ROCm/lib/Analysis/Allocation.cpp

#include "triton/Analysis/Allocation.h"
#include "mlir/Analysis/DataFlowFramework.h"
#include "mlir/Analysis/Liveness.h"
#include "mlir/Analysis/SliceAnalysis.h"
#include "mlir/Dialect/Tensor/IR/Tensor.h"
#include "triton/Analysis/Alias.h"
#include "triton/Dialect/TritonGPU/IR/Dialect.h"
#include "llvm/ADT/SmallVector.h"

#include <algorithm>
#include <limits>
#include <numeric>

using ::mlir::triton::gpu::BlockedEncodingAttr;
using ::mlir::triton::gpu::DotOperandEncodingAttr;
using ::mlir::triton::gpu::getContigPerThread;
using ::mlir::triton::gpu::getOrder;
using ::mlir::triton::gpu::getShapePerCTA;
using ::mlir::triton::gpu::getShapePerCTATile;
using ::mlir::triton::gpu::getSizePerThread;
using ::mlir::triton::gpu::MfmaEncodingAttr;
using ::mlir::triton::gpu::getUniqueContigPerThread;
using ::mlir::triton::gpu::MmaEncodingAttr;
using ::mlir::triton::gpu::SharedEncodingAttr;
using ::mlir::triton::gpu::SliceEncodingAttr;

namespace mlir {

//===----------------------------------------------------------------------===//
// Shared Memory Allocation Analysis
//===----------------------------------------------------------------------===//
namespace triton {

// Bitwidth of pointers
constexpr int kPtrBitWidth = 64;

static std::pair<SmallVector<unsigned>, SmallVector<unsigned>>
getCvtOrder(Attribute srcLayout, Attribute dstLayout) {
  auto srcMmaLayout = srcLayout.dyn_cast<MmaEncodingAttr>();
  auto srcDotLayout = srcLayout.dyn_cast<DotOperandEncodingAttr>();
  auto dstMmaLayout = dstLayout.dyn_cast<MmaEncodingAttr>();
  auto dstDotLayout = dstLayout.dyn_cast<DotOperandEncodingAttr>();
  assert(!(srcMmaLayout && dstMmaLayout) &&
         "Unexpected mma -> mma layout conversion");
  // mma or dot layout does not have an order, so the order depends on the
  // layout of the other operand.
  auto inOrd = (srcMmaLayout || srcDotLayout) ? getOrder(dstLayout)
                                              : getOrder(srcLayout);
  auto outOrd = (dstMmaLayout || dstDotLayout) ? getOrder(srcLayout)
                                               : getOrder(dstLayout);

  return {inOrd, outOrd};
}

SmallVector<unsigned> getRepShapeForCvtLayout(triton::gpu::ConvertLayoutOp op) {
  auto srcTy = op.getSrc().getType().cast<RankedTensorType>();
  auto dstTy = op.getResult().getType().cast<RankedTensorType>();
  Attribute srcLayout = srcTy.getEncoding();
  Attribute dstLayout = dstTy.getEncoding();

  if (shouldUseDistSmem(srcLayout, dstLayout)) {
    // TODO: padding to avoid bank conflicts
    return convertType<unsigned, int64_t>(getShapePerCTA(srcTy));
  }

  // MmaToDotShortcut and MmaToMmaShortcut doesn't use shared mem
  if (auto srcMmaLayout = srcLayout.dyn_cast<MmaEncodingAttr>()) {
    if (dstLayout.isa<DotOperandEncodingAttr>()) {
      if (isMmaToDotShortcut(srcTy, dstTy)) {
        return {};
      }
    } else if (auto dstMmaLayout = dstLayout.dyn_cast<MmaEncodingAttr>()) {
      if (isMmaToMmaShortcut(srcTy, dstTy)) {
        return {};
      }
    }
  }

#ifdef USE_ROCM
  if (srcLayout.isa<MfmaEncodingAttr>() &&
      srcLayout.dyn_cast<MfmaEncodingAttr>().getIsTransposed() &&
      dstLayout.isa<DotOperandEncodingAttr>())
    if (isMfmaToDotShortcut(srcTy, dstTy))
      return {};
#endif

  assert(srcLayout && dstLayout && "Unexpected layout in getRepShape()");

  auto srcShapePerCTA = getShapePerCTA(srcTy);
  auto dstShapePerCTA = getShapePerCTA(dstTy);
  auto srcShapePerCTATile = getShapePerCTATile(srcLayout, srcTy.getShape());
  auto dstShapePerCTATile = getShapePerCTATile(dstLayout, dstTy.getShape());

  unsigned rank = dstTy.getRank();
  SmallVector<unsigned> repShape(rank);
  for (unsigned d = 0; d < rank; ++d) {
    repShape[d] =
        std::max(std::min<unsigned>(srcShapePerCTA[d], srcShapePerCTATile[d]),
                 std::min<unsigned>(dstShapePerCTA[d], dstShapePerCTATile[d]));
  }
  return repShape;
}

SmallVector<unsigned>
getScratchConfigForCvtLayout(triton::gpu::ConvertLayoutOp op, unsigned &inVec,
                             unsigned &outVec) {
  auto repShape = getRepShapeForCvtLayout(op);
  if (repShape.empty())
    return repShape;

  auto srcTy = op.getSrc().getType().cast<RankedTensorType>();
  auto dstTy = op.getResult().getType().cast<RankedTensorType>();
  Attribute srcLayout = srcTy.getEncoding();
  Attribute dstLayout = dstTy.getEncoding();

  auto [inOrd, outOrd] = getCvtOrder(srcLayout, dstLayout);
  unsigned srcContigPerThread =
      getUniqueContigPerThread(srcLayout, srcTy.getShape())[inOrd[0]];
  unsigned dstContigPerThread =
      getUniqueContigPerThread(dstLayout, dstTy.getShape())[outOrd[0]];
  // TODO: Fix the legacy issue that ourOrd[0] == 0 always means
  //       that we cannot do vectorization.
  inVec = outOrd[0] == 0 ? 1 : inOrd[0] == 0 ? 1 : srcContigPerThread;
  outVec = outOrd[0] == 0 ? 1 : dstContigPerThread;

  if (repShape.size() <= 1)
    return repShape;
  unsigned paddedDim = 1;
  if (auto dstBlockedLayout = dstLayout.dyn_cast<BlockedEncodingAttr>()) {
    paddedDim = dstBlockedLayout.getOrder()[0];
  }
  unsigned pad = std::max(inVec, outVec);
  repShape[paddedDim] += pad;
  return repShape;
}

SmallVector<unsigned>
getScratchConfigForStoreAsync(triton::nvidia_gpu::StoreAsyncOp op) {
  auto srcTy = op.getSrc().getType().cast<RankedTensorType>();
  return convertType<unsigned, int64_t>(getShapePerCTA(srcTy));
}

// TODO: extend beyond scalars
SmallVector<unsigned> getScratchConfigForAtomicRMW(triton::AtomicRMWOp op) {
  SmallVector<unsigned> smemShape;
  if (op.getPtr().getType().isa<RankedTensorType>()) {
    // do nothing or just assert because shared memory is not used in tensor up
    // to now
  } else {
    // need only bytes for scalar
    // always vec = 1 and elemsPerThread = 1 for scalar?
    smemShape.push_back(1);
  }
  return smemShape;
}

SmallVector<unsigned> getScratchConfigForAtomicCAS(triton::AtomicCASOp op) {
  return SmallVector<unsigned>{1};
}

class AllocationAnalysis {
public:
  AllocationAnalysis(Operation *operation,
                     Allocation::FuncAllocMapT *funcAllocMap,
                     Allocation *allocation)
      : operation(operation), funcAllocMap(funcAllocMap),
        allocation(allocation) {
    run();
  }

private:
  using BufferT = Allocation::BufferT;

  /// Value -> Liveness Range
  using IntervalT = Interval<size_t>;
  /// Use MapVector to ensure determinism.
  using BufferRangeMapT = llvm::MapVector<BufferT *, IntervalT>;
  /// Nodes -> Nodes
  using GraphT = DenseMap<BufferT *, DenseSet<BufferT *>>;

  /// Set of Liveness Intervals
  class LivenessR : public SmallVector<IntervalT, 4> {
  public:
    LivenessR() = default;
    LivenessR(const LivenessR &) = default;

    /// Disjointness
    bool isDisjoint() const {
      if (size() < 2)
        return false;
      // sorted so the first OOB proves disjoint
      auto maxId = (*this)[0].end();
      for (auto rng : *this) {
        if (rng.start() <= maxId) {
          // adjoining
          maxId = std::max(maxId, rng.end());
        } else
          return true;
      }
      return false;
    }

    void sort() {
      llvm::sort(*this, [](const auto &lhs, const auto &rhs) {
                          return lhs.start() <= rhs.start();
                        });
    }

    bool addAdjacent(size_t id) {
      bool isAdjacent = false;
      for (auto &interval : *this) {
        if (interval.adjacent(id)) {
          isAdjacent = true;
          interval = interval.merge(IntervalT(id));
        }
      }
      return isAdjacent;
    }

    void add(size_t id) {
      if (!addAdjacent(id))
        push_back(IntervalT(id));
    }
    IntervalT unionize() const {
      IntervalT res;
      if (size()) {
        res = front();
        for (auto &I : *this)
          res = res.merge(I);
      }
      return res;
    }
  };
  
  typedef function_ref<LivenessR(Value value)> LivenessF;

  void run() {
    getValuesAndSizes();
    resolveLiveness();
    computeOffsets();
  }

  /// Initializes explicitly defined shared memory values for a given operation.
  void getExplicitValueSize(Operation *op) {
    // Values returned from scf.yield will not be allocated even though they
    // have the shared encoding.
    // For example: %a = scf.if -> yield
    // %a must be allocated elsewhere by other operations.
    // FIXME(Keren): extract and insert are always alias for now
    if (!maybeSharedAllocationOp(op) || maybeAliasOp(op))
      return;

    // XXX(Keren): Why this hard-coded alignment?
    size_t kAlignment = 8;
    for (Value result : op->getResults()) {
      if (triton::gpu::isSharedEncoding(result)) {
        // Bytes could be a different value once we support padding or other
        // allocation policies.
        auto tensorType = result.getType().dyn_cast<RankedTensorType>();
        auto shapePerCTA = triton::gpu::getShapePerCTA(tensorType);
        auto bytes = product<int64_t>(shapePerCTA) *
                     tensorType.getElementTypeBitWidth() / 8;

        // XXX(Keren): magic numbers 256 and 1024
        // benzh@maybe alignment should be passed in.
        // Software swizzling calculates phase based on offset, while hardware
        // swizzling do that based on physical address. Thus only by setting the
        // alignment to 1024 can ensure the correctness. 
        if (bytes > 256)
          kAlignment = 1024;
        allocation->addBuffer<BufferT::BufferKind::Explicit>(result, bytes,
                                                             kAlignment);
      }
    }
    if (isa<triton::nvidia_gpu::AllocMBarrierOp>(op)) {
      Value result = op->getResult(0);
      if (!result.getType().isa<RankedTensorType>())
        // In case AllocMBarrierOp is allocating scalar mbarriers
        allocation->addBuffer<BufferT::BufferKind::Explicit>(result, 8,
                                                             kAlignment);
    }
  }

  template <BufferT::BufferKind T>
  void maybeAddScratchBuffer(Operation *op, unsigned bytes,
                             unsigned alignment) {
    if (bytes > 0)
      allocation->addBuffer<T>(op, bytes, alignment);
  }

  template <BufferT::BufferKind T>
  void maybeAddScratchBuffer(Operation *op, unsigned bytes) {
    if (bytes > 0)
      allocation->addBuffer<T>(op, bytes);
  }

  /// Initializes temporary shared memory for a given operation.
  void getScratchValueSize(Operation *op) {
    const size_t scratchAlignment = 128;
    if (auto reduceOp = dyn_cast<triton::ReduceOp>(op)) {
      ReduceOpHelper helper(reduceOp);
      unsigned bytes = helper.getScratchSizeInBytes();
      maybeAddScratchBuffer<BufferT::BufferKind::Scratch>(op, bytes,
                                                          scratchAlignment);
    } else if (auto scanOp = dyn_cast<triton::ScanOp>(op)) {
      ScanLoweringHelper helper(scanOp);
      unsigned bytes = helper.getScratchSizeInBytes();
      maybeAddScratchBuffer<BufferT::BufferKind::Scratch>(op, bytes,
                                                          scratchAlignment);
    } else if (auto cvtLayout = dyn_cast<triton::gpu::ConvertLayoutOp>(op)) {
      auto srcTy = cvtLayout.getSrc().getType().cast<RankedTensorType>();
      auto dstTy = cvtLayout.getResult().getType().cast<RankedTensorType>();
      auto srcEncoding = srcTy.getEncoding();
      auto dstEncoding = dstTy.getEncoding();
      if (srcEncoding.isa<SharedEncodingAttr>() ||
          dstEncoding.isa<SharedEncodingAttr>()) {
        // Conversions from/to shared memory do not need scratch memory.
        return;
      }
      // ConvertLayoutOp with both input/output non-shared_layout
      // TODO: Besides of implementing ConvertLayoutOp via shared memory, it's
      //       also possible to realize it with other approaches in restricted
      //       conditions, such as warp-shuffle
      unsigned inVec = 0;
      unsigned outVec = 0;
      auto smemShape = getScratchConfigForCvtLayout(cvtLayout, inVec, outVec);
      unsigned elems = std::accumulate(smemShape.begin(), smemShape.end(), 1,
                                       std::multiplies{});
      auto bytes =
          srcTy.getElementType().isa<triton::PointerType>()
              ? elems * kPtrBitWidth / 8
              : elems * std::max<int>(8, srcTy.getElementTypeBitWidth()) / 8;
      maybeAddScratchBuffer<BufferT::BufferKind::Scratch>(op, bytes,
                                                          scratchAlignment);
    } else if (auto storeAsyncOp =
                   dyn_cast<triton::nvidia_gpu::StoreAsyncOp>(op)) {
      auto srcTy = storeAsyncOp.getSrc().getType().cast<RankedTensorType>();
      auto srcEncoding = srcTy.getEncoding();
      if (!srcEncoding.isa<MmaEncodingAttr>()) {
        return;
      }
      auto smemShape = getScratchConfigForStoreAsync(storeAsyncOp);
      unsigned elems = std::accumulate(smemShape.begin(), smemShape.end(), 1,
                                       std::multiplies{});
      auto bytes = elems * std::max<int>(8, srcTy.getElementTypeBitWidth()) / 8;
      maybeAddScratchBuffer<BufferT::BufferKind::Scratch>(op, bytes, 1024);
    } else if (auto atomicRMWOp = dyn_cast<triton::AtomicRMWOp>(op)) {
      auto value = op->getOperand(0);
      // only scalar requires scratch memory
      // make it explicit for readability
      if (value.getType().dyn_cast<RankedTensorType>()) {
        // nothing to do
      } else {
        auto smemShape = getScratchConfigForAtomicRMW(atomicRMWOp);
        unsigned elems = std::accumulate(smemShape.begin(), smemShape.end(), 1,
                                         std::multiplies{});
        auto elemTy =
            value.getType().cast<triton::PointerType>().getPointeeType();
        auto bytes =
            elemTy.isa<triton::PointerType>()
                ? elems * kPtrBitWidth / 8
                : elems * std::max<int>(8, elemTy.getIntOrFloatBitWidth()) / 8;
        maybeAddScratchBuffer<BufferT::BufferKind::Scratch>(op, bytes,
                                                            scratchAlignment);
      }
    } else if (auto atomicCASOp = dyn_cast<triton::AtomicCASOp>(op)) {
      // only scalar requires scratch memory
      // make it explicit for readability
      auto value = op->getOperand(0);
      if (value.getType().dyn_cast<RankedTensorType>()) {
        // nothing to do
      } else {
        auto smemShape = getScratchConfigForAtomicCAS(atomicCASOp);
        unsigned elems = std::accumulate(smemShape.begin(), smemShape.end(), 1,
                                         std::multiplies{});
        auto elemTy =
            value.getType().cast<triton::PointerType>().getPointeeType();
        auto bytes = elemTy.isa<triton::PointerType>()
                         ? elems * kPtrBitWidth / 8
                         : elems * elemTy.getIntOrFloatBitWidth() / 8;
        maybeAddScratchBuffer<BufferT::BufferKind::Scratch>(op, bytes,
                                                            scratchAlignment);
      }
    } else if (auto callOp = dyn_cast<CallOpInterface>(op)) {
      auto callable = callOp.resolveCallable();
      auto funcOp = dyn_cast<FunctionOpInterface>(callable);
      auto *funcAlloc = &(*funcAllocMap)[funcOp];
      auto bytes = funcAlloc->getSharedMemorySize();
      maybeAddScratchBuffer<BufferT::BufferKind::Virtual>(op, bytes,
                                                          scratchAlignment);
    }
  }

  void getValueAlias(Value value, SharedMemoryAliasAnalysis &analysis) {
    dataflow::Lattice<AliasInfo> *latticeElement =
        analysis.getLatticeElement(value);
    if (latticeElement) {
      AliasInfo &info = latticeElement->getValue();
      if (!info.getAllocs().empty()) {
        for (auto alloc : info.getAllocs()) {
          allocation->addAlias(value, alloc);
        }
      }
    }
  }

  /// Extract all shared memory values and their sizes
  void getValuesAndSizes() {
    // Get the alloc values
    operation->walk<WalkOrder::PreOrder>([&](Operation *op) {
      getExplicitValueSize(op);
      getScratchValueSize(op);
    });
    // Get the alias values
    std::unique_ptr<DataFlowSolver> solver = createDataFlowSolver();
    SharedMemoryAliasAnalysis *aliasAnalysis =
        solver->load<SharedMemoryAliasAnalysis>();
    if (failed(solver->initializeAndRun(operation))) {
      // TODO: return error instead of bailing out..
      llvm_unreachable("failed to run SharedMemoryAliasAnalysis");
    }
    operation->walk<WalkOrder::PreOrder>([&](Operation *op) {
      for (auto operand : op->getOperands()) {
        getValueAlias(operand, *aliasAnalysis);
      }
      for (auto value : op->getResults()) {
        getValueAlias(value, *aliasAnalysis);
      }
    });
  }

  /// Computes the liveness range of the allocated value.
  /// Each buffer is allocated only once.
  void resolveExplicitBufferLiveness(LivenessF getLiveness) {
    for (auto valueBufferIter : allocation->valueBuffer) {
      auto value = valueBufferIter.first;
      auto *buffer = valueBufferIter.second;
      auto ranges = getLiveness(value);
      bufferRange[buffer] = ranges.unionize();
    }
  }

  /// Extends the liveness range by unionizing the liveness range of the aliased
  /// values because each allocated buffer could be an alias of others, if block
  /// arguments are involved.
  /// Only unionize adjacent live ranges to account for loop-carried buffers that
  /// are mutually exclusive.
  /// Example from stream pipeliner:
  ///  3    %b0 = convert_layout %g0                  -+
  ///  4    %fr = for (.., %arg0 = %b0) {              |
  ///  5        %gn = load %pc                         |
  ///  6        %bc = convert_layout %arg0            -+
  ///  7        %v = add %bc, ...
  ///  8        %bn = convert_layout %gn              -+
  ///  9        %pn = addptr %pc, %cst                 |
  ///  10   }                                          |
  ///  11   %be = convert_layout %fr#1                -+
  ///  12   %ve = add %be
  void resolveAliasBufferLiveness(LivenessF getLiveness) {
    for (auto aliasBufferIter : allocation->aliasBuffer) {
      auto value = aliasBufferIter.first;
      auto buffers = aliasBufferIter.second;
      auto aranges = getLiveness(value);
      bool disjoint = aranges.isDisjoint();
      for (auto *buffer : buffers) {
        auto range = aranges[0];
        if (bufferRange.count(buffer)) {
          auto brange = bufferRange[buffer];
          if (disjoint) {
            // find adjacent/intersecting
            for (auto arange : aranges) {
              if (arange.adjacent(brange) ||
                  arange.intersects(brange))
                brange = arange.merge(brange);
            }
            range = brange;
          } else {
            // Extend the allocated buffer's range
            range = range.merge(brange);
          }
        }
        bufferRange[buffer] = range;
      }
    }
  }

  /// Computes the liveness range of scratched buffers.
  /// Some operations may have a temporary buffer that is not explicitly
  /// allocated, but is used to store intermediate results.
  void resolveScratchBufferLiveness(
      const DenseMap<Operation *, size_t> &operationId) {
    // Analyze liveness of scratch buffers and vritual buffers.
    auto processScratchMemory = [&](const auto &container) {
      for (auto opScratchIter : container) {
        // Any scratch memory's live range is the current operation's live
        // range.
        auto *op = opScratchIter.first;
        auto *buffer = opScratchIter.second;
        bufferRange.insert({buffer, Interval(operationId.lookup(op),
                                             operationId.lookup(op) + 1)});
      }
    };
    processScratchMemory(allocation->opScratch);
    processScratchMemory(allocation->opVirtual);
  }

  /// Resolves liveness of all values involved under the root operation.
  void resolveLiveness() {
    // Assign an ID to each operation using post-order traversal.
    // To achieve the correct liveness range, the parent operation's ID
    // should be greater than each of its child operation's ID .
    // Example:
    //     ...
    //     %5 = triton.convert_layout %4
    //     %6 = scf.for ... iter_args(%arg0 = %0) -> (i32) {
    //       %2 = triton.convert_layout %5
    //       ...
    //       scf.yield %arg0
    //     }
    // For example, %5 is defined in the parent region and used in
    // the child region, and is not passed as a block argument.
    // %6 should should have an ID greater than its child operations,
    // otherwise %5 liveness range ends before the child operation's liveness
    // range ends.
    DenseMap<Operation *, size_t> operationId;
    operation->walk<WalkOrder::PostOrder>(
        [&](Operation *op) { operationId[op] = operationId.size(); });

    // Analyze liveness of explicit buffers
    Liveness liveness(operation);
    auto getValueLivenessRange = [&](Value value) {
      LivenessR ranges;
      // Shared memory allocated by mbarrier cannot be reused
      if (value.getDefiningOp() &&
          isa<triton::nvidia_gpu::AllocMBarrierOp>(value.getDefiningOp())) {
        ranges.push_back(Interval(std::numeric_limits<size_t>::min(),
                        std::numeric_limits<size_t>::max()));
        return ranges;
      }

      auto liveOperations = liveness.resolveLiveness(value);
      std::for_each(liveOperations.begin(), liveOperations.end(),
                    [&](Operation *liveOp) {
                      ranges.add(operationId[liveOp]);
                    });
      ranges.sort();
      return ranges;
    };

    resolveExplicitBufferLiveness(getValueLivenessRange);
    resolveAliasBufferLiveness(getValueLivenessRange);
    resolveScratchBufferLiveness(operationId);
  }

  /// Computes the shared memory offsets for all related values.
  /// Paper: Algorithms for Compile-Time Memory Optimization
  /// (https://www.cs.utexas.edu/users/harrison/papers/compile-time.pdf)
  void computeOffsets() {
    SmallVector<BufferT *> buffers;
    for (auto bufferIter : bufferRange) {
      buffers.emplace_back(bufferIter.first);
    }

    DenseMap<BufferT *, size_t> bufferStart;
    calculateStarts(buffers, bufferStart);

    // NOTE: The original paper doesn't consider interference between
    // the bumped ranges. Buffers that previously do not interfere with
    // could interfere after offset bumping if their liveness ranges overlap.
    // Therefore, we rerun the interference graph algorithm after bumping so
    // that we regroup the buffers and color them again. Since we always
    // increase the buffer offset and keep reducing conflicts, we will
    // eventually reach a fixed point.
    GraphT interference;
    buildInterferenceGraph(buffers, bufferStart, interference);
    do {
      allocate(buffers, interference, bufferStart);
      buildInterferenceGraph(buffers, bufferStart, interference);
    } while (!interference.empty());
  }

  /// Computes the initial shared memory offsets.
  void calculateStarts(const SmallVector<BufferT *> &buffers,
                       DenseMap<BufferT *, size_t> &bufferStart) {
    //  v = values in shared memory
    //  t = triplet of (size, start, end)
    //  shared memory space
    //  -
    //  |         *******t4
    //  | /|\ v2 inserts t4, t5, and t6
    //  |  |
    //  | ******t5         ************t6
    //  | ^^^^^v2^^^^^^
    //  |  |      *********************t2
    //  | \|/ v2 erases t1
    //  | ******t1 ^^^^^^^^^v1^^^^^^^^^ ************t3
    //  |---------------------------------------------| liveness range
    //    1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
    // If the available triple's range is less than a given buffer range,
    // we won't know if there has been an overlap without using graph coloring.
    // Start -> Liveness Range
    using TripleMapT = std::multimap<size_t, IntervalT>;
    TripleMapT tripleMap;
    tripleMap.insert(std::make_pair(0, IntervalT()));
    SmallVector<BufferT *> xBuffers = buffers;
    while (!xBuffers.empty()) {
      auto tripleIt = tripleMap.begin();
      auto size = tripleIt->first;
      auto range = tripleIt->second;
      tripleMap.erase(tripleIt);
      auto bufferIt =
          std::find_if(xBuffers.begin(), xBuffers.end(), [&](auto *buffer) {
            auto xRange = bufferRange[buffer];
            bool res = xRange.intersects(range);
            for (auto val : tripleMap)
              res = res &&
                    !val.second.intersects(xRange); // only one buffer intersect
            return res;
          });
      if (bufferIt != xBuffers.end()) {
        auto buffer = *bufferIt;
        auto xSize = buffer->size;
        auto xRange = bufferRange.lookup(buffer);
        // TODO(Keren): A buffer's size shouldn't be determined here, have to
        // clean it up
        size_t alignment = buffer->alignment;
        size_t alignSize = ((size + alignment - 1) / alignment) * alignment;
        bufferStart[buffer] = alignSize;
        tripleMap.insert({alignSize + xSize,
                          Interval{std::max(range.start(), xRange.start()),
                                   std::min(range.end(), xRange.end())}});
        // We could either insert (range.start, xRange.start) or (range.start,
        // xRange.end), both are correct and determine the potential buffer
        // offset, and the graph coloring algorithm will solve the interference,
        // if any
        if (range.start() < xRange.start())
          tripleMap.insert({size, Interval{range.start(), xRange.end()}});
        if (xRange.end() < range.end())
          tripleMap.insert({size, Interval{xRange.start(), range.end()}});
        xBuffers.erase(bufferIt);
      }
    }
  }

  /// Builds a graph of all shared memory values. Edges are created between
  /// shared memory values that are overlapping.
  void buildInterferenceGraph(const SmallVector<BufferT *> &buffers,
                              const DenseMap<BufferT *, size_t> &bufferStart,
                              GraphT &interference) {
    // Reset interference graph
    interference.clear();
    for (auto x : buffers) {
      for (auto y : buffers) {
        if (x == y)
          continue;
        auto xStart = bufferStart.lookup(x);
        auto yStart = bufferStart.lookup(y);
        auto xSize = x->size;
        auto ySize = y->size;
        Interval xSizeRange = {xStart, xStart + xSize};
        Interval ySizeRange = {yStart, yStart + ySize};
        auto xOpRange = bufferRange.lookup(x);
        auto yOpRange = bufferRange.lookup(y);
        if (xOpRange.intersects(yOpRange) &&
            xSizeRange.intersects(ySizeRange)) {
          interference[x].insert(y);
        }
      }
    }
  }

  /// Finalizes shared memory offsets considering interference.
  void allocate(const SmallVector<BufferT *> &buffers,
                const GraphT &interference,
                DenseMap<BufferT *, size_t> &bufferStart) {
    // Reset shared memory size
    allocation->sharedMemorySize = 0;
    // First-fit graph coloring
    // Neighbors are nodes that interfere with each other.
    // We color a node by finding the index of the first available
    // non-neighboring node or the first neighboring node without any color.
    // Nodes with the same color do not interfere with each other.
    DenseMap<BufferT *, int> colors;
    for (auto value : buffers) {
      colors[value] = (value == buffers[0]) ? 0 : -1;
    }
    SmallVector<bool> available(buffers.size());
    for (auto x : buffers) {
      std::fill(available.begin(), available.end(), true);
      for (auto y : interference.lookup(x)) {
        int color = colors[y];
        if (color >= 0) {
          available[color] = false;
        }
      }
      auto it = std::find(available.begin(), available.end(), true);
      colors[x] = std::distance(available.begin(), it);
    }
    // Finalize allocation
    // color0: [0, 7), [0, 8), [0, 15) -> [0, 7), [0, 8), [0, 15)
    // color1: [7, 9) -> [0 + 1 * 15, 9 + 1 * 15) -> [15, 24)
    // color2: [8, 12) -> [8 + 2 * 15, 12 + 2 * 15) -> [38, 42)
    // TODO(Keren): We are wasting memory here.
    // Nodes with color2 can actually start with 24.
    for (auto x : buffers) {
      size_t adj = 0;
      for (auto y : interference.lookup(x)) {
        adj = std::max(adj, bufferStart.lookup(y) + y->size);
      }
      x->offset = bufferStart.lookup(x) + colors.lookup(x) * adj;
      bufferStart[x] = x->offset;
      allocation->sharedMemorySize =
          std::max(allocation->sharedMemorySize, x->offset + x->size);
    }
  }

  void dump() const {
    llvm::outs() << "DUMP: " << "\n";
    for (auto bufferIter : bufferRange) {
      
      llvm::outs() << "ID= " << bufferIter.first->id << "\n";
      // llvm::outs() << "             Kind= " << kind << "\n";
      llvm::outs() << "     Size= " << bufferIter.first->size << "\n";
      llvm::outs() << "     Offs= " << bufferIter.first->offset << "\n";
      llvm::outs() << "  -> " << bufferIter.second.start() << "\n";
      llvm::outs() << "  -> " << bufferIter.second.end() << "\n";
    }
  }

private:
  Operation *operation;
  Allocation::FuncAllocMapT *funcAllocMap;
  Allocation *allocation;
  BufferRangeMapT bufferRange;
};

} // namespace triton

void Allocation::run(FuncAllocMapT &funcAllocMap) {
  triton::AllocationAnalysis(getOperation(), &funcAllocMap, this);
}

} // namespace mlir