//===- polly/ScopInfo.h -----------------------------------------*- C++ -*-===//
//
//                     The LLVM Compiler Infrastructure
//
// This file is distributed under the University of Illinois Open Source
// License. See LICENSE.TXT for details.
//
//===----------------------------------------------------------------------===//
//
// Store the polyhedral model representation of a static control flow region,
// also called SCoP (Static Control Part).
//
// This representation is shared among several tools in the polyhedral
// community, which are e.g. CLooG, Pluto, Loopo, Graphite.
//
//===----------------------------------------------------------------------===//

#ifndef POLLY_SCOPINFO_H
#define POLLY_SCOPINFO_H

#include "polly/ScopDetection.h"
#include "polly/Support/SCEVAffinator.h"
#include "polly/Support/ScopHelper.h"
#include "llvm/ADT/ArrayRef.h"
#include "llvm/ADT/DenseMap.h"
#include "llvm/ADT/DenseSet.h"
#include "llvm/ADT/MapVector.h"
#include "llvm/ADT/SetVector.h"
#include "llvm/ADT/SmallVector.h"
#include "llvm/ADT/StringMap.h"
#include "llvm/ADT/StringRef.h"
#include "llvm/ADT/iterator_range.h"
#include "llvm/Analysis/RegionPass.h"
#include "llvm/Analysis/ScalarEvolutionExpressions.h"
#include "llvm/IR/DebugLoc.h"
#include "llvm/IR/Instruction.h"
#include "llvm/IR/Instructions.h"
#include "llvm/IR/IntrinsicInst.h"
#include "llvm/IR/PassManager.h"
#include "llvm/IR/ValueHandle.h"
#include "llvm/Pass.h"
#include "llvm/Support/Casting.h"
#include "isl/isl-noexceptions.h"
#include <algorithm>
#include <cassert>
#include <cstddef>
#include <forward_list>
#include <functional>
#include <list>
#include <map>
#include <memory>
#include <string>
#include <tuple>
#include <utility>
#include <vector>

using namespace llvm;

namespace llvm {

class AssumptionCache;
class BasicBlock;
class DataLayout;
class DominatorTree;
class Function;
class Loop;
class LoopInfo;
class OptimizationRemarkEmitter;
class PassRegistry;
class raw_ostream;
class ScalarEvolution;
class SCEV;
class Type;
class Value;

void initializeScopInfoRegionPassPass(PassRegistry &);
void initializeScopInfoWrapperPassPass(PassRegistry &);

} // end namespace llvm

struct isl_map;
struct isl_pw_multi_aff;
struct isl_schedule;
struct isl_set;
struct isl_union_map;

namespace polly {

class MemoryAccess;
class Scop;
class ScopStmt;

//===---------------------------------------------------------------------===//

extern bool UseInstructionNames;

/// Enumeration of assumptions Polly can take.
enum AssumptionKind {
  ALIASING,
  INBOUNDS,
  WRAPPING,
  UNSIGNED,
  PROFITABLE,
  ERRORBLOCK,
  COMPLEXITY,
  INFINITELOOP,
  INVARIANTLOAD,
  DELINEARIZATION,
};

/// Enum to distinguish between assumptions and restrictions.
enum AssumptionSign { AS_ASSUMPTION, AS_RESTRICTION };

/// The different memory kinds used in Polly.
///
/// We distinguish between arrays and various scalar memory objects. We use
/// the term ``array'' to describe memory objects that consist of a set of
/// individual data elements arranged in a multi-dimensional grid. A scalar
/// memory object describes an individual data element and is used to model
/// the definition and uses of llvm::Values.
///
/// The polyhedral model does traditionally not reason about SSA values. To
/// reason about llvm::Values we model them "as if" they were zero-dimensional
/// memory objects, even though they were not actually allocated in (main)
/// memory.  Memory for such objects is only alloca[ed] at CodeGeneration
/// time. To relate the memory slots used during code generation with the
/// llvm::Values they belong to the new names for these corresponding stack
/// slots are derived by appending suffixes (currently ".s2a" and ".phiops")
/// to the name of the original llvm::Value. To describe how def/uses are
/// modeled exactly we use these suffixes here as well.
///
/// There are currently four different kinds of memory objects:
enum class MemoryKind {
  /// MemoryKind::Array: Models a one or multi-dimensional array
  ///
  /// A memory object that can be described by a multi-dimensional array.
  /// Memory objects of this type are used to model actual multi-dimensional
  /// arrays as they exist in LLVM-IR, but they are also used to describe
  /// other objects:
  ///   - A single data element allocated on the stack using 'alloca' is
  ///     modeled as a one-dimensional, single-element array.
  ///   - A single data element allocated as a global variable is modeled as
  ///     one-dimensional, single-element array.
  ///   - Certain multi-dimensional arrays with variable size, which in
  ///     LLVM-IR are commonly expressed as a single-dimensional access with a
  ///     complicated access function, are modeled as multi-dimensional
  ///     memory objects (grep for "delinearization").
  Array,

  /// MemoryKind::Value: Models an llvm::Value
  ///
  /// Memory objects of type MemoryKind::Value are used to model the data flow
  /// induced by llvm::Values. For each llvm::Value that is used across
  /// BasicBlocks, one ScopArrayInfo object is created. A single memory WRITE
  /// stores the llvm::Value at its definition into the memory object and at
  /// each use of the llvm::Value (ignoring trivial intra-block uses) a
  /// corresponding READ is added. For instance, the use/def chain of a
  /// llvm::Value %V depicted below
  ///              ______________________
  ///              |DefBB:              |
  ///              |  %V = float op ... |
  ///              ----------------------
  ///               |                  |
  /// _________________               _________________
  /// |UseBB1:        |               |UseBB2:        |
  /// |  use float %V |               |  use float %V |
  /// -----------------               -----------------
  ///
  /// is modeled as if the following memory accesses occurred:
  ///
  ///                        __________________________
  ///                        |entry:                  |
  ///                        |  %V.s2a = alloca float |
  ///                        --------------------------
  ///                                     |
  ///                    ___________________________________
  ///                    |DefBB:                           |
  ///                    |  store %float %V, float* %V.s2a |
  ///                    -----------------------------------
  ///                           |                   |
  /// ____________________________________ ___________________________________
  /// |UseBB1:                           | |UseBB2:                          |
  /// |  %V.reload1 = load float* %V.s2a | |  %V.reload2 = load float* %V.s2a|
  /// |  use float %V.reload1            | |  use float %V.reload2           |
  /// ------------------------------------ -----------------------------------
  ///
  Value,

  /// MemoryKind::PHI: Models PHI nodes within the SCoP
  ///
  /// Besides the MemoryKind::Value memory object used to model the normal
  /// llvm::Value dependences described above, PHI nodes require an additional
  /// memory object of type MemoryKind::PHI to describe the forwarding of values
  /// to
  /// the PHI node.
  ///
  /// As an example, a PHIInst instructions
  ///
  /// %PHI = phi float [ %Val1, %IncomingBlock1 ], [ %Val2, %IncomingBlock2 ]
  ///
  /// is modeled as if the accesses occurred this way:
  ///
  ///                    _______________________________
  ///                    |entry:                       |
  ///                    |  %PHI.phiops = alloca float |
  ///                    -------------------------------
  ///                           |              |
  /// __________________________________  __________________________________
  /// |IncomingBlock1:                 |  |IncomingBlock2:                 |
  /// |  ...                           |  |  ...                           |
  /// |  store float %Val1 %PHI.phiops |  |  store float %Val2 %PHI.phiops |
  /// |  br label % JoinBlock          |  |  br label %JoinBlock           |
  /// ----------------------------------  ----------------------------------
  ///                             \            /
  ///                              \          /
  ///               _________________________________________
  ///               |JoinBlock:                             |
  ///               |  %PHI = load float, float* PHI.phiops |
  ///               -----------------------------------------
  ///
  /// Note that there can also be a scalar write access for %PHI if used in a
  /// different BasicBlock, i.e. there can be a memory object %PHI.phiops as
  /// well as a memory object %PHI.s2a.
  PHI,

  /// MemoryKind::ExitPHI: Models PHI nodes in the SCoP's exit block
  ///
  /// For PHI nodes in the Scop's exit block a special memory object kind is
  /// used. The modeling used is identical to MemoryKind::PHI, with the
  /// exception
  /// that there are no READs from these memory objects. The PHINode's
  /// llvm::Value is treated as a value escaping the SCoP. WRITE accesses
  /// write directly to the escaping value's ".s2a" alloca.
  ExitPHI
};

/// Maps from a loop to the affine function expressing its backedge taken count.
/// The backedge taken count already enough to express iteration domain as we
/// only allow loops with canonical induction variable.
/// A canonical induction variable is:
/// an integer recurrence that starts at 0 and increments by one each time
/// through the loop.
using LoopBoundMapType = std::map<const Loop *, const SCEV *>;

using AccFuncVector = std::vector<std::unique_ptr<MemoryAccess>>;

/// A class to store information about arrays in the SCoP.
///
/// Objects are accessible via the ScoP, MemoryAccess or the id associated with
/// the MemoryAccess access function.
///
class ScopArrayInfo {
public:
  /// Construct a ScopArrayInfo object.
  ///
  /// @param BasePtr        The array base pointer.
  /// @param ElementType    The type of the elements stored in the array.
  /// @param IslCtx         The isl context used to create the base pointer id.
  /// @param DimensionSizes A vector containing the size of each dimension.
  /// @param Kind           The kind of the array object.
  /// @param DL             The data layout of the module.
  /// @param S              The scop this array object belongs to.
  /// @param BaseName       The optional name of this memory reference.
  ScopArrayInfo(Value *BasePtr, Type *ElementType, isl::ctx IslCtx,
                ArrayRef<const SCEV *> DimensionSizes, MemoryKind Kind,
                const DataLayout &DL, Scop *S, const char *BaseName = nullptr);

  /// Destructor to free the isl id of the base pointer.
  ~ScopArrayInfo();

  ///  Update the element type of the ScopArrayInfo object.
  ///
  ///  Memory accesses referencing this ScopArrayInfo object may use
  ///  different element sizes. This function ensures the canonical element type
  ///  stored is small enough to model accesses to the current element type as
  ///  well as to @p NewElementType.
  ///
  ///  @param NewElementType An element type that is used to access this array.
  void updateElementType(Type *NewElementType);

  ///  Update the sizes of the ScopArrayInfo object.
  ///
  ///  A ScopArrayInfo object may be created without all outer dimensions being
  ///  available. This function is called when new memory accesses are added for
  ///  this ScopArrayInfo object. It verifies that sizes are compatible and adds
  ///  additional outer array dimensions, if needed.
  ///
  ///  @param Sizes       A vector of array sizes where the rightmost array
  ///                     sizes need to match the innermost array sizes already
  ///                     defined in SAI.
  ///  @param CheckConsistency Update sizes, even if new sizes are inconsistent
  ///                          with old sizes
  bool updateSizes(ArrayRef<const SCEV *> Sizes, bool CheckConsistency = true);

  /// Make the ScopArrayInfo model a Fortran array.
  /// It receives the Fortran array descriptor and stores this.
  /// It also adds a piecewise expression for the outermost dimension
  /// since this information is available for Fortran arrays at runtime.
  void applyAndSetFAD(Value *FAD);

  /// Get the FortranArrayDescriptor corresponding to this array if it exists,
  /// nullptr otherwise.
  Value *getFortranArrayDescriptor() const { return this->FAD; }

  /// Set the base pointer to @p BP.
  void setBasePtr(Value *BP) { BasePtr = BP; }

  /// Return the base pointer.
  Value *getBasePtr() const { return BasePtr; }

  // Set IsOnHeap to the value in parameter.
  void setIsOnHeap(bool value) { IsOnHeap = value; }

  /// For indirect accesses return the origin SAI of the BP, else null.
  const ScopArrayInfo *getBasePtrOriginSAI() const { return BasePtrOriginSAI; }

  /// The set of derived indirect SAIs for this origin SAI.
  const SmallSetVector<ScopArrayInfo *, 2> &getDerivedSAIs() const {
    return DerivedSAIs;
  }

  /// Return the number of dimensions.
  unsigned getNumberOfDimensions() const {
    if (Kind == MemoryKind::PHI || Kind == MemoryKind::ExitPHI ||
        Kind == MemoryKind::Value)
      return 0;
    return DimensionSizes.size();
  }

  /// Return the size of dimension @p dim as SCEV*.
  //
  //  Scalars do not have array dimensions and the first dimension of
  //  a (possibly multi-dimensional) array also does not carry any size
  //  information, in case the array is not newly created.
  const SCEV *getDimensionSize(unsigned Dim) const {
    assert(Dim < getNumberOfDimensions() && "Invalid dimension");
    return DimensionSizes[Dim];
  }

  /// Return the size of dimension @p dim as isl::pw_aff.
  //
  //  Scalars do not have array dimensions and the first dimension of
  //  a (possibly multi-dimensional) array also does not carry any size
  //  information, in case the array is not newly created.
  isl::pw_aff getDimensionSizePw(unsigned Dim) const {
    assert(Dim < getNumberOfDimensions() && "Invalid dimension");
    return DimensionSizesPw[Dim];
  }

  /// Get the canonical element type of this array.
  ///
  /// @returns The canonical element type of this array.
  Type *getElementType() const { return ElementType; }

  /// Get element size in bytes.
  int getElemSizeInBytes() const;

  /// Get the name of this memory reference.
  std::string getName() const;

  /// Return the isl id for the base pointer.
  isl::id getBasePtrId() const;

  /// Return what kind of memory this represents.
  MemoryKind getKind() const { return Kind; }

  /// Is this array info modeling an llvm::Value?
  bool isValueKind() const { return Kind == MemoryKind::Value; }

  /// Is this array info modeling special PHI node memory?
  ///
  /// During code generation of PHI nodes, there is a need for two kinds of
  /// virtual storage. The normal one as it is used for all scalar dependences,
  /// where the result of the PHI node is stored and later loaded from as well
  /// as a second one where the incoming values of the PHI nodes are stored
  /// into and reloaded when the PHI is executed. As both memories use the
  /// original PHI node as virtual base pointer, we have this additional
  /// attribute to distinguish the PHI node specific array modeling from the
  /// normal scalar array modeling.
  bool isPHIKind() const { return Kind == MemoryKind::PHI; }

  /// Is this array info modeling an MemoryKind::ExitPHI?
  bool isExitPHIKind() const { return Kind == MemoryKind::ExitPHI; }

  /// Is this array info modeling an array?
  bool isArrayKind() const { return Kind == MemoryKind::Array; }

  /// Is this array allocated on heap
  ///
  /// This property is only relevant if the array is allocated by Polly instead
  /// of pre-existing. If false, it is allocated using alloca instead malloca.
  bool isOnHeap() const { return IsOnHeap; }

#if !defined(NDEBUG) || defined(LLVM_ENABLE_DUMP)
  /// Dump a readable representation to stderr.
  void dump() const;
#endif

  /// Print a readable representation to @p OS.
  ///
  /// @param SizeAsPwAff Print the size as isl::pw_aff
  void print(raw_ostream &OS, bool SizeAsPwAff = false) const;

  /// Access the ScopArrayInfo associated with an access function.
  static const ScopArrayInfo *getFromAccessFunction(isl::pw_multi_aff PMA);

  /// Access the ScopArrayInfo associated with an isl Id.
  static const ScopArrayInfo *getFromId(isl::id Id);

  /// Get the space of this array access.
  isl::space getSpace() const;

  /// If the array is read only
  bool isReadOnly();

  /// Verify that @p Array is compatible to this ScopArrayInfo.
  ///
  /// Two arrays are compatible if their dimensionality, the sizes of their
  /// dimensions, and their element sizes match.
  ///
  /// @param Array The array to compare against.
  ///
  /// @returns True, if the arrays are compatible, False otherwise.
  bool isCompatibleWith(const ScopArrayInfo *Array) const;

private:
  void addDerivedSAI(ScopArrayInfo *DerivedSAI) {
    DerivedSAIs.insert(DerivedSAI);
  }

  /// For indirect accesses this is the SAI of the BP origin.
  const ScopArrayInfo *BasePtrOriginSAI;

  /// For origin SAIs the set of derived indirect SAIs.
  SmallSetVector<ScopArrayInfo *, 2> DerivedSAIs;

  /// The base pointer.
  AssertingVH<Value> BasePtr;

  /// The canonical element type of this array.
  ///
  /// The canonical element type describes the minimal accessible element in
  /// this array. Not all elements accessed, need to be of the very same type,
  /// but the allocation size of the type of the elements loaded/stored from/to
  /// this array needs to be a multiple of the allocation size of the canonical
  /// type.
  Type *ElementType;

  /// The isl id for the base pointer.
  isl::id Id;

  /// True if the newly allocated array is on heap.
  bool IsOnHeap = false;

  /// The sizes of each dimension as SCEV*.
  SmallVector<const SCEV *, 4> DimensionSizes;

  /// The sizes of each dimension as isl::pw_aff.
  SmallVector<isl::pw_aff, 4> DimensionSizesPw;

  /// The type of this scop array info object.
  ///
  /// We distinguish between SCALAR, PHI and ARRAY objects.
  MemoryKind Kind;

  /// The data layout of the module.
  const DataLayout &DL;

  /// The scop this SAI object belongs to.
  Scop &S;

  /// If this array models a Fortran array, then this points
  /// to the Fortran array descriptor.
  Value *FAD = nullptr;
};

/// Represent memory accesses in statements.
class MemoryAccess {
  friend class Scop;
  friend class ScopStmt;
  friend class ScopBuilder;

public:
  /// The access type of a memory access
  ///
  /// There are three kind of access types:
  ///
  /// * A read access
  ///
  /// A certain set of memory locations are read and may be used for internal
  /// calculations.
  ///
  /// * A must-write access
  ///
  /// A certain set of memory locations is definitely written. The old value is
  /// replaced by a newly calculated value. The old value is not read or used at
  /// all.
  ///
  /// * A may-write access
  ///
  /// A certain set of memory locations may be written. The memory location may
  /// contain a new value if there is actually a write or the old value may
  /// remain, if no write happens.
  enum AccessType {
    READ = 0x1,
    MUST_WRITE = 0x2,
    MAY_WRITE = 0x3,
  };

  /// Reduction access type
  ///
  /// Commutative and associative binary operations suitable for reductions
  enum ReductionType {
    RT_NONE, ///< Indicate no reduction at all
    RT_ADD,  ///< Addition
    RT_MUL,  ///< Multiplication
    RT_BOR,  ///< Bitwise Or
    RT_BXOR, ///< Bitwise XOr
    RT_BAND, ///< Bitwise And
  };

private:
  /// A unique identifier for this memory access.
  ///
  /// The identifier is unique between all memory accesses belonging to the same
  /// scop statement.
  isl::id Id;

  /// What is modeled by this MemoryAccess.
  /// @see MemoryKind
  MemoryKind Kind;

  /// Whether it a reading or writing access, and if writing, whether it
  /// is conditional (MAY_WRITE).
  enum AccessType AccType;

  /// Reduction type for reduction like accesses, RT_NONE otherwise
  ///
  /// An access is reduction like if it is part of a load-store chain in which
  /// both access the same memory location (use the same LLVM-IR value
  /// as pointer reference). Furthermore, between the load and the store there
  /// is exactly one binary operator which is known to be associative and
  /// commutative.
  ///
  /// TODO:
  ///
  /// We can later lift the constraint that the same LLVM-IR value defines the
  /// memory location to handle scops such as the following:
  ///
  ///    for i
  ///      for j
  ///        sum[i+j] = sum[i] + 3;
  ///
  /// Here not all iterations access the same memory location, but iterations
  /// for which j = 0 holds do. After lifting the equality check in ScopBuilder,
  /// subsequent transformations do not only need check if a statement is
  /// reduction like, but they also need to verify that that the reduction
  /// property is only exploited for statement instances that load from and
  /// store to the same data location. Doing so at dependence analysis time
  /// could allow us to handle the above example.
  ReductionType RedType = RT_NONE;

  /// Parent ScopStmt of this access.
  ScopStmt *Statement;

  /// The domain under which this access is not modeled precisely.
  ///
  /// The invalid domain for an access describes all parameter combinations
  /// under which the statement looks to be executed but is in fact not because
  /// some assumption/restriction makes the access invalid.
  isl::set InvalidDomain;

  // Properties describing the accessed array.
  // TODO: It might be possible to move them to ScopArrayInfo.
  // @{

  /// The base address (e.g., A for A[i+j]).
  ///
  /// The #BaseAddr of a memory access of kind MemoryKind::Array is the base
  /// pointer of the memory access.
  /// The #BaseAddr of a memory access of kind MemoryKind::PHI or
  /// MemoryKind::ExitPHI is the PHI node itself.
  /// The #BaseAddr of a memory access of kind MemoryKind::Value is the
  /// instruction defining the value.
  AssertingVH<Value> BaseAddr;

  /// Type a single array element wrt. this access.
  Type *ElementType;

  /// Size of each dimension of the accessed array.
  SmallVector<const SCEV *, 4> Sizes;
  // @}

  // Properties describing the accessed element.
  // @{

  /// The access instruction of this memory access.
  ///
  /// For memory accesses of kind MemoryKind::Array the access instruction is
  /// the Load or Store instruction performing the access.
  ///
  /// For memory accesses of kind MemoryKind::PHI or MemoryKind::ExitPHI the
  /// access instruction of a load access is the PHI instruction. The access
  /// instruction of a PHI-store is the incoming's block's terminator
  /// instruction.
  ///
  /// For memory accesses of kind MemoryKind::Value the access instruction of a
  /// load access is nullptr because generally there can be multiple
  /// instructions in the statement using the same llvm::Value. The access
  /// instruction of a write access is the instruction that defines the
  /// llvm::Value.
  Instruction *AccessInstruction = nullptr;

  /// Incoming block and value of a PHINode.
  SmallVector<std::pair<BasicBlock *, Value *>, 4> Incoming;

  /// The value associated with this memory access.
  ///
  ///  - For array memory accesses (MemoryKind::Array) it is the loaded result
  ///    or the stored value. If the access instruction is a memory intrinsic it
  ///    the access value is also the memory intrinsic.
  ///  - For accesses of kind MemoryKind::Value it is the access instruction
  ///    itself.
  ///  - For accesses of kind MemoryKind::PHI or MemoryKind::ExitPHI it is the
  ///    PHI node itself (for both, READ and WRITE accesses).
  ///
  AssertingVH<Value> AccessValue;

  /// Are all the subscripts affine expression?
  bool IsAffine = true;

  /// Subscript expression for each dimension.
  SmallVector<const SCEV *, 4> Subscripts;

  /// Relation from statement instances to the accessed array elements.
  ///
  /// In the common case this relation is a function that maps a set of loop
  /// indices to the memory address from which a value is loaded/stored:
  ///
  ///      for i
  ///        for j
  ///    S:     A[i + 3 j] = ...
  ///
  ///    => { S[i,j] -> A[i + 3j] }
  ///
  /// In case the exact access function is not known, the access relation may
  /// also be a one to all mapping { S[i,j] -> A[o] } describing that any
  /// element accessible through A might be accessed.
  ///
  /// In case of an access to a larger element belonging to an array that also
  /// contains smaller elements, the access relation models the larger access
  /// with multiple smaller accesses of the size of the minimal array element
  /// type:
  ///
  ///      short *A;
  ///
  ///      for i
  ///    S:     A[i] = *((double*)&A[4 * i]);
  ///
  ///    => { S[i] -> A[i]; S[i] -> A[o] : 4i <= o <= 4i + 3 }
  isl::map AccessRelation;

  /// Updated access relation read from JSCOP file.
  isl::map NewAccessRelation;

  /// Fortran arrays whose sizes are not statically known are stored in terms
  /// of a descriptor struct. This maintains a raw pointer to the memory,
  /// along with auxiliary fields with information such as dimensions.
  /// We hold a reference to the descriptor corresponding to a MemoryAccess
  /// into a Fortran array. FAD for "Fortran Array Descriptor"
  AssertingVH<Value> FAD;
  // @}

  isl::basic_map createBasicAccessMap(ScopStmt *Statement);

  void assumeNoOutOfBound();

  /// Compute bounds on an over approximated  access relation.
  ///
  /// @param ElementSize The size of one element accessed.
  void computeBoundsOnAccessRelation(unsigned ElementSize);

  /// Get the original access function as read from IR.
  isl::map getOriginalAccessRelation() const;

  /// Return the space in which the access relation lives in.
  isl::space getOriginalAccessRelationSpace() const;

  /// Get the new access function imported or set by a pass
  isl::map getNewAccessRelation() const;

  /// Fold the memory access to consider parametric offsets
  ///
  /// To recover memory accesses with array size parameters in the subscript
  /// expression we post-process the delinearization results.
  ///
  /// We would normally recover from an access A[exp0(i) * N + exp1(i)] into an
  /// array A[][N] the 2D access A[exp0(i)][exp1(i)]. However, another valid
  /// delinearization is A[exp0(i) - 1][exp1(i) + N] which - depending on the
  /// range of exp1(i) - may be preferable. Specifically, for cases where we
  /// know exp1(i) is negative, we want to choose the latter expression.
  ///
  /// As we commonly do not have any information about the range of exp1(i),
  /// we do not choose one of the two options, but instead create a piecewise
  /// access function that adds the (-1, N) offsets as soon as exp1(i) becomes
  /// negative. For a 2D array such an access function is created by applying
  /// the piecewise map:
  ///
  /// [i,j] -> [i, j] :      j >= 0
  /// [i,j] -> [i-1, j+N] :  j <  0
  ///
  /// We can generalize this mapping to arbitrary dimensions by applying this
  /// piecewise mapping pairwise from the rightmost to the leftmost access
  /// dimension. It would also be possible to cover a wider range by introducing
  /// more cases and adding multiple of Ns to these cases. However, this has
  /// not yet been necessary.
  /// The introduction of different cases necessarily complicates the memory
  /// access function, but cases that can be statically proven to not happen
  /// will be eliminated later on.
  void foldAccessRelation();

  /// Create the access relation for the underlying memory intrinsic.
  void buildMemIntrinsicAccessRelation();

  /// Assemble the access relation from all available information.
  ///
  /// In particular, used the information passes in the constructor and the
  /// parent ScopStmt set by setStatment().
  ///
  /// @param SAI Info object for the accessed array.
  void buildAccessRelation(const ScopArrayInfo *SAI);

  /// Carry index overflows of dimensions with constant size to the next higher
  /// dimension.
  ///
  /// For dimensions that have constant size, modulo the index by the size and
  /// add up the carry (floored division) to the next higher dimension. This is
  /// how overflow is defined in row-major order.
  /// It happens e.g. when ScalarEvolution computes the offset to the base
  /// pointer and would algebraically sum up all lower dimensions' indices of
  /// constant size.
  ///
  /// Example:
  ///   float (*A)[4];
  ///   A[1][6] -> A[2][2]
  void wrapConstantDimensions();

public:
  /// Create a new MemoryAccess.
  ///
  /// @param Stmt       The parent statement.
  /// @param AccessInst The instruction doing the access.
  /// @param BaseAddr   The accessed array's address.
  /// @param ElemType   The type of the accessed array elements.
  /// @param AccType    Whether read or write access.
  /// @param IsAffine   Whether the subscripts are affine expressions.
  /// @param Kind       The kind of memory accessed.
  /// @param Subscripts Subscript expressions
  /// @param Sizes      Dimension lengths of the accessed array.
  MemoryAccess(ScopStmt *Stmt, Instruction *AccessInst, AccessType AccType,
               Value *BaseAddress, Type *ElemType, bool Affine,
               ArrayRef<const SCEV *> Subscripts, ArrayRef<const SCEV *> Sizes,
               Value *AccessValue, MemoryKind Kind);

  /// Create a new MemoryAccess that corresponds to @p AccRel.
  ///
  /// Along with @p Stmt and @p AccType it uses information about dimension
  /// lengths of the accessed array, the type of the accessed array elements,
  /// the name of the accessed array that is derived from the object accessible
  /// via @p AccRel.
  ///
  /// @param Stmt       The parent statement.
  /// @param AccType    Whether read or write access.
  /// @param AccRel     The access relation that describes the memory access.
  MemoryAccess(ScopStmt *Stmt, AccessType AccType, isl::map AccRel);

  MemoryAccess(const MemoryAccess &) = delete;
  MemoryAccess &operator=(const MemoryAccess &) = delete;
  ~MemoryAccess();

  /// Add a new incoming block/value pairs for this PHI/ExitPHI access.
  ///
  /// @param IncomingBlock The PHI's incoming block.
  /// @param IncomingValue The value when reaching the PHI from the @p
  ///                      IncomingBlock.
  void addIncoming(BasicBlock *IncomingBlock, Value *IncomingValue) {
    assert(!isRead());
    assert(isAnyPHIKind());
    Incoming.emplace_back(std::make_pair(IncomingBlock, IncomingValue));
  }

  /// Return the list of possible PHI/ExitPHI values.
  ///
  /// After code generation moves some PHIs around during region simplification,
  /// we cannot reliably locate the original PHI node and its incoming values
  /// anymore. For this reason we remember these explicitly for all PHI-kind
  /// accesses.
  ArrayRef<std::pair<BasicBlock *, Value *>> getIncoming() const {
    assert(isAnyPHIKind());
    return Incoming;
  }

  /// Get the type of a memory access.
  enum AccessType getType() { return AccType; }

  /// Is this a reduction like access?
  bool isReductionLike() const { return RedType != RT_NONE; }

  /// Is this a read memory access?
  bool isRead() const { return AccType == MemoryAccess::READ; }

  /// Is this a must-write memory access?
  bool isMustWrite() const { return AccType == MemoryAccess::MUST_WRITE; }

  /// Is this a may-write memory access?
  bool isMayWrite() const { return AccType == MemoryAccess::MAY_WRITE; }

  /// Is this a write memory access?
  bool isWrite() const { return isMustWrite() || isMayWrite(); }

  /// Is this a memory intrinsic access (memcpy, memset, memmove)?
  bool isMemoryIntrinsic() const {
    return isa<MemIntrinsic>(getAccessInstruction());
  }

  /// Check if a new access relation was imported or set by a pass.
  bool hasNewAccessRelation() const { return !NewAccessRelation.is_null(); }

  /// Return the newest access relation of this access.
  ///
  /// There are two possibilities:
  ///   1) The original access relation read from the LLVM-IR.
  ///   2) A new access relation imported from a json file or set by another
  ///      pass (e.g., for privatization).
  ///
  /// As 2) is by construction "newer" than 1) we return the new access
  /// relation if present.
  ///
  isl::map getLatestAccessRelation() const {
    return hasNewAccessRelation() ? getNewAccessRelation()
                                  : getOriginalAccessRelation();
  }

  /// Old name of getLatestAccessRelation().
  isl::map getAccessRelation() const { return getLatestAccessRelation(); }

  /// Get an isl map describing the memory address accessed.
  ///
  /// In most cases the memory address accessed is well described by the access
  /// relation obtained with getAccessRelation. However, in case of arrays
  /// accessed with types of different size the access relation maps one access
  /// to multiple smaller address locations. This method returns an isl map that
  /// relates each dynamic statement instance to the unique memory location
  /// that is loaded from / stored to.
  ///
  /// For an access relation { S[i] -> A[o] : 4i <= o <= 4i + 3 } this method
  /// will return the address function { S[i] -> A[4i] }.
  ///
  /// @returns The address function for this memory access.
  isl::map getAddressFunction() const;

  /// Return the access relation after the schedule was applied.
  isl::pw_multi_aff
  applyScheduleToAccessRelation(isl::union_map Schedule) const;

  /// Get an isl string representing the access function read from IR.
  std::string getOriginalAccessRelationStr() const;

  /// Get an isl string representing a new access function, if available.
  std::string getNewAccessRelationStr() const;

  /// Get an isl string representing the latest access relation.
  std::string getAccessRelationStr() const;

  /// Get the original base address of this access (e.g. A for A[i+j]) when
  /// detected.
  ///
  /// This address may differ from the base address referenced by the original
  /// ScopArrayInfo to which this array belongs, as this memory access may
  /// have been canonicalized to a ScopArrayInfo which has a different but
  /// identically-valued base pointer in case invariant load hoisting is
  /// enabled.
  Value *getOriginalBaseAddr() const { return BaseAddr; }

  /// Get the detection-time base array isl::id for this access.
  isl::id getOriginalArrayId() const;

  /// Get the base array isl::id for this access, modifiable through
  /// setNewAccessRelation().
  isl::id getLatestArrayId() const;

  /// Old name of getOriginalArrayId().
  isl::id getArrayId() const { return getOriginalArrayId(); }

  /// Get the detection-time ScopArrayInfo object for the base address.
  const ScopArrayInfo *getOriginalScopArrayInfo() const;

  /// Get the ScopArrayInfo object for the base address, or the one set
  /// by setNewAccessRelation().
  const ScopArrayInfo *getLatestScopArrayInfo() const;

  /// Legacy name of getOriginalScopArrayInfo().
  const ScopArrayInfo *getScopArrayInfo() const {
    return getOriginalScopArrayInfo();
  }

  /// Return a string representation of the access's reduction type.
  const std::string getReductionOperatorStr() const;

  /// Return a string representation of the reduction type @p RT.
  static const std::string getReductionOperatorStr(ReductionType RT);

  /// Return the element type of the accessed array wrt. this access.
  Type *getElementType() const { return ElementType; }

  /// Return the access value of this memory access.
  Value *getAccessValue() const { return AccessValue; }

  /// Return llvm::Value that is stored by this access, if available.
  ///
  /// PHI nodes may not have a unique value available that is stored, as in
  /// case of region statements one out of possibly several llvm::Values
  /// might be stored. In this case nullptr is returned.
  Value *tryGetValueStored() {
    assert(isWrite() && "Only write statement store values");
    if (isAnyPHIKind()) {
      if (Incoming.size() == 1)
        return Incoming[0].second;
      return nullptr;
    }
    return AccessValue;
  }

  /// Return the access instruction of this memory access.
  Instruction *getAccessInstruction() const { return AccessInstruction; }

  /// Return the number of access function subscript.
  unsigned getNumSubscripts() const { return Subscripts.size(); }

  /// Return the access function subscript in the dimension @p Dim.
  const SCEV *getSubscript(unsigned Dim) const { return Subscripts[Dim]; }

  /// Compute the isl representation for the SCEV @p E wrt. this access.
  ///
  /// Note that this function will also adjust the invalid context accordingly.
  isl::pw_aff getPwAff(const SCEV *E);

  /// Get the invalid domain for this access.
  isl::set getInvalidDomain() const { return InvalidDomain; }

  /// Get the invalid context for this access.
  isl::set getInvalidContext() const { return getInvalidDomain().params(); }

  /// Get the stride of this memory access in the specified Schedule. Schedule
  /// is a map from the statement to a schedule where the innermost dimension is
  /// the dimension of the innermost loop containing the statement.
  isl::set getStride(isl::map Schedule) const;

  /// Get the FortranArrayDescriptor corresponding to this memory access if
  /// it exists, and nullptr otherwise.
  Value *getFortranArrayDescriptor() const { return this->FAD; }

  /// Is the stride of the access equal to a certain width? Schedule is a map
  /// from the statement to a schedule where the innermost dimension is the
  /// dimension of the innermost loop containing the statement.
  bool isStrideX(isl::map Schedule, int StrideWidth) const;

  /// Is consecutive memory accessed for a given statement instance set?
  /// Schedule is a map from the statement to a schedule where the innermost
  /// dimension is the dimension of the innermost loop containing the
  /// statement.
  bool isStrideOne(isl::map Schedule) const;

  /// Is always the same memory accessed for a given statement instance set?
  /// Schedule is a map from the statement to a schedule where the innermost
  /// dimension is the dimension of the innermost loop containing the
  /// statement.
  bool isStrideZero(isl::map Schedule) const;

  /// Return the kind when this access was first detected.
  MemoryKind getOriginalKind() const {
    assert(!getOriginalScopArrayInfo() /* not yet initialized */ ||
           getOriginalScopArrayInfo()->getKind() == Kind);
    return Kind;
  }

  /// Return the kind considering a potential setNewAccessRelation.
  MemoryKind getLatestKind() const {
    return getLatestScopArrayInfo()->getKind();
  }

  /// Whether this is an access of an explicit load or store in the IR.
  bool isOriginalArrayKind() const {
    return getOriginalKind() == MemoryKind::Array;
  }

  /// Whether storage memory is either an custom .s2a/.phiops alloca
  /// (false) or an existing pointer into an array (true).
  bool isLatestArrayKind() const {
    return getLatestKind() == MemoryKind::Array;
  }

  /// Old name of isOriginalArrayKind.
  bool isArrayKind() const { return isOriginalArrayKind(); }

  /// Whether this access is an array to a scalar memory object, without
  /// considering changes by setNewAccessRelation.
  ///
  /// Scalar accesses are accesses to MemoryKind::Value, MemoryKind::PHI or
  /// MemoryKind::ExitPHI.
  bool isOriginalScalarKind() const {
    return getOriginalKind() != MemoryKind::Array;
  }

  /// Whether this access is an array to a scalar memory object, also
  /// considering changes by setNewAccessRelation.
  bool isLatestScalarKind() const {
    return getLatestKind() != MemoryKind::Array;
  }

  /// Old name of isOriginalScalarKind.
  bool isScalarKind() const { return isOriginalScalarKind(); }

  /// Was this MemoryAccess detected as a scalar dependences?
  bool isOriginalValueKind() const {
    return getOriginalKind() == MemoryKind::Value;
  }

  /// Is this MemoryAccess currently modeling scalar dependences?
  bool isLatestValueKind() const {
    return getLatestKind() == MemoryKind::Value;
  }

  /// Old name of isOriginalValueKind().
  bool isValueKind() const { return isOriginalValueKind(); }

  /// Was this MemoryAccess detected as a special PHI node access?
  bool isOriginalPHIKind() const {
    return getOriginalKind() == MemoryKind::PHI;
  }

  /// Is this MemoryAccess modeling special PHI node accesses, also
  /// considering a potential change by setNewAccessRelation?
  bool isLatestPHIKind() const { return getLatestKind() == MemoryKind::PHI; }

  /// Old name of isOriginalPHIKind.
  bool isPHIKind() const { return isOriginalPHIKind(); }

  /// Was this MemoryAccess detected as the accesses of a PHI node in the
  /// SCoP's exit block?
  bool isOriginalExitPHIKind() const {
    return getOriginalKind() == MemoryKind::ExitPHI;
  }

  /// Is this MemoryAccess modeling the accesses of a PHI node in the
  /// SCoP's exit block? Can be changed to an array access using
  /// setNewAccessRelation().
  bool isLatestExitPHIKind() const {
    return getLatestKind() == MemoryKind::ExitPHI;
  }

  /// Old name of isOriginalExitPHIKind().
  bool isExitPHIKind() const { return isOriginalExitPHIKind(); }

  /// Was this access detected as one of the two PHI types?
  bool isOriginalAnyPHIKind() const {
    return isOriginalPHIKind() || isOriginalExitPHIKind();
  }

  /// Does this access originate from one of the two PHI types? Can be
  /// changed to an array access using setNewAccessRelation().
  bool isLatestAnyPHIKind() const {
    return isLatestPHIKind() || isLatestExitPHIKind();
  }

  /// Old name of isOriginalAnyPHIKind().
  bool isAnyPHIKind() const { return isOriginalAnyPHIKind(); }

  /// Get the statement that contains this memory access.
  ScopStmt *getStatement() const { return Statement; }

  /// Get the reduction type of this access
  ReductionType getReductionType() const { return RedType; }

  /// Set the array descriptor corresponding to the Array on which the
  /// memory access is performed.
  void setFortranArrayDescriptor(Value *FAD);

  /// Update the original access relation.
  ///
  /// We need to update the original access relation during scop construction,
  /// when unifying the memory accesses that access the same scop array info
  /// object. After the scop has been constructed, the original access relation
  /// should not be changed any more. Instead setNewAccessRelation should
  /// be called.
  void setAccessRelation(isl::map AccessRelation);

  /// Set the updated access relation read from JSCOP file.
  void setNewAccessRelation(isl::map NewAccessRelation);

  /// Return whether the MemoryyAccess is a partial access. That is, the access
  /// is not executed in some instances of the parent statement's domain.
  bool isLatestPartialAccess() const;

  /// Mark this a reduction like access
  void markAsReductionLike(ReductionType RT) { RedType = RT; }

  /// Align the parameters in the access relation to the scop context
  void realignParams();

  /// Update the dimensionality of the memory access.
  ///
  /// During scop construction some memory accesses may not be constructed with
  /// their full dimensionality, but outer dimensions may have been omitted if
  /// they took the value 'zero'. By updating the dimensionality of the
  /// statement we add additional zero-valued dimensions to match the
  /// dimensionality of the ScopArrayInfo object that belongs to this memory
  /// access.
  void updateDimensionality();

  /// Get identifier for the memory access.
  ///
  /// This identifier is unique for all accesses that belong to the same scop
  /// statement.
  isl::id getId() const;

  /// Print the MemoryAccess.
  ///
  /// @param OS The output stream the MemoryAccess is printed to.
  void print(raw_ostream &OS) const;

#if !defined(NDEBUG) || defined(LLVM_ENABLE_DUMP)
  /// Print the MemoryAccess to stderr.
  void dump() const;
#endif

  /// Is the memory access affine?
  bool isAffine() const { return IsAffine; }
};

raw_ostream &operator<<(raw_ostream &OS, MemoryAccess::ReductionType RT);

/// Ordered list type to hold accesses.
using MemoryAccessList = std::forward_list<MemoryAccess *>;

/// Helper structure for invariant memory accesses.
struct InvariantAccess {
  /// The memory access that is (partially) invariant.
  MemoryAccess *MA;

  /// The context under which the access is not invariant.
  isl::set NonHoistableCtx;
};

/// Ordered container type to hold invariant accesses.
using InvariantAccessesTy = SmallVector<InvariantAccess, 8>;

/// Type for equivalent invariant accesses and their domain context.
struct InvariantEquivClassTy {
  /// The pointer that identifies this equivalence class
  const SCEV *IdentifyingPointer;

  /// Memory accesses now treated invariant
  ///
  /// These memory accesses access the pointer location that identifies
  /// this equivalence class. They are treated as invariant and hoisted during
  /// code generation.
  MemoryAccessList InvariantAccesses;

  /// The execution context under which the memory location is accessed
  ///
  /// It is the union of the execution domains of the memory accesses in the
  /// InvariantAccesses list.
  isl::set ExecutionContext;

  /// The type of the invariant access
  ///
  /// It is used to differentiate between differently typed invariant loads from
  /// the same location.
  Type *AccessType;
};

/// Type for invariant accesses equivalence classes.
using InvariantEquivClassesTy = SmallVector<InvariantEquivClassTy, 8>;

/// Statement of the Scop
///
/// A Scop statement represents an instruction in the Scop.
///
/// It is further described by its iteration domain, its schedule and its data
/// accesses.
/// At the moment every statement represents a single basic block of LLVM-IR.
class ScopStmt {
  friend class ScopBuilder;

public:
  /// Create the ScopStmt from a BasicBlock.
  ScopStmt(Scop &parent, BasicBlock &bb, Loop *SurroundingLoop,
           std::vector<Instruction *> Instructions, int Count);

  /// Create an overapproximating ScopStmt for the region @p R.
  ///
  /// @param EntryBlockInstructions The list of instructions that belong to the
  ///                               entry block of the region statement.
  ///                               Instructions are only tracked for entry
  ///                               blocks for now. We currently do not allow
  ///                               to modify the instructions of blocks later
  ///                               in the region statement.
  ScopStmt(Scop &parent, Region &R, Loop *SurroundingLoop,
           std::vector<Instruction *> EntryBlockInstructions);

  /// Create a copy statement.
  ///
  /// @param Stmt       The parent statement.
  /// @param SourceRel  The source location.
  /// @param TargetRel  The target location.
  /// @param Domain     The original domain under which the copy statement would
  ///                   be executed.
  ScopStmt(Scop &parent, isl::map SourceRel, isl::map TargetRel,
           isl::set Domain);

  ScopStmt(const ScopStmt &) = delete;
  const ScopStmt &operator=(const ScopStmt &) = delete;
  ~ScopStmt();

private:
  /// Polyhedral description
  //@{

  /// The Scop containing this ScopStmt.
  Scop &Parent;

  /// The domain under which this statement is not modeled precisely.
  ///
  /// The invalid domain for a statement describes all parameter combinations
  /// under which the statement looks to be executed but is in fact not because
  /// some assumption/restriction makes the statement/scop invalid.
  isl::set InvalidDomain;

  /// The iteration domain describes the set of iterations for which this
  /// statement is executed.
  ///
  /// Example:
  ///     for (i = 0; i < 100 + b; ++i)
  ///       for (j = 0; j < i; ++j)
  ///         S(i,j);
  ///
  /// 'S' is executed for different values of i and j. A vector of all
  /// induction variables around S (i, j) is called iteration vector.
  /// The domain describes the set of possible iteration vectors.
  ///
  /// In this case it is:
  ///
  ///     Domain: 0 <= i <= 100 + b
  ///             0 <= j <= i
  ///
  /// A pair of statement and iteration vector (S, (5,3)) is called statement
  /// instance.
  isl::set Domain;

  /// The memory accesses of this statement.
  ///
  /// The only side effects of a statement are its memory accesses.
  using MemoryAccessVec = SmallVector<MemoryAccess *, 8>;
  MemoryAccessVec MemAccs;

  /// Mapping from instructions to (scalar) memory accesses.
  DenseMap<const Instruction *, MemoryAccessList> InstructionToAccess;

  /// The set of values defined elsewhere required in this ScopStmt and
  ///        their MemoryKind::Value READ MemoryAccesses.
  DenseMap<Value *, MemoryAccess *> ValueReads;

  /// The set of values defined in this ScopStmt that are required
  ///        elsewhere, mapped to their MemoryKind::Value WRITE MemoryAccesses.
  DenseMap<Instruction *, MemoryAccess *> ValueWrites;

  /// Map from PHI nodes to its incoming value when coming from this
  ///        statement.
  ///
  /// Non-affine subregions can have multiple exiting blocks that are incoming
  /// blocks of the PHI nodes. This map ensures that there is only one write
  /// operation for the complete subregion. A PHI selecting the relevant value
  /// will be inserted.
  DenseMap<PHINode *, MemoryAccess *> PHIWrites;

  /// Map from PHI nodes to its read access in this statement.
  DenseMap<PHINode *, MemoryAccess *> PHIReads;

  //@}

  /// A SCoP statement represents either a basic block (affine/precise case) or
  /// a whole region (non-affine case).
  ///
  /// Only one of the following two members will therefore be set and indicate
  /// which kind of statement this is.
  ///
  ///{

  /// The BasicBlock represented by this statement (in the affine case).
  BasicBlock *BB = nullptr;

  /// The region represented by this statement (in the non-affine case).
  Region *R = nullptr;

  ///}

  /// The isl AST build for the new generated AST.
  isl::ast_build Build;

  SmallVector<Loop *, 4> NestLoops;

  std::string BaseName;

  /// The closest loop that contains this statement.
  Loop *SurroundingLoop;

  /// Vector for Instructions in this statement.
  std::vector<Instruction *> Instructions;

  /// Remove @p MA from dictionaries pointing to them.
  void removeAccessData(MemoryAccess *MA);

public:
  /// Get an isl_ctx pointer.
  isl::ctx getIslCtx() const;

  /// Get the iteration domain of this ScopStmt.
  ///
  /// @return The iteration domain of this ScopStmt.
  isl::set getDomain() const;

  /// Get the space of the iteration domain
  ///
  /// @return The space of the iteration domain
  isl::space getDomainSpace() const;

  /// Get the id of the iteration domain space
  ///
  /// @return The id of the iteration domain space
  isl::id getDomainId() const;

  /// Get an isl string representing this domain.
  std::string getDomainStr() const;

  /// Get the schedule function of this ScopStmt.
  ///
  /// @return The schedule function of this ScopStmt, if it does not contain
  /// extension nodes, and nullptr, otherwise.
  isl::map getSchedule() const;

  /// Get an isl string representing this schedule.
  ///
  /// @return An isl string representing this schedule, if it does not contain
  /// extension nodes, and an empty string, otherwise.
  std::string getScheduleStr() const;

  /// Get the invalid domain for this statement.
  isl::set getInvalidDomain() const { return InvalidDomain; }

  /// Get the invalid context for this statement.
  isl::set getInvalidContext() const { return getInvalidDomain().params(); }

  /// Set the invalid context for this statement to @p ID.
  void setInvalidDomain(isl::set ID);

  /// Get the BasicBlock represented by this ScopStmt (if any).
  ///
  /// @return The BasicBlock represented by this ScopStmt, or null if the
  ///         statement represents a region.
  BasicBlock *getBasicBlock() const { return BB; }

  /// Return true if this statement represents a single basic block.
  bool isBlockStmt() const { return BB != nullptr; }

  /// Return true if this is a copy statement.
  bool isCopyStmt() const { return BB == nullptr && R == nullptr; }

  /// Get the region represented by this ScopStmt (if any).
  ///
  /// @return The region represented by this ScopStmt, or null if the statement
  ///         represents a basic block.
  Region *getRegion() const { return R; }

  /// Return true if this statement represents a whole region.
  bool isRegionStmt() const { return R != nullptr; }

  /// Return a BasicBlock from this statement.
  ///
  /// For block statements, it returns the BasicBlock itself. For subregion
  /// statements, return its entry block.
  BasicBlock *getEntryBlock() const;

  /// Return whether @p L is boxed within this statement.
  bool contains(const Loop *L) const {
    // Block statements never contain loops.
    if (isBlockStmt())
      return false;

    return getRegion()->contains(L);
  }

  /// Return whether this statement represents @p BB.
  bool represents(BasicBlock *BB) const {
    if (isCopyStmt())
      return false;
    if (isBlockStmt())
      return BB == getBasicBlock();
    return getRegion()->contains(BB);
  }

  /// Return whether this statement contains @p Inst.
  bool contains(Instruction *Inst) const {
    if (!Inst)
      return false;
    if (isBlockStmt())
      return std::find(Instructions.begin(), Instructions.end(), Inst) !=
             Instructions.end();
    return represents(Inst->getParent());
  }

  /// Return the closest innermost loop that contains this statement, but is not
  /// contained in it.
  ///
  /// For block statement, this is just the loop that contains the block. Region
  /// statements can contain boxed loops, so getting the loop of one of the
  /// region's BBs might return such an inner loop. For instance, the region's
  /// entry could be a header of a loop, but the region might extend to BBs
  /// after the loop exit. Similarly, the region might only contain parts of the
  /// loop body and still include the loop header.
  ///
  /// Most of the time the surrounding loop is the top element of #NestLoops,
  /// except when it is empty. In that case it return the loop that the whole
  /// SCoP is contained in. That can be nullptr if there is no such loop.
  Loop *getSurroundingLoop() const {
    assert(!isCopyStmt() &&
           "No surrounding loop for artificially created statements");
    return SurroundingLoop;
  }

  /// Return true if this statement does not contain any accesses.
  bool isEmpty() const { return MemAccs.empty(); }

  /// Find all array accesses for @p Inst.
  ///
  /// @param Inst The instruction accessing an array.
  ///
  /// @return A list of array accesses (MemoryKind::Array) accessed by @p Inst.
  ///         If there is no such access, it returns nullptr.
  const MemoryAccessList *
  lookupArrayAccessesFor(const Instruction *Inst) const {
    auto It = InstructionToAccess.find(Inst);
    if (It == InstructionToAccess.end())
      return nullptr;
    if (It->second.empty())
      return nullptr;
    return &It->second;
  }

  /// Return the only array access for @p Inst, if existing.
  ///
  /// @param Inst The instruction for which to look up the access.
  /// @returns The unique array memory access related to Inst or nullptr if
  ///          no array access exists
  MemoryAccess *getArrayAccessOrNULLFor(const Instruction *Inst) const {
    auto It = InstructionToAccess.find(Inst);
    if (It == InstructionToAccess.end())
      return nullptr;

    MemoryAccess *ArrayAccess = nullptr;

    for (auto Access : It->getSecond()) {
      if (!Access->isArrayKind())
        continue;

      assert(!ArrayAccess && "More then one array access for instruction");

      ArrayAccess = Access;
    }

    return ArrayAccess;
  }

  /// Return the only array access for @p Inst.
  ///
  /// @param Inst The instruction for which to look up the access.
  /// @returns The unique array memory access related to Inst.
  MemoryAccess &getArrayAccessFor(const Instruction *Inst) const {
    MemoryAccess *ArrayAccess = getArrayAccessOrNULLFor(Inst);

    assert(ArrayAccess && "No array access found for instruction!");
    return *ArrayAccess;
  }

  /// Return the MemoryAccess that writes the value of an instruction
  ///        defined in this statement, or nullptr if not existing, respectively
  ///        not yet added.
  MemoryAccess *lookupValueWriteOf(Instruction *Inst) const {
    assert((isRegionStmt() && R->contains(Inst)) ||
           (!isRegionStmt() && Inst->getParent() == BB));
    return ValueWrites.lookup(Inst);
  }

  /// Return the MemoryAccess that reloads a value, or nullptr if not
  ///        existing, respectively not yet added.
  MemoryAccess *lookupValueReadOf(Value *Inst) const {
    return ValueReads.lookup(Inst);
  }

  /// Return the MemoryAccess that loads a PHINode value, or nullptr if not
  /// existing, respectively not yet added.
  MemoryAccess *lookupPHIReadOf(PHINode *PHI) const {
    return PHIReads.lookup(PHI);
  }

  /// Return the PHI write MemoryAccess for the incoming values from any
  ///        basic block in this ScopStmt, or nullptr if not existing,
  ///        respectively not yet added.
  MemoryAccess *lookupPHIWriteOf(PHINode *PHI) const {
    assert(isBlockStmt() || R->getExit() == PHI->getParent());
    return PHIWrites.lookup(PHI);
  }

  /// Return the input access of the value, or null if no such MemoryAccess
  /// exists.
  ///
  /// The input access is the MemoryAccess that makes an inter-statement value
  /// available in this statement by reading it at the start of this statement.
  /// This can be a MemoryKind::Value if defined in another statement or a
  /// MemoryKind::PHI if the value is a PHINode in this statement.
  MemoryAccess *lookupInputAccessOf(Value *Val) const {
    if (isa<PHINode>(Val))
      if (auto InputMA = lookupPHIReadOf(cast<PHINode>(Val))) {
        assert(!lookupValueReadOf(Val) && "input accesses must be unique; a "
                                          "statement cannot read a .s2a and "
                                          ".phiops simultaneously");
        return InputMA;
      }

    if (auto *InputMA = lookupValueReadOf(Val))
      return InputMA;

    return nullptr;
  }

  /// Add @p Access to this statement's list of accesses.
  ///
  /// @param Access  The access to add.
  /// @param Prepend If true, will add @p Access before all other instructions
  ///                (instead of appending it).
  void addAccess(MemoryAccess *Access, bool Preprend = false);

  /// Remove a MemoryAccess from this statement.
  ///
  /// Note that scalar accesses that are caused by MA will
  /// be eliminated too.
  void removeMemoryAccess(MemoryAccess *MA);

  /// Remove @p MA from this statement.
  ///
  /// In contrast to removeMemoryAccess(), no other access will be eliminated.
  void removeSingleMemoryAccess(MemoryAccess *MA);

  using iterator = MemoryAccessVec::iterator;
  using const_iterator = MemoryAccessVec::const_iterator;

  iterator begin() { return MemAccs.begin(); }
  iterator end() { return MemAccs.end(); }
  const_iterator begin() const { return MemAccs.begin(); }
  const_iterator end() const { return MemAccs.end(); }
  size_t size() const { return MemAccs.size(); }

  unsigned getNumIterators() const;

  Scop *getParent() { return &Parent; }
  const Scop *getParent() const { return &Parent; }

  const std::vector<Instruction *> &getInstructions() const {
    return Instructions;
  }

  /// Set the list of instructions for this statement. It replaces the current
  /// list.
  void setInstructions(ArrayRef<Instruction *> Range) {
    Instructions.assign(Range.begin(), Range.end());
  }

  std::vector<Instruction *>::const_iterator insts_begin() const {
    return Instructions.begin();
  }

  std::vector<Instruction *>::const_iterator insts_end() const {
    return Instructions.end();
  }

  /// The range of instructions in this statement.
  iterator_range<std::vector<Instruction *>::const_iterator> insts() const {
    return {insts_begin(), insts_end()};
  }

  /// Insert an instruction before all other instructions in this statement.
  void prependInstruction(Instruction *Inst) {
    Instructions.insert(Instructions.begin(), Inst);
  }

  const char *getBaseName() const;

  /// Set the isl AST build.
  void setAstBuild(isl::ast_build B) { Build = B; }

  /// Get the isl AST build.
  isl::ast_build getAstBuild() const { return Build; }

  /// Restrict the domain of the statement.
  ///
  /// @param NewDomain The new statement domain.
  void restrictDomain(isl::set NewDomain);

  /// Get the loop for a dimension.
  ///
  /// @param Dimension The dimension of the induction variable
  /// @return The loop at a certain dimension.
  Loop *getLoopForDimension(unsigned Dimension) const;

  /// Align the parameters in the statement to the scop context
  void realignParams();

  /// Print the ScopStmt.
  ///
  /// @param OS                The output stream the ScopStmt is printed to.
  /// @param PrintInstructions Whether to print the statement's instructions as
  ///                          well.
  void print(raw_ostream &OS, bool PrintInstructions) const;

  /// Print the instructions in ScopStmt.
  ///
  void printInstructions(raw_ostream &OS) const;

  /// Check whether there is a value read access for @p V in this statement, and
  /// if not, create one.
  ///
  /// This allows to add MemoryAccesses after the initial creation of the Scop
  /// by ScopBuilder.
  ///
  /// @return The already existing or newly created MemoryKind::Value READ
  /// MemoryAccess.
  ///
  /// @see ScopBuilder::ensureValueRead(Value*,ScopStmt*)
  MemoryAccess *ensureValueRead(Value *V);

#if !defined(NDEBUG) || defined(LLVM_ENABLE_DUMP)
  /// Print the ScopStmt to stderr.
  void dump() const;
#endif
};

/// Print ScopStmt S to raw_ostream OS.
raw_ostream &operator<<(raw_ostream &OS, const ScopStmt &S);

/// Static Control Part
///
/// A Scop is the polyhedral representation of a control flow region detected
/// by the Scop detection. It is generated by translating the LLVM-IR and
/// abstracting its effects.
///
/// A Scop consists of a set of:
///
///   * A set of statements executed in the Scop.
///
///   * A set of global parameters
///   Those parameters are scalar integer values, which are constant during
///   execution.
///
///   * A context
///   This context contains information about the values the parameters
///   can take and relations between different parameters.
class Scop {
public:
  /// Type to represent a pair of minimal/maximal access to an array.
  using MinMaxAccessTy = std::pair<isl::pw_multi_aff, isl::pw_multi_aff>;

  /// Vector of minimal/maximal accesses to different arrays.
  using MinMaxVectorTy = SmallVector<MinMaxAccessTy, 4>;

  /// Pair of minimal/maximal access vectors representing
  /// read write and read only accesses
  using MinMaxVectorPairTy = std::pair<MinMaxVectorTy, MinMaxVectorTy>;

  /// Vector of pair of minimal/maximal access vectors representing
  /// non read only and read only accesses for each alias group.
  using MinMaxVectorPairVectorTy = SmallVector<MinMaxVectorPairTy, 4>;

private:
  friend class ScopBuilder;

  /// Isl context.
  ///
  /// We need a shared_ptr with reference counter to delete the context when all
  /// isl objects are deleted. We will distribute the shared_ptr to all objects
  /// that use the context to create isl objects, and increase the reference
  /// counter. By doing this, we guarantee that the context is deleted when we
  /// delete the last object that creates isl objects with the context. This
  /// declaration needs to be the first in class to gracefully destroy all isl
  /// objects before the context.
  std::shared_ptr<isl_ctx> IslCtx;

  ScalarEvolution *SE;
  DominatorTree *DT;

  /// The underlying Region.
  Region &R;

  /// The name of the SCoP (identical to the regions name)
  std::string name;

  /// The ID to be assigned to the next Scop in a function
  static int NextScopID;

  /// The name of the function currently under consideration
  static std::string CurrentFunc;

  // Access functions of the SCoP.
  //
  // This owns all the MemoryAccess objects of the Scop created in this pass.
  AccFuncVector AccessFunctions;

  /// Flag to indicate that the scheduler actually optimized the SCoP.
  bool IsOptimized = false;

  /// True if the underlying region has a single exiting block.
  bool HasSingleExitEdge;

  /// Flag to remember if the SCoP contained an error block or not.
  bool HasErrorBlock = false;

  /// Max loop depth.
  unsigned MaxLoopDepth = 0;

  /// Number of copy statements.
  unsigned CopyStmtsNum = 0;

  /// Flag to indicate if the Scop is to be skipped.
  bool SkipScop = false;

  using StmtSet = std::list<ScopStmt>;

  /// The statements in this Scop.
  StmtSet Stmts;

  /// Parameters of this Scop
  ParameterSetTy Parameters;

  /// Mapping from parameters to their ids.
  DenseMap<const SCEV *, isl::id> ParameterIds;

  /// The context of the SCoP created during SCoP detection.
  ScopDetection::DetectionContext &DC;

  /// OptimizationRemarkEmitter object for displaying diagnostic remarks
  OptimizationRemarkEmitter &ORE;

  /// A map from basic blocks to vector of SCoP statements. Currently this
  /// vector comprises only of a single statement.
  DenseMap<BasicBlock *, std::vector<ScopStmt *>> StmtMap;

  /// A map from instructions to SCoP statements.
  DenseMap<Instruction *, ScopStmt *> InstStmtMap;

  /// A map from basic blocks to their domains.
  DenseMap<BasicBlock *, isl::set> DomainMap;

  /// Constraints on parameters.
  isl::set Context = nullptr;

  /// The affinator used to translate SCEVs to isl expressions.
  SCEVAffinator Affinator;

  using ArrayInfoMapTy =
      std::map<std::pair<AssertingVH<const Value>, MemoryKind>,
               std::unique_ptr<ScopArrayInfo>>;

  using ArrayNameMapTy = StringMap<std::unique_ptr<ScopArrayInfo>>;

  using ArrayInfoSetTy = SetVector<ScopArrayInfo *>;

  /// A map to remember ScopArrayInfo objects for all base pointers.
  ///
  /// As PHI nodes may have two array info objects associated, we add a flag
  /// that distinguishes between the PHI node specific ArrayInfo object
  /// and the normal one.
  ArrayInfoMapTy ScopArrayInfoMap;

  /// A map to remember ScopArrayInfo objects for all names of memory
  ///        references.
  ArrayNameMapTy ScopArrayNameMap;

  /// A set to remember ScopArrayInfo objects.
  /// @see Scop::ScopArrayInfoMap
  ArrayInfoSetTy ScopArrayInfoSet;

  /// The assumptions under which this scop was built.
  ///
  /// When constructing a scop sometimes the exact representation of a statement
  /// or condition would be very complex, but there is a common case which is a
  /// lot simpler, but which is only valid under certain assumptions. The
  /// assumed context records the assumptions taken during the construction of
  /// this scop and that need to be code generated as a run-time test.
  isl::set AssumedContext;

  /// The restrictions under which this SCoP was built.
  ///
  /// The invalid context is similar to the assumed context as it contains
  /// constraints over the parameters. However, while we need the constraints
  /// in the assumed context to be "true" the constraints in the invalid context
  /// need to be "false". Otherwise they behave the same.
  isl::set InvalidContext;

  /// Helper struct to remember assumptions.
  struct Assumption {
    /// The kind of the assumption (e.g., WRAPPING).
    AssumptionKind Kind;

    /// Flag to distinguish assumptions and restrictions.
    AssumptionSign Sign;

    /// The valid/invalid context if this is an assumption/restriction.
    isl::set Set;

    /// The location that caused this assumption.
    DebugLoc Loc;

    /// An optional block whose domain can simplify the assumption.
    BasicBlock *BB;
  };

  /// Collection to hold taken assumptions.
  ///
  /// There are two reasons why we want to record assumptions first before we
  /// add them to the assumed/invalid context:
  ///   1) If the SCoP is not profitable or otherwise invalid without the
  ///      assumed/invalid context we do not have to compute it.
  ///   2) Information about the context are gathered rather late in the SCoP
  ///      construction (basically after we know all parameters), thus the user
  ///      might see overly complicated assumptions to be taken while they will
  ///      only be simplified later on.
  SmallVector<Assumption, 8> RecordedAssumptions;

  /// The schedule of the SCoP
  ///
  /// The schedule of the SCoP describes the execution order of the statements
  /// in the scop by assigning each statement instance a possibly
  /// multi-dimensional execution time. The schedule is stored as a tree of
  /// schedule nodes.
  ///
  /// The most common nodes in a schedule tree are so-called band nodes. Band
  /// nodes map statement instances into a multi dimensional schedule space.
  /// This space can be seen as a multi-dimensional clock.
  ///
  /// Example:
  ///
  /// <S,(5,4)>  may be mapped to (5,4) by this schedule:
  ///
  /// s0 = i (Year of execution)
  /// s1 = j (Day of execution)
  ///
  /// or to (9, 20) by this schedule:
  ///
  /// s0 = i + j (Year of execution)
  /// s1 = 20 (Day of execution)
  ///
  /// The order statement instances are executed is defined by the
  /// schedule vectors they are mapped to. A statement instance
  /// <A, (i, j, ..)> is executed before a statement instance <B, (i', ..)>, if
  /// the schedule vector of A is lexicographic smaller than the schedule
  /// vector of B.
  ///
  /// Besides band nodes, schedule trees contain additional nodes that specify
  /// a textual ordering between two subtrees or filter nodes that filter the
  /// set of statement instances that will be scheduled in a subtree. There
  /// are also several other nodes. A full description of the different nodes
  /// in a schedule tree is given in the isl manual.
  isl::schedule Schedule = nullptr;

  /// The set of minimal/maximal accesses for each alias group.
  ///
  /// When building runtime alias checks we look at all memory instructions and
  /// build so called alias groups. Each group contains a set of accesses to
  /// different base arrays which might alias with each other. However, between
  /// alias groups there is no aliasing possible.
  ///
  /// In a program with int and float pointers annotated with tbaa information
  /// we would probably generate two alias groups, one for the int pointers and
  /// one for the float pointers.
  ///
  /// During code generation we will create a runtime alias check for each alias
  /// group to ensure the SCoP is executed in an alias free environment.
  MinMaxVectorPairVectorTy MinMaxAliasGroups;

  /// Mapping from invariant loads to the representing invariant load of
  ///        their equivalence class.
  ValueToValueMap InvEquivClassVMap;

  /// List of invariant accesses.
  InvariantEquivClassesTy InvariantEquivClasses;

  /// The smallest array index not yet assigned.
  long ArrayIdx = 0;

  /// The smallest statement index not yet assigned.
  long StmtIdx = 0;

  /// A number that uniquely represents a Scop within its function
  const int ID;

  /// Map of values to the MemoryAccess that writes its definition.
  ///
  /// There must be at most one definition per llvm::Instruction in a SCoP.
  DenseMap<Value *, MemoryAccess *> ValueDefAccs;

  /// Map of values to the MemoryAccess that reads a PHI.
  DenseMap<PHINode *, MemoryAccess *> PHIReadAccs;

  /// List of all uses (i.e. read MemoryAccesses) for a MemoryKind::Value
  /// scalar.
  DenseMap<const ScopArrayInfo *, SmallVector<MemoryAccess *, 4>> ValueUseAccs;

  /// List of all incoming values (write MemoryAccess) of a MemoryKind::PHI or
  /// MemoryKind::ExitPHI scalar.
  DenseMap<const ScopArrayInfo *, SmallVector<MemoryAccess *, 4>>
      PHIIncomingAccs;

  /// Return the ID for a new Scop within a function
  static int getNextID(std::string ParentFunc);

  /// Scop constructor; invoked from ScopBuilder::buildScop.
  Scop(Region &R, ScalarEvolution &SE, LoopInfo &LI, DominatorTree &DT,
       ScopDetection::DetectionContext &DC, OptimizationRemarkEmitter &ORE);

  //@}

  /// Initialize this ScopBuilder.
  void init(AliasAnalysis &AA, AssumptionCache &AC, DominatorTree &DT,
            LoopInfo &LI);

  /// Propagate domains that are known due to graph properties.
  ///
  /// As a CFG is mostly structured we use the graph properties to propagate
  /// domains without the need to compute all path conditions. In particular, if
  /// a block A dominates a block B and B post-dominates A we know that the
  /// domain of B is a superset of the domain of A. As we do not have
  /// post-dominator information available here we use the less precise region
  /// information. Given a region R, we know that the exit is always executed if
  /// the entry was executed, thus the domain of the exit is a superset of the
  /// domain of the entry. In case the exit can only be reached from within the
  /// region the domains are in fact equal. This function will use this property
  /// to avoid the generation of condition constraints that determine when a
  /// branch is taken. If @p BB is a region entry block we will propagate its
  /// domain to the region exit block. Additionally, we put the region exit
  /// block in the @p FinishedExitBlocks set so we can later skip edges from
  /// within the region to that block.
  ///
  /// @param BB                 The block for which the domain is currently
  ///                           propagated.
  /// @param BBLoop             The innermost affine loop surrounding @p BB.
  /// @param FinishedExitBlocks Set of region exits the domain was set for.
  /// @param LI                 The LoopInfo for the current function.
  /// @param InvalidDomainMap   BB to InvalidDomain map for the BB of current
  ///                           region.
  void propagateDomainConstraintsToRegionExit(
      BasicBlock *BB, Loop *BBLoop,
      SmallPtrSetImpl<BasicBlock *> &FinishedExitBlocks, LoopInfo &LI,
      DenseMap<BasicBlock *, isl::set> &InvalidDomainMap);

  /// Compute the union of predecessor domains for @p BB.
  ///
  /// To compute the union of all domains of predecessors of @p BB this
  /// function applies similar reasoning on the CFG structure as described for
  ///   @see propagateDomainConstraintsToRegionExit
  ///
  /// @param BB     The block for which the predecessor domains are collected.
  /// @param Domain The domain under which BB is executed.
  /// @param DT     The DominatorTree for the current function.
  /// @param LI     The LoopInfo for the current function.
  ///
  /// @returns The domain under which @p BB is executed.
  isl::set getPredecessorDomainConstraints(BasicBlock *BB, isl::set Domain,
                                           DominatorTree &DT, LoopInfo &LI);

  /// Add loop carried constraints to the header block of the loop @p L.
  ///
  /// @param L                The loop to process.
  /// @param LI               The LoopInfo for the current function.
  /// @param InvalidDomainMap BB to InvalidDomain map for the BB of current
  ///                         region.
  ///
  /// @returns True if there was no problem and false otherwise.
  bool addLoopBoundsToHeaderDomain(
      Loop *L, LoopInfo &LI,
      DenseMap<BasicBlock *, isl::set> &InvalidDomainMap);

  /// Compute the branching constraints for each basic block in @p R.
  ///
  /// @param R                The region we currently build branching conditions
  ///                         for.
  /// @param DT               The DominatorTree for the current function.
  /// @param LI               The LoopInfo for the current function.
  /// @param InvalidDomainMap BB to InvalidDomain map for the BB of current
  ///                         region.
  ///
  /// @returns True if there was no problem and false otherwise.
  bool buildDomainsWithBranchConstraints(
      Region *R, DominatorTree &DT, LoopInfo &LI,
      DenseMap<BasicBlock *, isl::set> &InvalidDomainMap);

  /// Propagate the domain constraints through the region @p R.
  ///
  /// @param R                The region we currently build branching conditions
  /// for.
  /// @param DT               The DominatorTree for the current function.
  /// @param LI               The LoopInfo for the current function.
  /// @param InvalidDomainMap BB to InvalidDomain map for the BB of current
  ///                         region.
  ///
  /// @returns True if there was no problem and false otherwise.
  bool propagateDomainConstraints(
      Region *R, DominatorTree &DT, LoopInfo &LI,
      DenseMap<BasicBlock *, isl::set> &InvalidDomainMap);

  /// Propagate invalid domains of statements through @p R.
  ///
  /// This method will propagate invalid statement domains through @p R and at
  /// the same time add error block domains to them. Additionally, the domains
  /// of error statements and those only reachable via error statements will be
  /// replaced by an empty set. Later those will be removed completely.
  ///
  /// @param R                The currently traversed region.
  /// @param DT               The DominatorTree for the current function.
  /// @param LI               The LoopInfo for the current function.
  /// @param InvalidDomainMap BB to InvalidDomain map for the BB of current
  ///                         region.
  //
  /// @returns True if there was no problem and false otherwise.
  bool propagateInvalidStmtDomains(
      Region *R, DominatorTree &DT, LoopInfo &LI,
      DenseMap<BasicBlock *, isl::set> &InvalidDomainMap);

  /// Compute the domain for each basic block in @p R.
  ///
  /// @param R                The region we currently traverse.
  /// @param DT               The DominatorTree for the current function.
  /// @param LI               The LoopInfo for the current function.
  /// @param InvalidDomainMap BB to InvalidDomain map for the BB of current
  ///                         region.
  ///
  /// @returns True if there was no problem and false otherwise.
  bool buildDomains(Region *R, DominatorTree &DT, LoopInfo &LI,
                    DenseMap<BasicBlock *, isl::set> &InvalidDomainMap);

  /// Add parameter constraints to @p C that imply a non-empty domain.
  isl::set addNonEmptyDomainConstraints(isl::set C) const;

  /// Return the access for the base ptr of @p MA if any.
  MemoryAccess *lookupBasePtrAccess(MemoryAccess *MA);

  /// Check if the base ptr of @p MA is in the SCoP but not hoistable.
  bool hasNonHoistableBasePtrInScop(MemoryAccess *MA, isl::union_map Writes);

  /// Create equivalence classes for required invariant accesses.
  ///
  /// These classes will consolidate multiple required invariant loads from the
  /// same address in order to keep the number of dimensions in the SCoP
  /// description small. For each such class equivalence class only one
  /// representing element, hence one required invariant load, will be chosen
  /// and modeled as parameter. The method
  /// Scop::getRepresentingInvariantLoadSCEV() will replace each element from an
  /// equivalence class with the representing element that is modeled. As a
  /// consequence Scop::getIdForParam() will only return an id for the
  /// representing element of each equivalence class, thus for each required
  /// invariant location.
  void buildInvariantEquivalenceClasses();

  /// Return the context under which the access cannot be hoisted.
  ///
  /// @param Access The access to check.
  /// @param Writes The set of all memory writes in the scop.
  ///
  /// @return Return the context under which the access cannot be hoisted or a
  ///         nullptr if it cannot be hoisted at all.
  isl::set getNonHoistableCtx(MemoryAccess *Access, isl::union_map Writes);

  /// Verify that all required invariant loads have been hoisted.
  ///
  /// Invariant load hoisting is not guaranteed to hoist all loads that were
  /// assumed to be scop invariant during scop detection. This function checks
  /// for cases where the hoisting failed, but where it would have been
  /// necessary for our scop modeling to be correct. In case of insufficient
  /// hoisting the scop is marked as invalid.
  ///
  /// In the example below Bound[1] is required to be invariant:
  ///
  /// for (int i = 1; i < Bound[0]; i++)
  ///   for (int j = 1; j < Bound[1]; j++)
  ///     ...
  void verifyInvariantLoads();

  /// Hoist invariant memory loads and check for required ones.
  ///
  /// We first identify "common" invariant loads, thus loads that are invariant
  /// and can be hoisted. Then we check if all required invariant loads have
  /// been identified as (common) invariant. A load is a required invariant load
  /// if it was assumed to be invariant during SCoP detection, e.g., to assume
  /// loop bounds to be affine or runtime alias checks to be placeable. In case
  /// a required invariant load was not identified as (common) invariant we will
  /// drop this SCoP. An example for both "common" as well as required invariant
  /// loads is given below:
  ///
  /// for (int i = 1; i < *LB[0]; i++)
  ///   for (int j = 1; j < *LB[1]; j++)
  ///     A[i][j] += A[0][0] + (*V);
  ///
  /// Common inv. loads: V, A[0][0], LB[0], LB[1]
  /// Required inv. loads: LB[0], LB[1], (V, if it may alias with A or LB)
  void hoistInvariantLoads();

  /// Canonicalize arrays with base pointers from the same equivalence class.
  ///
  /// Some context: in our normal model we assume that each base pointer is
  /// related to a single specific memory region, where memory regions
  /// associated with different base pointers are disjoint. Consequently we do
  /// not need to compute additional data dependences that model possible
  /// overlaps of these memory regions. To verify our assumption we compute
  /// alias checks that verify that modeled arrays indeed do not overlap. In
  /// case an overlap is detected the runtime check fails and we fall back to
  /// the original code.
  ///
  /// In case of arrays where the base pointers are know to be identical,
  /// because they are dynamically loaded by accesses that are in the same
  /// invariant load equivalence class, such run-time alias check would always
  /// be false.
  ///
  /// This function makes sure that we do not generate consistently failing
  /// run-time checks for code that contains distinct arrays with known
  /// equivalent base pointers. It identifies for each invariant load
  /// equivalence class a single canonical array and canonicalizes all memory
  /// accesses that reference arrays that have base pointers that are known to
  /// be equal to the base pointer of such a canonical array to this canonical
  /// array.
  ///
  /// We currently do not canonicalize arrays for which certain memory accesses
  /// have been hoisted as loop invariant.
  void canonicalizeDynamicBasePtrs();

  /// Check if @p MA can always be hoisted without execution context.
  bool canAlwaysBeHoisted(MemoryAccess *MA, bool StmtInvalidCtxIsEmpty,
                          bool MAInvalidCtxIsEmpty,
                          bool NonHoistableCtxIsEmpty);

  /// Add invariant loads listed in @p InvMAs with the domain of @p Stmt.
  void addInvariantLoads(ScopStmt &Stmt, InvariantAccessesTy &InvMAs);

  /// Create an id for @p Param and store it in the ParameterIds map.
  void createParameterId(const SCEV *Param);

  /// Build the Context of the Scop.
  void buildContext();

  /// Add user provided parameter constraints to context (source code).
  void addUserAssumptions(AssumptionCache &AC, DominatorTree &DT, LoopInfo &LI,
                          DenseMap<BasicBlock *, isl::set> &InvalidDomainMap);

  /// Add user provided parameter constraints to context (command line).
  void addUserContext();

  /// Add the bounds of the parameters to the context.
  void addParameterBounds();

  /// Simplify the assumed and invalid context.
  void simplifyContexts();

  /// Get the representing SCEV for @p S if applicable, otherwise @p S.
  ///
  /// Invariant loads of the same location are put in an equivalence class and
  /// only one of them is chosen as a representing element that will be
  /// modeled as a parameter. The others have to be normalized, i.e.,
  /// replaced by the representing element of their equivalence class, in order
  /// to get the correct parameter value, e.g., in the SCEVAffinator.
  ///
  /// @param S The SCEV to normalize.
  ///
  /// @return The representing SCEV for invariant loads or @p S if none.
  const SCEV *getRepresentingInvariantLoadSCEV(const SCEV *S) const;

  /// Create a new SCoP statement for @p BB.
  ///
  /// A new statement for @p BB will be created and added to the statement
  /// vector
  /// and map.
  ///
  /// @param BB              The basic block we build the statement for.
  /// @param SurroundingLoop The loop the created statement is contained in.
  /// @param Instructions    The instructions in the statement.
  /// @param Count           The index of the created statement in @p BB.
  void addScopStmt(BasicBlock *BB, Loop *SurroundingLoop,
                   std::vector<Instruction *> Instructions, int Count);

  /// Create a new SCoP statement for @p R.
  ///
  /// A new statement for @p R will be created and added to the statement vector
  /// and map.
  ///
  /// @param R                      The region we build the statement for.
  /// @param SurroundingLoop        The loop the created statement is contained
  ///                               in.
  /// @param EntryBlockInstructions The (interesting) instructions in the
  ///                               entry block of the region statement.
  void addScopStmt(Region *R, Loop *SurroundingLoop,
                   std::vector<Instruction *> EntryBlockInstructions);

  /// Update access dimensionalities.
  ///
  /// When detecting memory accesses different accesses to the same array may
  /// have built with different dimensionality, as outer zero-values dimensions
  /// may not have been recognized as separate dimensions. This function goes
  /// again over all memory accesses and updates their dimensionality to match
  /// the dimensionality of the underlying ScopArrayInfo object.
  void updateAccessDimensionality();

  /// Fold size constants to the right.
  ///
  /// In case all memory accesses in a given dimension are multiplied with a
  /// common constant, we can remove this constant from the individual access
  /// functions and move it to the size of the memory access. We do this as this
  /// increases the size of the innermost dimension, consequently widens the
  /// valid range the array subscript in this dimension can evaluate to, and
  /// as a result increases the likelihood that our delinearization is
  /// correct.
  ///
  /// Example:
  ///
  ///    A[][n]
  ///    S[i,j] -> A[2i][2j+1]
  ///    S[i,j] -> A[2i][2j]
  ///
  ///    =>
  ///
  ///    A[][2n]
  ///    S[i,j] -> A[i][2j+1]
  ///    S[i,j] -> A[i][2j]
  ///
  /// Constants in outer dimensions can arise when the elements of a parametric
  /// multi-dimensional array are not elementary data types, but e.g.,
  /// structures.
  void foldSizeConstantsToRight();

  /// Fold memory accesses to handle parametric offset.
  ///
  /// As a post-processing step, we 'fold' memory accesses to parametric
  /// offsets in the access functions. @see MemoryAccess::foldAccess for
  /// details.
  void foldAccessRelations();

  /// Assume that all memory accesses are within bounds.
  ///
  /// After we have built a model of all memory accesses, we need to assume
  /// that the model we built matches reality -- aka. all modeled memory
  /// accesses always remain within bounds. We do this as last step, after
  /// all memory accesses have been modeled and canonicalized.
  void assumeNoOutOfBounds();

  /// Remove statements from the list of scop statements.
  ///
  /// @param ShouldDelete A function that returns true if the statement passed
  ///                     to it should be deleted.
  void removeStmts(std::function<bool(ScopStmt &)> ShouldDelete);

  /// Removes @p Stmt from the StmtMap.
  void removeFromStmtMap(ScopStmt &Stmt);

  /// Removes all statements where the entry block of the statement does not
  /// have a corresponding domain in the domain map.
  void removeStmtNotInDomainMap();

  /// Mark arrays that have memory accesses with FortranArrayDescriptor.
  void markFortranArrays();

  /// Finalize all access relations.
  ///
  /// When building up access relations, temporary access relations that
  /// correctly represent each individual access are constructed. However, these
  /// access relations can be inconsistent or non-optimal when looking at the
  /// set of accesses as a whole. This function finalizes the memory accesses
  /// and constructs a globally consistent state.
  void finalizeAccesses();

  /// Construct the schedule of this SCoP.
  ///
  /// @param LI The LoopInfo for the current function.
  void buildSchedule(LoopInfo &LI);

  /// A loop stack element to keep track of per-loop information during
  ///        schedule construction.
  using LoopStackElementTy = struct LoopStackElement {
    // The loop for which we keep information.
    Loop *L;

    // The (possibly incomplete) schedule for this loop.
    isl::schedule Schedule;

    // The number of basic blocks in the current loop, for which a schedule has
    // already been constructed.
    unsigned NumBlocksProcessed;

    LoopStackElement(Loop *L, isl::schedule S, unsigned NumBlocksProcessed)
        : L(L), Schedule(S), NumBlocksProcessed(NumBlocksProcessed) {}
  };

  /// The loop stack used for schedule construction.
  ///
  /// The loop stack keeps track of schedule information for a set of nested
  /// loops as well as an (optional) 'nullptr' loop that models the outermost
  /// schedule dimension. The loops in a loop stack always have a parent-child
  /// relation where the loop at position n is the parent of the loop at
  /// position n + 1.
  using LoopStackTy = SmallVector<LoopStackElementTy, 4>;

  /// Construct schedule information for a given Region and add the
  ///        derived information to @p LoopStack.
  ///
  /// Given a Region we derive schedule information for all RegionNodes
  /// contained in this region ensuring that the assigned execution times
  /// correctly model the existing control flow relations.
  ///
  /// @param R              The region which to process.
  /// @param LoopStack      A stack of loops that are currently under
  ///                       construction.
  /// @param LI The LoopInfo for the current function.
  void buildSchedule(Region *R, LoopStackTy &LoopStack, LoopInfo &LI);

  /// Build Schedule for the region node @p RN and add the derived
  ///        information to @p LoopStack.
  ///
  /// In case @p RN is a BasicBlock or a non-affine Region, we construct the
  /// schedule for this @p RN and also finalize loop schedules in case the
  /// current @p RN completes the loop.
  ///
  /// In case @p RN is a not-non-affine Region, we delegate the construction to
  /// buildSchedule(Region *R, ...).
  ///
  /// @param RN             The RegionNode region traversed.
  /// @param LoopStack      A stack of loops that are currently under
  ///                       construction.
  /// @param LI The LoopInfo for the current function.
  void buildSchedule(RegionNode *RN, LoopStackTy &LoopStack, LoopInfo &LI);

  /// Collect all memory access relations of a given type.
  ///
  /// @param Predicate A predicate function that returns true if an access is
  ///                  of a given type.
  ///
  /// @returns The set of memory accesses in the scop that match the predicate.
  isl::union_map
  getAccessesOfType(std::function<bool(MemoryAccess &)> Predicate);

  /// @name Helper functions for printing the Scop.
  ///
  //@{
  void printContext(raw_ostream &OS) const;
  void printArrayInfo(raw_ostream &OS) const;
  void printStatements(raw_ostream &OS, bool PrintInstructions) const;
  void printAliasAssumptions(raw_ostream &OS) const;
  //@}

public:
  Scop(const Scop &) = delete;
  Scop &operator=(const Scop &) = delete;
  ~Scop();

  /// Get the count of copy statements added to this Scop.
  ///
  /// @return The count of copy statements added to this Scop.
  unsigned getCopyStmtsNum() { return CopyStmtsNum; }

  /// Create a new copy statement.
  ///
  /// A new statement will be created and added to the statement vector.
  ///
  /// @param Stmt       The parent statement.
  /// @param SourceRel  The source location.
  /// @param TargetRel  The target location.
  /// @param Domain     The original domain under which the copy statement would
  ///                   be executed.
  ScopStmt *addScopStmt(isl::map SourceRel, isl::map TargetRel,
                        isl::set Domain);

  /// Add the access function to all MemoryAccess objects of the Scop
  ///        created in this pass.
  void addAccessFunction(MemoryAccess *Access) {
    AccessFunctions.emplace_back(Access);

    // Register value definitions.
    if (Access->isWrite() && Access->isOriginalValueKind()) {
      assert(!ValueDefAccs.count(Access->getAccessValue()) &&
             "there can be just one definition per value");
      ValueDefAccs[Access->getAccessValue()] = Access;
    } else if (Access->isRead() && Access->isOriginalPHIKind()) {
      PHINode *PHI = cast<PHINode>(Access->getAccessInstruction());
      assert(!PHIReadAccs.count(PHI) &&
             "there can be just one PHI read per PHINode");
      PHIReadAccs[PHI] = Access;
    }
  }

  /// Add metadata for @p Access.
  void addAccessData(MemoryAccess *Access);

  /// Remove the metadata stored for @p Access.
  void removeAccessData(MemoryAccess *Access);

  /// Return the scalar evolution.
  ScalarEvolution *getSE() const;

  /// Return the dominator tree.
  DominatorTree *getDT() const { return DT; }

  /// Return the LoopInfo used for this Scop.
  LoopInfo *getLI() const { return Affinator.getLI(); }

  /// Get the count of parameters used in this Scop.
  ///
  /// @return The count of parameters used in this Scop.
  size_t getNumParams() const { return Parameters.size(); }

  /// Take a list of parameters and add the new ones to the scop.
  void addParams(const ParameterSetTy &NewParameters);

  /// Return an iterator range containing the scop parameters.
  iterator_range<ParameterSetTy::iterator> parameters() const {
    return make_range(Parameters.begin(), Parameters.end());
  }

  /// Return whether this scop is empty, i.e. contains no statements that
  /// could be executed.
  bool isEmpty() const { return Stmts.empty(); }

  const StringRef getName() const { return name; }

  using array_iterator = ArrayInfoSetTy::iterator;
  using const_array_iterator = ArrayInfoSetTy::const_iterator;
  using array_range = iterator_range<ArrayInfoSetTy::iterator>;
  using const_array_range = iterator_range<ArrayInfoSetTy::const_iterator>;

  inline array_iterator array_begin() { return ScopArrayInfoSet.begin(); }

  inline array_iterator array_end() { return ScopArrayInfoSet.end(); }

  inline const_array_iterator array_begin() const {
    return ScopArrayInfoSet.begin();
  }

  inline const_array_iterator array_end() const {
    return ScopArrayInfoSet.end();
  }

  inline array_range arrays() {
    return array_range(array_begin(), array_end());
  }

  inline const_array_range arrays() const {
    return const_array_range(array_begin(), array_end());
  }

  /// Return the isl_id that represents a certain parameter.
  ///
  /// @param Parameter A SCEV that was recognized as a Parameter.
  ///
  /// @return The corresponding isl_id or NULL otherwise.
  isl::id getIdForParam(const SCEV *Parameter) const;

  /// Get the maximum region of this static control part.
  ///
  /// @return The maximum region of this static control part.
  inline const Region &getRegion() const { return R; }
  inline Region &getRegion() { return R; }

  /// Return the function this SCoP is in.
  Function &getFunction() const { return *R.getEntry()->getParent(); }

  /// Check if @p L is contained in the SCoP.
  bool contains(const Loop *L) const { return R.contains(L); }

  /// Check if @p BB is contained in the SCoP.
  bool contains(const BasicBlock *BB) const { return R.contains(BB); }

  /// Check if @p I is contained in the SCoP.
  bool contains(const Instruction *I) const { return R.contains(I); }

  /// Return the unique exit block of the SCoP.
  BasicBlock *getExit() const { return R.getExit(); }

  /// Return the unique exiting block of the SCoP if any.
  BasicBlock *getExitingBlock() const { return R.getExitingBlock(); }

  /// Return the unique entry block of the SCoP.
  BasicBlock *getEntry() const { return R.getEntry(); }

  /// Return the unique entering block of the SCoP if any.
  BasicBlock *getEnteringBlock() const { return R.getEnteringBlock(); }

  /// Return true if @p BB is the exit block of the SCoP.
  bool isExit(BasicBlock *BB) const { return getExit() == BB; }

  /// Return a range of all basic blocks in the SCoP.
  Region::block_range blocks() const { return R.blocks(); }

  /// Return true if and only if @p BB dominates the SCoP.
  bool isDominatedBy(const DominatorTree &DT, BasicBlock *BB) const;

  /// Get the maximum depth of the loop.
  ///
  /// @return The maximum depth of the loop.
  inline unsigned getMaxLoopDepth() const { return MaxLoopDepth; }

  /// Return the invariant equivalence class for @p Val if any.
  InvariantEquivClassTy *lookupInvariantEquivClass(Value *Val);

  /// Return the set of invariant accesses.
  InvariantEquivClassesTy &getInvariantAccesses() {
    return InvariantEquivClasses;
  }

  /// Check if the scop has any invariant access.
  bool hasInvariantAccesses() { return !InvariantEquivClasses.empty(); }

  /// Mark the SCoP as optimized by the scheduler.
  void markAsOptimized() { IsOptimized = true; }

  /// Check if the SCoP has been optimized by the scheduler.
  bool isOptimized() const { return IsOptimized; }

  /// Mark the SCoP to be skipped by ScopPass passes.
  void markAsToBeSkipped() { SkipScop = true; }

  /// Check if the SCoP is to be skipped by ScopPass passes.
  bool isToBeSkipped() const { return SkipScop; }

  /// Return the ID of the Scop
  int getID() const { return ID; }

  /// Get the name of the entry and exit blocks of this Scop.
  ///
  /// These along with the function name can uniquely identify a Scop.
  ///
  /// @return std::pair whose first element is the entry name & second element
  ///         is the exit name.
  std::pair<std::string, std::string> getEntryExitStr() const;

  /// Get the name of this Scop.
  std::string getNameStr() const;

  /// Get the constraint on parameter of this Scop.
  ///
  /// @return The constraint on parameter of this Scop.
  isl::set getContext() const;

  /// Return space of isl context parameters.
  ///
  /// Returns the set of context parameters that are currently constrained. In
  /// case the full set of parameters is needed, see @getFullParamSpace.
  isl::space getParamSpace() const;

  /// Return the full space of parameters.
  ///
  /// getParamSpace will only return the parameters of the context that are
  /// actually constrained, whereas getFullParamSpace will return all
  //  parameters. This is useful in cases, where we need to ensure all
  //  parameters are available, as certain isl functions will abort if this is
  //  not the case.
  isl::space getFullParamSpace() const;

  /// Get the assumed context for this Scop.
  ///
  /// @return The assumed context of this Scop.
  isl::set getAssumedContext() const;

  /// Return true if the optimized SCoP can be executed.
  ///
  /// In addition to the runtime check context this will also utilize the domain
  /// constraints to decide it the optimized version can actually be executed.
  ///
  /// @returns True if the optimized SCoP can be executed.
  bool hasFeasibleRuntimeContext() const;

  /// Check if the assumption in @p Set is trivial or not.
  ///
  /// @param Set  The relations between parameters that are assumed to hold.
  /// @param Sign Enum to indicate if the assumptions in @p Set are positive
  ///             (needed/assumptions) or negative (invalid/restrictions).
  ///
  /// @returns True if the assumption @p Set is not trivial.
  bool isEffectiveAssumption(isl::set Set, AssumptionSign Sign);

  /// Track and report an assumption.
  ///
  /// Use 'clang -Rpass-analysis=polly-scops' or 'opt
  /// -pass-remarks-analysis=polly-scops' to output the assumptions.
  ///
  /// @param Kind The assumption kind describing the underlying cause.
  /// @param Set  The relations between parameters that are assumed to hold.
  /// @param Loc  The location in the source that caused this assumption.
  /// @param Sign Enum to indicate if the assumptions in @p Set are positive
  ///             (needed/assumptions) or negative (invalid/restrictions).
  /// @param BB   The block in which this assumption was taken. Used to
  ///             calculate hotness when emitting remark.
  ///
  /// @returns True if the assumption is not trivial.
  bool trackAssumption(AssumptionKind Kind, isl::set Set, DebugLoc Loc,
                       AssumptionSign Sign, BasicBlock *BB);

  /// Add assumptions to assumed context.
  ///
  /// The assumptions added will be assumed to hold during the execution of the
  /// scop. However, as they are generally not statically provable, at code
  /// generation time run-time checks will be generated that ensure the
  /// assumptions hold.
  ///
  /// WARNING: We currently exploit in simplifyAssumedContext the knowledge
  ///          that assumptions do not change the set of statement instances
  ///          executed.
  ///
  /// @param Kind The assumption kind describing the underlying cause.
  /// @param Set  The relations between parameters that are assumed to hold.
  /// @param Loc  The location in the source that caused this assumption.
  /// @param Sign Enum to indicate if the assumptions in @p Set are positive
  ///             (needed/assumptions) or negative (invalid/restrictions).
  /// @param BB   The block in which this assumption was taken. Used to
  ///             calculate hotness when emitting remark.
  void addAssumption(AssumptionKind Kind, isl::set Set, DebugLoc Loc,
                     AssumptionSign Sign, BasicBlock *BB);

  /// Record an assumption for later addition to the assumed context.
  ///
  /// This function will add the assumption to the RecordedAssumptions. This
  /// collection will be added (@see addAssumption) to the assumed context once
  /// all paramaters are known and the context is fully built.
  ///
  /// @param Kind The assumption kind describing the underlying cause.
  /// @param Set  The relations between parameters that are assumed to hold.
  /// @param Loc  The location in the source that caused this assumption.
  /// @param Sign Enum to indicate if the assumptions in @p Set are positive
  ///             (needed/assumptions) or negative (invalid/restrictions).
  /// @param BB   The block in which this assumption was taken. If it is
  ///             set, the domain of that block will be used to simplify the
  ///             actual assumption in @p Set once it is added. This is useful
  ///             if the assumption was created prior to the domain.
  void recordAssumption(AssumptionKind Kind, isl::set Set, DebugLoc Loc,
                        AssumptionSign Sign, BasicBlock *BB = nullptr);

  /// Add all recorded assumptions to the assumed context.
  void addRecordedAssumptions();

  /// Mark the scop as invalid.
  ///
  /// This method adds an assumption to the scop that is always invalid. As a
  /// result, the scop will not be optimized later on. This function is commonly
  /// called when a condition makes it impossible (or too compile time
  /// expensive) to process this scop any further.
  ///
  /// @param Kind The assumption kind describing the underlying cause.
  /// @param Loc  The location in the source that triggered .
  /// @param BB   The BasicBlock where it was triggered.
  void invalidate(AssumptionKind Kind, DebugLoc Loc, BasicBlock *BB = nullptr);

  /// Get the invalid context for this Scop.
  ///
  /// @return The invalid context of this Scop.
  isl::set getInvalidContext() const;

  /// Return true if and only if the InvalidContext is trivial (=empty).
  bool hasTrivialInvalidContext() const { return InvalidContext.is_empty(); }

  /// A vector of memory accesses that belong to an alias group.
  using AliasGroupTy = SmallVector<MemoryAccess *, 4>;

  /// A vector of alias groups.
  using AliasGroupVectorTy = SmallVector<Scop::AliasGroupTy, 4>;

  /// Build the alias checks for this SCoP.
  bool buildAliasChecks(AliasAnalysis &AA);

  /// Build all alias groups for this SCoP.
  ///
  /// @returns True if __no__ error occurred, false otherwise.
  bool buildAliasGroups(AliasAnalysis &AA);

  /// Build alias groups for all memory accesses in the Scop.
  ///
  /// Using the alias analysis and an alias set tracker we build alias sets
  /// for all memory accesses inside the Scop. For each alias set we then map
  /// the aliasing pointers back to the memory accesses we know, thus obtain
  /// groups of memory accesses which might alias. We also collect the set of
  /// arrays through which memory is written.
  ///
  /// @param AA A reference to the alias analysis.
  ///
  /// @returns A pair consistent of a vector of alias groups and a set of arrays
  ///          through which memory is written.
  std::tuple<AliasGroupVectorTy, DenseSet<const ScopArrayInfo *>>
  buildAliasGroupsForAccesses(AliasAnalysis &AA);

  ///  Split alias groups by iteration domains.
  ///
  ///  We split each group based on the domains of the minimal/maximal accesses.
  ///  That means two minimal/maximal accesses are only in a group if their
  ///  access domains intersect. Otherwise, they are in different groups.
  ///
  ///  @param AliasGroups The alias groups to split
  void splitAliasGroupsByDomain(AliasGroupVectorTy &AliasGroups);

  /// Build a given alias group and its access data.
  ///
  /// @param AliasGroup     The alias group to build.
  /// @param HasWriteAccess A set of arrays through which memory is not only
  ///                       read, but also written.
  ///
  /// @returns True if __no__ error occurred, false otherwise.
  bool buildAliasGroup(Scop::AliasGroupTy &AliasGroup,
                       DenseSet<const ScopArrayInfo *> HasWriteAccess);

  /// Return all alias groups for this SCoP.
  const MinMaxVectorPairVectorTy &getAliasGroups() const {
    return MinMaxAliasGroups;
  }

  /// Get an isl string representing the context.
  std::string getContextStr() const;

  /// Get an isl string representing the assumed context.
  std::string getAssumedContextStr() const;

  /// Get an isl string representing the invalid context.
  std::string getInvalidContextStr() const;

  /// Return the list of ScopStmts that represent the given @p BB.
  ArrayRef<ScopStmt *> getStmtListFor(BasicBlock *BB) const;

  /// Return the last statement representing @p BB.
  ///
  /// Of the sequence of statements that represent a @p BB, this is the last one
  /// to be executed. It is typically used to determine which instruction to add
  /// a MemoryKind::PHI WRITE to. For this purpose, it is not strictly required
  /// to be executed last, only that the incoming value is available in it.
  ScopStmt *getLastStmtFor(BasicBlock *BB) const;

  /// Return the ScopStmts that represents the Region @p R, or nullptr if
  ///        it is not represented by any statement in this Scop.
  ArrayRef<ScopStmt *> getStmtListFor(Region *R) const;

  /// Return the ScopStmts that represents @p RN; can return nullptr if
  ///        the RegionNode is not within the SCoP or has been removed due to
  ///        simplifications.
  ArrayRef<ScopStmt *> getStmtListFor(RegionNode *RN) const;

  /// Return the ScopStmt an instruction belongs to, or nullptr if it
  ///        does not belong to any statement in this Scop.
  ScopStmt *getStmtFor(Instruction *Inst) const {
    return InstStmtMap.lookup(Inst);
  }

  /// Return the number of statements in the SCoP.
  size_t getSize() const { return Stmts.size(); }

  /// @name Statements Iterators
  ///
  /// These iterators iterate over all statements of this Scop.
  //@{
  using iterator = StmtSet::iterator;
  using const_iterator = StmtSet::const_iterator;

  iterator begin() { return Stmts.begin(); }
  iterator end() { return Stmts.end(); }
  const_iterator begin() const { return Stmts.begin(); }
  const_iterator end() const { return Stmts.end(); }

  using reverse_iterator = StmtSet::reverse_iterator;
  using const_reverse_iterator = StmtSet::const_reverse_iterator;

  reverse_iterator rbegin() { return Stmts.rbegin(); }
  reverse_iterator rend() { return Stmts.rend(); }
  const_reverse_iterator rbegin() const { return Stmts.rbegin(); }
  const_reverse_iterator rend() const { return Stmts.rend(); }
  //@}

  /// Return the set of required invariant loads.
  const InvariantLoadsSetTy &getRequiredInvariantLoads() const {
    return DC.RequiredILS;
  }

  /// Add @p LI to the set of required invariant loads.
  void addRequiredInvariantLoad(LoadInst *LI) { DC.RequiredILS.insert(LI); }

  /// Return true if and only if @p LI is a required invariant load.
  bool isRequiredInvariantLoad(LoadInst *LI) const {
    return getRequiredInvariantLoads().count(LI);
  }

  /// Return the set of boxed (thus overapproximated) loops.
  const BoxedLoopsSetTy &getBoxedLoops() const { return DC.BoxedLoopsSet; }

  /// Return true if and only if @p R is a non-affine subregion.
  bool isNonAffineSubRegion(const Region *R) {
    return DC.NonAffineSubRegionSet.count(R);
  }

  const MapInsnToMemAcc &getInsnToMemAccMap() const { return DC.InsnToMemAcc; }

  /// Return the (possibly new) ScopArrayInfo object for @p Access.
  ///
  /// @param ElementType The type of the elements stored in this array.
  /// @param Kind        The kind of the array info object.
  /// @param BaseName    The optional name of this memory reference.
  ScopArrayInfo *getOrCreateScopArrayInfo(Value *BasePtr, Type *ElementType,
                                          ArrayRef<const SCEV *> Sizes,
                                          MemoryKind Kind,
                                          const char *BaseName = nullptr);

  /// Create an array and return the corresponding ScopArrayInfo object.
  ///
  /// @param ElementType The type of the elements stored in this array.
  /// @param BaseName    The name of this memory reference.
  /// @param Sizes       The sizes of dimensions.
  ScopArrayInfo *createScopArrayInfo(Type *ElementType,
                                     const std::string &BaseName,
                                     const std::vector<unsigned> &Sizes);

  /// Return the cached ScopArrayInfo object for @p BasePtr.
  ///
  /// @param BasePtr   The base pointer the object has been stored for.
  /// @param Kind      The kind of array info object.
  ///
  /// @returns The ScopArrayInfo pointer or NULL if no such pointer is
  ///          available.
  const ScopArrayInfo *getScopArrayInfoOrNull(Value *BasePtr, MemoryKind Kind);

  /// Return the cached ScopArrayInfo object for @p BasePtr.
  ///
  /// @param BasePtr   The base pointer the object has been stored for.
  /// @param Kind      The kind of array info object.
  ///
  /// @returns The ScopArrayInfo pointer (may assert if no such pointer is
  ///          available).
  const ScopArrayInfo *getScopArrayInfo(Value *BasePtr, MemoryKind Kind);

  /// Invalidate ScopArrayInfo object for base address.
  ///
  /// @param BasePtr The base pointer of the ScopArrayInfo object to invalidate.
  /// @param Kind    The Kind of the ScopArrayInfo object.
  void invalidateScopArrayInfo(Value *BasePtr, MemoryKind Kind) {
    auto It = ScopArrayInfoMap.find(std::make_pair(BasePtr, Kind));
    if (It == ScopArrayInfoMap.end())
      return;
    ScopArrayInfoSet.remove(It->second.get());
    ScopArrayInfoMap.erase(It);
  }

  void setContext(isl::set NewContext);

  /// Align the parameters in the statement to the scop context
  void realignParams();

  /// Return true if this SCoP can be profitably optimized.
  ///
  /// @param ScalarsAreUnprofitable Never consider statements with scalar writes
  ///                               as profitably optimizable.
  ///
  /// @return Whether this SCoP can be profitably optimized.
  bool isProfitable(bool ScalarsAreUnprofitable) const;

  /// Return true if the SCoP contained at least one error block.
  bool hasErrorBlock() const { return HasErrorBlock; }

  /// Return true if the underlying region has a single exiting block.
  bool hasSingleExitEdge() const { return HasSingleExitEdge; }

  /// Print the static control part.
  ///
  /// @param OS The output stream the static control part is printed to.
  /// @param PrintInstructions Whether to print the statement's instructions as
  ///                          well.
  void print(raw_ostream &OS, bool PrintInstructions) const;

#if !defined(NDEBUG) || defined(LLVM_ENABLE_DUMP)
  /// Print the ScopStmt to stderr.
  void dump() const;
#endif

  /// Get the isl context of this static control part.
  ///
  /// @return The isl context of this static control part.
  isl::ctx getIslCtx() const;

  /// Directly return the shared_ptr of the context.
  const std::shared_ptr<isl_ctx> &getSharedIslCtx() const { return IslCtx; }

  /// Compute the isl representation for the SCEV @p E
  ///
  /// @param E  The SCEV that should be translated.
  /// @param BB An (optional) basic block in which the isl_pw_aff is computed.
  ///           SCEVs known to not reference any loops in the SCoP can be
  ///           passed without a @p BB.
  /// @param NonNegative Flag to indicate the @p E has to be non-negative.
  ///
  /// Note that this function will always return a valid isl_pw_aff. However, if
  /// the translation of @p E was deemed to complex the SCoP is invalidated and
  /// a dummy value of appropriate dimension is returned. This allows to bail
  /// for complex cases without "error handling code" needed on the users side.
  PWACtx getPwAff(const SCEV *E, BasicBlock *BB = nullptr,
                  bool NonNegative = false);

  /// Compute the isl representation for the SCEV @p E
  ///
  /// This function is like @see Scop::getPwAff() but strips away the invalid
  /// domain part associated with the piecewise affine function.
  isl::pw_aff getPwAffOnly(const SCEV *E, BasicBlock *BB = nullptr);

  /// Return the domain of @p Stmt.
  ///
  /// @param Stmt The statement for which the conditions should be returned.
  isl::set getDomainConditions(const ScopStmt *Stmt) const;

  /// Return the domain of @p BB.
  ///
  /// @param BB The block for which the conditions should be returned.
  isl::set getDomainConditions(BasicBlock *BB) const;

  /// Get a union set containing the iteration domains of all statements.
  isl::union_set getDomains() const;

  /// Get a union map of all may-writes performed in the SCoP.
  isl::union_map getMayWrites();

  /// Get a union map of all must-writes performed in the SCoP.
  isl::union_map getMustWrites();

  /// Get a union map of all writes performed in the SCoP.
  isl::union_map getWrites();

  /// Get a union map of all reads performed in the SCoP.
  isl::union_map getReads();

  /// Get a union map of all memory accesses performed in the SCoP.
  isl::union_map getAccesses();

  /// Get a union map of all memory accesses performed in the SCoP.
  ///
  /// @param Array The array to which the accesses should belong.
  isl::union_map getAccesses(ScopArrayInfo *Array);

  /// Get the schedule of all the statements in the SCoP.
  ///
  /// @return The schedule of all the statements in the SCoP, if the schedule of
  /// the Scop does not contain extension nodes, and nullptr, otherwise.
  isl::union_map getSchedule() const;

  /// Get a schedule tree describing the schedule of all statements.
  isl::schedule getScheduleTree() const;

  /// Update the current schedule
  ///
  /// NewSchedule The new schedule (given as a flat union-map).
  void setSchedule(isl::union_map NewSchedule);

  /// Update the current schedule
  ///
  /// NewSchedule The new schedule (given as schedule tree).
  void setScheduleTree(isl::schedule NewSchedule);

  /// Intersects the domains of all statements in the SCoP.
  ///
  /// @return true if a change was made
  bool restrictDomains(isl::union_set Domain);

  /// Get the depth of a loop relative to the outermost loop in the Scop.
  ///
  /// This will return
  ///    0 if @p L is an outermost loop in the SCoP
  ///   >0 for other loops in the SCoP
  ///   -1 if @p L is nullptr or there is no outermost loop in the SCoP
  int getRelativeLoopDepth(const Loop *L) const;

  /// Find the ScopArrayInfo associated with an isl Id
  ///        that has name @p Name.
  ScopArrayInfo *getArrayInfoByName(const std::string BaseName);

  /// Check whether @p Schedule contains extension nodes.
  ///
  /// @return true if @p Schedule contains extension nodes.
  static bool containsExtensionNode(isl::schedule Schedule);

  /// Simplify the SCoP representation.
  ///
  /// @param AfterHoisting Whether it is called after invariant load hoisting.
  ///                      When true, also removes statements without
  ///                      side-effects.
  void simplifySCoP(bool AfterHoisting);

  /// Get the next free array index.
  ///
  /// This function returns a unique index which can be used to identify an
  /// array.
  long getNextArrayIdx() { return ArrayIdx++; }

  /// Get the next free statement index.
  ///
  /// This function returns a unique index which can be used to identify a
  /// statement.
  long getNextStmtIdx() { return StmtIdx++; }

  /// Return the MemoryAccess that writes an llvm::Value, represented by a
  /// ScopArrayInfo.
  ///
  /// There can be at most one such MemoryAccess per llvm::Value in the SCoP.
  /// Zero is possible for read-only values.
  MemoryAccess *getValueDef(const ScopArrayInfo *SAI) const;

  /// Return all MemoryAccesses that us an llvm::Value, represented by a
  /// ScopArrayInfo.
  ArrayRef<MemoryAccess *> getValueUses(const ScopArrayInfo *SAI) const;

  /// Return the MemoryAccess that represents an llvm::PHINode.
  ///
  /// ExitPHIs's PHINode is not within the SCoPs. This function returns nullptr
  /// for them.
  MemoryAccess *getPHIRead(const ScopArrayInfo *SAI) const;

  /// Return all MemoryAccesses for all incoming statements of a PHINode,
  /// represented by a ScopArrayInfo.
  ArrayRef<MemoryAccess *> getPHIIncomings(const ScopArrayInfo *SAI) const;

  /// Return whether @p Inst has a use outside of this SCoP.
  bool isEscaping(Instruction *Inst);

  struct ScopStatistics {
    int NumAffineLoops = 0;
    int NumBoxedLoops = 0;

    int NumValueWrites = 0;
    int NumValueWritesInLoops = 0;
    int NumPHIWrites = 0;
    int NumPHIWritesInLoops = 0;
    int NumSingletonWrites = 0;
    int NumSingletonWritesInLoops = 0;
  };

  /// Collect statistic about this SCoP.
  ///
  /// These are most commonly used for LLVM's static counters (Statistic.h) in
  /// various places. If statistics are disabled, only zeros are returned to
  /// avoid the overhead.
  ScopStatistics getStatistics() const;
};

/// Print Scop scop to raw_ostream OS.
raw_ostream &operator<<(raw_ostream &OS, const Scop &scop);

/// The legacy pass manager's analysis pass to compute scop information
///        for a region.
class ScopInfoRegionPass : public RegionPass {
  /// The Scop pointer which is used to construct a Scop.
  std::unique_ptr<Scop> S;

public:
  static char ID; // Pass identification, replacement for typeid

  ScopInfoRegionPass() : RegionPass(ID) {}
  ~ScopInfoRegionPass() override = default;

  /// Build Scop object, the Polly IR of static control
  ///        part for the current SESE-Region.
  ///
  /// @return If the current region is a valid for a static control part,
  ///         return the Polly IR representing this static control part,
  ///         return null otherwise.
  Scop *getScop() { return S.get(); }
  const Scop *getScop() const { return S.get(); }

  /// Calculate the polyhedral scop information for a given Region.
  bool runOnRegion(Region *R, RGPassManager &RGM) override;

  void releaseMemory() override { S.reset(); }

  void print(raw_ostream &O, const Module *M = nullptr) const override;

  void getAnalysisUsage(AnalysisUsage &AU) const override;
};

class ScopInfo {
public:
  using RegionToScopMapTy = MapVector<Region *, std::unique_ptr<Scop>>;
  using reverse_iterator = RegionToScopMapTy::reverse_iterator;
  using const_reverse_iterator = RegionToScopMapTy::const_reverse_iterator;
  using iterator = RegionToScopMapTy::iterator;
  using const_iterator = RegionToScopMapTy::const_iterator;

private:
  /// A map of Region to its Scop object containing
  ///        Polly IR of static control part.
  RegionToScopMapTy RegionToScopMap;
  const DataLayout &DL;
  ScopDetection &SD;
  ScalarEvolution &SE;
  LoopInfo &LI;
  AliasAnalysis &AA;
  DominatorTree &DT;
  AssumptionCache &AC;
  OptimizationRemarkEmitter &ORE;

public:
  ScopInfo(const DataLayout &DL, ScopDetection &SD, ScalarEvolution &SE,
           LoopInfo &LI, AliasAnalysis &AA, DominatorTree &DT,
           AssumptionCache &AC, OptimizationRemarkEmitter &ORE);

  /// Get the Scop object for the given Region.
  ///
  /// @return If the given region is the maximal region within a scop, return
  ///         the scop object. If the given region is a subregion, return a
  ///         nullptr. Top level region containing the entry block of a function
  ///         is not considered in the scop creation.
  Scop *getScop(Region *R) const {
    auto MapIt = RegionToScopMap.find(R);
    if (MapIt != RegionToScopMap.end())
      return MapIt->second.get();
    return nullptr;
  }

  /// Recompute the Scop-Information for a function.
  ///
  /// This invalidates any iterators.
  void recompute();

  /// Handle invalidation explicitly
  bool invalidate(Function &F, const PreservedAnalyses &PA,
                  FunctionAnalysisManager::Invalidator &Inv);

  iterator begin() { return RegionToScopMap.begin(); }
  iterator end() { return RegionToScopMap.end(); }
  const_iterator begin() const { return RegionToScopMap.begin(); }
  const_iterator end() const { return RegionToScopMap.end(); }
  reverse_iterator rbegin() { return RegionToScopMap.rbegin(); }
  reverse_iterator rend() { return RegionToScopMap.rend(); }
  const_reverse_iterator rbegin() const { return RegionToScopMap.rbegin(); }
  const_reverse_iterator rend() const { return RegionToScopMap.rend(); }
  bool empty() const { return RegionToScopMap.empty(); }
};

struct ScopInfoAnalysis : public AnalysisInfoMixin<ScopInfoAnalysis> {
  static AnalysisKey Key;

  using Result = ScopInfo;

  Result run(Function &, FunctionAnalysisManager &);
};

struct ScopInfoPrinterPass : public PassInfoMixin<ScopInfoPrinterPass> {
  ScopInfoPrinterPass(raw_ostream &OS) : Stream(OS) {}

  PreservedAnalyses run(Function &, FunctionAnalysisManager &);

  raw_ostream &Stream;
};

//===----------------------------------------------------------------------===//
/// The legacy pass manager's analysis pass to compute scop information
///        for the whole function.
///
/// This pass will maintain a map of the maximal region within a scop to its
/// scop object for all the feasible scops present in a function.
/// This pass is an alternative to the ScopInfoRegionPass in order to avoid a
/// region pass manager.
class ScopInfoWrapperPass : public FunctionPass {
  std::unique_ptr<ScopInfo> Result;

public:
  ScopInfoWrapperPass() : FunctionPass(ID) {}
  ~ScopInfoWrapperPass() override = default;

  static char ID; // Pass identification, replacement for typeid

  ScopInfo *getSI() { return Result.get(); }
  const ScopInfo *getSI() const { return Result.get(); }

  /// Calculate all the polyhedral scops for a given function.
  bool runOnFunction(Function &F) override;

  void releaseMemory() override { Result.reset(); }

  void print(raw_ostream &O, const Module *M = nullptr) const override;

  void getAnalysisUsage(AnalysisUsage &AU) const override;
};

} // end namespace polly

#endif // POLLY_SCOPINFO_H