lib/Dialect/Linalg/Transforms/FusionOnTensors.cpp - llvm-project/mlir - Git at Google

 //===- Fusion.cpp - Implementation of linalg Fusion -----------------------===//
 //
 // 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 file implements linalg fusion on tensors
 //
 //===----------------------------------------------------------------------===//

 #include "PassDetail.h"
 #include "mlir/Analysis/SliceAnalysis.h"
 #include "mlir/Dialect/Affine/IR/AffineOps.h"
 #include "mlir/Dialect/Linalg/IR/LinalgOps.h"
 #include "mlir/Dialect/Linalg/IR/LinalgTypes.h"
 #include "mlir/Dialect/Linalg/Passes.h"
 #include "mlir/Dialect/Linalg/Transforms/Transforms.h"
 #include "mlir/Dialect/Linalg/Utils/Utils.h"
 #include "mlir/Dialect/Tensor/IR/Tensor.h"
 #include "mlir/IR/AffineExpr.h"
 #include "mlir/IR/AffineMap.h"
 #include "mlir/Support/LLVM.h"

 using namespace mlir;
 using namespace linalg;

 //===----------------------------------------------------------------------===//
 // StructuredOp specific helpers.
 //===----------------------------------------------------------------------===//

 /// Returns the tiled slice dimensions given the tiled consumer loop dimensions.
 /// The slice defines a hyper rectangular iteration space and fusing the
 /// producer is always possible. However, depending on the consumer indexing
 /// map, not all slice elements may be consumed and the tiles may overlap. In
 /// these cases, fusion introduces redundant computation.
 SmallVector<int64_t> getTiledSliceDims(OpOperand *consumerOperand,
                                        ArrayRef<int64_t> tiledLoopDims) {
   // Get the consumer operand indexing map.
   LinalgOp consumerOp = consumerOperand->getOwner();
   AffineMap indexingMap = consumerOp.getTiedIndexingMap(consumerOperand);

   // Search the slice dimensions tiled by a tile loop dimension.
   DenseSet<int64_t> tiledSliceDims;
   for (auto en : enumerate(indexingMap.getResults())) {
     for (auto tiledLoopDim : tiledLoopDims) {
       if (en.value().isFunctionOfDim(tiledLoopDim))
         tiledSliceDims.insert(en.index());
     }
   }
   return {tiledSliceDims.begin(), tiledSliceDims.end()};
 }

 /// Returns the producer fused in place of `sliceOp`. Tile the producer operands
 /// along the `tiledSliceDims` and clone the producer. Consider the case of
 /// fusion of an output tensor:
 /// ```
 /// %1 = producer ins(...) outs(%0)
 /// %2 = consumer ins(...) outs(%1)
 /// ```
 /// When consumer is tiled, %1 appears in the loop iter_args:
 /// ```
 /// %1 = producer ins(...) outs(%0)
 /// %2 = scf.for ... iter_args(%1) .. (%bbarg) {
 ///   %t1 = tensor.extract_slice %bbarg[..]
 ///   %t2 = consumer ins(...) outs(%t1)
 ///   %r = tensor.insert_slice %t2, %bbarg[...]
 /// }
 /// ```
 /// Fusing %1 into the loop requires updating iter_args(%1) to iter_args(%0):
 /// ```
 /// %2 = scf.for ... iter_args(%0) .. (%bbarg) {
 ///   %t0 = tensor.extract_slice %bbarg[..]
 ///   %t1 = producer ins(...) outs(%t0)
 ///   %t2 = consumer ins(...) outs(%t1)
 ///   %r = tensor.insert_slice %t2, %bbarg[...]
 /// }
 /// ```
 /// This transformation is only valid if %bbarg is exclusively used by the
 /// output ExtractSliceOp / InsertSliceOp pair, which is checked by the
 /// `fuseProducer` method.
 /// TODO: instead of check and failure, insert new iter_args each time a
 /// producer is fused into a consumer and fold away unused iter_args.
 static LinalgOp getTiledProducer(OpBuilder &b, OpResult producerResult,
                                  tensor::ExtractSliceOp sliceOp,
                                  ArrayRef<int64_t> tiledSliceDims,
                                  OpOperand *iterArg) {
   // Clone the producer after `sliceOp` since the slice may be reused to pass in
   // the producer result.
   OpBuilder::InsertionGuard guard(b);
   b.setInsertionPointAfter(sliceOp);

   // Get the producer.
   LinalgOp producerOp = producerResult.getOwner();
   Location loc = producerOp.getLoc();

   // Obtain the `producerOp` loop bounds and the `sliceOp` ranges.
   SmallVector<Value> producerLoopBounds;
   transform(producerOp.createLoopRanges(b, loc),
             std::back_inserter(producerLoopBounds),
             [](Range range) { return range.size; });
   SmallVector<Range> sliceOpRanges = sliceOp.getOrCreateRanges(b, loc);

   // Get the producer result indexing map.
   AffineMap producerIndexingMap = producerOp.getTiedIndexingMap(
       producerOp.getOutputOperand(producerResult.getResultNumber()));

   // Tile the producer operands given the `sliceOp` ranges. Iterate the
   // `tiledSliceDims` and store the tile offset and size for the tiled slice
   // dimension. Assumes the mapping from slice dimensions to producer loops is a
   // permutation.
   auto zero = b.create<ConstantIndexOp>(loc, 0);
   SmallVector<Value> tileIvs(producerOp.getNumLoops(), nullptr);
   SmallVector<Value> tileSizes(producerOp.getNumLoops(), zero);
   SmallVector<Value> allIvs(producerOp.getNumLoops(), nullptr);
   for (int64_t tiledSliceDim : tiledSliceDims) {
     AffineExpr result = producerIndexingMap.getResults()[tiledSliceDim];
     assert(result.isa<AffineDimExpr>() &&
            "expect producer indexing map is a projected permutation");
     int64_t tiledProducerLoop = result.cast<AffineDimExpr>().getPosition();
     tileIvs[tiledProducerLoop] = sliceOpRanges[tiledSliceDim].offset;
     tileSizes[tiledProducerLoop] = sliceOpRanges[tiledSliceDim].size;
     allIvs[tiledProducerLoop] = tileIvs[tiledProducerLoop];
   }
   erase_value(tileIvs, nullptr);
   SmallVector<Value> tiledOperands = producerOp.getInputAndOutputOperands();
   tiledOperands = makeTiledShapes(b, loc, producerOp, tiledOperands, tileIvs,
                                   tileSizes, producerLoopBounds);

   // Output fusion has to update the iteration arguments of the tile loop nest.
   // In particular, the iteration argument of the outermost tile loop needs to
   // be set to the producer output instead of the producer result and `clonedOp`
   // shall use the existing `sliceOp` result instead of the tiled producer
   // output operand.
   if (iterArg) {
     OpOperand *outputOperand =
         producerOp.getOutputOperand(producerResult.getResultNumber());
     iterArg->set(outputOperand->get());
     tiledOperands[outputOperand->getOperandNumber()] = sliceOp.getResult();
   }

   // Clone the producer using the tiled producer operands.
   TypeRange resultTypes = ValueRange(tiledOperands)
                               .take_back(producerOp.getNumOutputs())
                               .getTypes();
   LinalgOp clonedOp = producerOp.clone(b, loc, resultTypes, tiledOperands);

   // Shift all IndexOp results by the tile offset.
   addTileLoopIvsToIndexOpResults(b, clonedOp, allIvs);

   return clonedOp;
 }

 //===----------------------------------------------------------------------===//
 // TileLoopNest specific helpers.
 //===----------------------------------------------------------------------===//

 bool TileLoopNest::isEmpty() { return loopOps.empty(); }

 bool TileLoopNest::isValid() {
   // Check if the number of `tileLoopOps` and `tileLoopDims` match.
   if (loopOps.size() != loopDims.size())
     return false;

   // Check if the innermost tile loop is the parent of `tiledOp`.
   if (rootOp->getParentOp() != loopOps.back())
     return false;

   // Check if the tile loops are directly nested.
   return std::adjacent_find(loopOps.begin(), loopOps.end(),
                             [](Operation *op1, Operation *op2) {
                               return op1 != op2->getParentOp();
                             }) == loopOps.end();
 }

 SmallVector<BlockArgument> TileLoopNest::getTiedBBArgs(BlockArgument bbArg) {
   assert(bbArg && "expect the block argument to be non-zero");
   SmallVector<BlockArgument> bbArgs;

   // Search all tile loop block arguments from inner to outer.
   for (auto tileLoop : reverse(loopOps)) {
     if (bbArg.getOwner()->getParentOp() != tileLoop)
       return {};
     bbArgs.push_back(bbArg);
     OpOperand *iterArg = &tileLoop.getOpOperandForRegionIterArg(bbArg);
     bbArg = iterArg->get().dyn_cast<BlockArgument>();
   }

   // Reverse the block arguments to order them from outer to inner.
   return {bbArgs.rbegin(), bbArgs.rend()};
 }

 OpOperand *TileLoopNest::getTiedIterArg(BlockArgument bbArg) {
   // Search all block arguments and return the matching iteration argument.
   SmallVector<BlockArgument> bbArgs = getTiedBBArgs(bbArg);
   if (bbArgs.size() != loopOps.size())
     return nullptr;
   return &loopOps.front().getOpOperandForRegionIterArg(bbArgs.front());
 }

 bool TileLoopNest::hasOtherUses(BlockArgument bbArg,
                                 tensor::ExtractSliceOp sliceOp) {
   // Check the innermost block argument is either used by the ExtractSliceOp
   // `sliceOp`, the matching InsertSliceOp, or by a DimOp. Handle other uses
   // conservatively.
   for (Operation *op : bbArg.getUsers()) {
     if (!isa<tensor::DimOp, tensor::InsertSliceOp, tensor::ExtractSliceOp>(op))
       return false;
     if (auto extractSliceOp = dyn_cast<tensor::ExtractSliceOp>(op)) {
       if (extractSliceOp != sliceOp)
         return false;
     }
     if (auto insertSliceOp = dyn_cast<tensor::InsertSliceOp>(op)) {
       SetVector<Operation *> backwardSlice;
       getBackwardSlice(insertSliceOp.source(), &backwardSlice,
                        [](Operation *op) {
                          return isa<LinalgOp, tensor::InsertSliceOp>(op);
                        });
       if (backwardSlice.empty() || backwardSlice.front() != sliceOp)
         return false;
     }
   }

   // Check the block arguments, except for the innermost one, have one use.
   SmallVector<BlockArgument> bbArgs = getTiedBBArgs(bbArg);
   return !all_of(bbArgs, [&](BlockArgument bbArg) {
     return bbArg.hasOneUse() || bbArg == bbArgs.back();
   });
 }

 LogicalResult TileLoopNest::tileRootOp(OpBuilder &b,
                                        ArrayRef<int64_t> tileSizes,
                                        ArrayRef<int64_t> tileInterchange) {
   // Exit if all tile sizes are zero.
   if (tileSizes.size() == static_cast<size_t>(count(tileSizes, 0)))
     return success();

   // Tile the root operation.
   LinalgTilingOptions tilingOptions;
   tilingOptions = tilingOptions
                       .setInterchange(SmallVector<unsigned>(
                           tileInterchange.begin(), tileInterchange.end()))
                       .setTileSizes(tileSizes)
                       .setLoopType(LinalgTilingLoopType::Loops);
   Optional<TiledLinalgOp> tiledRootOp = tileLinalgOp(b, rootOp, tilingOptions);

   // Exit if tiling the root operation fails.
   if (!tiledRootOp.hasValue())
     return failure();

   // Replace all uses of the root operation if it has been tiled before. All
   // uses of the original untiled root operation are updated by the calling pass
   // or pattern.
   if (!isEmpty())
     rootOp->replaceAllUsesWith(tiledRootOp->tensorResults);

   // Update the root operation and append the loops and tile loop dimensions.
   rootOp = tiledRootOp->op;
   loopOps.append(tiledRootOp->loops.begin(), tiledRootOp->loops.end());
   for (auto en : enumerate(tileSizes)) {
     // Copy only the tiled loop dimensions with non-zero tile size.
     if (en.value() == 0)
       continue;
     loopDims.push_back(tileInterchange[en.index()]);
   }
   assert(isValid() && "expect tile loop nest to be valid after tiling");

   return success();
 }

 FailureOr<LinalgOp> TileLoopNest::fuseProducer(OpBuilder &b,
                                                OpOperand *rootOpOperand) {
   assert(rootOpOperand->getOwner() == rootOp &&
          "expect the root op to be the owner of the operand to fuse");
   assert(this->isValid() &&
          "expect the tile loop nest to satisfy all invariants");

   // Check the tile loop nest is non-empty.
   if (isEmpty())
     return failure();

   // Check `rootOpOperand` is defined by an ExtractSliceOp.
   auto sliceOp = rootOpOperand->get().getDefiningOp<tensor::ExtractSliceOp>();
   if (!sliceOp)
     return failure();

   // Check `sliceOp` is tiled by the tile loop nest.
   if (sliceOp->getParentOp() != rootOp->getParentOp())
     return failure();

   // Check if the producer is a LinalgOp possibly passed by iteration argument.
   OpOperand *iterArg = nullptr;
   auto producerResult = sliceOp.source().dyn_cast<OpResult>();
   if (auto bbArg = sliceOp.source().dyn_cast<BlockArgument>()) {
     iterArg = getTiedIterArg(bbArg);
     // Check the iteration argument may be used to pass in the producer output.
     if (!iterArg || hasOtherUses(bbArg, sliceOp))
       return failure();
     producerResult = iterArg->get().dyn_cast<OpResult>();
   }
   if (!producerResult || !isa<LinalgOp>(producerResult.getOwner()))
     return failure();

   // Compute the tiled producer slice dimensions given the tiled root operation
   // loop dimensions `loopDims`.
   SmallVector<int64_t> tiledSliceDims =
       getTiledSliceDims(rootOpOperand, loopDims);
   if (tiledSliceDims.empty())
     return failure();

   // Tile the producer operands and clone the producer in place of `sliceOp`.
   LinalgOp clonedOp =
       getTiledProducer(b, producerResult, sliceOp, tiledSliceDims, iterArg);

   // Cast the `clonedOp` result to gap type mismatches before canonicalization.
   Type consumerOperandType = rootOpOperand->get().getType();
   Value newResult = clonedOp->getResult(producerResult.getResultNumber());
   if (newResult.getType() != consumerOperandType) {
     OpBuilder::InsertionGuard guard(b);
     b.setInsertionPointAfter(clonedOp);
     newResult = b.create<tensor::CastOp>(producerResult.getLoc(),
                                          consumerOperandType, newResult);
   }

   // Replace the `sliceOp` uses except for the `clonedOp` output uses.
   sliceOp.getResult().replaceAllUsesExcept(newResult, clonedOp);
   return clonedOp;
 }

 ValueRange TileLoopNest::getRootOpReplacementResults() {
   assert(!isEmpty() && "expect tile loop nest to be non-empty");
   return loopOps.front()->getOpResults();
 }

 //===----------------------------------------------------------------------===//
 // Tile and fuse entry-points.
 //===----------------------------------------------------------------------===//

 FailureOr<TileLoopNest>
 mlir::linalg::tileConsumerAndFuseProducers(OpBuilder &b, LinalgOp consumerOp,
                                            ArrayRef<int64_t> tileSizes,
                                            ArrayRef<int64_t> tileInterchange) {
   assert(tileSizes.size() == tileInterchange.size() &&
          "expect the number of tile sizes and interchange dims to match");
   assert(isPermutation(tileInterchange) &&
          "expect tile interchange is a permutation");

   // Create an empty tile loop nest.
   TileLoopNest tileLoopNest(consumerOp);

   // Search the number of outer parallel loops to separate them from possible
   // inner reduction dimensions.
   SmallVector<StringAttr> iterTypes =
       llvm::to_vector<6>(consumerOp.iterator_types().getAsRange<StringAttr>());
   applyPermutationToVector(iterTypes, tileInterchange);
   auto *it = find_if(iterTypes, [&](StringAttr iterType) {
     return !isParallelIterator(iterType);
   });
   int64_t split = std::distance(iterTypes.begin(), it);

   // Tile the outer parallel loops and fuse the output operands.
   SmallVector<int64_t> outerTileSizes;
   outerTileSizes.append(tileSizes.begin(), tileSizes.begin() + split);
   outerTileSizes.append(tileSizes.size() - split, 0);
   if (failed(tileLoopNest.tileRootOp(b, outerTileSizes, tileInterchange)))
     return failure();
   for (OpOperand *opOperand : tileLoopNest.getRootOp().getOutputOperands())
     (void)tileLoopNest.fuseProducer(b, opOperand);

   // Tile the remaining loops and fuse the input operands.
   SmallVector<int64_t> innerTileSizes;
   innerTileSizes.append(split, 0);
   innerTileSizes.append(tileSizes.begin() + split, tileSizes.end());
   if (failed(tileLoopNest.tileRootOp(b, innerTileSizes, tileInterchange)))
     return failure();
   SmallVector<OpOperand *> inputOperands =
       tileLoopNest.getRootOp().getInputOperands();
   for (OpOperand *opOperand : tileLoopNest.getRootOp().getInputOperands())
     (void)tileLoopNest.fuseProducer(b, opOperand);

   return tileLoopNest;
 }

 namespace {
 struct LinalgTileAndFuseTensorOps
     : public LinalgTileAndFuseTensorOpsBase<LinalgTileAndFuseTensorOps> {

   void notifyFailure(StringRef message) {
     llvm::errs() << " - LinalgTileAndFuseTensorOps: " << message << "\n";
     signalPassFailure();
   }

   void runOnFunction() override {
     FuncOp funcOp = getFunction();
     OpBuilder b(funcOp.getContext());

     // Heuristic to find a good operation to tile and start fusion. Walk all
     // operations and select the one with the maximal backward slice of fusion
     // candidates.
     LinalgOp rootOp = nullptr;
     int64_t numFusionCandidates = -1;
     funcOp.walk([&](LinalgOp linalgOp) {
       SetVector<Operation *> backwardSlice;
       getBackwardSlice(linalgOp, &backwardSlice);
       int64_t backwardSliceSize = count_if(
           backwardSlice, [](Operation *op) { return isa<LinalgOp>(op); });
       if (backwardSliceSize > numFusionCandidates) {
         rootOp = linalgOp;
         numFusionCandidates = backwardSliceSize;
       }
     });
     if (!rootOp)
       return notifyFailure("expect to find a root operation");

     // Check `tileSizes` contains a tile size for every `rootOp` loop dimension.
     if (tileSizes.size() < rootOp.getNumLoops())
       return notifyFailure("expect #tile sizes >= #loops");

     // Check `tileInterchange` contains no entries or as many as `tileSizes`.
     if (!tileInterchange.empty() &&
         tileInterchange.size() != tileSizes.size()) {
       return notifyFailure(
           "expect the number of tile sizes and interchange dims to match");
     }

     // Copy the `tileSizes` and `tileInterchange` prefixes needed to tile
     // `rootOp` or use the identity interchange if `tileInterchange` is empty.
     SmallVector<int64_t> rootTileSizes(
         tileSizes.begin(), tileSizes.begin() + rootOp.getNumLoops());
     SmallVector<int64_t> rootInterchange =
         tileInterchange.empty()
             ? llvm::to_vector<6>(llvm::seq<int64_t>(0, rootOp.getNumLoops()))
             : SmallVector<int64_t>(tileInterchange.begin(),
                                    tileInterchange.begin() +
                                        rootOp.getNumLoops());

     // Check `rootInterchange` is a permutation of the `rootOp` loop dimensions.
     // It has to be a permutation since the tiling cannot tile the same loop
     // dimension multiple times.
     if (!isPermutation(rootInterchange))
       return notifyFailure(
           "expect the tile interchange permutes the root loops");

     // Tile `rootOp` and fuse its producers.
     FailureOr<TileLoopNest> tileLoopNest =
         tileConsumerAndFuseProducers(b, rootOp, rootTileSizes, rootInterchange);
     if (failed(tileLoopNest))
       return notifyFailure("tileConsumerAndFuseProducers failed unexpectedly");

     // Replace all uses of the tiled loop operation.
     rootOp->replaceAllUsesWith(tileLoopNest->getRootOpReplacementResults());
   }
 };
 } // namespace

 std::unique_ptr<OperationPass<FuncOp>>
 mlir::createLinalgTileAndFuseTensorOpsPass() {
   return std::make_unique<LinalgTileAndFuseTensorOps>();
 }
	//===- Fusion.cpp - Implementation of linalg Fusion -----------------------===//
	//
	// 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 file implements linalg fusion on tensors
	//
	//===----------------------------------------------------------------------===//

	#include "PassDetail.h"
	#include "mlir/Analysis/SliceAnalysis.h"
	#include "mlir/Dialect/Affine/IR/AffineOps.h"
	#include "mlir/Dialect/Linalg/IR/LinalgOps.h"
	#include "mlir/Dialect/Linalg/IR/LinalgTypes.h"
	#include "mlir/Dialect/Linalg/Passes.h"
	#include "mlir/Dialect/Linalg/Transforms/Transforms.h"
	#include "mlir/Dialect/Linalg/Utils/Utils.h"
	#include "mlir/Dialect/Tensor/IR/Tensor.h"
	#include "mlir/IR/AffineExpr.h"
	#include "mlir/IR/AffineMap.h"
	#include "mlir/Support/LLVM.h"

	using namespace mlir;
	using namespace linalg;

	//===----------------------------------------------------------------------===//
	// StructuredOp specific helpers.
	//===----------------------------------------------------------------------===//

	/// Returns the tiled slice dimensions given the tiled consumer loop dimensions.
	/// The slice defines a hyper rectangular iteration space and fusing the
	/// producer is always possible. However, depending on the consumer indexing
	/// map, not all slice elements may be consumed and the tiles may overlap. In
	/// these cases, fusion introduces redundant computation.
	SmallVector<int64_t> getTiledSliceDims(OpOperand *consumerOperand,
	ArrayRef<int64_t> tiledLoopDims) {
	// Get the consumer operand indexing map.
	LinalgOp consumerOp = consumerOperand->getOwner();
	AffineMap indexingMap = consumerOp.getTiedIndexingMap(consumerOperand);

	// Search the slice dimensions tiled by a tile loop dimension.
	DenseSet<int64_t> tiledSliceDims;
	for (auto en : enumerate(indexingMap.getResults())) {
	for (auto tiledLoopDim : tiledLoopDims) {
	if (en.value().isFunctionOfDim(tiledLoopDim))
	tiledSliceDims.insert(en.index());
	}
	}
	return {tiledSliceDims.begin(), tiledSliceDims.end()};
	}

	/// Returns the producer fused in place of `sliceOp`. Tile the producer operands
	/// along the `tiledSliceDims` and clone the producer. Consider the case of
	/// fusion of an output tensor:
	/// ```
	/// %1 = producer ins(...) outs(%0)
	/// %2 = consumer ins(...) outs(%1)
	/// ```
	/// When consumer is tiled, %1 appears in the loop iter_args:
	/// ```
	/// %1 = producer ins(...) outs(%0)
	/// %2 = scf.for ... iter_args(%1) .. (%bbarg) {
	/// %t1 = tensor.extract_slice %bbarg[..]
	/// %t2 = consumer ins(...) outs(%t1)
	/// %r = tensor.insert_slice %t2, %bbarg[...]
	/// }
	/// ```
	/// Fusing %1 into the loop requires updating iter_args(%1) to iter_args(%0):
	/// ```
	/// %2 = scf.for ... iter_args(%0) .. (%bbarg) {
	/// %t0 = tensor.extract_slice %bbarg[..]
	/// %t1 = producer ins(...) outs(%t0)
	/// %t2 = consumer ins(...) outs(%t1)
	/// %r = tensor.insert_slice %t2, %bbarg[...]
	/// }
	/// ```
	/// This transformation is only valid if %bbarg is exclusively used by the
	/// output ExtractSliceOp / InsertSliceOp pair, which is checked by the
	/// `fuseProducer` method.
	/// TODO: instead of check and failure, insert new iter_args each time a
	/// producer is fused into a consumer and fold away unused iter_args.
	static LinalgOp getTiledProducer(OpBuilder &b, OpResult producerResult,
	tensor::ExtractSliceOp sliceOp,
	ArrayRef<int64_t> tiledSliceDims,
	OpOperand *iterArg) {
	// Clone the producer after `sliceOp` since the slice may be reused to pass in
	// the producer result.
	OpBuilder::InsertionGuard guard(b);
	b.setInsertionPointAfter(sliceOp);

	// Get the producer.
	LinalgOp producerOp = producerResult.getOwner();
	Location loc = producerOp.getLoc();

	// Obtain the `producerOp` loop bounds and the `sliceOp` ranges.
	SmallVector<Value> producerLoopBounds;
	transform(producerOp.createLoopRanges(b, loc),
	std::back_inserter(producerLoopBounds),
	[](Range range) { return range.size; });
	SmallVector<Range> sliceOpRanges = sliceOp.getOrCreateRanges(b, loc);

	// Get the producer result indexing map.
	AffineMap producerIndexingMap = producerOp.getTiedIndexingMap(
	producerOp.getOutputOperand(producerResult.getResultNumber()));

	// Tile the producer operands given the `sliceOp` ranges. Iterate the
	// `tiledSliceDims` and store the tile offset and size for the tiled slice
	// dimension. Assumes the mapping from slice dimensions to producer loops is a
	// permutation.
	auto zero = b.create<ConstantIndexOp>(loc, 0);
	SmallVector<Value> tileIvs(producerOp.getNumLoops(), nullptr);
	SmallVector<Value> tileSizes(producerOp.getNumLoops(), zero);
	SmallVector<Value> allIvs(producerOp.getNumLoops(), nullptr);
	for (int64_t tiledSliceDim : tiledSliceDims) {
	AffineExpr result = producerIndexingMap.getResults()[tiledSliceDim];
	assert(result.isa<AffineDimExpr>() &&
	"expect producer indexing map is a projected permutation");
	int64_t tiledProducerLoop = result.cast<AffineDimExpr>().getPosition();
	tileIvs[tiledProducerLoop] = sliceOpRanges[tiledSliceDim].offset;
	tileSizes[tiledProducerLoop] = sliceOpRanges[tiledSliceDim].size;
	allIvs[tiledProducerLoop] = tileIvs[tiledProducerLoop];
	}
	erase_value(tileIvs, nullptr);
	SmallVector<Value> tiledOperands = producerOp.getInputAndOutputOperands();
	tiledOperands = makeTiledShapes(b, loc, producerOp, tiledOperands, tileIvs,
	tileSizes, producerLoopBounds);

	// Output fusion has to update the iteration arguments of the tile loop nest.
	// In particular, the iteration argument of the outermost tile loop needs to
	// be set to the producer output instead of the producer result and `clonedOp`
	// shall use the existing `sliceOp` result instead of the tiled producer
	// output operand.
	if (iterArg) {
	OpOperand *outputOperand =
	producerOp.getOutputOperand(producerResult.getResultNumber());
	iterArg->set(outputOperand->get());
	tiledOperands[outputOperand->getOperandNumber()] = sliceOp.getResult();
	}

	// Clone the producer using the tiled producer operands.
	TypeRange resultTypes = ValueRange(tiledOperands)
	.take_back(producerOp.getNumOutputs())
	.getTypes();
	LinalgOp clonedOp = producerOp.clone(b, loc, resultTypes, tiledOperands);

	// Shift all IndexOp results by the tile offset.
	addTileLoopIvsToIndexOpResults(b, clonedOp, allIvs);

	return clonedOp;
	}

	//===----------------------------------------------------------------------===//
	// TileLoopNest specific helpers.
	//===----------------------------------------------------------------------===//

	bool TileLoopNest::isEmpty() { return loopOps.empty(); }

	bool TileLoopNest::isValid() {
	// Check if the number of `tileLoopOps` and `tileLoopDims` match.
	if (loopOps.size() != loopDims.size())
	return false;

	// Check if the innermost tile loop is the parent of `tiledOp`.
	if (rootOp->getParentOp() != loopOps.back())
	return false;

	// Check if the tile loops are directly nested.
	return std::adjacent_find(loopOps.begin(), loopOps.end(),
	[](Operation op1, Operation op2) {
	return op1 != op2->getParentOp();
	}) == loopOps.end();
	}

	SmallVector<BlockArgument> TileLoopNest::getTiedBBArgs(BlockArgument bbArg) {
	assert(bbArg && "expect the block argument to be non-zero");
	SmallVector<BlockArgument> bbArgs;

	// Search all tile loop block arguments from inner to outer.
	for (auto tileLoop : reverse(loopOps)) {
	if (bbArg.getOwner()->getParentOp() != tileLoop)
	return {};
	bbArgs.push_back(bbArg);
	OpOperand *iterArg = &tileLoop.getOpOperandForRegionIterArg(bbArg);
	bbArg = iterArg->get().dyn_cast<BlockArgument>();
	}

	// Reverse the block arguments to order them from outer to inner.
	return {bbArgs.rbegin(), bbArgs.rend()};
	}

	OpOperand *TileLoopNest::getTiedIterArg(BlockArgument bbArg) {
	// Search all block arguments and return the matching iteration argument.
	SmallVector<BlockArgument> bbArgs = getTiedBBArgs(bbArg);
	if (bbArgs.size() != loopOps.size())
	return nullptr;
	return &loopOps.front().getOpOperandForRegionIterArg(bbArgs.front());
	}

	bool TileLoopNest::hasOtherUses(BlockArgument bbArg,
	tensor::ExtractSliceOp sliceOp) {
	// Check the innermost block argument is either used by the ExtractSliceOp
	// `sliceOp`, the matching InsertSliceOp, or by a DimOp. Handle other uses
	// conservatively.
	for (Operation *op : bbArg.getUsers()) {
	if (!isa<tensor::DimOp, tensor::InsertSliceOp, tensor::ExtractSliceOp>(op))
	return false;
	if (auto extractSliceOp = dyn_cast<tensor::ExtractSliceOp>(op)) {
	if (extractSliceOp != sliceOp)
	return false;
	}
	if (auto insertSliceOp = dyn_cast<tensor::InsertSliceOp>(op)) {
	SetVector<Operation *> backwardSlice;
	getBackwardSlice(insertSliceOp.source(), &backwardSlice,
	[](Operation *op) {
	return isa<LinalgOp, tensor::InsertSliceOp>(op);
	});
	if (backwardSlice.empty() \|\| backwardSlice.front() != sliceOp)
	return false;
	}
	}

	// Check the block arguments, except for the innermost one, have one use.
	SmallVector<BlockArgument> bbArgs = getTiedBBArgs(bbArg);
	return !all_of(bbArgs, [&](BlockArgument bbArg) {
	return bbArg.hasOneUse() \|\| bbArg == bbArgs.back();
	});
	}

	LogicalResult TileLoopNest::tileRootOp(OpBuilder &b,
	ArrayRef<int64_t> tileSizes,
	ArrayRef<int64_t> tileInterchange) {
	// Exit if all tile sizes are zero.
	if (tileSizes.size() == static_cast<size_t>(count(tileSizes, 0)))
	return success();

	// Tile the root operation.
	LinalgTilingOptions tilingOptions;
	tilingOptions = tilingOptions
	.setInterchange(SmallVector<unsigned>(
	tileInterchange.begin(), tileInterchange.end()))
	.setTileSizes(tileSizes)
	.setLoopType(LinalgTilingLoopType::Loops);
	Optional<TiledLinalgOp> tiledRootOp = tileLinalgOp(b, rootOp, tilingOptions);

	// Exit if tiling the root operation fails.
	if (!tiledRootOp.hasValue())
	return failure();

	// Replace all uses of the root operation if it has been tiled before. All
	// uses of the original untiled root operation are updated by the calling pass
	// or pattern.
	if (!isEmpty())
	rootOp->replaceAllUsesWith(tiledRootOp->tensorResults);

	// Update the root operation and append the loops and tile loop dimensions.
	rootOp = tiledRootOp->op;
	loopOps.append(tiledRootOp->loops.begin(), tiledRootOp->loops.end());
	for (auto en : enumerate(tileSizes)) {
	// Copy only the tiled loop dimensions with non-zero tile size.
	if (en.value() == 0)
	continue;
	loopDims.push_back(tileInterchange[en.index()]);
	}
	assert(isValid() && "expect tile loop nest to be valid after tiling");

	return success();
	}

	FailureOr<LinalgOp> TileLoopNest::fuseProducer(OpBuilder &b,
	OpOperand *rootOpOperand) {
	assert(rootOpOperand->getOwner() == rootOp &&
	"expect the root op to be the owner of the operand to fuse");
	assert(this->isValid() &&
	"expect the tile loop nest to satisfy all invariants");

	// Check the tile loop nest is non-empty.
	if (isEmpty())
	return failure();

	// Check `rootOpOperand` is defined by an ExtractSliceOp.
	auto sliceOp = rootOpOperand->get().getDefiningOp<tensor::ExtractSliceOp>();
	if (!sliceOp)
	return failure();

	// Check `sliceOp` is tiled by the tile loop nest.
	if (sliceOp->getParentOp() != rootOp->getParentOp())
	return failure();

	// Check if the producer is a LinalgOp possibly passed by iteration argument.
	OpOperand *iterArg = nullptr;
	auto producerResult = sliceOp.source().dyn_cast<OpResult>();
	if (auto bbArg = sliceOp.source().dyn_cast<BlockArgument>()) {
	iterArg = getTiedIterArg(bbArg);
	// Check the iteration argument may be used to pass in the producer output.
	if (!iterArg \|\| hasOtherUses(bbArg, sliceOp))
	return failure();
	producerResult = iterArg->get().dyn_cast<OpResult>();
	}
	if (!producerResult \|\| !isa<LinalgOp>(producerResult.getOwner()))
	return failure();

	// Compute the tiled producer slice dimensions given the tiled root operation
	// loop dimensions `loopDims`.
	SmallVector<int64_t> tiledSliceDims =
	getTiledSliceDims(rootOpOperand, loopDims);
	if (tiledSliceDims.empty())
	return failure();

	// Tile the producer operands and clone the producer in place of `sliceOp`.
	LinalgOp clonedOp =
	getTiledProducer(b, producerResult, sliceOp, tiledSliceDims, iterArg);

	// Cast the `clonedOp` result to gap type mismatches before canonicalization.
	Type consumerOperandType = rootOpOperand->get().getType();
	Value newResult = clonedOp->getResult(producerResult.getResultNumber());
	if (newResult.getType() != consumerOperandType) {
	OpBuilder::InsertionGuard guard(b);
	b.setInsertionPointAfter(clonedOp);
	newResult = b.create<tensor::CastOp>(producerResult.getLoc(),
	consumerOperandType, newResult);
	}

	// Replace the `sliceOp` uses except for the `clonedOp` output uses.
	sliceOp.getResult().replaceAllUsesExcept(newResult, clonedOp);
	return clonedOp;
	}

	ValueRange TileLoopNest::getRootOpReplacementResults() {
	assert(!isEmpty() && "expect tile loop nest to be non-empty");
	return loopOps.front()->getOpResults();
	}

	//===----------------------------------------------------------------------===//
	// Tile and fuse entry-points.
	//===----------------------------------------------------------------------===//

	FailureOr<TileLoopNest>
	mlir::linalg::tileConsumerAndFuseProducers(OpBuilder &b, LinalgOp consumerOp,
	ArrayRef<int64_t> tileSizes,
	ArrayRef<int64_t> tileInterchange) {
	assert(tileSizes.size() == tileInterchange.size() &&
	"expect the number of tile sizes and interchange dims to match");
	assert(isPermutation(tileInterchange) &&
	"expect tile interchange is a permutation");

	// Create an empty tile loop nest.
	TileLoopNest tileLoopNest(consumerOp);

	// Search the number of outer parallel loops to separate them from possible
	// inner reduction dimensions.
	SmallVector<StringAttr> iterTypes =
	llvm::to_vector<6>(consumerOp.iterator_types().getAsRange<StringAttr>());
	applyPermutationToVector(iterTypes, tileInterchange);
	auto *it = find_if(iterTypes, [&](StringAttr iterType) {
	return !isParallelIterator(iterType);
	});
	int64_t split = std::distance(iterTypes.begin(), it);

	// Tile the outer parallel loops and fuse the output operands.
	SmallVector<int64_t> outerTileSizes;
	outerTileSizes.append(tileSizes.begin(), tileSizes.begin() + split);
	outerTileSizes.append(tileSizes.size() - split, 0);
	if (failed(tileLoopNest.tileRootOp(b, outerTileSizes, tileInterchange)))
	return failure();
	for (OpOperand *opOperand : tileLoopNest.getRootOp().getOutputOperands())
	(void)tileLoopNest.fuseProducer(b, opOperand);

	// Tile the remaining loops and fuse the input operands.
	SmallVector<int64_t> innerTileSizes;
	innerTileSizes.append(split, 0);
	innerTileSizes.append(tileSizes.begin() + split, tileSizes.end());
	if (failed(tileLoopNest.tileRootOp(b, innerTileSizes, tileInterchange)))
	return failure();
	SmallVector<OpOperand *> inputOperands =
	tileLoopNest.getRootOp().getInputOperands();
	for (OpOperand *opOperand : tileLoopNest.getRootOp().getInputOperands())
	(void)tileLoopNest.fuseProducer(b, opOperand);

	return tileLoopNest;
	}

	namespace {
	struct LinalgTileAndFuseTensorOps
	: public LinalgTileAndFuseTensorOpsBase<LinalgTileAndFuseTensorOps> {

	void notifyFailure(StringRef message) {
	llvm::errs() << " - LinalgTileAndFuseTensorOps: " << message << "\n";
	signalPassFailure();
	}

	void runOnFunction() override {
	FuncOp funcOp = getFunction();
	OpBuilder b(funcOp.getContext());

	// Heuristic to find a good operation to tile and start fusion. Walk all
	// operations and select the one with the maximal backward slice of fusion
	// candidates.
	LinalgOp rootOp = nullptr;
	int64_t numFusionCandidates = -1;
	funcOp.walk([&](LinalgOp linalgOp) {
	SetVector<Operation *> backwardSlice;
	getBackwardSlice(linalgOp, &backwardSlice);
	int64_t backwardSliceSize = count_if(
	backwardSlice, [](Operation *op) { return isa<LinalgOp>(op); });
	if (backwardSliceSize > numFusionCandidates) {
	rootOp = linalgOp;
	numFusionCandidates = backwardSliceSize;
	}
	});
	if (!rootOp)
	return notifyFailure("expect to find a root operation");

	// Check `tileSizes` contains a tile size for every `rootOp` loop dimension.
	if (tileSizes.size() < rootOp.getNumLoops())
	return notifyFailure("expect #tile sizes >= #loops");

	// Check `tileInterchange` contains no entries or as many as `tileSizes`.
	if (!tileInterchange.empty() &&
	tileInterchange.size() != tileSizes.size()) {
	return notifyFailure(
	"expect the number of tile sizes and interchange dims to match");
	}

	// Copy the `tileSizes` and `tileInterchange` prefixes needed to tile
	// `rootOp` or use the identity interchange if `tileInterchange` is empty.
	SmallVector<int64_t> rootTileSizes(
	tileSizes.begin(), tileSizes.begin() + rootOp.getNumLoops());
	SmallVector<int64_t> rootInterchange =
	tileInterchange.empty()
	? llvm::to_vector<6>(llvm::seq<int64_t>(0, rootOp.getNumLoops()))
	: SmallVector<int64_t>(tileInterchange.begin(),
	tileInterchange.begin() +
	rootOp.getNumLoops());

	// Check `rootInterchange` is a permutation of the `rootOp` loop dimensions.
	// It has to be a permutation since the tiling cannot tile the same loop
	// dimension multiple times.
	if (!isPermutation(rootInterchange))
	return notifyFailure(
	"expect the tile interchange permutes the root loops");

	// Tile `rootOp` and fuse its producers.
	FailureOr<TileLoopNest> tileLoopNest =
	tileConsumerAndFuseProducers(b, rootOp, rootTileSizes, rootInterchange);
	if (failed(tileLoopNest))
	return notifyFailure("tileConsumerAndFuseProducers failed unexpectedly");

	// Replace all uses of the tiled loop operation.
	rootOp->replaceAllUsesWith(tileLoopNest->getRootOpReplacementResults());
	}
	};
	} // namespace

	std::unique_ptr<OperationPass<FuncOp>>
	mlir::createLinalgTileAndFuseTensorOpsPass() {
	return std::make_unique<LinalgTileAndFuseTensorOps>();
	}