[MLIR][Presburger] Implement computation of generating function for unimodular cones

[Abhinav271828]

We implement a function that computes the generating function corresponding to a unimodular cone.
The generating function for a polytope is obtained by summing these generating functions over all tangent cones.

[llvmbot]

@llvm/pr-subscribers-mlir-presburger

@llvm/pr-subscribers-mlir

Author: None (Abhinav271828)

Changes

We implement a function that computes the generating function corresponding to a unimodular cone.
The generating function for a polytope is obtained by summing these generating functions over all tangent cones.

---

Full diff: https://github.com/llvm/llvm-project/pull/77235.diff

3 Files Affected:

• (modified) mlir/include/mlir/Analysis/Presburger/Barvinok.h (+6)
• (modified) mlir/lib/Analysis/Presburger/Barvinok.cpp (+68)
• (modified) mlir/unittests/Analysis/Presburger/BarvinokTest.cpp (+36)

diff --git a/mlir/include/mlir/Analysis/Presburger/Barvinok.h b/mlir/include/mlir/Analysis/Presburger/Barvinok.h
index 15e805860db237..93b29e2d718e59 100644
--- a/mlir/include/mlir/Analysis/Presburger/Barvinok.h
+++ b/mlir/include/mlir/Analysis/Presburger/Barvinok.h
@@ -24,6 +24,7 @@
 #ifndef MLIR_ANALYSIS_PRESBURGER_BARVINOK_H
 #define MLIR_ANALYSIS_PRESBURGER_BARVINOK_H
 
+#include "mlir/Analysis/Presburger/GeneratingFunction.h"
 #include "mlir/Analysis/Presburger/IntegerRelation.h"
 #include "mlir/Analysis/Presburger/Matrix.h"
 #include <optional>
@@ -77,6 +78,11 @@ ConeV getDual(ConeH cone);
 /// The returned cone is pointed at the origin.
 ConeH getDual(ConeV cone);
 
+/// Compute the generating function for a unimodular cone.
+/// It assert-fails if the input cone is not unimodular.
+GeneratingFunction unimodularConeGeneratingFunction(ParamPoint vertex, int sign,
+                                                    ConeH cone);
+
 } // namespace detail
 } // namespace presburger
 } // namespace mlir
diff --git a/mlir/lib/Analysis/Presburger/Barvinok.cpp b/mlir/lib/Analysis/Presburger/Barvinok.cpp
index 9152b66968a1f5..f0cabc36e537a9 100644
--- a/mlir/lib/Analysis/Presburger/Barvinok.cpp
+++ b/mlir/lib/Analysis/Presburger/Barvinok.cpp
@@ -63,3 +63,71 @@ MPInt mlir::presburger::detail::getIndex(ConeV cone) {
 
   return cone.determinant();
 }
+
+/// Compute the generating function for a unimodular cone.
+GeneratingFunction mlir::presburger::detail::unimodularConeGeneratingFunction(
+    ParamPoint vertex, int sign, ConeH cone) {
+  // `cone` is assumed to be unimodular.
+  assert(getIndex(getDual(cone)) == 1 && "input cone is not unimodular!");
+
+  unsigned numVar = cone.getNumVars();
+  unsigned numIneq = cone.getNumInequalities();
+
+  // Thus its ray matrix, U, is the inverse of the
+  // transpose of its inequality matrix, `cone`.
+  FracMatrix transp(numVar, numIneq);
+  for (unsigned i = 0; i < numVar; i++)
+    for (unsigned j = 0; j < numIneq; j++)
+      transp(j, i) = Fraction(cone.atIneq(i, j), 1);
+
+  FracMatrix generators(numVar, numIneq);
+  transp.determinant(&generators); // This is the U-matrix.
+
+  // The denominators of the generating function
+  // are given by the generators of the cone, i.e.,
+  // the rows of the matrix U.
+  std::vector<Point> denominator(numIneq);
+  ArrayRef<Fraction> row;
+  for (unsigned i = 0; i < numVar; i++) {
+    row = generators.getRow(i);
+    denominator[i] = Point(row);
+  }
+
+  // The vertex is v : [d, n+1].
+  // We need to find affine functions of parameters λi(p)
+  // such that v = Σ λi(p)*ui.
+  // The λi are given by the columns of Λ = v^T @ U^{-1} = v^T @ transp.
+  // Then the numerator will be Σ -floor(-λi(p))*u_i.
+  // Thus we store the numerator as the affine function -Λ,
+  // since the generators are already stored in the denominator.
+  // Note that the outer -1 will have to be accounted for, as it is not stored.
+  // See end for an example.
+
+  unsigned numColumns = vertex.getNumColumns();
+  unsigned numRows = vertex.getNumRows();
+  ParamPoint numerator(numColumns, numRows);
+  SmallVector<Fraction> ithCol(numRows);
+  for (unsigned i = 0; i < numColumns; i++) {
+    for (unsigned j = 0; j < numRows; j++)
+      ithCol[j] = vertex(j, i);
+    numerator.setRow(i, transp.preMultiplyWithRow(ithCol));
+    numerator.negateRow(i);
+  }
+
+  return GeneratingFunction(numColumns - 1, SmallVector<int>(1, sign),
+                            std::vector({numerator}),
+                            std::vector({denominator}));
+
+  // Suppose the vertex is given by the matrix [ 2  2   0], with 2 params
+  //                                           [-1 -1/2 1]
+  // and the cone has H-representation [0  -1] => U-matrix [ 2 -1]
+  //                                   [-1 -2]             [-1  0]
+  // Therefore Λ will be given by [ 1    0 ] and the negation of this will be
+  // stored as the numerator.
+  //                              [ 1/2 -1 ]
+  //                              [ -1  -2 ]
+
+  // Algebraically, the numerator exponent is
+  // [ -2 ⌊ -N - M/2 +1 ⌋ + 1 ⌊ 0 +M +2 ⌋ ] -> first  COLUMN of U is [2, -1]
+  // [  1 ⌊ -N - M/2 +1 ⌋ + 0 ⌊ 0 +M +2 ⌋ ] -> second COLUMN of U is [-1, 0]
+}
diff --git a/mlir/unittests/Analysis/Presburger/BarvinokTest.cpp b/mlir/unittests/Analysis/Presburger/BarvinokTest.cpp
index b88baa6c6b48a4..2936d95c802e9c 100644
--- a/mlir/unittests/Analysis/Presburger/BarvinokTest.cpp
+++ b/mlir/unittests/Analysis/Presburger/BarvinokTest.cpp
@@ -46,3 +46,39 @@ TEST(BarvinokTest, getIndex) {
       4, 4, {{4, 2, 5, 1}, {4, 1, 3, 6}, {8, 2, 5, 6}, {5, 2, 5, 7}});
   EXPECT_EQ(getIndex(cone), cone.determinant());
 }
+
+// The following cones and vertices are randomly generated
+// (s.t. the cones are unimodular) and the generating functions
+// are computed. We check that the results contain the correct
+// matrices.
+TEST(BarvinokTest, unimodularConeGeneratingFunction) {
+  ConeH cone = defineHRep(2);
+  cone.addInequality({0, -1, 0});
+  cone.addInequality({-1, -2, 0});
+
+  ParamPoint vertex =
+      makeFracMatrix(2, 3, {{2, 2, 0}, {-1, -Fraction(1, 2), 1}});
+
+  GeneratingFunction gf = unimodularConeGeneratingFunction(vertex, 1, cone);
+
+  EXPECT_EQ_REPR_GENERATINGFUNCTION(
+      gf, GeneratingFunction(
+              2, {1},
+              {makeFracMatrix(3, 2, {{-1, 0}, {-Fraction(1, 2), 1}, {1, 2}})},
+              {{{2, -1}, {-1, 0}}}));
+
+  cone = defineHRep(3);
+  cone.addInequality({7, 1, 6, 0});
+  cone.addInequality({9, 1, 7, 0});
+  cone.addInequality({8, -1, 1, 0});
+
+  vertex = makeFracMatrix(3, 2, {{5, 2}, {6, 2}, {7, 1}});
+
+  gf = unimodularConeGeneratingFunction(vertex, 1, cone);
+
+  EXPECT_EQ_REPR_GENERATINGFUNCTION(
+      gf,
+      GeneratingFunction(
+          1, {1}, {makeFracMatrix(2, 3, {{-83, -100, -41}, {-22, -27, -15}})},
+          {{{8, 47, -17}, {-7, -41, 15}, {1, 5, -2}}}));
+}