[MLIR][Sparse] Refactor lattice code into its own file

[gussmith23]

Summary

Moves iteration lattice/merger code into new SparseTensor/Utils directory. A follow-up CL will add lattice/merger unit tests.

[aartbik]

mlir/include/mlir/Dialect/SparseTensor/Utils/Lattice.h

70

For classes we often keep the 1:1 with header file and class name, so perhaps (in a next CL) we should rename Merger to "Lattice"? What do you think

The TensorExpr and LatPoint structs are smaller, and can easily live here too, but could perhaps even be nested inside Lattice at a future, date, but due to namespaces, there is less urgency to that.

121

you moved all code from header to cpp (good), but starting from here down, you keep it "inline".

was this on purpose? some are small, but some also have loops in them, so just curious what your reasoning was

[aartbik]

mlir/include/mlir/Dialect/SparseTensor/Utils/Lattice.h

14

TRANSFORMS -> UTILS

[aartbik]

mlir/lib/Dialect/SparseTensor/Utils/CMakeLists.txt

9

the utils alone does not depend on linalg, right?

[aartbik]

mlir/include/mlir/Dialect/SparseTensor/Utils/Lattice.h

17

why are we pulling this in?

[gussmith23]

Updates from Aart reviews on Critique

[gussmith23]

Address Aart's comments

[gussmith23]

mlir/include/mlir/Dialect/SparseTensor/Utils/Lattice.h

70

Hm, it depends -- does a Merger truly represent a single lattice? It seems to me like it actually represents the multiple lattices present in a single kernel, unless all of them collectively are actually referred to as a single lattice. There's probably a clearer name than "merger", probably involving "lattice"...could be as simple as "Lattices", or maybe "ExpressionLattices", or "LatticeSet"...

121

Should I move all impls to the cpp? My general thinking was just "keep short impls inline" because that's what I've seen others do. I can move the hasAnyDimOf impl over to Lattice.cpp, or I can just move all of them over! Let me know what you'd prefer.

[aartbik]

mlir/include/mlir/Dialect/SparseTensor/Utils/Lattice.h

70

I am fine with original name, but was just thinking of the filename Lattice vs. class name Merger.

But I don't want to overthink it either, since this does not seem to be a strict llvm style anyway.

121

Let's move hasAnyDimOf and onlyDenseDiff over, keep all others here.
Agreed?

[gussmith23]

Address Aart's comments

[gussmith23]

Address Aart's comments

[aartbik]

mlir/lib/Dialect/SparseTensor/Transforms/Sparsification.cpp

44–50

One last question, we seem to remove more includes here then the new Merger.h brings back in.
Is this as a result of auto cleanup, since I expect some to be necessary (like linalg at least).

[gussmith23]

Re-introduce includes, cmake fix

[gussmith23]

mlir/lib/Dialect/SparseTensor/Transforms/Sparsification.cpp

44–50

Brought them back!

mlir/include/mlir/Dialect/SparseTensor/Utils/Lattice.h

+//===- Lattice.h - Utilities for defining lattices --------------*- C++ -*-===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+//
+// This header file defines utilities for dealing with iteration lattices.
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef MLIR_DIALECT_SPARSETENSOR_TRANSFORMS_LATTICE_H_
+#define MLIR_DIALECT_SPARSETENSOR_TRANSFORMS_LATTICE_H_
+
+#include "mlir/IR/Value.h"
+#include "llvm/ADT/BitVector.h"
+
+namespace mlir {
+namespace sparse_tensor {
+
+enum class Kind { kTensor, kInvariant, kMulF, kMulI, kAddF, kAddI };
+enum class Dim { kSparse, kDense, kSingle, kUndef };
+
+/// Tensor expression. Represents a MLIR expression in tensor index notation.
+/// For tensors, e0 denotes the tensor index. For invariants, the IR value is
+/// stored directly. For binary operations, e0 and e1 denote the index of the
+/// children tensor expressions.
+struct TensorExp {
+  TensorExp(Kind k, unsigned x, unsigned y, Value v)
+      : kind(k), e0(x), e1(y), val(v) {
+    assert((kind == Kind::kTensor && e0 != -1u && e1 == -1u && !val) ||
+           (kind == Kind::kInvariant && e0 == -1u && e1 == -1u && val) ||
+           (kind >= Kind::kMulF && e0 != -1u && e1 != -1u && !val));
+  }
+  Kind kind;
+  /// Indices of children expression(s).
+  unsigned e0;
+  unsigned e1;
+  /// Direct link to IR for an invariant. During code generation,
+  /// field is used to cache "hoisted" loop invariant tensor loads.
+  Value val;
+};
+
+/// Lattice point. Each lattice point consists of a conjunction of tensor
+/// loop indices (encoded in a bitvector) and the index of the corresponding
+/// tensor expression.
+struct LatPoint {
+  LatPoint(unsigned n, unsigned e, unsigned b) : bits(n, false), exp(e) {
+    bits.set(b);
+  }
+  LatPoint(const llvm::BitVector &b, unsigned e) : bits(b), exp(e) {}
+  /// Conjunction of tensor loop indices as bitvector. This represents
+  /// all indices involved in the tensor expression
+  llvm::BitVector bits;
+  /// Simplified conjunction of tensor loop indices as bitvector. This
+  /// represents a simplified condition under which this tensor expression
+  /// must execute. Pre-computed during codegen to avoid repeated eval.
+  llvm::BitVector simple;
+  /// Index of the tensor expresssion.
+  unsigned exp;
+};
+
+/// A class to handle all iteration lattice operations. This class abstracts
+/// away from some implementation details of storing iteration lattices and
+/// tensor expressions. This allows for fine-tuning performance characteristics
+/// independently from the basic algorithm if bottlenecks are identified.
+class Merger {
+public:
+  /// Constructs a merger for the given number of tensors and loops. The
+  /// user supplies the number of tensors involved in the kernel, with the
+  /// last tensor in this set denoting the output tensor. The merger adds an
+  /// additional synthetic tensor at the end of this set to represent all
+  /// invariant expressions in the kernel.
+  Merger(unsigned t, unsigned l)
+      : outTensor(t - 1), numTensors(t + 1), numLoops(l),
+        dims(t + 1, std::vector<Dim>(l, Dim::kUndef)) {}
+
+  /// Adds a tensor expression. Returns its index.
+  unsigned addExp(Kind k, unsigned e0, unsigned e1 = -1u, Value v = Value());
+  unsigned addExp(Kind k, Value v) { return addExp(k, -1u, -1u, v); }
+
+  /// Adds an iteration lattice point. Returns its index.
+  unsigned addLat(unsigned t, unsigned i, unsigned e);
+
+  /// Adds a new, initially empty, set. Returns its index.
+  unsigned addSet();
+
+  /// Computes a single conjunction of two lattice points by taking the "union"
+  /// of loop indices (effectively constructing a larger "intersection" of those
+  /// indices) with a newly constructed tensor (sub)expression of given kind.
+  /// Returns the index of the new lattice point.
+  unsigned conjLatPoint(Kind kind, unsigned p0, unsigned p1);
+
+  /// Conjunctive merge of two lattice sets L0 and L1 is conjunction of
+  /// cartesian product. Returns the index of the new set.
+  unsigned takeConj(Kind kind, unsigned s0, unsigned s1);
+
+  /// Disjunctive merge of two lattice sets L0 and L1 is (L0 /\_op L1, L0, L1).
+  /// Returns the index of the new set.
+  unsigned takeDisj(Kind kind, unsigned s0, unsigned s1);
+
+  /// Optimizes the iteration lattice points in the given set. This
+  /// method should be called right before code generation to avoid
+  /// generating redundant loops and conditions.
+  unsigned optimizeSet(unsigned s0);
+
+  /// Simplifies the conditions in a conjunction of a given lattice point
+  /// within the given set using just two basic rules:
+  /// (1) multiple dense conditions are reduced to single dense, and
+  /// (2) a *singleton* sparse/dense is reduced to sparse/random access.
+  llvm::BitVector simplifyCond(unsigned s, unsigned p0);
+
+  /// Returns true if Li > Lj.
+  bool latGT(unsigned i, unsigned j) const;
+
+  /// Returns true if Li and Lj only differ in dense.
+  bool onlyDenseDiff(unsigned i, unsigned j) {
+    llvm::BitVector tmp = latPoints[j].bits;
+    tmp ^= latPoints[i].bits;
+    return !hasAnyDimOf(tmp, Dim::kSparse);
+  }
+
+  /// Bit translation.
+  unsigned tensor(unsigned b) const { return b % numTensors; }
+  unsigned index(unsigned b) const { return b / numTensors; }
+
+  /// Returns true if bit corresponds to queried dim.
+  bool isDim(unsigned b, Dim d) const { return isDim(tensor(b), index(b), d); }
+
+  /// Returns true if bit corresponds to index of output tensor.
+  bool isOutTensor(unsigned b, unsigned i) const {
+    return tensor(b) == outTensor && index(b) == i;
+  }
+
+  /// Returns true if tensor access at given index has queried dim.
+  bool isDim(unsigned t, unsigned i, Dim d) const {
+    assert(t < numTensors && i < numLoops);
+    return dims[t][i] == d;
+  }
+
+  /// Returns true if any set bit corresponds to queried dim.
+  bool hasAnyDimOf(const llvm::BitVector &bits, Dim d) const {
+    for (unsigned b = 0, be = bits.size(); b < be; b++)
+      if (bits[b] && isDim(b, d))
+        return true;
+    return false;
+  }
+
+  /// Setter
+  void setDim(unsigned t, unsigned i, Dim d) { dims[t][i] = d; }
+
+  /// Getters.
+  TensorExp &exp(unsigned e) { return tensorExps[e]; }
+  LatPoint &lat(unsigned l) { return latPoints[l]; }
+  SmallVector<unsigned, 16> &set(unsigned s) { return latSets[s]; }
+
+private:
+  const unsigned outTensor;
+  const unsigned numTensors;
+  const unsigned numLoops;
+
+  std::vector<std::vector<Dim>> dims;
+  llvm::SmallVector<TensorExp, 32> tensorExps;
+  llvm::SmallVector<LatPoint, 16> latPoints;
+  llvm::SmallVector<SmallVector<unsigned, 16>, 8> latSets;
+};
+
+} // namespace sparse_tensor
+} // namespace mlir
+
+#endif // MLIR_DIALECT_SPARSETENSOR_TRANSFORMS_LATTICE_H_

mlir/lib/Dialect/SparseTensor/CMakeLists.txt

 add_subdirectory(IR)
 add_subdirectory(Transforms)
+add_subdirectory(Utils)

mlir/lib/Dialect/SparseTensor/Transforms/Sparsification.cpp

 // Proceedings of the ACM on Programming Languages, October 2017.
 // [Kjolstad20] Fredrik Berg Kjolstad. Sparse Tensor Algebra Compilation.
 // PhD thesis, MIT, February, 2020 (tensor-compiler.org).
 //
 // Implementation detail: We use llvm::SmallVector for vectors with
 // variable lengths and std::vector for vectors with fixed lengths.
 //===----------------------------------------------------------------------===//
 
-#include "mlir/Dialect/Linalg/IR/LinalgOps.h"
 #include "mlir/Dialect/Linalg/Utils/Utils.h"
-#include "mlir/Dialect/MemRef/IR/MemRef.h"
-#include "mlir/Dialect/SCF/SCF.h"
 #include "mlir/Dialect/SparseTensor/IR/SparseTensor.h"
 #include "mlir/Dialect/SparseTensor/Transforms/Passes.h"
-#include "mlir/Dialect/StandardOps/IR/Ops.h"
+#include "mlir/Dialect/SparseTensor/Utils/Lattice.h"
 #include "mlir/Dialect/Vector/VectorOps.h"
 #include "mlir/IR/Matchers.h"
-#include "mlir/IR/TensorEncoding.h"
-#include "llvm/ADT/SmallBitVector.h"
 
 using namespace mlir;
 using namespace mlir::sparse_tensor;
 
 namespace {
 
-enum class Kind { kTensor, kInvariant, kMulF, kMulI, kAddF, kAddI };
-enum class Dim { kSparse, kDense, kSingle, kUndef };
-
-/// Tensor expression. Represents a MLIR expression in tensor index notation.
-/// For tensors, e0 denotes the tensor index. For invariants, the IR value is
-/// stored directly. For binary operations, e0 and e1 denote the index of the
-/// children tensor expressions.
-struct TensorExp {
-  TensorExp(Kind k, unsigned x, unsigned y, Value v)
-      : kind(k), e0(x), e1(y), val(v) {
-    assert((kind == Kind::kTensor && e0 != -1u && e1 == -1u && !val) ||
-           (kind == Kind::kInvariant && e0 == -1u && e1 == -1u && val) ||
-           (kind >= Kind::kMulF && e0 != -1u && e1 != -1u && !val));
-  }
-  Kind kind;
-  /// Indices of children expression(s).
-  unsigned e0;
-  unsigned e1;
-  /// Direct link to IR for an invariant. During code generation,
-  /// field is used to cache "hoisted" loop invariant tensor loads.
-  Value val;
-};
-
-/// Lattice point. Each lattice point consists of a conjunction of tensor
-/// loop indices (encoded in a bitvector) and the index of the corresponding
-/// tensor expression.
-struct LatPoint {
-  LatPoint(unsigned n, unsigned e, unsigned b) : bits(n, false), exp(e) {
-    bits.set(b);
-  }
-  LatPoint(const llvm::BitVector &b, unsigned e) : bits(b), exp(e) {}
-  /// Conjunction of tensor loop indices as bitvector. This represents
-  /// all indices involved in the tensor expression
-  llvm::BitVector bits;
-  /// Simplified conjunction of tensor loop indices as bitvector. This
-  /// represents a simplified condition under which this tensor expression
-  /// must execute. Pre-computed during codegen to avoid repeated eval.
-  llvm::BitVector simple;
-  /// Index of the tensor expresssion.
-  unsigned exp;
-};
-
-/// A class to handle all iteration lattice operations. This class abstracts
-/// away from some implementation details of storing iteration lattices and
-/// tensor expressions. This allows for fine-tuning performance characteristics
-/// independently from the basic algorithm if bottlenecks are identified.
-class Merger {
-public:
-  /// Constructs a merger for the given number of tensors and loops. The
-  /// user supplies the number of tensors involved in the kernel, with the
-  /// last tensor in this set denoting the output tensor. The merger adds an
-  /// additional synthetic tensor at the end of this set to represent all
-  /// invariant expressions in the kernel.
-  Merger(unsigned t, unsigned l)
-      : outTensor(t - 1), numTensors(t + 1), numLoops(l),
-        dims(t + 1, std::vector<Dim>(l, Dim::kUndef)) {}
-
-  /// Adds a tensor expression. Returns its index.
-  unsigned addExp(Kind k, unsigned e0, unsigned e1 = -1u, Value v = Value()) {
-    unsigned e = tensorExps.size();
-    tensorExps.push_back(TensorExp(k, e0, e1, v));
-    return e;
-  }
-  unsigned addExp(Kind k, Value v) { return addExp(k, -1u, -1u, v); }
-
-  /// Adds an iteration lattice point. Returns its index.
-  unsigned addLat(unsigned t, unsigned i, unsigned e) {
-    assert(t < numTensors && i < numLoops);
-    unsigned p = latPoints.size();
-    latPoints.push_back(LatPoint(numLoops * numTensors, e, numTensors * i + t));
-    return p;
-  }
-
-  /// Adds a new, initially empty, set. Returns its index.
-  unsigned addSet() {
-    unsigned s = latSets.size();
-    latSets.emplace_back(SmallVector<unsigned, 16>());
-    return s;
-  }
-
-  /// Computes a single conjunction of two lattice points by taking the "union"
-  /// of loop indices (effectively constructing a larger "intersection" of those
-  /// indices) with a newly constructed tensor (sub)expression of given kind.
-  /// Returns the index of the new lattice point.
-  unsigned conjLatPoint(Kind kind, unsigned p0, unsigned p1) {
-    unsigned p = latPoints.size();
-    llvm::BitVector nb = llvm::BitVector(latPoints[p0].bits);
-    nb |= latPoints[p1].bits;
-    unsigned e = addExp(kind, latPoints[p0].exp, latPoints[p1].exp);
-    latPoints.push_back(LatPoint(nb, e));
-    return p;
-  }
-
-  /// Conjunctive merge of two lattice sets L0 and L1 is conjunction of
-  /// cartesian product. Returns the index of the new set.
-  unsigned takeConj(Kind kind, unsigned s0, unsigned s1) {
-    unsigned s = addSet();
-    for (unsigned p0 : latSets[s0])
-      for (unsigned p1 : latSets[s1])
-        latSets[s].push_back(conjLatPoint(kind, p0, p1));
-    return s;
-  }
-
-  /// Disjunctive merge of two lattice sets L0 and L1 is (L0 /\_op L1, L0, L1).
-  /// Returns the index of the new set.
-  unsigned takeDisj(Kind kind, unsigned s0, unsigned s1) {
-    unsigned s = takeConj(kind, s0, s1);
-    for (unsigned p : latSets[s0])
-      latSets[s].push_back(p);
-    for (unsigned p : latSets[s1])
-      latSets[s].push_back(p);
-    return s;
-  }
-
-  /// Optimizes the iteration lattice points in the given set. This
-  /// method should be called right before code generation to avoid
-  /// generating redundant loops and conditions.
-  unsigned optimizeSet(unsigned s0) {
-    unsigned s = addSet();
-    assert(latSets[s0].size() != 0);
-    unsigned p0 = latSets[s0][0];
-    for (unsigned p1 : latSets[s0]) {
-      bool add = true;
-      if (p0 != p1) {
-        // Is this a straightforward copy?
-        unsigned e = latPoints[p1].exp;
-        if (exp(e).kind == Kind::kTensor && exp(e).e0 == outTensor)
-          continue;
-        // Conjunction already covered?
-        for (unsigned p2 : latSets[s]) {
-          assert(!latGT(p1, p2)); // Lj => Li would be bad
-          if (onlyDenseDiff(p2, p1)) {
-            add = false;
-            break;
-          }
-        }
-        assert(!add || latGT(p0, p1));
-      }
-      if (add)
-        latSets[s].push_back(p1);
-    }
-    for (unsigned p : latSets[s])
-      latPoints[p].simple = simplifyCond(s, p);
-    return s;
-  }
-
-  /// Simplifies the conditions in a conjunction of a given lattice point
-  /// within the given set using just two basic rules:
-  /// (1) multiple dense conditions are reduced to single dense, and
-  /// (2) a *singleton* sparse/dense is reduced to sparse/random access.
-  llvm::BitVector simplifyCond(unsigned s, unsigned p0) {
-    // First determine if this lattice point is a *singleton*, i.e.,
-    // the last point in a lattice, no other is less than this one.
-    bool isSingleton = true;
-    for (unsigned p1 : latSets[s]) {
-      if (p0 != p1 && latGT(p0, p1)) {
-        isSingleton = false;
-        break;
-      }
-    }
-    // Now apply the two basic rules.
-    llvm::BitVector simple = latPoints[p0].bits;
-    bool reset = isSingleton && hasAnyDimOf(simple, Dim::kSparse);
-    for (unsigned b = 0, be = simple.size(); b < be; b++) {
-      if (simple[b] && !isDim(b, Dim::kSparse)) {
-        if (reset)
-          simple.reset(b);
-        reset = true;
-      }
-    }
-    return simple;
-  }
-
-  /// Returns true if Li > Lj.
-  bool latGT(unsigned i, unsigned j) const {
-    const llvm::BitVector &bitsi = latPoints[i].bits;
-    const llvm::BitVector &bitsj = latPoints[j].bits;
-    assert(bitsi.size() == bitsj.size());
-    if (bitsi.count() > bitsj.count()) {
-      for (unsigned b = 0, be = bitsj.size(); b < be; b++)
-        if (bitsj[b] && !bitsi[b])
-          return false;
-      return true;
-    }
-    return false;
-  }
-
-  /// Returns true if Li and Lj only differ in dense.
-  bool onlyDenseDiff(unsigned i, unsigned j) {
-    llvm::BitVector tmp = latPoints[j].bits;
-    tmp ^= latPoints[i].bits;
-    return !hasAnyDimOf(tmp, Dim::kSparse);
-  }
-
-  /// Bit translation.
-  unsigned tensor(unsigned b) const { return b % numTensors; }
-  unsigned index(unsigned b) const { return b / numTensors; }
-
-  /// Returns true if bit corresponds to queried dim.
-  bool isDim(unsigned b, Dim d) const { return isDim(tensor(b), index(b), d); }
-
-  /// Returns true if bit corresponds to index of output tensor.
-  bool isOutTensor(unsigned b, unsigned i) const {
-    return tensor(b) == outTensor && index(b) == i;
-  }
-
-  /// Returns true if tensor access at given index has queried dim.
-  bool isDim(unsigned t, unsigned i, Dim d) const {
-    assert(t < numTensors && i < numLoops);
-    return dims[t][i] == d;
-  }
-
-  /// Returns true if any set bit corresponds to queried dim.
-  bool hasAnyDimOf(const llvm::BitVector &bits, Dim d) const {
-    for (unsigned b = 0, be = bits.size(); b < be; b++)
-      if (bits[b] && isDim(b, d))
-        return true;
-    return false;
-  }
-
-  /// Setter
-  void setDim(unsigned t, unsigned i, Dim d) { dims[t][i] = d; }
-
-  /// Getters.
-  TensorExp &exp(unsigned e) { return tensorExps[e]; }
-  LatPoint &lat(unsigned l) { return latPoints[l]; }
-  SmallVector<unsigned, 16> &set(unsigned s) { return latSets[s]; }
-
-private:
-  const unsigned outTensor;
-  const unsigned numTensors;
-  const unsigned numLoops;
-
-  std::vector<std::vector<Dim>> dims;
-  llvm::SmallVector<TensorExp, 32> tensorExps;
-  llvm::SmallVector<LatPoint, 16> latPoints;
-  llvm::SmallVector<SmallVector<unsigned, 16>, 8> latSets;
-};
-
 // Code generation.
 struct CodeGen {
   CodeGen(SparsificationOptions o, unsigned numTensors, unsigned numLoops)
       : options(o), loops(numLoops), sizes(numLoops), buffers(numTensors),
         pointers(numTensors, std::vector<Value>(numLoops)),
         indices(numTensors, std::vector<Value>(numLoops)),
         highs(numTensors, std::vector<Value>(numLoops)),
         pidxs(numTensors, std::vector<Value>(numLoops)),

mlir/lib/Dialect/SparseTensor/Utils/CMakeLists.txt

+add_mlir_dialect_library(MLIRSparseTensorUtils
+  Lattice.cpp
+
+  ADDITIONAL_HEADER_DIRS
+  ${MLIR_MAIN_INCLUDE_DIR}/mlir/Dialect/SparseTensor
+)

mlir/lib/Dialect/SparseTensor/Utils/Lattice.cpp

+//===- Lattice.cpp - Implementation of iteration lattices -----------------===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
+//
+//===----------------------------------------------------------------------===//
+
+#include "mlir/Dialect/SparseTensor/Utils/Lattice.h"
+
+namespace mlir {
+namespace sparse_tensor {
+
+unsigned Merger::addExp(Kind k, unsigned e0, unsigned e1, Value v) {
+  unsigned e = tensorExps.size();
+  tensorExps.push_back(TensorExp(k, e0, e1, v));
+  return e;
+}
+
+unsigned Merger::addLat(unsigned t, unsigned i, unsigned e) {
+  assert(t < numTensors && i < numLoops);
+  unsigned p = latPoints.size();
+  latPoints.push_back(LatPoint(numLoops * numTensors, e, numTensors * i + t));
+  return p;
+}
+
+unsigned Merger::addSet() {
+  unsigned s = latSets.size();
+  latSets.emplace_back(SmallVector<unsigned, 16>());
+  return s;
+}
+
+unsigned Merger::conjLatPoint(Kind kind, unsigned p0, unsigned p1) {
+  unsigned p = latPoints.size();
+  llvm::BitVector nb = llvm::BitVector(latPoints[p0].bits);
+  nb |= latPoints[p1].bits;
+  unsigned e = addExp(kind, latPoints[p0].exp, latPoints[p1].exp);
+  latPoints.push_back(LatPoint(nb, e));
+  return p;
+}
+
+unsigned Merger::takeConj(Kind kind, unsigned s0, unsigned s1) {
+  unsigned s = addSet();
+  for (unsigned p0 : latSets[s0])
+    for (unsigned p1 : latSets[s1])
+      latSets[s].push_back(conjLatPoint(kind, p0, p1));
+  return s;
+}
+
+unsigned Merger::takeDisj(Kind kind, unsigned s0, unsigned s1) {
+  unsigned s = takeConj(kind, s0, s1);
+  for (unsigned p : latSets[s0])
+    latSets[s].push_back(p);
+  for (unsigned p : latSets[s1])
+    latSets[s].push_back(p);
+  return s;
+}
+
+unsigned Merger::optimizeSet(unsigned s0) {
+  unsigned s = addSet();
+  assert(latSets[s0].size() != 0);
+  unsigned p0 = latSets[s0][0];
+  for (unsigned p1 : latSets[s0]) {
+    bool add = true;
+    if (p0 != p1) {
+      // Is this a straightforward copy?
+      unsigned e = latPoints[p1].exp;
+      if (exp(e).kind == Kind::kTensor && exp(e).e0 == outTensor)
+        continue;
+      // Conjunction already covered?
+      for (unsigned p2 : latSets[s]) {
+        assert(!latGT(p1, p2)); // Lj => Li would be bad
+        if (onlyDenseDiff(p2, p1)) {
+          add = false;
+          break;
+        }
+      }
+      assert(!add || latGT(p0, p1));
+    }
+    if (add)
+      latSets[s].push_back(p1);
+  }
+  for (unsigned p : latSets[s])
+    latPoints[p].simple = simplifyCond(s, p);
+  return s;
+}
+
+llvm::BitVector Merger::simplifyCond(unsigned s, unsigned p0) {
+  // First determine if this lattice point is a *singleton*, i.e.,
+  // the last point in a lattice, no other is less than this one.
+  bool isSingleton = true;
+  for (unsigned p1 : latSets[s]) {
+    if (p0 != p1 && latGT(p0, p1)) {
+      isSingleton = false;
+      break;
+    }
+  }
+  // Now apply the two basic rules.
+  llvm::BitVector simple = latPoints[p0].bits;
+  bool reset = isSingleton && hasAnyDimOf(simple, Dim::kSparse);
+  for (unsigned b = 0, be = simple.size(); b < be; b++) {
+    if (simple[b] && !isDim(b, Dim::kSparse)) {
+      if (reset)
+        simple.reset(b);
+      reset = true;
+    }
+  }
+  return simple;
+}
+
+bool Merger::latGT(unsigned i, unsigned j) const {
+  const llvm::BitVector &bitsi = latPoints[i].bits;
+  const llvm::BitVector &bitsj = latPoints[j].bits;
+  assert(bitsi.size() == bitsj.size());
+  if (bitsi.count() > bitsj.count()) {
+    for (unsigned b = 0, be = bitsj.size(); b < be; b++)
+      if (bitsj[b] && !bitsi[b])
+        return false;
+    return true;
+  }
+  return false;
+}
+
+} // namespace sparse_tensor
+} // namespace mlir