[Mlir-commits] [mlir] [MLIR][Presburger] Implement computation of generating function for unimodular cones (PR #77235)
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Sun Jan 7 03:12:10 PST 2024
llvmbot wrote:
<!--LLVM PR SUMMARY COMMENT-->
@llvm/pr-subscribers-mlir-presburger
@llvm/pr-subscribers-mlir
Author: None (Abhinav271828)
<details>
<summary>Changes</summary>
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
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}}}));
+}
``````````
</details>
https://github.com/llvm/llvm-project/pull/77235
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