[MLIR][Presburger] Add LLL basis reduction (#75565)

Add a method for LLL basis reduction to the FracMatrix class.
This needs an abs() method for Fractions, which is added to Fraction.h.

Author: Abhinav271828

mlir/include/mlir/Analysis/Presburger/Fraction.h

@@ -101,6 +101,11 @@ inline bool operator>=(const Fraction &x, const Fraction &y) {
   return compare(x, y) >= 0;
 }
 
+inline Fraction abs(const Fraction &f) {
+  assert(f.den > 0 && "denominator of fraction must be positive!");
+  return Fraction(abs(f.num), f.den);
+}
+
 inline Fraction reduce(const Fraction &f) {
   if (f == Fraction(0))
     return Fraction(0, 1);
@@ -124,6 +129,12 @@ inline Fraction operator-(const Fraction &x, const Fraction &y) {
   return reduce(Fraction(x.num * y.den - x.den * y.num, x.den * y.den));
 }
 
+// Find the integer nearest to a given fraction.
+inline MPInt round(const Fraction &f) {
+  MPInt rem = f.num % f.den;
+  return (f.num / f.den) + (rem > f.den / 2);
+}
+
 inline Fraction &operator+=(Fraction &x, const Fraction &y) {
   x = x + y;
   return x;

mlir/include/mlir/Analysis/Presburger/Matrix.h

@@ -270,6 +270,12 @@ class FracMatrix : public Matrix<Fraction> {
   // of the rows of matrix (cubic time).
   // The rows of the matrix must be linearly independent.
   FracMatrix gramSchmidt() const;
+
+  // Run LLL basis reduction on the matrix, modifying it in-place.
+  // The parameter is what [the original
+  // paper](https://www.cs.cmu.edu/~avrim/451f11/lectures/lect1129_LLL.pdf)
+  // calls `y`, usually 3/4.
+  void LLL(Fraction delta);
 };
 
 } // namespace presburger

mlir/lib/Analysis/Presburger/Matrix.cpp

@@ -576,4 +576,71 @@ FracMatrix FracMatrix::gramSchmidt() const {
     }
   }
   return orth;
-}
+}
+
+// Convert the matrix, interpreted (row-wise) as a basis
+// to an LLL-reduced basis.
+//
+// This is an implementation of the algorithm described in
+// "Factoring polynomials with rational coefficients" by
+// A. K. Lenstra, H. W. Lenstra Jr., L. Lovasz.
+//
+// Let {b_1,  ..., b_n}  be the current basis and
+//     {b_1*, ..., b_n*} be the Gram-Schmidt orthogonalised
+//                          basis (unnormalized).
+// Define the Gram-Schmidt coefficients μ_ij as
+// (b_i • b_j*) / (b_j* • b_j*), where (•) represents the inner product.
+//
+// We iterate starting from the second row to the last row.
+//
+// For the kth row, we first check μ_kj for all rows j < k.
+// We subtract b_j (scaled by the integer nearest to μ_kj)
+// from b_k.
+//
+// Now, we update k.
+// If b_k and b_{k-1} satisfy the Lovasz condition
+//    |b_k|^2 ≥ (δ - μ_k{k-1}^2) |b_{k-1}|^2,
+// we are done and we increment k.
+// Otherwise, we swap b_k and b_{k-1} and decrement k.
+//
+// We repeat this until k = n and return.
+void FracMatrix::LLL(Fraction delta) {
+  MPInt nearest;
+  Fraction mu;
+
+  // `gsOrth` holds the Gram-Schmidt orthogonalisation
+  // of the matrix at all times. It is recomputed every
+  // time the matrix is modified during the algorithm.
+  // This is naive and can be optimised.
+  FracMatrix gsOrth = gramSchmidt();
+
+  // We start from the second row.
+  unsigned k = 1;
+  while (k < getNumRows()) {
+    for (unsigned j = k - 1; j < k; j--) {
+      // Compute the Gram-Schmidt coefficient μ_jk.
+      mu = dotProduct(getRow(k), gsOrth.getRow(j)) /
+           dotProduct(gsOrth.getRow(j), gsOrth.getRow(j));
+      nearest = round(mu);
+      // Subtract b_j scaled by the integer nearest to μ_jk from b_k.
+      addToRow(k, getRow(j), -Fraction(nearest, 1));
+      gsOrth = gramSchmidt(); // Update orthogonalization.
+    }
+    mu = dotProduct(getRow(k), gsOrth.getRow(k - 1)) /
+         dotProduct(gsOrth.getRow(k - 1), gsOrth.getRow(k - 1));
+    // Check the Lovasz condition for b_k and b_{k-1}.
+    if (dotProduct(gsOrth.getRow(k), gsOrth.getRow(k)) >
+        (delta - mu * mu) *
+            dotProduct(gsOrth.getRow(k - 1), gsOrth.getRow(k - 1))) {
+      // If it is satisfied, proceed to the next k.
+      k += 1;
+    } else {
+      // If it is not satisfied, decrement k (without
+      // going beyond the second row).
+      swapRows(k, k - 1);
+      gsOrth = gramSchmidt(); // Update orthogonalization.
+      k = k > 1 ? k - 1 : 1;
+    }
+  }
+  return;
+}

mlir/unittests/Analysis/Presburger/MatrixTest.cpp

@@ -377,4 +377,52 @@ TEST(MatrixTest, gramSchmidt) {
   gs = mat.gramSchmidt();
 
   EXPECT_EQ_FRAC_MATRIX(gs, FracMatrix::identity(10));
+}
+
+void checkReducedBasis(FracMatrix mat, Fraction delta) {
+  FracMatrix gsOrth = mat.gramSchmidt();
+
+  // Size-reduced check.
+  for (unsigned i = 0, e = mat.getNumRows(); i < e; i++) {
+    for (unsigned j = 0; j < i; j++) {
+      Fraction mu = dotProduct(mat.getRow(i), gsOrth.getRow(j)) /
+                    dotProduct(gsOrth.getRow(j), gsOrth.getRow(j));
+      EXPECT_TRUE(abs(mu) <= Fraction(1, 2));
+    }
+  }
+
+  // Lovasz condition check.
+  for (unsigned i = 1, e = mat.getNumRows(); i < e; i++) {
+    Fraction mu = dotProduct(mat.getRow(i), gsOrth.getRow(i - 1)) /
+                  dotProduct(gsOrth.getRow(i - 1), gsOrth.getRow(i - 1));
+    EXPECT_TRUE(dotProduct(mat.getRow(i), mat.getRow(i)) >
+                (delta - mu * mu) *
+                    dotProduct(gsOrth.getRow(i - 1), gsOrth.getRow(i - 1)));
+  }
+}
+
+TEST(MatrixTest, LLL) {
+  FracMatrix mat =
+      makeFracMatrix(3, 3,
+                     {{Fraction(1, 1), Fraction(1, 1), Fraction(1, 1)},
+                      {Fraction(-1, 1), Fraction(0, 1), Fraction(2, 1)},
+                      {Fraction(3, 1), Fraction(5, 1), Fraction(6, 1)}});
+  mat.LLL(Fraction(3, 4));
+
+  checkReducedBasis(mat, Fraction(3, 4));
+
+  mat = makeFracMatrix(
+      2, 2,
+      {{Fraction(12, 1), Fraction(2, 1)}, {Fraction(13, 1), Fraction(4, 1)}});
+  mat.LLL(Fraction(3, 4));
+
+  checkReducedBasis(mat, Fraction(3, 4));
+
+  mat = makeFracMatrix(3, 3,
+                       {{Fraction(1, 1), Fraction(0, 1), Fraction(2, 1)},
+                        {Fraction(0, 1), Fraction(1, 3), -Fraction(5, 3)},
+                        {Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)}});
+  mat.LLL(Fraction(3, 4));
+
+  checkReducedBasis(mat, Fraction(3, 4));
 }