedron in parameter
+ // space.
+ FracMatrix activeRegion(numIneqs - numVars, numParams + 1);
+
+ // These vectors store lists of
+ // subsets of inequalities,
+ // the vertices corresponding to them, and
+ // the active regions of the vertices, in order.
+ std::vector<IntMatrix> subsets;
+ std::vector<ParamPoint> vertices;
+ std::vector<PresburgerRelation> activeRegions;
+
+ FracMatrix a2(numIneqs - numVars, numVars);
+ FracMatrix b2c2(numIneqs - numVars, numParams + 1);
+
+ // We iterate over all subsets of inequalities with cardinality numVars,
+ // using bitsets up to 2^numIneqs to enumerate.
+ for (std::bitset<16> indicator(((1ul << numVars) - 1ul)
+ << (numIneqs - numVars));
+ indicator.to_ulong() <=
+ ((1ul << numVars) - 1ul)
+ << (numIneqs - numVars); // d 1's followed by n-numVars 0's
+ indicator = std::bitset<16>(indicator.to_ulong() - 1)) {
+
+ if (indicator.count() != numVars)
+ continue;
+
+ // Collect the inequalities corresponding to the bits which are set.
+ IntMatrix subset(numVars, numVars + numParams + 1);
+ unsigned j1 = 0, j2 = 0;
+ for (unsigned i = 0; i < numIneqs; i++)
+ if (indicator.test(i))
+ subset.setRow(j1++, poly.getInequality(i));
+
+ else {
+ // 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.
+ for (unsigned k = 0; k < numVars; k++)
+ a2(j2, k) = poly.atIneq(i, k);
+ for (unsigned k = numVars; k < numVars + numParams + 1; k++)
+ b2c2(j2, k - numVars) = poly.atIneq(i, k);
+ j2++;
+ }
+
+ // Find the vertex, if any, corresponding to the current subset of
+ // inequalities.
+ std::optional<ParamPoint> vertex = findVertex(subset); // d x (p+1)
+
+ if (vertex == std::nullopt)
+ continue;
+ // If this subset corresponds to a vertex, store it.
+ vertices.push_back(*vertex);
+ subsets.push_back(subset);
+
+ // 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].
+ // Thus, the coefficients of the parameters after substitution become
+ // (A2 • X + B2)
+ // and the constant terms become
+ // (A2 • y + c2).
+ // The region is therefore given by
+ // (A2 • X + B2) p + (A2 • y + c2) ≥ 0
+ // This is equivalent to A2 • [X | y] + [B2 | c2]
+ // Thus we premultiply [X | y] with each row of A2
+ // and add each row of [B2 | c2].
+ for (unsigned i = 0; i < numIneqs - numVars; i++) {
+ activeRegion.setRow(i, (*vertex).preMultiplyWithRow(a2.getRow(i)));
+ activeRegion.addToRow(i, b2c2.getRow(i), 1);
+ }
+
+ // We convert the representation of the active region to an integers-only
+ // form so as to store it as an PresburgerRelation.
+ // We do this by taking the LCM of the denominators of all the coefficients
+ // and multiplying by it throughout.
+ IntMatrix activeRegionNorm = IntMatrix(numIneqs - numVars, numParams + 1);
+ IntegerRelation activeRegionRel =
+ IntegerRelation(PresburgerSpace::getRelationSpace(0, numParams, 0, 0));
+ MPInt lcmDenoms = MPInt(1);
+ for (unsigned i = 0; i < numIneqs - numVars; i++) {
+ for (unsigned j = 0; j < numParams + 1; j++)
+ lcmDenoms = lcm(lcmDenoms, activeRegion(i, j).den);
+ for (unsigned j = 0; j < numParams + 1; j++)
+ activeRegionNorm(i, j) =
+ (activeRegion(i, j) * lcmDenoms).getAsInteger();
+
+ activeRegionRel.addInequality(activeRegionNorm.getRow(i));
+ }
+
+ activeRegions.push_back(PresburgerRelation(activeRegionRel));
+ }
+
+ // Now, we use Clauss-Loechner decomposition to identify regions in parameter
+ // space where each vertex is active. These regions (chambers) have the
+ // property that no two of them have a full-dimensional intersection, i.e.,
+ // they may share "faces" or "edges", but their intersection can only have
+ // up to numVars-1 dimensions.
+
+ // We maintain a list of regions and their associated vertex sets,
+ // initialized with the first vertex and its corresponding activity region.
+ std::vector<std::pair<PresburgerRelation, std::vector<unsigned>>> chambers = {
+ std::make_pair(activeRegions[0], std::vector({0u}))};
+ // Note that instead of storing lists of actual vertices, we store lists
+ // of indices. Thus the set {2, 3, 4} represents the vertex set
+ // {vertices[2], vertices[3], vertices[4]}.
+
+ std::vector<std::pair<PresburgerRelation, std::vector<unsigned>>> newChambers;
+
+ // We iterate over the vertex set.
+ // For each vertex v_j and its activity region R_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, where v_j is active;
+ // 2. the difference R_i - R_j, where v_j is inactive.
+ // Once we have examined all R_i, we add a final chamber
+ // R_j - (union of all existing chambers),
+ // in which only v_j is active.
+ for (unsigned j = 1; j < vertices.size(); j++) {
+ newChambers.clear();
+
+ PresburgerRelation r_j = activeRegions[j];
+ ParamPoint v_j = vertices[j];
+
+ for (unsigned i = 0; i < chambers.size(); i++) {
+ auto [r_i, v_i] = chambers[i];
+
+ // First, we check if the intersection of R_j and R_i.
+ // It is a disjoint union of convex regions in the parameter space,
+ // and so we know that it is full-dimensional if any of the disjuncts
+ // is full-dimensional.
+ PresburgerRelation intersection = r_i.intersect(r_j);
+ bool isFullDim = false;
+ for (auto disjunct : intersection.getAllDisjuncts())
+ if (disjunct.isFullDim()) {
+ isFullDim = true;
+ break;
+ }
+ isFullDim = (numParams == 0) || isFullDim;
+
+ // If the intersection is not full-dimensional, we do not modify
+ // the chamber list.
+ if (!isFullDim)
+ newChambers.push_back(chambers[i]);
+ else {
+ // If it is, we add the intersection and the difference as new chambers.
+ PresburgerRelation subtraction = r_i.subtract(r_j);
+ newChambers.push_back(std::make_pair(subtraction, v_i));
+
+ v_i.push_back(j);
+ newChambers.push_back(std::make_pair(intersection, v_i));
+ }
+ }
+
+ // Finally we compute the chamber where only v_j is active by subtracting
+ // all existing chambers from R_j.
+ for (auto chamber : newChambers)
+ r_j = r_j.subtract(chamber.first);
+ newChambers.push_back(std::make_pair(r_j, std::vector({j})));
+
+ // We filter `chambers` to remove empty regions.
+ chambers.clear();
+ for (auto chamber : newChambers) {
+ bool empty = true;
+ for (auto disjunct : chamber.first.getAllDisjuncts())
+ if (!disjunct.isEmpty()) {
+ empty = false;
+ break;
+ }
+ if (!empty)
+ chambers.push_back(chamber);
+ }
+ }
+
+ // Now, we compute the generating function. For each chamber, we iterate over
+ // the vertices active in it, and compute the generating function for each
+ // of them. The sum of these generating functions is the GF corresponding to
+ // the entire polytope.
+ SmallVector<MPInt> ineq(numVars + 1);
+ for (auto chamber : chambers) {
+ GeneratingFunction chamberGf(numParams, {}, {}, {});
+ for (unsigned i : chamber.second) {
+ // We collect the inequalities corresponding to each vertex.
+ // We only need the coefficients of the variables (NOT the parameters)
+ // as the generating function only depends on these.
+ ConeH tgtCone = defineHRep(numVars);
+ for (unsigned j = 0; j < numVars; j++) {
+ for (unsigned k = 0; k < numVars; k++)
+ ineq[k] = subsets[i](j, k);
+ ineq[numVars] = subsets[i](j, numVars + numParams);
+ tgtCone.addInequality(ineq);
+ }
+ // We assume that the tangent cone is unimodular.
+ SmallVector<std::pair<int, ConeH>, 4> unimodCones = {
+ std::make_pair(1, tgtCone)};
+ for (auto signedCone : unimodCones)
+ chamberGf =
+ chamberGf + unimodularConeGeneratingFunction(
+ vertices[i], signedCone.first, signedCone.second);
+ }
+ gf.push_back(std::make_pair(chamber.first, chamberGf));
+ }
+ return gf;
+}
+
/// 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.
diff --git a/mlir/lib/Analysis/Presburger/IntegerRelation.cpp b/mlir/lib/Analysis/Presburger/IntegerRelation.cpp
index 7d2a63d17676f5..b05075b2892ace 100644
--- a/mlir/lib/Analysis/Presburger/IntegerRelation.cpp
+++ b/mlir/lib/Analysis/Presburger/IntegerRelation.cpp
@@ -2498,6 +2498,48 @@ void IntegerRelation::printSpace(raw_ostream &os) const {
os << getNumConstraints() << " constraints\n";
}
+void IntegerRelation::removeTrivialEqualities() {
+ bool flag;
+ for (unsigned i = 0; i < getNumEqualities(); i++) {
+ flag = true;
+ for (unsigned j = 0; j < getNumVars() + 1; j++)
+ if (atEq(i, j) != 0)
+ flag = false;
+ if (flag)
+ removeEquality(i);
+ }
+}
+
+bool IntegerRelation::isFullDim() {
+ removeTrivialEqualities();
+
+ // If there is a non-trivial equality, the space cannot be full-dimensional.
+ if (getNumEqualities() > 0)
+ return false;
+
+ // If along the direction of any of the inequalities, the upper and lower
+ // optima are the same, then the region is not full-dimensional.
+ Simplex simplex(*this);
+ for (unsigned i = 0; i < getNumInequalities(); i++) {
+ auto ineq = inequalities.getRow(i);
+ auto upOpt = simplex.computeOptimum(Simplex::Direction::Up, ineq);
+ auto downOpt = simplex.computeOptimum(Simplex::Direction::Down, ineq);
+
+ if (upOpt.getKind() == OptimumKind::Unbounded ||
+ downOpt.getKind() == OptimumKind::Unbounded)
+ continue;
+
+ // Check if the upper and lower optima are equal.
+ if (upOpt.getKind() == OptimumKind::Bounded &&
+ downOpt.getKind() == OptimumKind::Bounded && *upOpt == *downOpt)
+ return false;
+ }
+ // If none of the inequalities were such that the upper and lower optima
+ // along their direction were equal, then we conclude that the region is full
+ // dimensional.
+ return true;
+}
+
void IntegerRelation::print(raw_ostream &os) const {
assert(hasConsistentState());
printSpace(os);
``````````
</details>
https://github.com/llvm/llvm-project/pull/78987
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