llvm/lib/Analysis/LoopAccessAnalysis.cpp

//===- LoopAccessAnalysis.cpp - Loop Access Analysis Implementation --------==//
//
//                     The LLVM Compiler Infrastructure
//
// This file is distributed under the University of Illinois Open Source
// License. See LICENSE.TXT for details.
//
//===----------------------------------------------------------------------===//
//
// The implementation for the loop memory dependence that was originally
// developed for the loop vectorizer.
//
//===----------------------------------------------------------------------===//

#include "llvm/Analysis/LoopAccessAnalysis.h"
#include "llvm/Analysis/LoopInfo.h"
#include "llvm/Analysis/ScalarEvolutionExpander.h"
#include "llvm/Analysis/TargetLibraryInfo.h"
#include "llvm/Analysis/ValueTracking.h"
#include "llvm/IR/DiagnosticInfo.h"
#include "llvm/IR/Dominators.h"
#include "llvm/IR/IRBuilder.h"
#include "llvm/Support/Debug.h"
#include "llvm/Support/raw_ostream.h"
#include "llvm/Analysis/VectorUtils.h"
using namespace llvm;

#define DEBUG_TYPE "loop-accesses"

static cl::opt<unsigned, true>
VectorizationFactor("force-vector-width", cl::Hidden,
                    cl::desc("Sets the SIMD width. Zero is autoselect."),
                    cl::location(VectorizerParams::VectorizationFactor));
unsigned VectorizerParams::VectorizationFactor;

static cl::opt<unsigned, true>
VectorizationInterleave("force-vector-interleave", cl::Hidden,
                        cl::desc("Sets the vectorization interleave count. "
                                 "Zero is autoselect."),
                        cl::location(
                            VectorizerParams::VectorizationInterleave));
unsigned VectorizerParams::VectorizationInterleave;

static cl::opt<unsigned, true> RuntimeMemoryCheckThreshold(
    "runtime-memory-check-threshold", cl::Hidden,
    cl::desc("When performing memory disambiguation checks at runtime do not "
             "generate more than this number of comparisons (default = 8)."),
    cl::location(VectorizerParams::RuntimeMemoryCheckThreshold), cl::init(8));
unsigned VectorizerParams::RuntimeMemoryCheckThreshold;

/// \brief The maximum iterations used to merge memory checks
static cl::opt<unsigned> MemoryCheckMergeThreshold(
    "memory-check-merge-threshold", cl::Hidden,
    cl::desc("Maximum number of comparisons done when trying to merge "
             "runtime memory checks. (default = 100)"),
    cl::init(100));

/// Maximum SIMD width.
const unsigned VectorizerParams::MaxVectorWidth = 64;

/// \brief We collect interesting dependences up to this threshold.
static cl::opt<unsigned> MaxInterestingDependence(
    "max-interesting-dependences", cl::Hidden,
    cl::desc("Maximum number of interesting dependences collected by "
             "loop-access analysis (default = 100)"),
    cl::init(100));

bool VectorizerParams::isInterleaveForced() {
  return ::VectorizationInterleave.getNumOccurrences() > 0;
}

void LoopAccessReport::emitAnalysis(const LoopAccessReport &Message,
                                    const Function *TheFunction,
                                    const Loop *TheLoop,
                                    const char *PassName) {
  DebugLoc DL = TheLoop->getStartLoc();
  if (const Instruction *I = Message.getInstr())
    DL = I->getDebugLoc();
  emitOptimizationRemarkAnalysis(TheFunction->getContext(), PassName,
                                 *TheFunction, DL, Message.str());
}

Value *llvm::stripIntegerCast(Value *V) {
  if (CastInst *CI = dyn_cast<CastInst>(V))
    if (CI->getOperand(0)->getType()->isIntegerTy())
      return CI->getOperand(0);
  return V;
}

const SCEV *llvm::replaceSymbolicStrideSCEV(ScalarEvolution *SE,
                                            const ValueToValueMap &PtrToStride,
                                            Value *Ptr, Value *OrigPtr) {

  const SCEV *OrigSCEV = SE->getSCEV(Ptr);

  // If there is an entry in the map return the SCEV of the pointer with the
  // symbolic stride replaced by one.
  ValueToValueMap::const_iterator SI =
      PtrToStride.find(OrigPtr ? OrigPtr : Ptr);
  if (SI != PtrToStride.end()) {
    Value *StrideVal = SI->second;

    // Strip casts.
    StrideVal = stripIntegerCast(StrideVal);

    // Replace symbolic stride by one.
    Value *One = ConstantInt::get(StrideVal->getType(), 1);
    ValueToValueMap RewriteMap;
    RewriteMap[StrideVal] = One;

    const SCEV *ByOne =
        SCEVParameterRewriter::rewrite(OrigSCEV, *SE, RewriteMap, true);
    DEBUG(dbgs() << "LAA: Replacing SCEV: " << *OrigSCEV << " by: " << *ByOne
                 << "\n");
    return ByOne;
  }

  // Otherwise, just return the SCEV of the original pointer.
  return SE->getSCEV(Ptr);
}

void RuntimePointerChecking::insert(Loop *Lp, Value *Ptr, bool WritePtr,
                                    unsigned DepSetId, unsigned ASId,
                                    const ValueToValueMap &Strides) {
  // Get the stride replaced scev.
  const SCEV *Sc = replaceSymbolicStrideSCEV(SE, Strides, Ptr);
  const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(Sc);
  assert(AR && "Invalid addrec expression");
  const SCEV *Ex = SE->getBackedgeTakenCount(Lp);

  const SCEV *ScStart = AR->getStart();
  const SCEV *ScEnd = AR->evaluateAtIteration(Ex, *SE);
  const SCEV *Step = AR->getStepRecurrence(*SE);

  // For expressions with negative step, the upper bound is ScStart and the
  // lower bound is ScEnd.
  if (const SCEVConstant *CStep = dyn_cast<const SCEVConstant>(Step)) {
    if (CStep->getValue()->isNegative())
      std::swap(ScStart, ScEnd);
  } else {
    // Fallback case: the step is not constant, but the we can still
    // get the upper and lower bounds of the interval by using min/max
    // expressions.
    ScStart = SE->getUMinExpr(ScStart, ScEnd);
    ScEnd = SE->getUMaxExpr(AR->getStart(), ScEnd);
  }

  Pointers.emplace_back(Ptr, ScStart, ScEnd, WritePtr, DepSetId, ASId, Sc);
}

SmallVector<RuntimePointerChecking::PointerCheck, 4>
RuntimePointerChecking::generateChecks(
    const SmallVectorImpl<int> *PtrPartition) const {
  SmallVector<PointerCheck, 4> Checks;

  for (unsigned I = 0; I < CheckingGroups.size(); ++I) {
    for (unsigned J = I + 1; J < CheckingGroups.size(); ++J) {
      const RuntimePointerChecking::CheckingPtrGroup &CGI = CheckingGroups[I];
      const RuntimePointerChecking::CheckingPtrGroup &CGJ = CheckingGroups[J];

      if (needsChecking(CGI, CGJ, PtrPartition))
        Checks.push_back(std::make_pair(&CGI, &CGJ));
    }
  }
  return Checks;
}

bool RuntimePointerChecking::needsChecking(
    const CheckingPtrGroup &M, const CheckingPtrGroup &N,
    const SmallVectorImpl<int> *PtrPartition) const {
  for (unsigned I = 0, EI = M.Members.size(); EI != I; ++I)
    for (unsigned J = 0, EJ = N.Members.size(); EJ != J; ++J)
      if (needsChecking(M.Members[I], N.Members[J], PtrPartition))
        return true;
  return false;
}

/// Compare \p I and \p J and return the minimum.
/// Return nullptr in case we couldn't find an answer.
static const SCEV *getMinFromExprs(const SCEV *I, const SCEV *J,
                                   ScalarEvolution *SE) {
  const SCEV *Diff = SE->getMinusSCEV(J, I);
  const SCEVConstant *C = dyn_cast<const SCEVConstant>(Diff);

  if (!C)
    return nullptr;
  if (C->getValue()->isNegative())
    return J;
  return I;
}

bool RuntimePointerChecking::CheckingPtrGroup::addPointer(unsigned Index) {
  const SCEV *Start = RtCheck.Pointers[Index].Start;
  const SCEV *End = RtCheck.Pointers[Index].End;

  // Compare the starts and ends with the known minimum and maximum
  // of this set. We need to know how we compare against the min/max
  // of the set in order to be able to emit memchecks.
  const SCEV *Min0 = getMinFromExprs(Start, Low, RtCheck.SE);
  if (!Min0)
    return false;

  const SCEV *Min1 = getMinFromExprs(End, High, RtCheck.SE);
  if (!Min1)
    return false;

  // Update the low bound  expression if we've found a new min value.
  if (Min0 == Start)
    Low = Start;

  // Update the high bound expression if we've found a new max value.
  if (Min1 != End)
    High = End;

  Members.push_back(Index);
  return true;
}

void RuntimePointerChecking::groupChecks(
    MemoryDepChecker::DepCandidates &DepCands, bool UseDependencies) {
  // We build the groups from dependency candidates equivalence classes
  // because:
  //    - We know that pointers in the same equivalence class share
  //      the same underlying object and therefore there is a chance
  //      that we can compare pointers
  //    - We wouldn't be able to merge two pointers for which we need
  //      to emit a memcheck. The classes in DepCands are already
  //      conveniently built such that no two pointers in the same
  //      class need checking against each other.

  // We use the following (greedy) algorithm to construct the groups
  // For every pointer in the equivalence class:
  //   For each existing group:
  //   - if the difference between this pointer and the min/max bounds
  //     of the group is a constant, then make the pointer part of the
  //     group and update the min/max bounds of that group as required.

  CheckingGroups.clear();

  // If we need to check two pointers to the same underlying object
  // with a non-constant difference, we shouldn't perform any pointer
  // grouping with those pointers. This is because we can easily get
  // into cases where the resulting check would return false, even when
  // the accesses are safe.
  //
  // The following example shows this:
  // for (i = 0; i < 1000; ++i)
  //   a[5000 + i * m] = a[i] + a[i + 9000]
  //
  // Here grouping gives a check of (5000, 5000 + 1000 * m) against
  // (0, 10000) which is always false. However, if m is 1, there is no
  // dependence. Not grouping the checks for a[i] and a[i + 9000] allows
  // us to perform an accurate check in this case.
  //
  // The above case requires that we have an UnknownDependence between
  // accesses to the same underlying object. This cannot happen unless
  // ShouldRetryWithRuntimeCheck is set, and therefore UseDependencies
  // is also false. In this case we will use the fallback path and create
  // separate checking groups for all pointers.
 
  // If we don't have the dependency partitions, construct a new
  // checking pointer group for each pointer. This is also required
  // for correctness, because in this case we can have checking between
  // pointers to the same underlying object.
  if (!UseDependencies) {
    for (unsigned I = 0; I < Pointers.size(); ++I)
      CheckingGroups.push_back(CheckingPtrGroup(I, *this));
    return;
  }

  unsigned TotalComparisons = 0;

  DenseMap<Value *, unsigned> PositionMap;
  for (unsigned Index = 0; Index < Pointers.size(); ++Index)
    PositionMap[Pointers[Index].PointerValue] = Index;

  // We need to keep track of what pointers we've already seen so we
  // don't process them twice.
  SmallSet<unsigned, 2> Seen;

  // Go through all equivalence classes, get the the "pointer check groups"
  // and add them to the overall solution. We use the order in which accesses
  // appear in 'Pointers' to enforce determinism.
  for (unsigned I = 0; I < Pointers.size(); ++I) {
    // We've seen this pointer before, and therefore already processed
    // its equivalence class.
    if (Seen.count(I))
      continue;

    MemoryDepChecker::MemAccessInfo Access(Pointers[I].PointerValue,
                                           Pointers[I].IsWritePtr);

    SmallVector<CheckingPtrGroup, 2> Groups;
    auto LeaderI = DepCands.findValue(DepCands.getLeaderValue(Access));

    // Because DepCands is constructed by visiting accesses in the order in
    // which they appear in alias sets (which is deterministic) and the
    // iteration order within an equivalence class member is only dependent on
    // the order in which unions and insertions are performed on the
    // equivalence class, the iteration order is deterministic.
    for (auto MI = DepCands.member_begin(LeaderI), ME = DepCands.member_end();
         MI != ME; ++MI) {
      unsigned Pointer = PositionMap[MI->getPointer()];
      bool Merged = false;
      // Mark this pointer as seen.
      Seen.insert(Pointer);

      // Go through all the existing sets and see if we can find one
      // which can include this pointer.
      for (CheckingPtrGroup &Group : Groups) {
        // Don't perform more than a certain amount of comparisons.
        // This should limit the cost of grouping the pointers to something
        // reasonable.  If we do end up hitting this threshold, the algorithm
        // will create separate groups for all remaining pointers.
        if (TotalComparisons > MemoryCheckMergeThreshold)
          break;

        TotalComparisons++;

        if (Group.addPointer(Pointer)) {
          Merged = true;
          break;
        }
      }

      if (!Merged)
        // We couldn't add this pointer to any existing set or the threshold
        // for the number of comparisons has been reached. Create a new group
        // to hold the current pointer.
        Groups.push_back(CheckingPtrGroup(Pointer, *this));
    }

    // We've computed the grouped checks for this partition.
    // Save the results and continue with the next one.
    std::copy(Groups.begin(), Groups.end(), std::back_inserter(CheckingGroups));
  }
}

bool RuntimePointerChecking::arePointersInSamePartition(
    const SmallVectorImpl<int> &PtrToPartition, unsigned PtrIdx1,
    unsigned PtrIdx2) {
  return (PtrToPartition[PtrIdx1] != -1 &&
          PtrToPartition[PtrIdx1] == PtrToPartition[PtrIdx2]);
}

bool RuntimePointerChecking::needsChecking(
    unsigned I, unsigned J, const SmallVectorImpl<int> *PtrPartition) const {
  const PointerInfo &PointerI = Pointers[I];
  const PointerInfo &PointerJ = Pointers[J];

  // No need to check if two readonly pointers intersect.
  if (!PointerI.IsWritePtr && !PointerJ.IsWritePtr)
    return false;

  // Only need to check pointers between two different dependency sets.
  if (PointerI.DependencySetId == PointerJ.DependencySetId)
    return false;

  // Only need to check pointers in the same alias set.
  if (PointerI.AliasSetId != PointerJ.AliasSetId)
    return false;

  // If PtrPartition is set omit checks between pointers of the same partition.
  if (PtrPartition && arePointersInSamePartition(*PtrPartition, I, J))
    return false;

  return true;
}

void RuntimePointerChecking::printChecks(
    raw_ostream &OS, const SmallVectorImpl<PointerCheck> &Checks,
    unsigned Depth) const {
  unsigned N = 0;
  for (const auto &Check : Checks) {
    const auto &First = Check.first->Members, &Second = Check.second->Members;

    OS.indent(Depth) << "Check " << N++ << ":\n";

    OS.indent(Depth + 2) << "Comparing group (" << Check.first << "):\n";
    for (unsigned K = 0; K < First.size(); ++K)
      OS.indent(Depth + 2) << *Pointers[First[K]].PointerValue << "\n";

    OS.indent(Depth + 2) << "Against group (" << Check.second << "):\n";
    for (unsigned K = 0; K < Second.size(); ++K)
      OS.indent(Depth + 2) << *Pointers[Second[K]].PointerValue << "\n";
  }
}

void RuntimePointerChecking::print(
    raw_ostream &OS, unsigned Depth,
    const SmallVectorImpl<int> *PtrPartition) const {

  OS.indent(Depth) << "Run-time memory checks:\n";
  printChecks(OS, generateChecks(PtrPartition), Depth);

  OS.indent(Depth) << "Grouped accesses:\n";
  for (unsigned I = 0; I < CheckingGroups.size(); ++I) {
    const auto &CG = CheckingGroups[I];

    OS.indent(Depth + 2) << "Group " << &CG << ":\n";
    OS.indent(Depth + 4) << "(Low: " << *CG.Low << " High: " << *CG.High
                         << ")\n";
    for (unsigned J = 0; J < CG.Members.size(); ++J) {
      OS.indent(Depth + 6) << "Member: " << *Pointers[CG.Members[J]].Expr
                           << "\n";
    }
  }
}

unsigned RuntimePointerChecking::getNumberOfChecks(
    const SmallVectorImpl<int> *PtrPartition) const {

  unsigned NumPartitions = CheckingGroups.size();
  unsigned CheckCount = 0;

  for (unsigned I = 0; I < NumPartitions; ++I)
    for (unsigned J = I + 1; J < NumPartitions; ++J)
      if (needsChecking(CheckingGroups[I], CheckingGroups[J], PtrPartition))
        CheckCount++;
  return CheckCount;
}

namespace {
/// \brief Analyses memory accesses in a loop.
///
/// Checks whether run time pointer checks are needed and builds sets for data
/// dependence checking.
class AccessAnalysis {
public:
  /// \brief Read or write access location.
  typedef PointerIntPair<Value *, 1, bool> MemAccessInfo;
  typedef SmallPtrSet<MemAccessInfo, 8> MemAccessInfoSet;

  AccessAnalysis(const DataLayout &Dl, AliasAnalysis *AA, LoopInfo *LI,
                 MemoryDepChecker::DepCandidates &DA)
      : DL(Dl), AST(*AA), LI(LI), DepCands(DA),
        IsRTCheckAnalysisNeeded(false) {}

  /// \brief Register a load  and whether it is only read from.
  void addLoad(MemoryLocation &Loc, bool IsReadOnly) {
    Value *Ptr = const_cast<Value*>(Loc.Ptr);
    AST.add(Ptr, MemoryLocation::UnknownSize, Loc.AATags);
    Accesses.insert(MemAccessInfo(Ptr, false));
    if (IsReadOnly)
      ReadOnlyPtr.insert(Ptr);
  }

  /// \brief Register a store.
  void addStore(MemoryLocation &Loc) {
    Value *Ptr = const_cast<Value*>(Loc.Ptr);
    AST.add(Ptr, MemoryLocation::UnknownSize, Loc.AATags);
    Accesses.insert(MemAccessInfo(Ptr, true));
  }

  /// \brief Check whether we can check the pointers at runtime for
  /// non-intersection.
  ///
  /// Returns true if we need no check or if we do and we can generate them
  /// (i.e. the pointers have computable bounds).
  bool canCheckPtrAtRT(RuntimePointerChecking &RtCheck, ScalarEvolution *SE,
                       Loop *TheLoop, const ValueToValueMap &Strides,
                       bool ShouldCheckStride = false);

  /// \brief Goes over all memory accesses, checks whether a RT check is needed
  /// and builds sets of dependent accesses.
  void buildDependenceSets() {
    processMemAccesses();
  }

  /// \brief Initial processing of memory accesses determined that we need to
  /// perform dependency checking.
  ///
  /// Note that this can later be cleared if we retry memcheck analysis without
  /// dependency checking (i.e. ShouldRetryWithRuntimeCheck).
  bool isDependencyCheckNeeded() { return !CheckDeps.empty(); }

  /// We decided that no dependence analysis would be used.  Reset the state.
  void resetDepChecks(MemoryDepChecker &DepChecker) {
    CheckDeps.clear();
    DepChecker.clearInterestingDependences();
  }

  MemAccessInfoSet &getDependenciesToCheck() { return CheckDeps; }

private:
  typedef SetVector<MemAccessInfo> PtrAccessSet;

  /// \brief Go over all memory access and check whether runtime pointer checks
  /// are needed and build sets of dependency check candidates.
  void processMemAccesses();

  /// Set of all accesses.
  PtrAccessSet Accesses;

  const DataLayout &DL;

  /// Set of accesses that need a further dependence check.
  MemAccessInfoSet CheckDeps;

  /// Set of pointers that are read only.
  SmallPtrSet<Value*, 16> ReadOnlyPtr;

  /// An alias set tracker to partition the access set by underlying object and
  //intrinsic property (such as TBAA metadata).
  AliasSetTracker AST;

  LoopInfo *LI;

  /// Sets of potentially dependent accesses - members of one set share an
  /// underlying pointer. The set "CheckDeps" identfies which sets really need a
  /// dependence check.
  MemoryDepChecker::DepCandidates &DepCands;

  /// \brief Initial processing of memory accesses determined that we may need
  /// to add memchecks.  Perform the analysis to determine the necessary checks.
  ///
  /// Note that, this is different from isDependencyCheckNeeded.  When we retry
  /// memcheck analysis without dependency checking
  /// (i.e. ShouldRetryWithRuntimeCheck), isDependencyCheckNeeded is cleared
  /// while this remains set if we have potentially dependent accesses.
  bool IsRTCheckAnalysisNeeded;
};

} // end anonymous namespace

/// \brief Check whether a pointer can participate in a runtime bounds check.
static bool hasComputableBounds(ScalarEvolution *SE,
                                const ValueToValueMap &Strides, Value *Ptr) {
  const SCEV *PtrScev = replaceSymbolicStrideSCEV(SE, Strides, Ptr);
  const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(PtrScev);
  if (!AR)
    return false;

  return AR->isAffine();
}

bool AccessAnalysis::canCheckPtrAtRT(RuntimePointerChecking &RtCheck,
                                     ScalarEvolution *SE, Loop *TheLoop,
                                     const ValueToValueMap &StridesMap,
                                     bool ShouldCheckStride) {
  // Find pointers with computable bounds. We are going to use this information
  // to place a runtime bound check.
  bool CanDoRT = true;

  bool NeedRTCheck = false;
  if (!IsRTCheckAnalysisNeeded) return true;

  bool IsDepCheckNeeded = isDependencyCheckNeeded();

  // We assign a consecutive id to access from different alias sets.
  // Accesses between different groups doesn't need to be checked.
  unsigned ASId = 1;
  for (auto &AS : AST) {
    int NumReadPtrChecks = 0;
    int NumWritePtrChecks = 0;

    // We assign consecutive id to access from different dependence sets.
    // Accesses within the same set don't need a runtime check.
    unsigned RunningDepId = 1;
    DenseMap<Value *, unsigned> DepSetId;

    for (auto A : AS) {
      Value *Ptr = A.getValue();
      bool IsWrite = Accesses.count(MemAccessInfo(Ptr, true));
      MemAccessInfo Access(Ptr, IsWrite);

      if (IsWrite)
        ++NumWritePtrChecks;
      else
        ++NumReadPtrChecks;

      if (hasComputableBounds(SE, StridesMap, Ptr) &&
          // When we run after a failing dependency check we have to make sure
          // we don't have wrapping pointers.
          (!ShouldCheckStride ||
           isStridedPtr(SE, Ptr, TheLoop, StridesMap) == 1)) {
        // The id of the dependence set.
        unsigned DepId;

        if (IsDepCheckNeeded) {
          Value *Leader = DepCands.getLeaderValue(Access).getPointer();
          unsigned &LeaderId = DepSetId[Leader];
          if (!LeaderId)
            LeaderId = RunningDepId++;
          DepId = LeaderId;
        } else
          // Each access has its own dependence set.
          DepId = RunningDepId++;

        RtCheck.insert(TheLoop, Ptr, IsWrite, DepId, ASId, StridesMap);

        DEBUG(dbgs() << "LAA: Found a runtime check ptr:" << *Ptr << '\n');
      } else {
        DEBUG(dbgs() << "LAA: Can't find bounds for ptr:" << *Ptr << '\n');
        CanDoRT = false;
      }
    }

    // If we have at least two writes or one write and a read then we need to
    // check them.  But there is no need to checks if there is only one
    // dependence set for this alias set.
    //
    // Note that this function computes CanDoRT and NeedRTCheck independently.
    // For example CanDoRT=false, NeedRTCheck=false means that we have a pointer
    // for which we couldn't find the bounds but we don't actually need to emit
    // any checks so it does not matter.
    if (!(IsDepCheckNeeded && CanDoRT && RunningDepId == 2))
      NeedRTCheck |= (NumWritePtrChecks >= 2 || (NumReadPtrChecks >= 1 &&
                                                 NumWritePtrChecks >= 1));

    ++ASId;
  }

  // If the pointers that we would use for the bounds comparison have different
  // address spaces, assume the values aren't directly comparable, so we can't
  // use them for the runtime check. We also have to assume they could
  // overlap. In the future there should be metadata for whether address spaces
  // are disjoint.
  unsigned NumPointers = RtCheck.Pointers.size();
  for (unsigned i = 0; i < NumPointers; ++i) {
    for (unsigned j = i + 1; j < NumPointers; ++j) {
      // Only need to check pointers between two different dependency sets.
      if (RtCheck.Pointers[i].DependencySetId ==
          RtCheck.Pointers[j].DependencySetId)
       continue;
      // Only need to check pointers in the same alias set.
      if (RtCheck.Pointers[i].AliasSetId != RtCheck.Pointers[j].AliasSetId)
        continue;

      Value *PtrI = RtCheck.Pointers[i].PointerValue;
      Value *PtrJ = RtCheck.Pointers[j].PointerValue;

      unsigned ASi = PtrI->getType()->getPointerAddressSpace();
      unsigned ASj = PtrJ->getType()->getPointerAddressSpace();
      if (ASi != ASj) {
        DEBUG(dbgs() << "LAA: Runtime check would require comparison between"
                       " different address spaces\n");
        return false;
      }
    }
  }

  if (NeedRTCheck && CanDoRT)
    RtCheck.groupChecks(DepCands, IsDepCheckNeeded);

  DEBUG(dbgs() << "LAA: We need to do " << RtCheck.getNumberOfChecks(nullptr)
               << " pointer comparisons.\n");

  RtCheck.Need = NeedRTCheck;

  bool CanDoRTIfNeeded = !NeedRTCheck || CanDoRT;
  if (!CanDoRTIfNeeded)
    RtCheck.reset();
  return CanDoRTIfNeeded;
}

void AccessAnalysis::processMemAccesses() {
  // We process the set twice: first we process read-write pointers, last we
  // process read-only pointers. This allows us to skip dependence tests for
  // read-only pointers.

  DEBUG(dbgs() << "LAA: Processing memory accesses...\n");
  DEBUG(dbgs() << "  AST: "; AST.dump());
  DEBUG(dbgs() << "LAA:   Accesses(" << Accesses.size() << "):\n");
  DEBUG({
    for (auto A : Accesses)
      dbgs() << "\t" << *A.getPointer() << " (" <<
                (A.getInt() ? "write" : (ReadOnlyPtr.count(A.getPointer()) ?
                                         "read-only" : "read")) << ")\n";
  });

  // The AliasSetTracker has nicely partitioned our pointers by metadata
  // compatibility and potential for underlying-object overlap. As a result, we
  // only need to check for potential pointer dependencies within each alias
  // set.
  for (auto &AS : AST) {
    // Note that both the alias-set tracker and the alias sets themselves used
    // linked lists internally and so the iteration order here is deterministic
    // (matching the original instruction order within each set).

    bool SetHasWrite = false;

    // Map of pointers to last access encountered.
    typedef DenseMap<Value*, MemAccessInfo> UnderlyingObjToAccessMap;
    UnderlyingObjToAccessMap ObjToLastAccess;

    // Set of access to check after all writes have been processed.
    PtrAccessSet DeferredAccesses;

    // Iterate over each alias set twice, once to process read/write pointers,
    // and then to process read-only pointers.
    for (int SetIteration = 0; SetIteration < 2; ++SetIteration) {
      bool UseDeferred = SetIteration > 0;
      PtrAccessSet &S = UseDeferred ? DeferredAccesses : Accesses;

      for (auto AV : AS) {
        Value *Ptr = AV.getValue();

        // For a single memory access in AliasSetTracker, Accesses may contain
        // both read and write, and they both need to be handled for CheckDeps.
        for (auto AC : S) {
          if (AC.getPointer() != Ptr)
            continue;

          bool IsWrite = AC.getInt();

          // If we're using the deferred access set, then it contains only
          // reads.
          bool IsReadOnlyPtr = ReadOnlyPtr.count(Ptr) && !IsWrite;
          if (UseDeferred && !IsReadOnlyPtr)
            continue;
          // Otherwise, the pointer must be in the PtrAccessSet, either as a
          // read or a write.
          assert(((IsReadOnlyPtr && UseDeferred) || IsWrite ||
                  S.count(MemAccessInfo(Ptr, false))) &&
                 "Alias-set pointer not in the access set?");

          MemAccessInfo Access(Ptr, IsWrite);
          DepCands.insert(Access);

          // Memorize read-only pointers for later processing and skip them in
          // the first round (they need to be checked after we have seen all
          // write pointers). Note: we also mark pointer that are not
          // consecutive as "read-only" pointers (so that we check
          // "a[b[i]] +="). Hence, we need the second check for "!IsWrite".
          if (!UseDeferred && IsReadOnlyPtr) {
            DeferredAccesses.insert(Access);
            continue;
          }

          // If this is a write - check other reads and writes for conflicts. If
          // this is a read only check other writes for conflicts (but only if
          // there is no other write to the ptr - this is an optimization to
          // catch "a[i] = a[i] + " without having to do a dependence check).
          if ((IsWrite || IsReadOnlyPtr) && SetHasWrite) {
            CheckDeps.insert(Access);
            IsRTCheckAnalysisNeeded = true;
          }

          if (IsWrite)
            SetHasWrite = true;

          // Create sets of pointers connected by a shared alias set and
          // underlying object.
          typedef SmallVector<Value *, 16> ValueVector;
          ValueVector TempObjects;

          GetUnderlyingObjects(Ptr, TempObjects, DL, LI);
          DEBUG(dbgs() << "Underlying objects for pointer " << *Ptr << "\n");
          for (Value *UnderlyingObj : TempObjects) {
            UnderlyingObjToAccessMap::iterator Prev =
                ObjToLastAccess.find(UnderlyingObj);
            if (Prev != ObjToLastAccess.end())
              DepCands.unionSets(Access, Prev->second);

            ObjToLastAccess[UnderlyingObj] = Access;
            DEBUG(dbgs() << "  " << *UnderlyingObj << "\n");
          }
        }
      }
    }
  }
}

static bool isInBoundsGep(Value *Ptr) {
  if (GetElementPtrInst *GEP = dyn_cast<GetElementPtrInst>(Ptr))
    return GEP->isInBounds();
  return false;
}

/// \brief Return true if an AddRec pointer \p Ptr is unsigned non-wrapping,
/// i.e. monotonically increasing/decreasing.
static bool isNoWrapAddRec(Value *Ptr, const SCEVAddRecExpr *AR,
                           ScalarEvolution *SE, const Loop *L) {
  // FIXME: This should probably only return true for NUW.
  if (AR->getNoWrapFlags(SCEV::NoWrapMask))
    return true;

  // Scalar evolution does not propagate the non-wrapping flags to values that
  // are derived from a non-wrapping induction variable because non-wrapping
  // could be flow-sensitive.
  //
  // Look through the potentially overflowing instruction to try to prove
  // non-wrapping for the *specific* value of Ptr.

  // The arithmetic implied by an inbounds GEP can't overflow.
  auto *GEP = dyn_cast<GetElementPtrInst>(Ptr);
  if (!GEP || !GEP->isInBounds())
    return false;

  // Make sure there is only one non-const index and analyze that.
  Value *NonConstIndex = nullptr;
  for (auto Index = GEP->idx_begin(); Index != GEP->idx_end(); ++Index)
    if (!isa<ConstantInt>(*Index)) {
      if (NonConstIndex)
        return false;
      NonConstIndex = *Index;
    }
  if (!NonConstIndex)
    // The recurrence is on the pointer, ignore for now.
    return false;

  // The index in GEP is signed.  It is non-wrapping if it's derived from a NSW
  // AddRec using a NSW operation.
  if (auto *OBO = dyn_cast<OverflowingBinaryOperator>(NonConstIndex))
    if (OBO->hasNoSignedWrap() &&
        // Assume constant for other the operand so that the AddRec can be
        // easily found.
        isa<ConstantInt>(OBO->getOperand(1))) {
      auto *OpScev = SE->getSCEV(OBO->getOperand(0));

      if (auto *OpAR = dyn_cast<SCEVAddRecExpr>(OpScev))
        return OpAR->getLoop() == L && OpAR->getNoWrapFlags(SCEV::FlagNSW);
    }

  return false;
}

/// \brief Check whether the access through \p Ptr has a constant stride.
int llvm::isStridedPtr(ScalarEvolution *SE, Value *Ptr, const Loop *Lp,
                       const ValueToValueMap &StridesMap) {
  Type *Ty = Ptr->getType();
  assert(Ty->isPointerTy() && "Unexpected non-ptr");

  // Make sure that the pointer does not point to aggregate types.
  auto *PtrTy = cast<PointerType>(Ty);
  if (PtrTy->getElementType()->isAggregateType()) {
    DEBUG(dbgs() << "LAA: Bad stride - Not a pointer to a scalar type"
          << *Ptr << "\n");
    return 0;
  }

  const SCEV *PtrScev = replaceSymbolicStrideSCEV(SE, StridesMap, Ptr);

  const SCEVAddRecExpr *AR = dyn_cast<SCEVAddRecExpr>(PtrScev);
  if (!AR) {
    DEBUG(dbgs() << "LAA: Bad stride - Not an AddRecExpr pointer "
          << *Ptr << " SCEV: " << *PtrScev << "\n");
    return 0;
  }

  // The accesss function must stride over the innermost loop.
  if (Lp != AR->getLoop()) {
    DEBUG(dbgs() << "LAA: Bad stride - Not striding over innermost loop " <<
          *Ptr << " SCEV: " << *PtrScev << "\n");
  }

  // The address calculation must not wrap. Otherwise, a dependence could be
  // inverted.
  // An inbounds getelementptr that is a AddRec with a unit stride
  // cannot wrap per definition. The unit stride requirement is checked later.
  // An getelementptr without an inbounds attribute and unit stride would have
  // to access the pointer value "0" which is undefined behavior in address
  // space 0, therefore we can also vectorize this case.
  bool IsInBoundsGEP = isInBoundsGep(Ptr);
  bool IsNoWrapAddRec = isNoWrapAddRec(Ptr, AR, SE, Lp);
  bool IsInAddressSpaceZero = PtrTy->getAddressSpace() == 0;
  if (!IsNoWrapAddRec && !IsInBoundsGEP && !IsInAddressSpaceZero) {
    DEBUG(dbgs() << "LAA: Bad stride - Pointer may wrap in the address space "
          << *Ptr << " SCEV: " << *PtrScev << "\n");
    return 0;
  }

  // Check the step is constant.
  const SCEV *Step = AR->getStepRecurrence(*SE);

  // Calculate the pointer stride and check if it is constant.
  const SCEVConstant *C = dyn_cast<SCEVConstant>(Step);
  if (!C) {
    DEBUG(dbgs() << "LAA: Bad stride - Not a constant strided " << *Ptr <<
          " SCEV: " << *PtrScev << "\n");
    return 0;
  }

  auto &DL = Lp->getHeader()->getModule()->getDataLayout();
  int64_t Size = DL.getTypeAllocSize(PtrTy->getElementType());
  const APInt &APStepVal = C->getValue()->getValue();

  // Huge step value - give up.
  if (APStepVal.getBitWidth() > 64)
    return 0;

  int64_t StepVal = APStepVal.getSExtValue();

  // Strided access.
  int64_t Stride = StepVal / Size;
  int64_t Rem = StepVal % Size;
  if (Rem)
    return 0;

  // If the SCEV could wrap but we have an inbounds gep with a unit stride we
  // know we can't "wrap around the address space". In case of address space
  // zero we know that this won't happen without triggering undefined behavior.
  if (!IsNoWrapAddRec && (IsInBoundsGEP || IsInAddressSpaceZero) &&
      Stride != 1 && Stride != -1)
    return 0;

  return Stride;
}

bool MemoryDepChecker::Dependence::isSafeForVectorization(DepType Type) {
  switch (Type) {
  case NoDep:
  case Forward:
  case BackwardVectorizable:
    return true;

  case Unknown:
  case ForwardButPreventsForwarding:
  case Backward:
  case BackwardVectorizableButPreventsForwarding:
    return false;
  }
  llvm_unreachable("unexpected DepType!");
}

bool MemoryDepChecker::Dependence::isInterestingDependence(DepType Type) {
  switch (Type) {
  case NoDep:
  case Forward:
    return false;

  case BackwardVectorizable:
  case Unknown:
  case ForwardButPreventsForwarding:
  case Backward:
  case BackwardVectorizableButPreventsForwarding:
    return true;
  }
  llvm_unreachable("unexpected DepType!");
}

bool MemoryDepChecker::Dependence::isPossiblyBackward() const {
  switch (Type) {
  case NoDep:
  case Forward:
  case ForwardButPreventsForwarding:
    return false;

  case Unknown:
  case BackwardVectorizable:
  case Backward:
  case BackwardVectorizableButPreventsForwarding:
    return true;
  }
  llvm_unreachable("unexpected DepType!");
}

bool MemoryDepChecker::couldPreventStoreLoadForward(unsigned Distance,
                                                    unsigned TypeByteSize) {
  // If loads occur at a distance that is not a multiple of a feasible vector
  // factor store-load forwarding does not take place.
  // Positive dependences might cause troubles because vectorizing them might
  // prevent store-load forwarding making vectorized code run a lot slower.
  //   a[i] = a[i-3] ^ a[i-8];
  //   The stores to a[i:i+1] don't align with the stores to a[i-3:i-2] and
  //   hence on your typical architecture store-load forwarding does not take
  //   place. Vectorizing in such cases does not make sense.
  // Store-load forwarding distance.
  const unsigned NumCyclesForStoreLoadThroughMemory = 8*TypeByteSize;
  // Maximum vector factor.
  unsigned MaxVFWithoutSLForwardIssues =
    VectorizerParams::MaxVectorWidth * TypeByteSize;
  if(MaxSafeDepDistBytes < MaxVFWithoutSLForwardIssues)
    MaxVFWithoutSLForwardIssues = MaxSafeDepDistBytes;

  for (unsigned vf = 2*TypeByteSize; vf <= MaxVFWithoutSLForwardIssues;
       vf *= 2) {
    if (Distance % vf && Distance / vf < NumCyclesForStoreLoadThroughMemory) {
      MaxVFWithoutSLForwardIssues = (vf >>=1);
      break;
    }
  }

  if (MaxVFWithoutSLForwardIssues< 2*TypeByteSize) {
    DEBUG(dbgs() << "LAA: Distance " << Distance <<
          " that could cause a store-load forwarding conflict\n");
    return true;
  }

  if (MaxVFWithoutSLForwardIssues < MaxSafeDepDistBytes &&
      MaxVFWithoutSLForwardIssues !=
      VectorizerParams::MaxVectorWidth * TypeByteSize)
    MaxSafeDepDistBytes = MaxVFWithoutSLForwardIssues;
  return false;
}

/// \brief Check the dependence for two accesses with the same stride \p Stride.
/// \p Distance is the positive distance and \p TypeByteSize is type size in
/// bytes.
///
/// \returns true if they are independent.
static bool areStridedAccessesIndependent(unsigned Distance, unsigned Stride,
                                          unsigned TypeByteSize) {
  assert(Stride > 1 && "The stride must be greater than 1");
  assert(TypeByteSize > 0 && "The type size in byte must be non-zero");
  assert(Distance > 0 && "The distance must be non-zero");

  // Skip if the distance is not multiple of type byte size.
  if (Distance % TypeByteSize)
    return false;

  unsigned ScaledDist = Distance / TypeByteSize;

  // No dependence if the scaled distance is not multiple of the stride.
  // E.g.
  //      for (i = 0; i < 1024 ; i += 4)
  //        A[i+2] = A[i] + 1;
  //
  // Two accesses in memory (scaled distance is 2, stride is 4):
  //     | A[0] |      |      |      | A[4] |      |      |      |
  //     |      |      | A[2] |      |      |      | A[6] |      |
  //
  // E.g.
  //      for (i = 0; i < 1024 ; i += 3)
  //        A[i+4] = A[i] + 1;
  //
  // Two accesses in memory (scaled distance is 4, stride is 3):
  //     | A[0] |      |      | A[3] |      |      | A[6] |      |      |
  //     |      |      |      |      | A[4] |      |      | A[7] |      |
  return ScaledDist % Stride;
}

MemoryDepChecker::Dependence::DepType
MemoryDepChecker::isDependent(const MemAccessInfo &A, unsigned AIdx,
                              const MemAccessInfo &B, unsigned BIdx,
                              const ValueToValueMap &Strides) {
  assert (AIdx < BIdx && "Must pass arguments in program order");

  Value *APtr = A.getPointer();
  Value *BPtr = B.getPointer();
  bool AIsWrite = A.getInt();
  bool BIsWrite = B.getInt();

  // Two reads are independent.
  if (!AIsWrite && !BIsWrite)
    return Dependence::NoDep;

  // We cannot check pointers in different address spaces.
  if (APtr->getType()->getPointerAddressSpace() !=
      BPtr->getType()->getPointerAddressSpace())
    return Dependence::Unknown;

  const SCEV *AScev = replaceSymbolicStrideSCEV(SE, Strides, APtr);
  const SCEV *BScev = replaceSymbolicStrideSCEV(SE, Strides, BPtr);

  int StrideAPtr = isStridedPtr(SE, APtr, InnermostLoop, Strides);
  int StrideBPtr = isStridedPtr(SE, BPtr, InnermostLoop, Strides);

  const SCEV *Src = AScev;
  const SCEV *Sink = BScev;

  // If the induction step is negative we have to invert source and sink of the
  // dependence.
  if (StrideAPtr < 0) {
    //Src = BScev;
    //Sink = AScev;
    std::swap(APtr, BPtr);
    std::swap(Src, Sink);
    std::swap(AIsWrite, BIsWrite);
    std::swap(AIdx, BIdx);
    std::swap(StrideAPtr, StrideBPtr);
  }

  const SCEV *Dist = SE->getMinusSCEV(Sink, Src);

  DEBUG(dbgs() << "LAA: Src Scev: " << *Src << "Sink Scev: " << *Sink
        << "(Induction step: " << StrideAPtr <<  ")\n");
  DEBUG(dbgs() << "LAA: Distance for " << *InstMap[AIdx] << " to "
        << *InstMap[BIdx] << ": " << *Dist << "\n");

  // Need accesses with constant stride. We don't want to vectorize
  // "A[B[i]] += ..." and similar code or pointer arithmetic that could wrap in
  // the address space.
  if (!StrideAPtr || !StrideBPtr || StrideAPtr != StrideBPtr){
    DEBUG(dbgs() << "Pointer access with non-constant stride\n");
    return Dependence::Unknown;
  }

  const SCEVConstant *C = dyn_cast<SCEVConstant>(Dist);
  if (!C) {
    DEBUG(dbgs() << "LAA: Dependence because of non-constant distance\n");
    ShouldRetryWithRuntimeCheck = true;
    return Dependence::Unknown;
  }

  Type *ATy = APtr->getType()->getPointerElementType();
  Type *BTy = BPtr->getType()->getPointerElementType();
  auto &DL = InnermostLoop->getHeader()->getModule()->getDataLayout();
  unsigned TypeByteSize = DL.getTypeAllocSize(ATy);

  // Negative distances are not plausible dependencies.
  const APInt &Val = C->getValue()->getValue();
  if (Val.isNegative()) {
    bool IsTrueDataDependence = (AIsWrite && !BIsWrite);
    if (IsTrueDataDependence &&
        (couldPreventStoreLoadForward(Val.abs().getZExtValue(), TypeByteSize) ||
         ATy != BTy))
      return Dependence::ForwardButPreventsForwarding;

    DEBUG(dbgs() << "LAA: Dependence is negative: NoDep\n");
    return Dependence::Forward;
  }

  // Write to the same location with the same size.
  // Could be improved to assert type sizes are the same (i32 == float, etc).
  if (Val == 0) {
    if (ATy == BTy)
      return Dependence::NoDep;
    DEBUG(dbgs() << "LAA: Zero dependence difference but different types\n");
    return Dependence::Unknown;
  }

  assert(Val.isStrictlyPositive() && "Expect a positive value");

  if (ATy != BTy) {
    DEBUG(dbgs() <<
          "LAA: ReadWrite-Write positive dependency with different types\n");
    return Dependence::Unknown;
  }

  unsigned Distance = (unsigned) Val.getZExtValue();

  unsigned Stride = std::abs(StrideAPtr);
  if (Stride > 1 &&
      areStridedAccessesIndependent(Distance, Stride, TypeByteSize)) {
    DEBUG(dbgs() << "LAA: Strided accesses are independent\n");
    return Dependence::NoDep;
  }

  // Bail out early if passed-in parameters make vectorization not feasible.
  unsigned ForcedFactor = (VectorizerParams::VectorizationFactor ?
                           VectorizerParams::VectorizationFactor : 1);
  unsigned ForcedUnroll = (VectorizerParams::VectorizationInterleave ?
                           VectorizerParams::VectorizationInterleave : 1);
  // The minimum number of iterations for a vectorized/unrolled version.
  unsigned MinNumIter = std::max(ForcedFactor * ForcedUnroll, 2U);

  // It's not vectorizable if the distance is smaller than the minimum distance
  // needed for a vectroized/unrolled version. Vectorizing one iteration in
  // front needs TypeByteSize * Stride. Vectorizing the last iteration needs
  // TypeByteSize (No need to plus the last gap distance).
  //
  // E.g. Assume one char is 1 byte in memory and one int is 4 bytes.
  //      foo(int *A) {
  //        int *B = (int *)((char *)A + 14);
  //        for (i = 0 ; i < 1024 ; i += 2)
  //          B[i] = A[i] + 1;
  //      }
  //
  // Two accesses in memory (stride is 2):
  //     | A[0] |      | A[2] |      | A[4] |      | A[6] |      |
  //                              | B[0] |      | B[2] |      | B[4] |
  //
  // Distance needs for vectorizing iterations except the last iteration:
  // 4 * 2 * (MinNumIter - 1). Distance needs for the last iteration: 4.
  // So the minimum distance needed is: 4 * 2 * (MinNumIter - 1) + 4.
  //
  // If MinNumIter is 2, it is vectorizable as the minimum distance needed is
  // 12, which is less than distance.
  //
  // If MinNumIter is 4 (Say if a user forces the vectorization factor to be 4),
  // the minimum distance needed is 28, which is greater than distance. It is
  // not safe to do vectorization.
  unsigned MinDistanceNeeded =
      TypeByteSize * Stride * (MinNumIter - 1) + TypeByteSize;
  if (MinDistanceNeeded > Distance) {
    DEBUG(dbgs() << "LAA: Failure because of positive distance " << Distance
                 << '\n');
    return Dependence::Backward;
  }

  // Unsafe if the minimum distance needed is greater than max safe distance.
  if (MinDistanceNeeded > MaxSafeDepDistBytes) {
    DEBUG(dbgs() << "LAA: Failure because it needs at least "
                 << MinDistanceNeeded << " size in bytes");
    return Dependence::Backward;
  }

  // Positive distance bigger than max vectorization factor.
  // FIXME: Should use max factor instead of max distance in bytes, which could
  // not handle different types.
  // E.g. Assume one char is 1 byte in memory and one int is 4 bytes.
  //      void foo (int *A, char *B) {
  //        for (unsigned i = 0; i < 1024; i++) {
  //          A[i+2] = A[i] + 1;
  //          B[i+2] = B[i] + 1;
  //        }
  //      }
  //
  // This case is currently unsafe according to the max safe distance. If we
  // analyze the two accesses on array B, the max safe dependence distance
  // is 2. Then we analyze the accesses on array A, the minimum distance needed
  // is 8, which is less than 2 and forbidden vectorization, But actually
  // both A and B could be vectorized by 2 iterations.
  MaxSafeDepDistBytes =
      Distance < MaxSafeDepDistBytes ? Distance : MaxSafeDepDistBytes;

  bool IsTrueDataDependence = (!AIsWrite && BIsWrite);
  if (IsTrueDataDependence &&
      couldPreventStoreLoadForward(Distance, TypeByteSize))
    return Dependence::BackwardVectorizableButPreventsForwarding;

  DEBUG(dbgs() << "LAA: Positive distance " << Val.getSExtValue()
               << " with max VF = "
               << MaxSafeDepDistBytes / (TypeByteSize * Stride) << '\n');

  return Dependence::BackwardVectorizable;
}

bool MemoryDepChecker::areDepsSafe(DepCandidates &AccessSets,
                                   MemAccessInfoSet &CheckDeps,
                                   const ValueToValueMap &Strides) {

  MaxSafeDepDistBytes = -1U;
  while (!CheckDeps.empty()) {
    MemAccessInfo CurAccess = *CheckDeps.begin();

    // Get the relevant memory access set.
    EquivalenceClasses<MemAccessInfo>::iterator I =
      AccessSets.findValue(AccessSets.getLeaderValue(CurAccess));

    // Check accesses within this set.
    EquivalenceClasses<MemAccessInfo>::member_iterator AI, AE;
    AI = AccessSets.member_begin(I), AE = AccessSets.member_end();

    // Check every access pair.
    while (AI != AE) {
      CheckDeps.erase(*AI);
      EquivalenceClasses<MemAccessInfo>::member_iterator OI = std::next(AI);
      while (OI != AE) {
        // Check every accessing instruction pair in program order.
        for (std::vector<unsigned>::iterator I1 = Accesses[*AI].begin(),
             I1E = Accesses[*AI].end(); I1 != I1E; ++I1)
          for (std::vector<unsigned>::iterator I2 = Accesses[*OI].begin(),
               I2E = Accesses[*OI].end(); I2 != I2E; ++I2) {
            auto A = std::make_pair(&*AI, *I1);
            auto B = std::make_pair(&*OI, *I2);

            assert(*I1 != *I2);
            if (*I1 > *I2)
              std::swap(A, B);

            Dependence::DepType Type =
                isDependent(*A.first, A.second, *B.first, B.second, Strides);
            SafeForVectorization &= Dependence::isSafeForVectorization(Type);

            // Gather dependences unless we accumulated MaxInterestingDependence
            // dependences.  In that case return as soon as we find the first
            // unsafe dependence.  This puts a limit on this quadratic
            // algorithm.
            if (RecordInterestingDependences) {
              if (Dependence::isInterestingDependence(Type))
                InterestingDependences.push_back(
                    Dependence(A.second, B.second, Type));

              if (InterestingDependences.size() >= MaxInterestingDependence) {
                RecordInterestingDependences = false;
                InterestingDependences.clear();
                DEBUG(dbgs() << "Too many dependences, stopped recording\n");
              }
            }
            if (!RecordInterestingDependences && !SafeForVectorization)
              return false;
          }
        ++OI;
      }
      AI++;
    }
  }

  DEBUG(dbgs() << "Total Interesting Dependences: "
               << InterestingDependences.size() << "\n");
  return SafeForVectorization;
}

SmallVector<Instruction *, 4>
MemoryDepChecker::getInstructionsForAccess(Value *Ptr, bool isWrite) const {
  MemAccessInfo Access(Ptr, isWrite);
  auto &IndexVector = Accesses.find(Access)->second;

  SmallVector<Instruction *, 4> Insts;
  std::transform(IndexVector.begin(), IndexVector.end(),
                 std::back_inserter(Insts),
                 [&](unsigned Idx) { return this->InstMap[Idx]; });
  return Insts;
}

const char *MemoryDepChecker::Dependence::DepName[] = {
    "NoDep", "Unknown", "Forward", "ForwardButPreventsForwarding", "Backward",
    "BackwardVectorizable", "BackwardVectorizableButPreventsForwarding"};

void MemoryDepChecker::Dependence::print(
    raw_ostream &OS, unsigned Depth,
    const SmallVectorImpl<Instruction *> &Instrs) const {
  OS.indent(Depth) << DepName[Type] << ":\n";
  OS.indent(Depth + 2) << *Instrs[Source] << " -> \n";
  OS.indent(Depth + 2) << *Instrs[Destination] << "\n";
}

bool LoopAccessInfo::canAnalyzeLoop() {
  // We need to have a loop header.
  DEBUG(dbgs() << "LAA: Found a loop: " <<
        TheLoop->getHeader()->getName() << '\n');

    // We can only analyze innermost loops.
  if (!TheLoop->empty()) {
    DEBUG(dbgs() << "LAA: loop is not the innermost loop\n");
    emitAnalysis(LoopAccessReport() << "loop is not the innermost loop");
    return false;
  }

  // We must have a single backedge.
  if (TheLoop->getNumBackEdges() != 1) {
    DEBUG(dbgs() << "LAA: loop control flow is not understood by analyzer\n");
    emitAnalysis(
        LoopAccessReport() <<
        "loop control flow is not understood by analyzer");
    return false;
  }

  // We must have a single exiting block.
  if (!TheLoop->getExitingBlock()) {
    DEBUG(dbgs() << "LAA: loop control flow is not understood by analyzer\n");
    emitAnalysis(
        LoopAccessReport() <<
        "loop control flow is not understood by analyzer");
    return false;
  }

  // We only handle bottom-tested loops, i.e. loop in which the condition is
  // checked at the end of each iteration. With that we can assume that all
  // instructions in the loop are executed the same number of times.
  if (TheLoop->getExitingBlock() != TheLoop->getLoopLatch()) {
    DEBUG(dbgs() << "LAA: loop control flow is not understood by analyzer\n");
    emitAnalysis(
        LoopAccessReport() <<
        "loop control flow is not understood by analyzer");
    return false;
  }

  // ScalarEvolution needs to be able to find the exit count.
  const SCEV *ExitCount = SE->getBackedgeTakenCount(TheLoop);
  if (ExitCount == SE->getCouldNotCompute()) {
    emitAnalysis(LoopAccessReport() <<
                 "could not determine number of loop iterations");
    DEBUG(dbgs() << "LAA: SCEV could not compute the loop exit count.\n");
    return false;
  }

  return true;
}

void LoopAccessInfo::analyzeLoop(const ValueToValueMap &Strides) {

  typedef SmallVector<Value*, 16> ValueVector;
  typedef SmallPtrSet<Value*, 16> ValueSet;

  // Holds the Load and Store *instructions*.
  ValueVector Loads;
  ValueVector Stores;

  // Holds all the different accesses in the loop.
  unsigned NumReads = 0;
  unsigned NumReadWrites = 0;

  PtrRtChecking.Pointers.clear();
  PtrRtChecking.Need = false;

  const bool IsAnnotatedParallel = TheLoop->isAnnotatedParallel();

  // For each block.
  for (Loop::block_iterator bb = TheLoop->block_begin(),
       be = TheLoop->block_end(); bb != be; ++bb) {

    // Scan the BB and collect legal loads and stores.
    for (BasicBlock::iterator it = (*bb)->begin(), e = (*bb)->end(); it != e;
         ++it) {

      // If this is a load, save it. If this instruction can read from memory
      // but is not a load, then we quit. Notice that we don't handle function
      // calls that read or write.
      if (it->mayReadFromMemory()) {
        // Many math library functions read the rounding mode. We will only
        // vectorize a loop if it contains known function calls that don't set
        // the flag. Therefore, it is safe to ignore this read from memory.
        CallInst *Call = dyn_cast<CallInst>(it);
        if (Call && getIntrinsicIDForCall(Call, TLI))
          continue;

        // If the function has an explicit vectorized counterpart, we can safely
        // assume that it can be vectorized.
        if (Call && !Call->isNoBuiltin() && Call->getCalledFunction() &&
            TLI->isFunctionVectorizable(Call->getCalledFunction()->getName()))
          continue;

        LoadInst *Ld = dyn_cast<LoadInst>(it);
        if (!Ld || (!Ld->isSimple() && !IsAnnotatedParallel)) {
          emitAnalysis(LoopAccessReport(Ld)
                       << "read with atomic ordering or volatile read");
          DEBUG(dbgs() << "LAA: Found a non-simple load.\n");
          CanVecMem = false;
          return;
        }
        NumLoads++;
        Loads.push_back(Ld);
        DepChecker.addAccess(Ld);
        continue;
      }

      // Save 'store' instructions. Abort if other instructions write to memory.
      if (it->mayWriteToMemory()) {
        StoreInst *St = dyn_cast<StoreInst>(it);
        if (!St) {
          emitAnalysis(LoopAccessReport(it) <<
                       "instruction cannot be vectorized");
          CanVecMem = false;
          return;
        }
        if (!St->isSimple() && !IsAnnotatedParallel) {
          emitAnalysis(LoopAccessReport(St)
                       << "write with atomic ordering or volatile write");
          DEBUG(dbgs() << "LAA: Found a non-simple store.\n");
          CanVecMem = false;
          return;
        }
        NumStores++;
        Stores.push_back(St);
        DepChecker.addAccess(St);
      }
    } // Next instr.
  } // Next block.

  // Now we have two lists that hold the loads and the stores.
  // Next, we find the pointers that they use.

  // Check if we see any stores. If there are no stores, then we don't
  // care if the pointers are *restrict*.
  if (!Stores.size()) {
    DEBUG(dbgs() << "LAA: Found a read-only loop!\n");
    CanVecMem = true;
    return;
  }

  MemoryDepChecker::DepCandidates DependentAccesses;
  AccessAnalysis Accesses(TheLoop->getHeader()->getModule()->getDataLayout(),
                          AA, LI, DependentAccesses);

  // Holds the analyzed pointers. We don't want to call GetUnderlyingObjects
  // multiple times on the same object. If the ptr is accessed twice, once
  // for read and once for write, it will only appear once (on the write
  // list). This is okay, since we are going to check for conflicts between
  // writes and between reads and writes, but not between reads and reads.
  ValueSet Seen;

  ValueVector::iterator I, IE;
  for (I = Stores.begin(), IE = Stores.end(); I != IE; ++I) {
    StoreInst *ST = cast<StoreInst>(*I);
    Value* Ptr = ST->getPointerOperand();
    // Check for store to loop invariant address.
    StoreToLoopInvariantAddress |= isUniform(Ptr);
    // If we did *not* see this pointer before, insert it to  the read-write
    // list. At this phase it is only a 'write' list.
    if (Seen.insert(Ptr).second) {
      ++NumReadWrites;

      MemoryLocation Loc = MemoryLocation::get(ST);
      // The TBAA metadata could have a control dependency on the predication
      // condition, so we cannot rely on it when determining whether or not we
      // need runtime pointer checks.
      if (blockNeedsPredication(ST->getParent(), TheLoop, DT))
        Loc.AATags.TBAA = nullptr;

      Accesses.addStore(Loc);
    }
  }

  if (IsAnnotatedParallel) {
    DEBUG(dbgs()
          << "LAA: A loop annotated parallel, ignore memory dependency "
          << "checks.\n");
    CanVecMem = true;
    return;
  }

  for (I = Loads.begin(), IE = Loads.end(); I != IE; ++I) {
    LoadInst *LD = cast<LoadInst>(*I);
    Value* Ptr = LD->getPointerOperand();
    // If we did *not* see this pointer before, insert it to the
    // read list. If we *did* see it before, then it is already in
    // the read-write list. This allows us to vectorize expressions
    // such as A[i] += x;  Because the address of A[i] is a read-write
    // pointer. This only works if the index of A[i] is consecutive.
    // If the address of i is unknown (for example A[B[i]]) then we may
    // read a few words, modify, and write a few words, and some of the
    // words may be written to the same address.
    bool IsReadOnlyPtr = false;
    if (Seen.insert(Ptr).second || !isStridedPtr(SE, Ptr, TheLoop, Strides)) {
      ++NumReads;
      IsReadOnlyPtr = true;
    }

    MemoryLocation Loc = MemoryLocation::get(LD);
    // The TBAA metadata could have a control dependency on the predication
    // condition, so we cannot rely on it when determining whether or not we
    // need runtime pointer checks.
    if (blockNeedsPredication(LD->getParent(), TheLoop, DT))
      Loc.AATags.TBAA = nullptr;

    Accesses.addLoad(Loc, IsReadOnlyPtr);
  }

  // If we write (or read-write) to a single destination and there are no
  // other reads in this loop then is it safe to vectorize.
  if (NumReadWrites == 1 && NumReads == 0) {
    DEBUG(dbgs() << "LAA: Found a write-only loop!\n");
    CanVecMem = true;
    return;
  }

  // Build dependence sets and check whether we need a runtime pointer bounds
  // check.
  Accesses.buildDependenceSets();

  // Find pointers with computable bounds. We are going to use this information
  // to place a runtime bound check.
  bool CanDoRTIfNeeded =
      Accesses.canCheckPtrAtRT(PtrRtChecking, SE, TheLoop, Strides);
  if (!CanDoRTIfNeeded) {
    emitAnalysis(LoopAccessReport() << "cannot identify array bounds");
    DEBUG(dbgs() << "LAA: We can't vectorize because we can't find "
                 << "the array bounds.\n");
    CanVecMem = false;
    return;
  }

  DEBUG(dbgs() << "LAA: We can perform a memory runtime check if needed.\n");

  CanVecMem = true;
  if (Accesses.isDependencyCheckNeeded()) {
    DEBUG(dbgs() << "LAA: Checking memory dependencies\n");
    CanVecMem = DepChecker.areDepsSafe(
        DependentAccesses, Accesses.getDependenciesToCheck(), Strides);
    MaxSafeDepDistBytes = DepChecker.getMaxSafeDepDistBytes();

    if (!CanVecMem && DepChecker.shouldRetryWithRuntimeCheck()) {
      DEBUG(dbgs() << "LAA: Retrying with memory checks\n");

      // Clear the dependency checks. We assume they are not needed.
      Accesses.resetDepChecks(DepChecker);

      PtrRtChecking.reset();
      PtrRtChecking.Need = true;

      CanDoRTIfNeeded =
          Accesses.canCheckPtrAtRT(PtrRtChecking, SE, TheLoop, Strides, true);

      // Check that we found the bounds for the pointer.
      if (!CanDoRTIfNeeded) {
        emitAnalysis(LoopAccessReport()
                     << "cannot check memory dependencies at runtime");
        DEBUG(dbgs() << "LAA: Can't vectorize with memory checks\n");
        CanVecMem = false;
        return;
      }

      CanVecMem = true;
    }
  }

  if (CanVecMem)
    DEBUG(dbgs() << "LAA: No unsafe dependent memory operations in loop.  We"
                 << (PtrRtChecking.Need ? "" : " don't")
                 << " need runtime memory checks.\n");
  else {
    emitAnalysis(LoopAccessReport() <<
                 "unsafe dependent memory operations in loop");
    DEBUG(dbgs() << "LAA: unsafe dependent memory operations in loop\n");
  }
}

bool LoopAccessInfo::blockNeedsPredication(BasicBlock *BB, Loop *TheLoop,
                                           DominatorTree *DT)  {
  assert(TheLoop->contains(BB) && "Unknown block used");

  // Blocks that do not dominate the latch need predication.
  BasicBlock* Latch = TheLoop->getLoopLatch();
  return !DT->dominates(BB, Latch);
}

void LoopAccessInfo::emitAnalysis(LoopAccessReport &Message) {
  assert(!Report && "Multiple reports generated");
  Report = Message;
}

bool LoopAccessInfo::isUniform(Value *V) const {
  return (SE->isLoopInvariant(SE->getSCEV(V), TheLoop));
}

// FIXME: this function is currently a duplicate of the one in
// LoopVectorize.cpp.
static Instruction *getFirstInst(Instruction *FirstInst, Value *V,
                                 Instruction *Loc) {
  if (FirstInst)
    return FirstInst;
  if (Instruction *I = dyn_cast<Instruction>(V))
    return I->getParent() == Loc->getParent() ? I : nullptr;
  return nullptr;
}

/// \brief IR Values for the lower and upper bounds of a pointer evolution.
struct PointerBounds {
  Value *Start;
  Value *End;
};

/// \brief Expand code for the lower and upper bound of the pointer group \p CG
/// in \p TheLoop.  \return the values for the bounds.
static PointerBounds
expandBounds(const RuntimePointerChecking::CheckingPtrGroup *CG, Loop *TheLoop,
             Instruction *Loc, SCEVExpander &Exp, ScalarEvolution *SE,
             const RuntimePointerChecking &PtrRtChecking) {
  Value *Ptr = PtrRtChecking.Pointers[CG->Members[0]].PointerValue;
  const SCEV *Sc = SE->getSCEV(Ptr);

  if (SE->isLoopInvariant(Sc, TheLoop)) {
    DEBUG(dbgs() << "LAA: Adding RT check for a loop invariant ptr:" << *Ptr
                 << "\n");
    return {Ptr, Ptr};
  } else {
    unsigned AS = Ptr->getType()->getPointerAddressSpace();
    LLVMContext &Ctx = Loc->getContext();

    // Use this type for pointer arithmetic.
    Type *PtrArithTy = Type::getInt8PtrTy(Ctx, AS);
    Value *Start = nullptr, *End = nullptr;

    DEBUG(dbgs() << "LAA: Adding RT check for range:\n");
    Start = Exp.expandCodeFor(CG->Low, PtrArithTy, Loc);
    End = Exp.expandCodeFor(CG->High, PtrArithTy, Loc);
    DEBUG(dbgs() << "Start: " << *CG->Low << " End: " << *CG->High << "\n");
    return {Start, End};
  }
}

/// \brief Turns a collection of checks into a collection of expanded upper and
/// lower bounds for both pointers in the check.
static SmallVector<std::pair<PointerBounds, PointerBounds>, 4> expandBounds(
    const SmallVectorImpl<RuntimePointerChecking::PointerCheck> &PointerChecks,
    Loop *L, Instruction *Loc, ScalarEvolution *SE, SCEVExpander &Exp,
    const RuntimePointerChecking &PtrRtChecking) {
  SmallVector<std::pair<PointerBounds, PointerBounds>, 4> ChecksWithBounds;

  // Here we're relying on the SCEV Expander's cache to only emit code for the
  // same bounds once.
  std::transform(
      PointerChecks.begin(), PointerChecks.end(),
      std::back_inserter(ChecksWithBounds),
      [&](const RuntimePointerChecking::PointerCheck &Check) {
        PointerBounds
          First = expandBounds(Check.first, L, Loc, Exp, SE, PtrRtChecking),
          Second = expandBounds(Check.second, L, Loc, Exp, SE, PtrRtChecking);
        return std::make_pair(First, Second);
      });

  return ChecksWithBounds;
}

std::pair<Instruction *, Instruction *> LoopAccessInfo::addRuntimeCheck(
    Instruction *Loc,
    const SmallVectorImpl<RuntimePointerChecking::PointerCheck> &PointerChecks)
    const {

  SCEVExpander Exp(*SE, DL, "induction");
  auto ExpandedChecks =
      expandBounds(PointerChecks, TheLoop, Loc, SE, Exp, PtrRtChecking);

  LLVMContext &Ctx = Loc->getContext();
  Instruction *FirstInst = nullptr;
  IRBuilder<> ChkBuilder(Loc);
  // Our instructions might fold to a constant.
  Value *MemoryRuntimeCheck = nullptr;

  for (const auto &Check : ExpandedChecks) {
    const PointerBounds &A = Check.first, &B = Check.second;
    unsigned AS0 = A.Start->getType()->getPointerAddressSpace();
    unsigned AS1 = B.Start->getType()->getPointerAddressSpace();

    assert((AS0 == B.End->getType()->getPointerAddressSpace()) &&
           (AS1 == A.End->getType()->getPointerAddressSpace()) &&
           "Trying to bounds check pointers with different address spaces");

    Type *PtrArithTy0 = Type::getInt8PtrTy(Ctx, AS0);
    Type *PtrArithTy1 = Type::getInt8PtrTy(Ctx, AS1);

    Value *Start0 = ChkBuilder.CreateBitCast(A.Start, PtrArithTy0, "bc");
    Value *Start1 = ChkBuilder.CreateBitCast(B.Start, PtrArithTy1, "bc");
    Value *End0 =   ChkBuilder.CreateBitCast(A.End,   PtrArithTy1, "bc");
    Value *End1 =   ChkBuilder.CreateBitCast(B.End,   PtrArithTy0, "bc");

    Value *Cmp0 = ChkBuilder.CreateICmpULE(Start0, End1, "bound0");
    FirstInst = getFirstInst(FirstInst, Cmp0, Loc);
    Value *Cmp1 = ChkBuilder.CreateICmpULE(Start1, End0, "bound1");
    FirstInst = getFirstInst(FirstInst, Cmp1, Loc);
    Value *IsConflict = ChkBuilder.CreateAnd(Cmp0, Cmp1, "found.conflict");
    FirstInst = getFirstInst(FirstInst, IsConflict, Loc);
    if (MemoryRuntimeCheck) {
      IsConflict =
          ChkBuilder.CreateOr(MemoryRuntimeCheck, IsConflict, "conflict.rdx");
      FirstInst = getFirstInst(FirstInst, IsConflict, Loc);
    }
    MemoryRuntimeCheck = IsConflict;
  }

  if (!MemoryRuntimeCheck)
    return std::make_pair(nullptr, nullptr);

  // We have to do this trickery because the IRBuilder might fold the check to a
  // constant expression in which case there is no Instruction anchored in a
  // the block.
  Instruction *Check = BinaryOperator::CreateAnd(MemoryRuntimeCheck,
                                                 ConstantInt::getTrue(Ctx));
  ChkBuilder.Insert(Check, "memcheck.conflict");
  FirstInst = getFirstInst(FirstInst, Check, Loc);
  return std::make_pair(FirstInst, Check);
}

std::pair<Instruction *, Instruction *> LoopAccessInfo::addRuntimeCheck(
    Instruction *Loc) const {
  if (!PtrRtChecking.Need)
    return std::make_pair(nullptr, nullptr);

  return addRuntimeCheck(Loc, PtrRtChecking.generateChecks());
}

LoopAccessInfo::LoopAccessInfo(Loop *L, ScalarEvolution *SE,
                               const DataLayout &DL,
                               const TargetLibraryInfo *TLI, AliasAnalysis *AA,
                               DominatorTree *DT, LoopInfo *LI,
                               const ValueToValueMap &Strides)
    : PtrRtChecking(SE), DepChecker(SE, L), TheLoop(L), SE(SE), DL(DL),
      TLI(TLI), AA(AA), DT(DT), LI(LI), NumLoads(0), NumStores(0),
      MaxSafeDepDistBytes(-1U), CanVecMem(false),
      StoreToLoopInvariantAddress(false) {
  if (canAnalyzeLoop())
    analyzeLoop(Strides);
}

void LoopAccessInfo::print(raw_ostream &OS, unsigned Depth) const {
  if (CanVecMem) {
    if (PtrRtChecking.Need)
      OS.indent(Depth) << "Memory dependences are safe with run-time checks\n";
    else
      OS.indent(Depth) << "Memory dependences are safe\n";
  }

  if (Report)
    OS.indent(Depth) << "Report: " << Report->str() << "\n";

  if (auto *InterestingDependences = DepChecker.getInterestingDependences()) {
    OS.indent(Depth) << "Interesting Dependences:\n";
    for (auto &Dep : *InterestingDependences) {
      Dep.print(OS, Depth + 2, DepChecker.getMemoryInstructions());
      OS << "\n";
    }
  } else
    OS.indent(Depth) << "Too many interesting dependences, not recorded\n";

  // List the pair of accesses need run-time checks to prove independence.
  PtrRtChecking.print(OS, Depth);
  OS << "\n";

  OS.indent(Depth) << "Store to invariant address was "
                   << (StoreToLoopInvariantAddress ? "" : "not ")
                   << "found in loop.\n";
}

const LoopAccessInfo &
LoopAccessAnalysis::getInfo(Loop *L, const ValueToValueMap &Strides) {
  auto &LAI = LoopAccessInfoMap[L];

#ifndef NDEBUG
  assert((!LAI || LAI->NumSymbolicStrides == Strides.size()) &&
         "Symbolic strides changed for loop");
#endif

  if (!LAI) {
    const DataLayout &DL = L->getHeader()->getModule()->getDataLayout();
    LAI = llvm::make_unique<LoopAccessInfo>(L, SE, DL, TLI, AA, DT, LI,
                                            Strides);
#ifndef NDEBUG
    LAI->NumSymbolicStrides = Strides.size();
#endif
  }
  return *LAI.get();
}

void LoopAccessAnalysis::print(raw_ostream &OS, const Module *M) const {
  LoopAccessAnalysis &LAA = *const_cast<LoopAccessAnalysis *>(this);

  ValueToValueMap NoSymbolicStrides;

  for (Loop *TopLevelLoop : *LI)
    for (Loop *L : depth_first(TopLevelLoop)) {
      OS.indent(2) << L->getHeader()->getName() << ":\n";
      auto &LAI = LAA.getInfo(L, NoSymbolicStrides);
      LAI.print(OS, 4);
    }
}

bool LoopAccessAnalysis::runOnFunction(Function &F) {
  SE = &getAnalysis<ScalarEvolution>();
  auto *TLIP = getAnalysisIfAvailable<TargetLibraryInfoWrapperPass>();
  TLI = TLIP ? &TLIP->getTLI() : nullptr;
  AA = &getAnalysis<AliasAnalysis>();
  DT = &getAnalysis<DominatorTreeWrapperPass>().getDomTree();
  LI = &getAnalysis<LoopInfoWrapperPass>().getLoopInfo();

  return false;
}

void LoopAccessAnalysis::getAnalysisUsage(AnalysisUsage &AU) const {
    AU.addRequired<ScalarEvolution>();
    AU.addRequired<AliasAnalysis>();
    AU.addRequired<DominatorTreeWrapperPass>();
    AU.addRequired<LoopInfoWrapperPass>();

    AU.setPreservesAll();
}

char LoopAccessAnalysis::ID = 0;
static const char laa_name[] = "Loop Access Analysis";
#define LAA_NAME "loop-accesses"

INITIALIZE_PASS_BEGIN(LoopAccessAnalysis, LAA_NAME, laa_name, false, true)
INITIALIZE_AG_DEPENDENCY(AliasAnalysis)
INITIALIZE_PASS_DEPENDENCY(ScalarEvolution)
INITIALIZE_PASS_DEPENDENCY(DominatorTreeWrapperPass)
INITIALIZE_PASS_DEPENDENCY(LoopInfoWrapperPass)
INITIALIZE_PASS_END(LoopAccessAnalysis, LAA_NAME, laa_name, false, true)

namespace llvm {
  Pass *createLAAPass() {
    return new LoopAccessAnalysis();
  }
}