polly/lib/Analysis/ScopInfo.cpp

//===--------- ScopInfo.cpp  - Create Scops from LLVM IR ------------------===//
//
//                     The LLVM Compiler Infrastructure
//
// This file is distributed under the University of Illinois Open Source
// License. See LICENSE.TXT for details.
//
//===----------------------------------------------------------------------===//
//
// Create a polyhedral description for a static control flow region.
//
// The pass creates a polyhedral description of the Scops detected by the Scop
// detection derived from their LLVM-IR code.
//
// This representation is shared among several tools in the polyhedral
// community, which are e.g. Cloog, Pluto, Loopo, Graphite.
//
//===----------------------------------------------------------------------===//

#include "polly/LinkAllPasses.h"
#include "polly/Options.h"
#include "polly/ScopInfo.h"
#include "polly/Support/GICHelper.h"
#include "polly/Support/SCEVValidator.h"
#include "polly/Support/ScopHelper.h"
#include "polly/CodeGen/BlockGenerators.h"
#include "llvm/ADT/MapVector.h"
#include "llvm/ADT/SetVector.h"
#include "llvm/ADT/Statistic.h"
#include "llvm/ADT/STLExtras.h"
#include "llvm/ADT/StringExtras.h"
#include "llvm/ADT/PostOrderIterator.h"
#include "llvm/Analysis/LoopIterator.h"
#include "llvm/Analysis/AliasAnalysis.h"
#include "llvm/Analysis/LoopInfo.h"
#include "llvm/Analysis/RegionIterator.h"
#include "llvm/Analysis/ScalarEvolutionExpressions.h"
#include "llvm/Support/Debug.h"
#include "isl/aff.h"
#include "isl/constraint.h"
#include "isl/local_space.h"
#include "isl/map.h"
#include "isl/options.h"
#include "isl/printer.h"
#include "isl/schedule.h"
#include "isl/schedule_node.h"
#include "isl/set.h"
#include "isl/union_map.h"
#include "isl/union_set.h"
#include "isl/val.h"
#include <sstream>
#include <string>
#include <vector>

using namespace llvm;
using namespace polly;

#define DEBUG_TYPE "polly-scops"

STATISTIC(ScopFound, "Number of valid Scops");
STATISTIC(RichScopFound, "Number of Scops containing a loop");

static cl::opt<bool> ModelReadOnlyScalars(
    "polly-analyze-read-only-scalars",
    cl::desc("Model read-only scalar values in the scop description"),
    cl::Hidden, cl::ZeroOrMore, cl::init(true), cl::cat(PollyCategory));

// Multiplicative reductions can be disabled separately as these kind of
// operations can overflow easily. Additive reductions and bit operations
// are in contrast pretty stable.
static cl::opt<bool> DisableMultiplicativeReductions(
    "polly-disable-multiplicative-reductions",
    cl::desc("Disable multiplicative reductions"), cl::Hidden, cl::ZeroOrMore,
    cl::init(false), cl::cat(PollyCategory));

static cl::opt<unsigned> RunTimeChecksMaxParameters(
    "polly-rtc-max-parameters",
    cl::desc("The maximal number of parameters allowed in RTCs."), cl::Hidden,
    cl::ZeroOrMore, cl::init(8), cl::cat(PollyCategory));

static cl::opt<unsigned> RunTimeChecksMaxArraysPerGroup(
    "polly-rtc-max-arrays-per-group",
    cl::desc("The maximal number of arrays to compare in each alias group."),
    cl::Hidden, cl::ZeroOrMore, cl::init(20), cl::cat(PollyCategory));
static cl::opt<std::string> UserContextStr(
    "polly-context", cl::value_desc("isl parameter set"),
    cl::desc("Provide additional constraints on the context parameters"),
    cl::init(""), cl::cat(PollyCategory));

static cl::opt<bool> DetectReductions("polly-detect-reductions",
                                      cl::desc("Detect and exploit reductions"),
                                      cl::Hidden, cl::ZeroOrMore,
                                      cl::init(true), cl::cat(PollyCategory));

//===----------------------------------------------------------------------===//
/// Helper Classes

void Comparison::print(raw_ostream &OS) const {
  // Not yet implemented.
}

// Create a sequence of two schedules. Either argument may be null and is
// interpreted as the empty schedule. Can also return null if both schedules are
// empty.
static __isl_give isl_schedule *
combineInSequence(__isl_take isl_schedule *Prev,
                  __isl_take isl_schedule *Succ) {
  if (!Prev)
    return Succ;
  if (!Succ)
    return Prev;

  return isl_schedule_sequence(Prev, Succ);
}

static __isl_give isl_set *addRangeBoundsToSet(__isl_take isl_set *S,
                                               const ConstantRange &Range,
                                               int dim,
                                               enum isl_dim_type type) {
  isl_val *V;
  isl_ctx *ctx = isl_set_get_ctx(S);

  bool useLowerUpperBound = Range.isSignWrappedSet() && !Range.isFullSet();
  const auto LB = useLowerUpperBound ? Range.getLower() : Range.getSignedMin();
  V = isl_valFromAPInt(ctx, LB, true);
  isl_set *SLB = isl_set_lower_bound_val(isl_set_copy(S), type, dim, V);

  const auto UB = useLowerUpperBound ? Range.getUpper() : Range.getSignedMax();
  V = isl_valFromAPInt(ctx, UB, true);
  if (useLowerUpperBound)
    V = isl_val_sub_ui(V, 1);
  isl_set *SUB = isl_set_upper_bound_val(S, type, dim, V);

  if (useLowerUpperBound)
    return isl_set_union(SLB, SUB);
  else
    return isl_set_intersect(SLB, SUB);
}

static const ScopArrayInfo *identifyBasePtrOriginSAI(Scop *S, Value *BasePtr) {
  LoadInst *BasePtrLI = dyn_cast<LoadInst>(BasePtr);
  if (!BasePtrLI)
    return nullptr;

  if (!S->getRegion().contains(BasePtrLI))
    return nullptr;

  ScalarEvolution &SE = *S->getSE();

  auto *OriginBaseSCEV =
      SE.getPointerBase(SE.getSCEV(BasePtrLI->getPointerOperand()));
  if (!OriginBaseSCEV)
    return nullptr;

  auto *OriginBaseSCEVUnknown = dyn_cast<SCEVUnknown>(OriginBaseSCEV);
  if (!OriginBaseSCEVUnknown)
    return nullptr;

  return S->getScopArrayInfo(OriginBaseSCEVUnknown->getValue());
}

ScopArrayInfo::ScopArrayInfo(Value *BasePtr, Type *ElementType, isl_ctx *Ctx,
                             const SmallVector<const SCEV *, 4> &DimensionSizes,
                             bool IsPHI, Scop *S)
    : BasePtr(BasePtr), ElementType(ElementType),
      DimensionSizes(DimensionSizes), IsPHI(IsPHI) {
  std::string BasePtrName =
      getIslCompatibleName("MemRef_", BasePtr, IsPHI ? "__phi" : "");
  Id = isl_id_alloc(Ctx, BasePtrName.c_str(), this);
  for (const SCEV *Expr : DimensionSizes) {
    isl_pw_aff *Size = S->getPwAff(Expr);
    DimensionSizesPw.push_back(Size);
  }

  BasePtrOriginSAI = identifyBasePtrOriginSAI(S, BasePtr);
  if (BasePtrOriginSAI)
    const_cast<ScopArrayInfo *>(BasePtrOriginSAI)->addDerivedSAI(this);
}

ScopArrayInfo::~ScopArrayInfo() {
  isl_id_free(Id);
  for (isl_pw_aff *Size : DimensionSizesPw)
    isl_pw_aff_free(Size);
}

std::string ScopArrayInfo::getName() const { return isl_id_get_name(Id); }

int ScopArrayInfo::getElemSizeInBytes() const {
  return ElementType->getPrimitiveSizeInBits() / 8;
}

isl_id *ScopArrayInfo::getBasePtrId() const { return isl_id_copy(Id); }

void ScopArrayInfo::dump() const { print(errs()); }

void ScopArrayInfo::print(raw_ostream &OS, bool SizeAsPwAff) const {
  OS.indent(8) << *getElementType() << " " << getName() << "[*]";
  for (unsigned u = 0; u < getNumberOfDimensions(); u++) {
    OS << "[";

    if (SizeAsPwAff)
      OS << " " << DimensionSizesPw[u] << " ";
    else
      OS << *DimensionSizes[u];

    OS << "]";
  }

  if (BasePtrOriginSAI)
    OS << " [BasePtrOrigin: " << BasePtrOriginSAI->getName() << "]";

  OS << " // Element size " << getElemSizeInBytes() << "\n";
}

const ScopArrayInfo *
ScopArrayInfo::getFromAccessFunction(__isl_keep isl_pw_multi_aff *PMA) {
  isl_id *Id = isl_pw_multi_aff_get_tuple_id(PMA, isl_dim_out);
  assert(Id && "Output dimension didn't have an ID");
  return getFromId(Id);
}

const ScopArrayInfo *ScopArrayInfo::getFromId(isl_id *Id) {
  void *User = isl_id_get_user(Id);
  const ScopArrayInfo *SAI = static_cast<ScopArrayInfo *>(User);
  isl_id_free(Id);
  return SAI;
}

void IRAccess::print(raw_ostream &OS) const {
  if (isRead())
    OS << "Read ";
  else {
    if (isMayWrite())
      OS << "May";
    OS << "Write ";
  }
  OS << BaseAddress->getName() << '[' << *Offset << "]\n";
}

const std::string
MemoryAccess::getReductionOperatorStr(MemoryAccess::ReductionType RT) {
  switch (RT) {
  case MemoryAccess::RT_NONE:
    llvm_unreachable("Requested a reduction operator string for a memory "
                     "access which isn't a reduction");
  case MemoryAccess::RT_ADD:
    return "+";
  case MemoryAccess::RT_MUL:
    return "*";
  case MemoryAccess::RT_BOR:
    return "|";
  case MemoryAccess::RT_BXOR:
    return "^";
  case MemoryAccess::RT_BAND:
    return "&";
  }
  llvm_unreachable("Unknown reduction type");
  return "";
}

/// @brief Return the reduction type for a given binary operator
static MemoryAccess::ReductionType getReductionType(const BinaryOperator *BinOp,
                                                    const Instruction *Load) {
  if (!BinOp)
    return MemoryAccess::RT_NONE;
  switch (BinOp->getOpcode()) {
  case Instruction::FAdd:
    if (!BinOp->hasUnsafeAlgebra())
      return MemoryAccess::RT_NONE;
  // Fall through
  case Instruction::Add:
    return MemoryAccess::RT_ADD;
  case Instruction::Or:
    return MemoryAccess::RT_BOR;
  case Instruction::Xor:
    return MemoryAccess::RT_BXOR;
  case Instruction::And:
    return MemoryAccess::RT_BAND;
  case Instruction::FMul:
    if (!BinOp->hasUnsafeAlgebra())
      return MemoryAccess::RT_NONE;
  // Fall through
  case Instruction::Mul:
    if (DisableMultiplicativeReductions)
      return MemoryAccess::RT_NONE;
    return MemoryAccess::RT_MUL;
  default:
    return MemoryAccess::RT_NONE;
  }
}
//===----------------------------------------------------------------------===//

MemoryAccess::~MemoryAccess() {
  isl_id_free(Id);
  isl_map_free(AccessRelation);
  isl_map_free(NewAccessRelation);
}

static MemoryAccess::AccessType getMemoryAccessType(const IRAccess &Access) {
  switch (Access.getType()) {
  case IRAccess::READ:
    return MemoryAccess::READ;
  case IRAccess::MUST_WRITE:
    return MemoryAccess::MUST_WRITE;
  case IRAccess::MAY_WRITE:
    return MemoryAccess::MAY_WRITE;
  }
  llvm_unreachable("Unknown IRAccess type!");
}

const ScopArrayInfo *MemoryAccess::getScopArrayInfo() const {
  isl_id *ArrayId = getArrayId();
  void *User = isl_id_get_user(ArrayId);
  const ScopArrayInfo *SAI = static_cast<ScopArrayInfo *>(User);
  isl_id_free(ArrayId);
  return SAI;
}

__isl_give isl_id *MemoryAccess::getArrayId() const {
  return isl_map_get_tuple_id(AccessRelation, isl_dim_out);
}

__isl_give isl_pw_multi_aff *MemoryAccess::applyScheduleToAccessRelation(
    __isl_take isl_union_map *USchedule) const {
  isl_map *Schedule, *ScheduledAccRel;
  isl_union_set *UDomain;

  UDomain = isl_union_set_from_set(getStatement()->getDomain());
  USchedule = isl_union_map_intersect_domain(USchedule, UDomain);
  Schedule = isl_map_from_union_map(USchedule);
  ScheduledAccRel = isl_map_apply_domain(getAccessRelation(), Schedule);
  return isl_pw_multi_aff_from_map(ScheduledAccRel);
}

__isl_give isl_map *MemoryAccess::getOriginalAccessRelation() const {
  return isl_map_copy(AccessRelation);
}

std::string MemoryAccess::getOriginalAccessRelationStr() const {
  return stringFromIslObj(AccessRelation);
}

__isl_give isl_space *MemoryAccess::getOriginalAccessRelationSpace() const {
  return isl_map_get_space(AccessRelation);
}

__isl_give isl_map *MemoryAccess::getNewAccessRelation() const {
  return isl_map_copy(NewAccessRelation);
}

std::string MemoryAccess::getNewAccessRelationStr() const {
  return stringFromIslObj(NewAccessRelation);
}

__isl_give isl_basic_map *
MemoryAccess::createBasicAccessMap(ScopStmt *Statement) {
  isl_space *Space = isl_space_set_alloc(Statement->getIslCtx(), 0, 1);
  Space = isl_space_align_params(Space, Statement->getDomainSpace());

  return isl_basic_map_from_domain_and_range(
      isl_basic_set_universe(Statement->getDomainSpace()),
      isl_basic_set_universe(Space));
}

// Formalize no out-of-bound access assumption
//
// When delinearizing array accesses we optimistically assume that the
// delinearized accesses do not access out of bound locations (the subscript
// expression of each array evaluates for each statement instance that is
// executed to a value that is larger than zero and strictly smaller than the
// size of the corresponding dimension). The only exception is the outermost
// dimension for which we do not need to assume any upper bound.  At this point
// we formalize this assumption to ensure that at code generation time the
// relevant run-time checks can be generated.
//
// To find the set of constraints necessary to avoid out of bound accesses, we
// first build the set of data locations that are not within array bounds. We
// then apply the reverse access relation to obtain the set of iterations that
// may contain invalid accesses and reduce this set of iterations to the ones
// that are actually executed by intersecting them with the domain of the
// statement. If we now project out all loop dimensions, we obtain a set of
// parameters that may cause statement instances to be executed that may
// possibly yield out of bound memory accesses. The complement of these
// constraints is the set of constraints that needs to be assumed to ensure such
// statement instances are never executed.
void MemoryAccess::assumeNoOutOfBound(const IRAccess &Access) {
  isl_space *Space = isl_space_range(getOriginalAccessRelationSpace());
  isl_set *Outside = isl_set_empty(isl_space_copy(Space));
  for (int i = 1, Size = Access.Subscripts.size(); i < Size; ++i) {
    isl_local_space *LS = isl_local_space_from_space(isl_space_copy(Space));
    isl_pw_aff *Var =
        isl_pw_aff_var_on_domain(isl_local_space_copy(LS), isl_dim_set, i);
    isl_pw_aff *Zero = isl_pw_aff_zero_on_domain(LS);

    isl_set *DimOutside;

    DimOutside = isl_pw_aff_lt_set(isl_pw_aff_copy(Var), Zero);
    isl_pw_aff *SizeE = Statement->getPwAff(Access.Sizes[i - 1]);

    SizeE = isl_pw_aff_drop_dims(SizeE, isl_dim_in, 0,
                                 Statement->getNumIterators());
    SizeE = isl_pw_aff_add_dims(SizeE, isl_dim_in,
                                isl_space_dim(Space, isl_dim_set));
    SizeE = isl_pw_aff_set_tuple_id(SizeE, isl_dim_in,
                                    isl_space_get_tuple_id(Space, isl_dim_set));

    DimOutside = isl_set_union(DimOutside, isl_pw_aff_le_set(SizeE, Var));

    Outside = isl_set_union(Outside, DimOutside);
  }

  Outside = isl_set_apply(Outside, isl_map_reverse(getAccessRelation()));
  Outside = isl_set_intersect(Outside, Statement->getDomain());
  Outside = isl_set_params(Outside);

  // Remove divs to avoid the construction of overly complicated assumptions.
  // Doing so increases the set of parameter combinations that are assumed to
  // not appear. This is always save, but may make the resulting run-time check
  // bail out more often than strictly necessary.
  Outside = isl_set_remove_divs(Outside);
  Outside = isl_set_complement(Outside);
  Statement->getParent()->addAssumption(Outside);
  isl_space_free(Space);
}

void MemoryAccess::computeBoundsOnAccessRelation(unsigned ElementSize) {
  ScalarEvolution *SE = Statement->getParent()->getSE();

  Value *Ptr = getPointerOperand(*getAccessInstruction());
  if (!Ptr || !SE->isSCEVable(Ptr->getType()))
    return;

  auto *PtrSCEV = SE->getSCEV(Ptr);
  if (isa<SCEVCouldNotCompute>(PtrSCEV))
    return;

  auto *BasePtrSCEV = SE->getPointerBase(PtrSCEV);
  if (BasePtrSCEV && !isa<SCEVCouldNotCompute>(BasePtrSCEV))
    PtrSCEV = SE->getMinusSCEV(PtrSCEV, BasePtrSCEV);

  const ConstantRange &Range = SE->getSignedRange(PtrSCEV);
  if (Range.isFullSet())
    return;

  bool isWrapping = Range.isSignWrappedSet();
  unsigned BW = Range.getBitWidth();
  const auto LB = isWrapping ? Range.getLower() : Range.getSignedMin();
  const auto UB = isWrapping ? Range.getUpper() : Range.getSignedMax();

  auto Min = LB.sdiv(APInt(BW, ElementSize));
  auto Max = (UB - APInt(BW, 1)).sdiv(APInt(BW, ElementSize));

  isl_set *AccessRange = isl_map_range(isl_map_copy(AccessRelation));
  AccessRange =
      addRangeBoundsToSet(AccessRange, ConstantRange(Min, Max), 0, isl_dim_set);
  AccessRelation = isl_map_intersect_range(AccessRelation, AccessRange);
}

__isl_give isl_map *MemoryAccess::foldAccess(const IRAccess &Access,
                                             __isl_take isl_map *AccessRelation,
                                             ScopStmt *Statement) {
  int Size = Access.Subscripts.size();

  for (int i = Size - 2; i >= 0; --i) {
    isl_space *Space;
    isl_map *MapOne, *MapTwo;
    isl_pw_aff *DimSize = Statement->getPwAff(Access.Sizes[i]);

    isl_space *SpaceSize = isl_pw_aff_get_space(DimSize);
    isl_pw_aff_free(DimSize);
    isl_id *ParamId = isl_space_get_dim_id(SpaceSize, isl_dim_param, 0);

    Space = isl_map_get_space(AccessRelation);
    Space = isl_space_map_from_set(isl_space_range(Space));
    Space = isl_space_align_params(Space, SpaceSize);

    int ParamLocation = isl_space_find_dim_by_id(Space, isl_dim_param, ParamId);
    isl_id_free(ParamId);

    MapOne = isl_map_universe(isl_space_copy(Space));
    for (int j = 0; j < Size; ++j)
      MapOne = isl_map_equate(MapOne, isl_dim_in, j, isl_dim_out, j);
    MapOne = isl_map_lower_bound_si(MapOne, isl_dim_in, i + 1, 0);

    MapTwo = isl_map_universe(isl_space_copy(Space));
    for (int j = 0; j < Size; ++j)
      if (j < i || j > i + 1)
        MapTwo = isl_map_equate(MapTwo, isl_dim_in, j, isl_dim_out, j);

    isl_local_space *LS = isl_local_space_from_space(Space);
    isl_constraint *C;
    C = isl_equality_alloc(isl_local_space_copy(LS));
    C = isl_constraint_set_constant_si(C, -1);
    C = isl_constraint_set_coefficient_si(C, isl_dim_in, i, 1);
    C = isl_constraint_set_coefficient_si(C, isl_dim_out, i, -1);
    MapTwo = isl_map_add_constraint(MapTwo, C);
    C = isl_equality_alloc(LS);
    C = isl_constraint_set_coefficient_si(C, isl_dim_in, i + 1, 1);
    C = isl_constraint_set_coefficient_si(C, isl_dim_out, i + 1, -1);
    C = isl_constraint_set_coefficient_si(C, isl_dim_param, ParamLocation, 1);
    MapTwo = isl_map_add_constraint(MapTwo, C);
    MapTwo = isl_map_upper_bound_si(MapTwo, isl_dim_in, i + 1, -1);

    MapOne = isl_map_union(MapOne, MapTwo);
    AccessRelation = isl_map_apply_range(AccessRelation, MapOne);
  }
  return AccessRelation;
}

MemoryAccess::MemoryAccess(const IRAccess &Access, Instruction *AccInst,
                           ScopStmt *Statement, const ScopArrayInfo *SAI,
                           int Identifier)
    : AccType(getMemoryAccessType(Access)), Statement(Statement),
      AccessInstruction(AccInst), AccessValue(Access.getAccessValue()),
      NewAccessRelation(nullptr) {

  isl_ctx *Ctx = Statement->getIslCtx();
  BaseAddr = Access.getBase();
  BaseName = getIslCompatibleName("MemRef_", getBaseAddr(), "");

  isl_id *BaseAddrId = SAI->getBasePtrId();

  auto IdName = "__polly_array_ref_" + std::to_string(Identifier);
  Id = isl_id_alloc(Ctx, IdName.c_str(), nullptr);

  if (!Access.isAffine()) {
    // We overapproximate non-affine accesses with a possible access to the
    // whole array. For read accesses it does not make a difference, if an
    // access must or may happen. However, for write accesses it is important to
    // differentiate between writes that must happen and writes that may happen.
    AccessRelation = isl_map_from_basic_map(createBasicAccessMap(Statement));
    AccessRelation =
        isl_map_set_tuple_id(AccessRelation, isl_dim_out, BaseAddrId);

    computeBoundsOnAccessRelation(Access.getElemSizeInBytes());
    return;
  }

  isl_space *Space = isl_space_alloc(Ctx, 0, Statement->getNumIterators(), 0);
  AccessRelation = isl_map_universe(Space);

  for (int i = 0, Size = Access.Subscripts.size(); i < Size; ++i) {
    isl_pw_aff *Affine = Statement->getPwAff(Access.Subscripts[i]);

    if (Size == 1) {
      // For the non delinearized arrays, divide the access function of the last
      // subscript by the size of the elements in the array.
      //
      // A stride one array access in C expressed as A[i] is expressed in
      // LLVM-IR as something like A[i * elementsize]. This hides the fact that
      // two subsequent values of 'i' index two values that are stored next to
      // each other in memory. By this division we make this characteristic
      // obvious again.
      isl_val *v = isl_val_int_from_si(Ctx, Access.getElemSizeInBytes());
      Affine = isl_pw_aff_scale_down_val(Affine, v);
    }

    isl_map *SubscriptMap = isl_map_from_pw_aff(Affine);

    AccessRelation = isl_map_flat_range_product(AccessRelation, SubscriptMap);
  }

  AccessRelation = foldAccess(Access, AccessRelation, Statement);

  Space = Statement->getDomainSpace();
  AccessRelation = isl_map_set_tuple_id(
      AccessRelation, isl_dim_in, isl_space_get_tuple_id(Space, isl_dim_set));
  AccessRelation =
      isl_map_set_tuple_id(AccessRelation, isl_dim_out, BaseAddrId);

  assumeNoOutOfBound(Access);
  AccessRelation = isl_map_gist_domain(AccessRelation, Statement->getDomain());
  isl_space_free(Space);
}

void MemoryAccess::realignParams() {
  isl_space *ParamSpace = Statement->getParent()->getParamSpace();
  AccessRelation = isl_map_align_params(AccessRelation, ParamSpace);
}

const std::string MemoryAccess::getReductionOperatorStr() const {
  return MemoryAccess::getReductionOperatorStr(getReductionType());
}

__isl_give isl_id *MemoryAccess::getId() const { return isl_id_copy(Id); }

raw_ostream &polly::operator<<(raw_ostream &OS,
                               MemoryAccess::ReductionType RT) {
  if (RT == MemoryAccess::RT_NONE)
    OS << "NONE";
  else
    OS << MemoryAccess::getReductionOperatorStr(RT);
  return OS;
}

void MemoryAccess::print(raw_ostream &OS) const {
  switch (AccType) {
  case READ:
    OS.indent(12) << "ReadAccess :=\t";
    break;
  case MUST_WRITE:
    OS.indent(12) << "MustWriteAccess :=\t";
    break;
  case MAY_WRITE:
    OS.indent(12) << "MayWriteAccess :=\t";
    break;
  }
  OS << "[Reduction Type: " << getReductionType() << "] ";
  OS << "[Scalar: " << isScalar() << "]\n";
  OS.indent(16) << getOriginalAccessRelationStr() << ";\n";
  if (hasNewAccessRelation())
    OS.indent(11) << "new: " << getNewAccessRelationStr() << ";\n";
}

void MemoryAccess::dump() const { print(errs()); }

// Create a map in the size of the provided set domain, that maps from the
// one element of the provided set domain to another element of the provided
// set domain.
// The mapping is limited to all points that are equal in all but the last
// dimension and for which the last dimension of the input is strict smaller
// than the last dimension of the output.
//
//   getEqualAndLarger(set[i0, i1, ..., iX]):
//
//   set[i0, i1, ..., iX] -> set[o0, o1, ..., oX]
//     : i0 = o0, i1 = o1, ..., i(X-1) = o(X-1), iX < oX
//
static isl_map *getEqualAndLarger(isl_space *setDomain) {
  isl_space *Space = isl_space_map_from_set(setDomain);
  isl_map *Map = isl_map_universe(Space);
  unsigned lastDimension = isl_map_dim(Map, isl_dim_in) - 1;

  // Set all but the last dimension to be equal for the input and output
  //
  //   input[i0, i1, ..., iX] -> output[o0, o1, ..., oX]
  //     : i0 = o0, i1 = o1, ..., i(X-1) = o(X-1)
  for (unsigned i = 0; i < lastDimension; ++i)
    Map = isl_map_equate(Map, isl_dim_in, i, isl_dim_out, i);

  // Set the last dimension of the input to be strict smaller than the
  // last dimension of the output.
  //
  //   input[?,?,?,...,iX] -> output[?,?,?,...,oX] : iX < oX
  Map = isl_map_order_lt(Map, isl_dim_in, lastDimension, isl_dim_out,
                         lastDimension);
  return Map;
}

__isl_give isl_set *
MemoryAccess::getStride(__isl_take const isl_map *Schedule) const {
  isl_map *S = const_cast<isl_map *>(Schedule);
  isl_map *AccessRelation = getAccessRelation();
  isl_space *Space = isl_space_range(isl_map_get_space(S));
  isl_map *NextScatt = getEqualAndLarger(Space);

  S = isl_map_reverse(S);
  NextScatt = isl_map_lexmin(NextScatt);

  NextScatt = isl_map_apply_range(NextScatt, isl_map_copy(S));
  NextScatt = isl_map_apply_range(NextScatt, isl_map_copy(AccessRelation));
  NextScatt = isl_map_apply_domain(NextScatt, S);
  NextScatt = isl_map_apply_domain(NextScatt, AccessRelation);

  isl_set *Deltas = isl_map_deltas(NextScatt);
  return Deltas;
}

bool MemoryAccess::isStrideX(__isl_take const isl_map *Schedule,
                             int StrideWidth) const {
  isl_set *Stride, *StrideX;
  bool IsStrideX;

  Stride = getStride(Schedule);
  StrideX = isl_set_universe(isl_set_get_space(Stride));
  for (unsigned i = 0; i < isl_set_dim(StrideX, isl_dim_set) - 1; i++)
    StrideX = isl_set_fix_si(StrideX, isl_dim_set, i, 0);
  StrideX = isl_set_fix_si(StrideX, isl_dim_set,
                           isl_set_dim(StrideX, isl_dim_set) - 1, StrideWidth);
  IsStrideX = isl_set_is_subset(Stride, StrideX);

  isl_set_free(StrideX);
  isl_set_free(Stride);

  return IsStrideX;
}

bool MemoryAccess::isStrideZero(const isl_map *Schedule) const {
  return isStrideX(Schedule, 0);
}

bool MemoryAccess::isScalar() const {
  return isl_map_n_out(AccessRelation) == 0;
}

bool MemoryAccess::isStrideOne(const isl_map *Schedule) const {
  return isStrideX(Schedule, 1);
}

void MemoryAccess::setNewAccessRelation(isl_map *NewAccess) {
  isl_map_free(NewAccessRelation);
  NewAccessRelation = NewAccess;
}

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

isl_map *ScopStmt::getSchedule() const {
  isl_set *Domain = getDomain();
  if (isl_set_is_empty(Domain)) {
    isl_set_free(Domain);
    return isl_map_from_aff(
        isl_aff_zero_on_domain(isl_local_space_from_space(getDomainSpace())));
  }
  auto *Schedule = getParent()->getSchedule();
  Schedule = isl_union_map_intersect_domain(
      Schedule, isl_union_set_from_set(isl_set_copy(Domain)));
  if (isl_union_map_is_empty(Schedule)) {
    isl_set_free(Domain);
    isl_union_map_free(Schedule);
    return isl_map_from_aff(
        isl_aff_zero_on_domain(isl_local_space_from_space(getDomainSpace())));
  }
  auto *M = isl_map_from_union_map(Schedule);
  M = isl_map_coalesce(M);
  M = isl_map_gist_domain(M, Domain);
  M = isl_map_coalesce(M);
  return M;
}

__isl_give isl_pw_aff *ScopStmt::getPwAff(const SCEV *E) {
  return getParent()->getPwAff(E, Domain);
}

void ScopStmt::restrictDomain(__isl_take isl_set *NewDomain) {
  assert(isl_set_is_subset(NewDomain, Domain) &&
         "New domain is not a subset of old domain!");
  isl_set_free(Domain);
  Domain = NewDomain;
}

void ScopStmt::buildAccesses(BasicBlock *Block, bool isApproximated) {
  AccFuncSetType *AFS = Parent.getAccessFunctions(Block);
  if (!AFS)
    return;

  for (auto &AccessPair : *AFS) {
    IRAccess &Access = AccessPair.first;
    Instruction *AccessInst = AccessPair.second;
    Type *ElementType = Access.getAccessValue()->getType();

    const ScopArrayInfo *SAI = getParent()->getOrCreateScopArrayInfo(
        Access.getBase(), ElementType, Access.Sizes, Access.isPHI());

    if (isApproximated && Access.isWrite())
      Access.setMayWrite();

    MemoryAccessList *&MAL = InstructionToAccess[AccessInst];
    if (!MAL)
      MAL = new MemoryAccessList();
    MAL->emplace_front(Access, AccessInst, this, SAI, MemAccs.size());
    MemAccs.push_back(&MAL->front());
  }
}

void ScopStmt::realignParams() {
  for (MemoryAccess *MA : *this)
    MA->realignParams();

  Domain = isl_set_align_params(Domain, Parent.getParamSpace());
}

/// @brief Add @p BSet to the set @p User if @p BSet is bounded.
static isl_stat collectBoundedParts(__isl_take isl_basic_set *BSet,
                                    void *User) {
  isl_set **BoundedParts = static_cast<isl_set **>(User);
  if (isl_basic_set_is_bounded(BSet))
    *BoundedParts = isl_set_union(*BoundedParts, isl_set_from_basic_set(BSet));
  else
    isl_basic_set_free(BSet);
  return isl_stat_ok;
}

/// @brief Return the bounded parts of @p S.
static __isl_give isl_set *collectBoundedParts(__isl_take isl_set *S) {
  isl_set *BoundedParts = isl_set_empty(isl_set_get_space(S));
  isl_set_foreach_basic_set(S, collectBoundedParts, &BoundedParts);
  isl_set_free(S);
  return BoundedParts;
}

/// @brief Compute the (un)bounded parts of @p S wrt. to dimension @p Dim.
///
/// @returns A separation of @p S into first an unbounded then a bounded subset,
///          both with regards to the dimension @p Dim.
static std::pair<__isl_give isl_set *, __isl_give isl_set *>
partitionSetParts(__isl_take isl_set *S, unsigned Dim) {

  for (unsigned u = 0, e = isl_set_n_dim(S); u < e; u++)
    S = isl_set_lower_bound_si(S, isl_dim_set, u, 0);

  unsigned NumDimsS = isl_set_n_dim(S);
  isl_set *OnlyDimS = isl_set_copy(S);

  // Remove dimensions that are greater than Dim as they are not interesting.
  assert(NumDimsS >= Dim + 1);
  OnlyDimS =
      isl_set_project_out(OnlyDimS, isl_dim_set, Dim + 1, NumDimsS - Dim - 1);

  // Create artificial parametric upper bounds for dimensions smaller than Dim
  // as we are not interested in them.
  OnlyDimS = isl_set_insert_dims(OnlyDimS, isl_dim_param, 0, Dim);
  for (unsigned u = 0; u < Dim; u++) {
    isl_constraint *C = isl_inequality_alloc(
        isl_local_space_from_space(isl_set_get_space(OnlyDimS)));
    C = isl_constraint_set_coefficient_si(C, isl_dim_param, u, 1);
    C = isl_constraint_set_coefficient_si(C, isl_dim_set, u, -1);
    OnlyDimS = isl_set_add_constraint(OnlyDimS, C);
  }

  // Collect all bounded parts of OnlyDimS.
  isl_set *BoundedParts = collectBoundedParts(OnlyDimS);

  // Create the dimensions greater than Dim again.
  BoundedParts = isl_set_insert_dims(BoundedParts, isl_dim_set, Dim + 1,
                                     NumDimsS - Dim - 1);

  // Remove the artificial upper bound parameters again.
  BoundedParts = isl_set_remove_dims(BoundedParts, isl_dim_param, 0, Dim);

  isl_set *UnboundedParts = isl_set_subtract(S, isl_set_copy(BoundedParts));
  return std::make_pair(UnboundedParts, BoundedParts);
}

static __isl_give isl_set *buildConditionSet(ICmpInst::Predicate Pred,
                                             isl_pw_aff *L, isl_pw_aff *R) {
  switch (Pred) {
  case ICmpInst::ICMP_EQ:
    return isl_pw_aff_eq_set(L, R);
  case ICmpInst::ICMP_NE:
    return isl_pw_aff_ne_set(L, R);
  case ICmpInst::ICMP_SLT:
    return isl_pw_aff_lt_set(L, R);
  case ICmpInst::ICMP_SLE:
    return isl_pw_aff_le_set(L, R);
  case ICmpInst::ICMP_SGT:
    return isl_pw_aff_gt_set(L, R);
  case ICmpInst::ICMP_SGE:
    return isl_pw_aff_ge_set(L, R);
  case ICmpInst::ICMP_ULT:
    return isl_pw_aff_lt_set(L, R);
  case ICmpInst::ICMP_UGT:
    return isl_pw_aff_gt_set(L, R);
  case ICmpInst::ICMP_ULE:
    return isl_pw_aff_le_set(L, R);
  case ICmpInst::ICMP_UGE:
    return isl_pw_aff_ge_set(L, R);
  default:
    llvm_unreachable("Non integer predicate not supported");
  }
}

/// @brief Build the conditions sets for the branch @p BI in the @p Domain.
///
/// This will fill @p ConditionSets with the conditions under which control
/// will be moved from @p BI to its successors. Hence, @p ConditionSets will
/// have as many elements as @p BI has successors.
static void
buildConditionSets(Scop &S, BranchInst *BI, Loop *L, __isl_keep isl_set *Domain,
                   SmallVectorImpl<__isl_give isl_set *> &ConditionSets) {

  if (BI->isUnconditional()) {
    ConditionSets.push_back(isl_set_copy(Domain));
    return;
  }

  Value *Condition = BI->getCondition();

  isl_set *ConsequenceCondSet = nullptr;
  if (auto *CCond = dyn_cast<ConstantInt>(Condition)) {
    if (CCond->isZero())
      ConsequenceCondSet = isl_set_empty(isl_set_get_space(Domain));
    else
      ConsequenceCondSet = isl_set_universe(isl_set_get_space(Domain));
  } else {
    auto *ICond = dyn_cast<ICmpInst>(Condition);
    assert(ICond &&
           "Condition of exiting branch was neither constant nor ICmp!");

    ScalarEvolution &SE = *S.getSE();
    isl_pw_aff *LHS, *RHS;
    LHS = S.getPwAff(SE.getSCEVAtScope(ICond->getOperand(0), L), Domain);
    RHS = S.getPwAff(SE.getSCEVAtScope(ICond->getOperand(1), L), Domain);
    ConsequenceCondSet = buildConditionSet(ICond->getPredicate(), LHS, RHS);

    for (unsigned u = 0, e = isl_set_n_dim(Domain); u < e; u++) {
      isl_id *DimId = isl_set_get_dim_id(Domain, isl_dim_set, u);
      ConsequenceCondSet =
          isl_set_set_dim_id(ConsequenceCondSet, isl_dim_set, u, DimId);
    }
  }

  assert(ConsequenceCondSet);
  isl_set *AlternativeCondSet =
      isl_set_complement(isl_set_copy(ConsequenceCondSet));

  ConditionSets.push_back(isl_set_coalesce(
      isl_set_intersect(ConsequenceCondSet, isl_set_copy(Domain))));
  ConditionSets.push_back(isl_set_coalesce(
      isl_set_intersect(AlternativeCondSet, isl_set_copy(Domain))));
}

void ScopStmt::buildDomain() {
  isl_id *Id;

  Id = isl_id_alloc(getIslCtx(), getBaseName(), this);

  Domain = getParent()->getDomainConditions(this);
  Domain = isl_set_coalesce(Domain);
  Domain = isl_set_set_tuple_id(Domain, Id);
}

void ScopStmt::deriveAssumptionsFromGEP(GetElementPtrInst *GEP) {
  int Dimension = 0;
  isl_ctx *Ctx = Parent.getIslCtx();
  isl_local_space *LSpace = isl_local_space_from_space(getDomainSpace());
  Type *Ty = GEP->getPointerOperandType();
  ScalarEvolution &SE = *Parent.getSE();

  if (auto *PtrTy = dyn_cast<PointerType>(Ty)) {
    Dimension = 1;
    Ty = PtrTy->getElementType();
  }

  while (auto ArrayTy = dyn_cast<ArrayType>(Ty)) {
    unsigned int Operand = 1 + Dimension;

    if (GEP->getNumOperands() <= Operand)
      break;

    const SCEV *Expr = SE.getSCEV(GEP->getOperand(Operand));

    if (isAffineExpr(&Parent.getRegion(), Expr, SE)) {
      isl_pw_aff *AccessOffset = getPwAff(Expr);
      AccessOffset =
          isl_pw_aff_set_tuple_id(AccessOffset, isl_dim_in, getDomainId());

      isl_pw_aff *DimSize = isl_pw_aff_from_aff(isl_aff_val_on_domain(
          isl_local_space_copy(LSpace),
          isl_val_int_from_si(Ctx, ArrayTy->getNumElements())));

      isl_set *OutOfBound = isl_pw_aff_ge_set(AccessOffset, DimSize);
      OutOfBound = isl_set_intersect(getDomain(), OutOfBound);
      OutOfBound = isl_set_params(OutOfBound);
      isl_set *InBound = isl_set_complement(OutOfBound);
      isl_set *Executed = isl_set_params(getDomain());

      // A => B == !A or B
      isl_set *InBoundIfExecuted =
          isl_set_union(isl_set_complement(Executed), InBound);

      Parent.addAssumption(InBoundIfExecuted);
    }

    Dimension += 1;
    Ty = ArrayTy->getElementType();
  }

  isl_local_space_free(LSpace);
}

void ScopStmt::deriveAssumptions(BasicBlock *Block) {
  for (Instruction &Inst : *Block)
    if (auto *GEP = dyn_cast<GetElementPtrInst>(&Inst))
      deriveAssumptionsFromGEP(GEP);
}

void ScopStmt::collectSurroundingLoops() {
  for (unsigned u = 0, e = isl_set_n_dim(Domain); u < e; u++) {
    isl_id *DimId = isl_set_get_dim_id(Domain, isl_dim_set, u);
    NestLoops.push_back(static_cast<Loop *>(isl_id_get_user(DimId)));
    isl_id_free(DimId);
  }
}

ScopStmt::ScopStmt(Scop &parent, Region &R)
    : Parent(parent), BB(nullptr), R(&R), Build(nullptr) {

  BaseName = getIslCompatibleName("Stmt_", R.getNameStr(), "");

  buildDomain();
  collectSurroundingLoops();

  BasicBlock *EntryBB = R.getEntry();
  for (BasicBlock *Block : R.blocks()) {
    buildAccesses(Block, Block != EntryBB);
    deriveAssumptions(Block);
  }
  if (DetectReductions)
    checkForReductions();
}

ScopStmt::ScopStmt(Scop &parent, BasicBlock &bb)
    : Parent(parent), BB(&bb), R(nullptr), Build(nullptr) {

  BaseName = getIslCompatibleName("Stmt_", &bb, "");

  buildDomain();
  collectSurroundingLoops();
  buildAccesses(BB);
  deriveAssumptions(BB);
  if (DetectReductions)
    checkForReductions();
}

/// @brief Collect loads which might form a reduction chain with @p StoreMA
///
/// Check if the stored value for @p StoreMA is a binary operator with one or
/// two loads as operands. If the binary operand is commutative & associative,
/// used only once (by @p StoreMA) and its load operands are also used only
/// once, we have found a possible reduction chain. It starts at an operand
/// load and includes the binary operator and @p StoreMA.
///
/// Note: We allow only one use to ensure the load and binary operator cannot
///       escape this block or into any other store except @p StoreMA.
void ScopStmt::collectCandiateReductionLoads(
    MemoryAccess *StoreMA, SmallVectorImpl<MemoryAccess *> &Loads) {
  auto *Store = dyn_cast<StoreInst>(StoreMA->getAccessInstruction());
  if (!Store)
    return;

  // Skip if there is not one binary operator between the load and the store
  auto *BinOp = dyn_cast<BinaryOperator>(Store->getValueOperand());
  if (!BinOp)
    return;

  // Skip if the binary operators has multiple uses
  if (BinOp->getNumUses() != 1)
    return;

  // Skip if the opcode of the binary operator is not commutative/associative
  if (!BinOp->isCommutative() || !BinOp->isAssociative())
    return;

  // Skip if the binary operator is outside the current SCoP
  if (BinOp->getParent() != Store->getParent())
    return;

  // Skip if it is a multiplicative reduction and we disabled them
  if (DisableMultiplicativeReductions &&
      (BinOp->getOpcode() == Instruction::Mul ||
       BinOp->getOpcode() == Instruction::FMul))
    return;

  // Check the binary operator operands for a candidate load
  auto *PossibleLoad0 = dyn_cast<LoadInst>(BinOp->getOperand(0));
  auto *PossibleLoad1 = dyn_cast<LoadInst>(BinOp->getOperand(1));
  if (!PossibleLoad0 && !PossibleLoad1)
    return;

  // A load is only a candidate if it cannot escape (thus has only this use)
  if (PossibleLoad0 && PossibleLoad0->getNumUses() == 1)
    if (PossibleLoad0->getParent() == Store->getParent())
      Loads.push_back(lookupAccessFor(PossibleLoad0));
  if (PossibleLoad1 && PossibleLoad1->getNumUses() == 1)
    if (PossibleLoad1->getParent() == Store->getParent())
      Loads.push_back(lookupAccessFor(PossibleLoad1));
}

/// @brief Check for reductions in this ScopStmt
///
/// Iterate over all store memory accesses and check for valid binary reduction
/// like chains. For all candidates we check if they have the same base address
/// and there are no other accesses which overlap with them. The base address
/// check rules out impossible reductions candidates early. The overlap check,
/// together with the "only one user" check in collectCandiateReductionLoads,
/// guarantees that none of the intermediate results will escape during
/// execution of the loop nest. We basically check here that no other memory
/// access can access the same memory as the potential reduction.
void ScopStmt::checkForReductions() {
  SmallVector<MemoryAccess *, 2> Loads;
  SmallVector<std::pair<MemoryAccess *, MemoryAccess *>, 4> Candidates;

  // First collect candidate load-store reduction chains by iterating over all
  // stores and collecting possible reduction loads.
  for (MemoryAccess *StoreMA : MemAccs) {
    if (StoreMA->isRead())
      continue;

    Loads.clear();
    collectCandiateReductionLoads(StoreMA, Loads);
    for (MemoryAccess *LoadMA : Loads)
      Candidates.push_back(std::make_pair(LoadMA, StoreMA));
  }

  // Then check each possible candidate pair.
  for (const auto &CandidatePair : Candidates) {
    bool Valid = true;
    isl_map *LoadAccs = CandidatePair.first->getAccessRelation();
    isl_map *StoreAccs = CandidatePair.second->getAccessRelation();

    // Skip those with obviously unequal base addresses.
    if (!isl_map_has_equal_space(LoadAccs, StoreAccs)) {
      isl_map_free(LoadAccs);
      isl_map_free(StoreAccs);
      continue;
    }

    // And check if the remaining for overlap with other memory accesses.
    isl_map *AllAccsRel = isl_map_union(LoadAccs, StoreAccs);
    AllAccsRel = isl_map_intersect_domain(AllAccsRel, getDomain());
    isl_set *AllAccs = isl_map_range(AllAccsRel);

    for (MemoryAccess *MA : MemAccs) {
      if (MA == CandidatePair.first || MA == CandidatePair.second)
        continue;

      isl_map *AccRel =
          isl_map_intersect_domain(MA->getAccessRelation(), getDomain());
      isl_set *Accs = isl_map_range(AccRel);

      if (isl_set_has_equal_space(AllAccs, Accs) || isl_set_free(Accs)) {
        isl_set *OverlapAccs = isl_set_intersect(Accs, isl_set_copy(AllAccs));
        Valid = Valid && isl_set_is_empty(OverlapAccs);
        isl_set_free(OverlapAccs);
      }
    }

    isl_set_free(AllAccs);
    if (!Valid)
      continue;

    const LoadInst *Load =
        dyn_cast<const LoadInst>(CandidatePair.first->getAccessInstruction());
    MemoryAccess::ReductionType RT =
        getReductionType(dyn_cast<BinaryOperator>(Load->user_back()), Load);

    // If no overlapping access was found we mark the load and store as
    // reduction like.
    CandidatePair.first->markAsReductionLike(RT);
    CandidatePair.second->markAsReductionLike(RT);
  }
}

std::string ScopStmt::getDomainStr() const { return stringFromIslObj(Domain); }

std::string ScopStmt::getScheduleStr() const {
  auto *S = getSchedule();
  auto Str = stringFromIslObj(S);
  isl_map_free(S);
  return Str;
}

unsigned ScopStmt::getNumParams() const { return Parent.getNumParams(); }

unsigned ScopStmt::getNumIterators() const { return NestLoops.size(); }

const char *ScopStmt::getBaseName() const { return BaseName.c_str(); }

const Loop *ScopStmt::getLoopForDimension(unsigned Dimension) const {
  return NestLoops[Dimension];
}

isl_ctx *ScopStmt::getIslCtx() const { return Parent.getIslCtx(); }

__isl_give isl_set *ScopStmt::getDomain() const { return isl_set_copy(Domain); }

__isl_give isl_space *ScopStmt::getDomainSpace() const {
  return isl_set_get_space(Domain);
}

__isl_give isl_id *ScopStmt::getDomainId() const {
  return isl_set_get_tuple_id(Domain);
}

ScopStmt::~ScopStmt() {
  DeleteContainerSeconds(InstructionToAccess);
  isl_set_free(Domain);
}

void ScopStmt::print(raw_ostream &OS) const {
  OS << "\t" << getBaseName() << "\n";
  OS.indent(12) << "Domain :=\n";

  if (Domain) {
    OS.indent(16) << getDomainStr() << ";\n";
  } else
    OS.indent(16) << "n/a\n";

  OS.indent(12) << "Schedule :=\n";

  if (Domain) {
    OS.indent(16) << getScheduleStr() << ";\n";
  } else
    OS.indent(16) << "n/a\n";

  for (MemoryAccess *Access : MemAccs)
    Access->print(OS);
}

void ScopStmt::dump() const { print(dbgs()); }

//===----------------------------------------------------------------------===//
/// Scop class implement

void Scop::setContext(__isl_take isl_set *NewContext) {
  NewContext = isl_set_align_params(NewContext, isl_set_get_space(Context));
  isl_set_free(Context);
  Context = NewContext;
}

void Scop::addParams(std::vector<const SCEV *> NewParameters) {
  for (const SCEV *Parameter : NewParameters) {
    Parameter = extractConstantFactor(Parameter, *SE).second;
    if (ParameterIds.find(Parameter) != ParameterIds.end())
      continue;

    int dimension = Parameters.size();

    Parameters.push_back(Parameter);
    ParameterIds[Parameter] = dimension;
  }
}

__isl_give isl_id *Scop::getIdForParam(const SCEV *Parameter) const {
  ParamIdType::const_iterator IdIter = ParameterIds.find(Parameter);

  if (IdIter == ParameterIds.end())
    return nullptr;

  std::string ParameterName;

  if (const SCEVUnknown *ValueParameter = dyn_cast<SCEVUnknown>(Parameter)) {
    Value *Val = ValueParameter->getValue();
    ParameterName = Val->getName();
  }

  if (ParameterName == "" || ParameterName.substr(0, 2) == "p_")
    ParameterName = "p_" + utostr_32(IdIter->second);

  return isl_id_alloc(getIslCtx(), ParameterName.c_str(),
                      const_cast<void *>((const void *)Parameter));
}

isl_set *Scop::addNonEmptyDomainConstraints(isl_set *C) const {
  isl_set *DomainContext = isl_union_set_params(getDomains());
  return isl_set_intersect_params(C, DomainContext);
}

void Scop::addUserContext() {
  if (UserContextStr.empty())
    return;

  isl_set *UserContext = isl_set_read_from_str(IslCtx, UserContextStr.c_str());
  isl_space *Space = getParamSpace();
  if (isl_space_dim(Space, isl_dim_param) !=
      isl_set_dim(UserContext, isl_dim_param)) {
    auto SpaceStr = isl_space_to_str(Space);
    errs() << "Error: the context provided in -polly-context has not the same "
           << "number of dimensions than the computed context. Due to this "
           << "mismatch, the -polly-context option is ignored. Please provide "
           << "the context in the parameter space: " << SpaceStr << ".\n";
    free(SpaceStr);
    isl_set_free(UserContext);
    isl_space_free(Space);
    return;
  }

  for (unsigned i = 0; i < isl_space_dim(Space, isl_dim_param); i++) {
    auto NameContext = isl_set_get_dim_name(Context, isl_dim_param, i);
    auto NameUserContext = isl_set_get_dim_name(UserContext, isl_dim_param, i);

    if (strcmp(NameContext, NameUserContext) != 0) {
      auto SpaceStr = isl_space_to_str(Space);
      errs() << "Error: the name of dimension " << i
             << " provided in -polly-context "
             << "is '" << NameUserContext << "', but the name in the computed "
             << "context is '" << NameContext
             << "'. Due to this name mismatch, "
             << "the -polly-context option is ignored. Please provide "
             << "the context in the parameter space: " << SpaceStr << ".\n";
      free(SpaceStr);
      isl_set_free(UserContext);
      isl_space_free(Space);
      return;
    }

    UserContext =
        isl_set_set_dim_id(UserContext, isl_dim_param, i,
                           isl_space_get_dim_id(Space, isl_dim_param, i));
  }

  Context = isl_set_intersect(Context, UserContext);
  isl_space_free(Space);
}

void Scop::buildContext() {
  isl_space *Space = isl_space_params_alloc(IslCtx, 0);
  Context = isl_set_universe(isl_space_copy(Space));
  AssumedContext = isl_set_universe(Space);
}

void Scop::addParameterBounds() {
  for (const auto &ParamID : ParameterIds) {
    int dim = ParamID.second;

    ConstantRange SRange = SE->getSignedRange(ParamID.first);

    Context = addRangeBoundsToSet(Context, SRange, dim, isl_dim_param);
  }
}

void Scop::realignParams() {
  // Add all parameters into a common model.
  isl_space *Space = isl_space_params_alloc(IslCtx, ParameterIds.size());

  for (const auto &ParamID : ParameterIds) {
    const SCEV *Parameter = ParamID.first;
    isl_id *id = getIdForParam(Parameter);
    Space = isl_space_set_dim_id(Space, isl_dim_param, ParamID.second, id);
  }

  // Align the parameters of all data structures to the model.
  Context = isl_set_align_params(Context, Space);

  for (ScopStmt &Stmt : *this)
    Stmt.realignParams();
}

void Scop::simplifyAssumedContext() {
  // The parameter constraints of the iteration domains give us a set of
  // constraints that need to hold for all cases where at least a single
  // statement iteration is executed in the whole scop. We now simplify the
  // assumed context under the assumption that such constraints hold and at
  // least a single statement iteration is executed. For cases where no
  // statement instances are executed, the assumptions we have taken about
  // the executed code do not matter and can be changed.
  //
  // WARNING: This only holds if the assumptions we have taken do not reduce
  //          the set of statement instances that are executed. Otherwise we
  //          may run into a case where the iteration domains suggest that
  //          for a certain set of parameter constraints no code is executed,
  //          but in the original program some computation would have been
  //          performed. In such a case, modifying the run-time conditions and
  //          possibly influencing the run-time check may cause certain scops
  //          to not be executed.
  //
  // Example:
  //
  //   When delinearizing the following code:
  //
  //     for (long i = 0; i < 100; i++)
  //       for (long j = 0; j < m; j++)
  //         A[i+p][j] = 1.0;
  //
  //   we assume that the condition m <= 0 or (m >= 1 and p >= 0) holds as
  //   otherwise we would access out of bound data. Now, knowing that code is
  //   only executed for the case m >= 0, it is sufficient to assume p >= 0.
  AssumedContext =
      isl_set_gist_params(AssumedContext, isl_union_set_params(getDomains()));
  AssumedContext = isl_set_gist_params(AssumedContext, getContext());
}

/// @brief Add the minimal/maximal access in @p Set to @p User.
static isl_stat buildMinMaxAccess(__isl_take isl_set *Set, void *User) {
  Scop::MinMaxVectorTy *MinMaxAccesses = (Scop::MinMaxVectorTy *)User;
  isl_pw_multi_aff *MinPMA, *MaxPMA;
  isl_pw_aff *LastDimAff;
  isl_aff *OneAff;
  unsigned Pos;

  // Restrict the number of parameters involved in the access as the lexmin/
  // lexmax computation will take too long if this number is high.
  //
  // Experiments with a simple test case using an i7 4800MQ:
  //
  //  #Parameters involved | Time (in sec)
  //            6          |     0.01
  //            7          |     0.04
  //            8          |     0.12
  //            9          |     0.40
  //           10          |     1.54
  //           11          |     6.78
  //           12          |    30.38
  //
  if (isl_set_n_param(Set) > RunTimeChecksMaxParameters) {
    unsigned InvolvedParams = 0;
    for (unsigned u = 0, e = isl_set_n_param(Set); u < e; u++)
      if (isl_set_involves_dims(Set, isl_dim_param, u, 1))
        InvolvedParams++;

    if (InvolvedParams > RunTimeChecksMaxParameters) {
      isl_set_free(Set);
      return isl_stat_error;
    }
  }

  Set = isl_set_remove_divs(Set);

  MinPMA = isl_set_lexmin_pw_multi_aff(isl_set_copy(Set));
  MaxPMA = isl_set_lexmax_pw_multi_aff(isl_set_copy(Set));

  MinPMA = isl_pw_multi_aff_coalesce(MinPMA);
  MaxPMA = isl_pw_multi_aff_coalesce(MaxPMA);

  // Adjust the last dimension of the maximal access by one as we want to
  // enclose the accessed memory region by MinPMA and MaxPMA. The pointer
  // we test during code generation might now point after the end of the
  // allocated array but we will never dereference it anyway.
  assert(isl_pw_multi_aff_dim(MaxPMA, isl_dim_out) &&
         "Assumed at least one output dimension");
  Pos = isl_pw_multi_aff_dim(MaxPMA, isl_dim_out) - 1;
  LastDimAff = isl_pw_multi_aff_get_pw_aff(MaxPMA, Pos);
  OneAff = isl_aff_zero_on_domain(
      isl_local_space_from_space(isl_pw_aff_get_domain_space(LastDimAff)));
  OneAff = isl_aff_add_constant_si(OneAff, 1);
  LastDimAff = isl_pw_aff_add(LastDimAff, isl_pw_aff_from_aff(OneAff));
  MaxPMA = isl_pw_multi_aff_set_pw_aff(MaxPMA, Pos, LastDimAff);

  MinMaxAccesses->push_back(std::make_pair(MinPMA, MaxPMA));

  isl_set_free(Set);
  return isl_stat_ok;
}

static __isl_give isl_set *getAccessDomain(MemoryAccess *MA) {
  isl_set *Domain = MA->getStatement()->getDomain();
  Domain = isl_set_project_out(Domain, isl_dim_set, 0, isl_set_n_dim(Domain));
  return isl_set_reset_tuple_id(Domain);
}

/// @brief Wrapper function to calculate minimal/maximal accesses to each array.
static bool calculateMinMaxAccess(__isl_take isl_union_map *Accesses,
                                  __isl_take isl_union_set *Domains,
                                  Scop::MinMaxVectorTy &MinMaxAccesses) {

  Accesses = isl_union_map_intersect_domain(Accesses, Domains);
  isl_union_set *Locations = isl_union_map_range(Accesses);
  Locations = isl_union_set_coalesce(Locations);
  Locations = isl_union_set_detect_equalities(Locations);
  bool Valid = (0 == isl_union_set_foreach_set(Locations, buildMinMaxAccess,
                                               &MinMaxAccesses));
  isl_union_set_free(Locations);
  return Valid;
}

/// @brief Helper to treat non-affine regions and basic blocks the same.
///
///{

/// @brief Return the block that is the representing block for @p RN.
static inline BasicBlock *getRegionNodeBasicBlock(RegionNode *RN) {
  return RN->isSubRegion() ? RN->getNodeAs<Region>()->getEntry()
                           : RN->getNodeAs<BasicBlock>();
}

/// @brief Return the @p idx'th block that is executed after @p RN.
static inline BasicBlock *getRegionNodeSuccessor(RegionNode *RN, BranchInst *BI,
                                                 unsigned idx) {
  if (RN->isSubRegion()) {
    assert(idx == 0);
    return RN->getNodeAs<Region>()->getExit();
  }
  return BI->getSuccessor(idx);
}

/// @brief Return the smallest loop surrounding @p RN.
static inline Loop *getRegionNodeLoop(RegionNode *RN, LoopInfo &LI) {
  if (!RN->isSubRegion())
    return LI.getLoopFor(RN->getNodeAs<BasicBlock>());

  Region *NonAffineSubRegion = RN->getNodeAs<Region>();
  Loop *L = LI.getLoopFor(NonAffineSubRegion->getEntry());
  while (L && NonAffineSubRegion->contains(L))
    L = L->getParentLoop();
  return L;
}

static inline unsigned getNumBlocksInRegionNode(RegionNode *RN) {
  if (!RN->isSubRegion())
    return 1;

  unsigned NumBlocks = 0;
  Region *R = RN->getNodeAs<Region>();
  for (auto BB : R->blocks()) {
    (void)BB;
    NumBlocks++;
  }
  return NumBlocks;
}

///}

static inline __isl_give isl_set *addDomainDimId(__isl_take isl_set *Domain,
                                                 unsigned Dim, Loop *L) {
  isl_id *DimId =
      isl_id_alloc(isl_set_get_ctx(Domain), nullptr, static_cast<void *>(L));
  return isl_set_set_dim_id(Domain, isl_dim_set, Dim, DimId);
}

isl_set *Scop::getDomainConditions(ScopStmt *Stmt) {
  BasicBlock *BB = Stmt->isBlockStmt() ? Stmt->getBasicBlock()
                                       : Stmt->getRegion()->getEntry();
  return isl_set_copy(DomainMap[BB]);
}

void Scop::buildDomains(Region *R, LoopInfo &LI, ScopDetection &SD,
                        DominatorTree &DT) {

  auto *EntryBB = R->getEntry();
  int LD = getRelativeLoopDepth(LI.getLoopFor(EntryBB));
  auto *S = isl_set_universe(isl_space_set_alloc(getIslCtx(), 0, LD + 1));

  Loop *L = LI.getLoopFor(EntryBB);
  while (LD-- >= 0) {
    S = addDomainDimId(S, LD + 1, L);
    L = L->getParentLoop();
  }

  DomainMap[EntryBB] = S;

  buildDomainsWithBranchConstraints(R, LI, SD, DT);
  addLoopBoundsToHeaderDomains(LI, SD, DT);
  propagateDomainConstraints(R, LI, SD, DT);
}

void Scop::buildDomainsWithBranchConstraints(Region *R, LoopInfo &LI,
                                             ScopDetection &SD,
                                             DominatorTree &DT) {
  RegionInfo &RI = *R->getRegionInfo();

  // To create the domain for each block in R we iterate over all blocks and
  // subregions in R and propagate the conditions under which the current region
  // element is executed. To this end we iterate in reverse post order over R as
  // it ensures that we first visit all predecessors of a region node (either a
  // basic block or a subregion) before we visit the region node itself.
  // Initially, only the domain for the SCoP region entry block is set and from
  // there we propagate the current domain to all successors, however we add the
  // condition that the successor is actually executed next.
  // As we are only interested in non-loop carried constraints here we can
  // simply skip loop back edges.

  ReversePostOrderTraversal<Region *> RTraversal(R);
  for (auto *RN : RTraversal) {

    // Recurse for affine subregions but go on for basic blocks and non-affine
    // subregions.
    if (RN->isSubRegion()) {
      Region *SubRegion = RN->getNodeAs<Region>();
      if (!SD.isNonAffineSubRegion(SubRegion, &getRegion())) {
        buildDomainsWithBranchConstraints(SubRegion, LI, SD, DT);
        continue;
      }
    }

    BasicBlock *BB = getRegionNodeBasicBlock(RN);
    TerminatorInst *TI = BB->getTerminator();

    // Unreachable instructions do not have successors so we can skip them.
    if (isa<UnreachableInst>(TI)) {
      // Assume unreachables only in error blocks.
      assert(isErrorBlock(*BB));
      continue;
    }

    isl_set *Domain = DomainMap[BB];
    DEBUG(dbgs() << "\tVisit: " << BB->getName() << " : " << Domain << "\n");
    assert(Domain && "Due to reverse post order traversal of the region all "
                     "predecessor of the current region node should have been "
                     "visited and a domain for this region node should have "
                     "been set.");

    Loop *BBLoop = getRegionNodeLoop(RN, LI);
    int BBLoopDepth = getRelativeLoopDepth(BBLoop);

    // Build the condition sets for the successor nodes of the current region
    // node. If it is a non-affine subregion we will always execute the single
    // exit node, hence the single entry node domain is the condition set. For
    // basic blocks we use the helper function buildConditionSets.
    SmallVector<isl_set *, 2> ConditionSets;
    BranchInst *BI = cast<BranchInst>(TI);
    if (RN->isSubRegion())
      ConditionSets.push_back(isl_set_copy(Domain));
    else
      buildConditionSets(*this, BI, BBLoop, Domain, ConditionSets);

    // Now iterate over the successors and set their initial domain based on
    // their condition set. We skip back edges here and have to be careful when
    // we leave a loop not to keep constraints over a dimension that doesn't
    // exist anymore.
    for (unsigned u = 0, e = ConditionSets.size(); u < e; u++) {
      BasicBlock *SuccBB = getRegionNodeSuccessor(RN, BI, u);
      isl_set *CondSet = ConditionSets[u];

      // Skip back edges.
      if (DT.dominates(SuccBB, BB)) {
        isl_set_free(CondSet);
        continue;
      }

      // Do not adjust the number of dimensions if we enter a boxed loop or are
      // in a non-affine subregion or if the surrounding loop stays the same.
      Loop *SuccBBLoop = LI.getLoopFor(SuccBB);
      Region *SuccRegion = RI.getRegionFor(SuccBB);
      if (BBLoop != SuccBBLoop && !RN->isSubRegion() &&
          !(SD.isNonAffineSubRegion(SuccRegion, &getRegion()) &&
            SuccRegion->contains(SuccBBLoop))) {

        // Check if the edge to SuccBB is a loop entry or exit edge. If so
        // adjust the dimensionality accordingly. Lastly, if we leave a loop
        // and enter a new one we need to drop the old constraints.
        int SuccBBLoopDepth = getRelativeLoopDepth(SuccBBLoop);
        unsigned LoopDepthDiff = std::abs(BBLoopDepth - SuccBBLoopDepth);
        if (BBLoopDepth > SuccBBLoopDepth) {
          CondSet = isl_set_project_out(CondSet, isl_dim_set,
                                        isl_set_n_dim(CondSet) - LoopDepthDiff,
                                        LoopDepthDiff);
        } else if (SuccBBLoopDepth > BBLoopDepth) {
          assert(LoopDepthDiff == 1);
          CondSet = isl_set_add_dims(CondSet, isl_dim_set, 1);
          CondSet = addDomainDimId(CondSet, SuccBBLoopDepth, SuccBBLoop);
        } else if (BBLoopDepth >= 0) {
          assert(LoopDepthDiff <= 1);
          CondSet = isl_set_project_out(CondSet, isl_dim_set, BBLoopDepth, 1);
          CondSet = isl_set_add_dims(CondSet, isl_dim_set, 1);
          CondSet = addDomainDimId(CondSet, SuccBBLoopDepth, SuccBBLoop);
        }
      }

      // Set the domain for the successor or merge it with an existing domain in
      // case there are multiple paths (without loop back edges) to the
      // successor block.
      isl_set *&SuccDomain = DomainMap[SuccBB];
      if (!SuccDomain)
        SuccDomain = CondSet;
      else
        SuccDomain = isl_set_union(SuccDomain, CondSet);

      SuccDomain = isl_set_coalesce(SuccDomain);
      DEBUG(dbgs() << "\tSet SuccBB: " << SuccBB->getName() << " : " << Domain
                   << "\n");
    }
  }
}

/// @brief Return the domain for @p BB wrt @p DomainMap.
///
/// This helper function will lookup @p BB in @p DomainMap but also handle the
/// case where @p BB is contained in a non-affine subregion using the region
/// tree obtained by @p RI.
static __isl_give isl_set *
getDomainForBlock(BasicBlock *BB, DenseMap<BasicBlock *, isl_set *> &DomainMap,
                  RegionInfo &RI) {
  auto DIt = DomainMap.find(BB);
  if (DIt != DomainMap.end())
    return isl_set_copy(DIt->getSecond());

  Region *R = RI.getRegionFor(BB);
  while (R->getEntry() == BB)
    R = R->getParent();
  return getDomainForBlock(R->getEntry(), DomainMap, RI);
}

void Scop::propagateDomainConstraints(Region *R, LoopInfo &LI,
                                      ScopDetection &SD, DominatorTree &DT) {
  // Iterate over the region R and propagate the domain constrains from the
  // predecessors to the current node. In contrast to the
  // buildDomainsWithBranchConstraints function, this one will pull the domain
  // information from the predecessors instead of pushing it to the successors.
  // Additionally, we assume the domains to be already present in the domain
  // map here. However, we iterate again in reverse post order so we know all
  // predecessors have been visited before a block or non-affine subregion is
  // visited.

  // The set of boxed loops (loops in non-affine subregions) for this SCoP.
  auto &BoxedLoops = *SD.getBoxedLoops(&getRegion());

  ReversePostOrderTraversal<Region *> RTraversal(R);
  for (auto *RN : RTraversal) {

    // Recurse for affine subregions but go on for basic blocks and non-affine
    // subregions.
    if (RN->isSubRegion()) {
      Region *SubRegion = RN->getNodeAs<Region>();
      if (!SD.isNonAffineSubRegion(SubRegion, &getRegion())) {
        propagateDomainConstraints(SubRegion, LI, SD, DT);
        continue;
      }
    }

    BasicBlock *BB = getRegionNodeBasicBlock(RN);
    Loop *BBLoop = getRegionNodeLoop(RN, LI);
    int BBLoopDepth = getRelativeLoopDepth(BBLoop);

    isl_set *&Domain = DomainMap[BB];
    assert(Domain && "Due to reverse post order traversal of the region all "
                     "predecessor of the current region node should have been "
                     "visited and a domain for this region node should have "
                     "been set.");

    isl_set *PredDom = isl_set_empty(isl_set_get_space(Domain));
    for (auto *PredBB : predecessors(BB)) {

      // Skip backedges
      if (DT.dominates(BB, PredBB))
        continue;

      isl_set *PredBBDom = nullptr;

      // Handle the SCoP entry block with its outside predecessors.
      if (!getRegion().contains(PredBB))
        PredBBDom = isl_set_universe(isl_set_get_space(PredDom));

      if (!PredBBDom) {
        // Determine the loop depth of the predecessor and adjust its domain to
        // the domain of the current block. This can mean we have to:
        //  o) Drop a dimension if this block is the exit of a loop, not the
        //     header of a new loop and the predecessor was part of the loop.
        //  o) Add an unconstrainted new dimension if this block is the header
        //     of a loop and the predecessor is not part of it.
        //  o) Drop the information about the innermost loop dimension when the
        //     predecessor and the current block are surrounded by different
        //     loops in the same depth.
        PredBBDom = getDomainForBlock(PredBB, DomainMap, *R->getRegionInfo());
        Loop *PredBBLoop = LI.getLoopFor(PredBB);
        while (BoxedLoops.count(PredBBLoop))
          PredBBLoop = PredBBLoop->getParentLoop();

        int PredBBLoopDepth = getRelativeLoopDepth(PredBBLoop);
        unsigned LoopDepthDiff = std::abs(BBLoopDepth - PredBBLoopDepth);
        if (BBLoopDepth < PredBBLoopDepth)
          PredBBDom = isl_set_project_out(
              PredBBDom, isl_dim_set, isl_set_n_dim(PredBBDom) - LoopDepthDiff,
              LoopDepthDiff);
        else if (PredBBLoopDepth < BBLoopDepth) {
          assert(LoopDepthDiff == 1);
          PredBBDom = isl_set_add_dims(PredBBDom, isl_dim_set, 1);
        } else if (BBLoop != PredBBLoop && BBLoopDepth >= 0) {
          assert(LoopDepthDiff <= 1);
          PredBBDom = isl_set_drop_constraints_involving_dims(
              PredBBDom, isl_dim_set, BBLoopDepth, 1);
        }
      }

      PredDom = isl_set_union(PredDom, PredBBDom);
    }

    // Under the union of all predecessor conditions we can reach this block.
    Domain = isl_set_intersect(Domain, PredDom);

    // Add assumptions for error blocks.
    if (isErrorBlock(*BB)) {
      IsOptimized = true;
      isl_set *DomPar = isl_set_params(isl_set_copy(Domain));
      addAssumption(isl_set_complement(DomPar));
    }
  }
}

/// @brief Create a map from SetSpace -> SetSpace where the dimensions @p Dim
///        is incremented by one and all other dimensions are equal, e.g.,
///             [i0, i1, i2, i3] -> [i0, i1, i2 + 1, i3]
///        if @p Dim is 2 and @p SetSpace has 4 dimensions.
static __isl_give isl_map *
createNextIterationMap(__isl_take isl_space *SetSpace, unsigned Dim) {
  auto *MapSpace = isl_space_map_from_set(SetSpace);
  auto *NextIterationMap = isl_map_universe(isl_space_copy(MapSpace));
  for (unsigned u = 0; u < isl_map_n_in(NextIterationMap); u++)
    if (u != Dim)
      NextIterationMap =
          isl_map_equate(NextIterationMap, isl_dim_in, u, isl_dim_out, u);
  auto *C = isl_constraint_alloc_equality(isl_local_space_from_space(MapSpace));
  C = isl_constraint_set_constant_si(C, 1);
  C = isl_constraint_set_coefficient_si(C, isl_dim_in, Dim, 1);
  C = isl_constraint_set_coefficient_si(C, isl_dim_out, Dim, -1);
  NextIterationMap = isl_map_add_constraint(NextIterationMap, C);
  return NextIterationMap;
}

/// @brief Add @p L & all children to @p Loops if they are not in @p BoxedLoops.
static inline void
addLoopAndSubloops(Loop *L, SmallVectorImpl<Loop *> &Loops,
                   const ScopDetection::BoxedLoopsSetTy &BoxedLoops) {
  if (BoxedLoops.count(L))
    return;

  Loops.push_back(L);
  for (Loop *Subloop : *L)
    addLoopAndSubloops(Subloop, Loops, BoxedLoops);
}

/// @brief Add loops in @p R to @p RegionLoops if they are not in @p BoxedLoops.
static inline void
collectLoopsInRegion(Region &R, LoopInfo &LI,
                     SmallVector<Loop *, 8> &RegionLoops,
                     const ScopDetection::BoxedLoopsSetTy &BoxedLoops) {

  SmallVector<Loop *, 8> Loops(LI.begin(), LI.end());
  while (!Loops.empty()) {
    Loop *L = Loops.pop_back_val();

    if (R.contains(L))
      addLoopAndSubloops(L, RegionLoops, BoxedLoops);
    else if (L->contains(R.getEntry()))
      Loops.append(L->begin(), L->end());
  }
}

/// @brief Create a set from @p Space with @p Dim fixed to 0.
static __isl_give isl_set *
createFirstIterationDomain(__isl_take isl_space *Space, unsigned Dim) {
  auto *Domain = isl_set_universe(Space);
  Domain = isl_set_fix_si(Domain, isl_dim_set, Dim, 0);
  return Domain;
}

void Scop::addLoopBoundsToHeaderDomains(LoopInfo &LI, ScopDetection &SD,
                                        DominatorTree &DT) {
  // We iterate over all loops in the SCoP, create the condition set under which
  // we will take the back edge, and then apply these restrictions to the
  // header.

  Region &R = getRegion();
  SmallVector<Loop *, 8> RegionLoops;
  collectLoopsInRegion(R, LI, RegionLoops, *SD.getBoxedLoops(&R));

  while (!RegionLoops.empty()) {
    Loop *L = RegionLoops.pop_back_val();
    int LoopDepth = getRelativeLoopDepth(L);
    assert(LoopDepth >= 0 && "Loop in region should have at least depth one");

    BasicBlock *LatchBB = L->getLoopLatch();
    assert(LatchBB && "TODO implement multiple exit loop handling");

    isl_set *LatchBBDom = DomainMap[LatchBB];
    isl_set *BackedgeCondition = nullptr;

    BasicBlock *HeaderBB = L->getHeader();

    BranchInst *BI = cast<BranchInst>(LatchBB->getTerminator());
    if (BI->isUnconditional())
      BackedgeCondition = isl_set_copy(LatchBBDom);
    else {
      SmallVector<isl_set *, 2> ConditionSets;
      int idx = BI->getSuccessor(0) != HeaderBB;
      buildConditionSets(*this, BI, L, LatchBBDom, ConditionSets);

      // Free the non back edge condition set as we do not need it.
      isl_set_free(ConditionSets[1 - idx]);

      BackedgeCondition = ConditionSets[idx];
    }

    isl_set *&HeaderBBDom = DomainMap[HeaderBB];
    isl_set *FirstIteration =
        createFirstIterationDomain(isl_set_get_space(HeaderBBDom), LoopDepth);

    isl_map *NextIterationMap =
        createNextIterationMap(isl_set_get_space(HeaderBBDom), LoopDepth);

    int LatchLoopDepth = getRelativeLoopDepth(LI.getLoopFor(LatchBB));
    assert(LatchLoopDepth >= LoopDepth);
    BackedgeCondition =
        isl_set_project_out(BackedgeCondition, isl_dim_set, LoopDepth + 1,
                            LatchLoopDepth - LoopDepth);

    auto Parts = partitionSetParts(BackedgeCondition, LoopDepth);

    // If a loop has an unbounded back edge condition part (here Parts.first)
    // we do not want to assume the header will even be executed for the first
    // iteration of an execution that will lead to an infinite loop. While it
    // would not be wrong to do so, it does not seem helpful.
    FirstIteration = isl_set_subtract(FirstIteration, Parts.first);

    BackedgeCondition = isl_set_apply(Parts.second, NextIterationMap);
    BackedgeCondition = isl_set_union(BackedgeCondition, FirstIteration);
    BackedgeCondition = isl_set_coalesce(BackedgeCondition);

    HeaderBBDom = isl_set_intersect(HeaderBBDom, BackedgeCondition);
  }
}

void Scop::buildAliasChecks(AliasAnalysis &AA) {
  if (!PollyUseRuntimeAliasChecks)
    return;

  if (buildAliasGroups(AA))
    return;

  // If a problem occurs while building the alias groups we need to delete
  // this SCoP and pretend it wasn't valid in the first place. To this end
  // we make the assumed context infeasible.
  addAssumption(isl_set_empty(getParamSpace()));

  DEBUG(dbgs() << "\n\nNOTE: Run time checks for " << getNameStr()
               << " could not be created as the number of parameters involved "
                  "is too high. The SCoP will be "
                  "dismissed.\nUse:\n\t--polly-rtc-max-parameters=X\nto adjust "
                  "the maximal number of parameters but be advised that the "
                  "compile time might increase exponentially.\n\n");
}

bool Scop::buildAliasGroups(AliasAnalysis &AA) {
  // To create sound alias checks we perform the following steps:
  //   o) Use the alias analysis and an alias set tracker to build alias sets
  //      for all memory accesses inside the SCoP.
  //   o) 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.
  //   o) We divide 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
  //      ones.
  //   o) We partition each group into read only and non read only accesses.
  //   o) For each group with more than one base pointer we then compute minimal
  //      and maximal accesses to each array of a group in read only and non
  //      read only partitions separately.
  using AliasGroupTy = SmallVector<MemoryAccess *, 4>;

  AliasSetTracker AST(AA);

  DenseMap<Value *, MemoryAccess *> PtrToAcc;
  DenseSet<Value *> HasWriteAccess;
  for (ScopStmt &Stmt : *this) {

    // Skip statements with an empty domain as they will never be executed.
    isl_set *StmtDomain = Stmt.getDomain();
    bool StmtDomainEmpty = isl_set_is_empty(StmtDomain);
    isl_set_free(StmtDomain);
    if (StmtDomainEmpty)
      continue;

    for (MemoryAccess *MA : Stmt) {
      if (MA->isScalar())
        continue;
      if (!MA->isRead())
        HasWriteAccess.insert(MA->getBaseAddr());
      Instruction *Acc = MA->getAccessInstruction();
      PtrToAcc[getPointerOperand(*Acc)] = MA;
      AST.add(Acc);
    }
  }

  SmallVector<AliasGroupTy, 4> AliasGroups;
  for (AliasSet &AS : AST) {
    if (AS.isMustAlias() || AS.isForwardingAliasSet())
      continue;
    AliasGroupTy AG;
    for (auto PR : AS)
      AG.push_back(PtrToAcc[PR.getValue()]);
    assert(AG.size() > 1 &&
           "Alias groups should contain at least two accesses");
    AliasGroups.push_back(std::move(AG));
  }

  // Split the alias groups based on their domain.
  for (unsigned u = 0; u < AliasGroups.size(); u++) {
    AliasGroupTy NewAG;
    AliasGroupTy &AG = AliasGroups[u];
    AliasGroupTy::iterator AGI = AG.begin();
    isl_set *AGDomain = getAccessDomain(*AGI);
    while (AGI != AG.end()) {
      MemoryAccess *MA = *AGI;
      isl_set *MADomain = getAccessDomain(MA);
      if (isl_set_is_disjoint(AGDomain, MADomain)) {
        NewAG.push_back(MA);
        AGI = AG.erase(AGI);
        isl_set_free(MADomain);
      } else {
        AGDomain = isl_set_union(AGDomain, MADomain);
        AGI++;
      }
    }
    if (NewAG.size() > 1)
      AliasGroups.push_back(std::move(NewAG));
    isl_set_free(AGDomain);
  }

  MapVector<const Value *, SmallPtrSet<MemoryAccess *, 8>> ReadOnlyPairs;
  SmallPtrSet<const Value *, 4> NonReadOnlyBaseValues;
  for (AliasGroupTy &AG : AliasGroups) {
    NonReadOnlyBaseValues.clear();
    ReadOnlyPairs.clear();

    if (AG.size() < 2) {
      AG.clear();
      continue;
    }

    for (auto II = AG.begin(); II != AG.end();) {
      Value *BaseAddr = (*II)->getBaseAddr();
      if (HasWriteAccess.count(BaseAddr)) {
        NonReadOnlyBaseValues.insert(BaseAddr);
        II++;
      } else {
        ReadOnlyPairs[BaseAddr].insert(*II);
        II = AG.erase(II);
      }
    }

    // If we don't have read only pointers check if there are at least two
    // non read only pointers, otherwise clear the alias group.
    if (ReadOnlyPairs.empty() && NonReadOnlyBaseValues.size() <= 1) {
      AG.clear();
      continue;
    }

    // If we don't have non read only pointers clear the alias group.
    if (NonReadOnlyBaseValues.empty()) {
      AG.clear();
      continue;
    }

    // Calculate minimal and maximal accesses for non read only accesses.
    MinMaxAliasGroups.emplace_back();
    MinMaxVectorPairTy &pair = MinMaxAliasGroups.back();
    MinMaxVectorTy &MinMaxAccessesNonReadOnly = pair.first;
    MinMaxVectorTy &MinMaxAccessesReadOnly = pair.second;
    MinMaxAccessesNonReadOnly.reserve(AG.size());

    isl_union_map *Accesses = isl_union_map_empty(getParamSpace());

    // AG contains only non read only accesses.
    for (MemoryAccess *MA : AG)
      Accesses = isl_union_map_add_map(Accesses, MA->getAccessRelation());

    bool Valid = calculateMinMaxAccess(Accesses, getDomains(),
                                       MinMaxAccessesNonReadOnly);

    // Bail out if the number of values we need to compare is too large.
    // This is important as the number of comparisions grows quadratically with
    // the number of values we need to compare.
    if (!Valid || (MinMaxAccessesNonReadOnly.size() + !ReadOnlyPairs.empty() >
                   RunTimeChecksMaxArraysPerGroup))
      return false;

    // Calculate minimal and maximal accesses for read only accesses.
    MinMaxAccessesReadOnly.reserve(ReadOnlyPairs.size());
    Accesses = isl_union_map_empty(getParamSpace());

    for (const auto &ReadOnlyPair : ReadOnlyPairs)
      for (MemoryAccess *MA : ReadOnlyPair.second)
        Accesses = isl_union_map_add_map(Accesses, MA->getAccessRelation());

    Valid =
        calculateMinMaxAccess(Accesses, getDomains(), MinMaxAccessesReadOnly);

    if (!Valid)
      return false;
  }

  return true;
}

static Loop *getLoopSurroundingRegion(Region &R, LoopInfo &LI) {
  Loop *L = LI.getLoopFor(R.getEntry());
  return L ? (R.contains(L) ? L->getParentLoop() : L) : nullptr;
}

static unsigned getMaxLoopDepthInRegion(const Region &R, LoopInfo &LI,
                                        ScopDetection &SD) {

  const ScopDetection::BoxedLoopsSetTy *BoxedLoops = SD.getBoxedLoops(&R);

  unsigned MinLD = INT_MAX, MaxLD = 0;
  for (BasicBlock *BB : R.blocks()) {
    if (Loop *L = LI.getLoopFor(BB)) {
      if (!R.contains(L))
        continue;
      if (BoxedLoops && BoxedLoops->count(L))
        continue;
      unsigned LD = L->getLoopDepth();
      MinLD = std::min(MinLD, LD);
      MaxLD = std::max(MaxLD, LD);
    }
  }

  // Handle the case that there is no loop in the SCoP first.
  if (MaxLD == 0)
    return 1;

  assert(MinLD >= 1 && "Minimal loop depth should be at least one");
  assert(MaxLD >= MinLD &&
         "Maximal loop depth was smaller than mininaml loop depth?");
  return MaxLD - MinLD + 1;
}

Scop::Scop(Region &R, AccFuncMapType &AccFuncMap,
           ScalarEvolution &ScalarEvolution, DominatorTree &DT,
           isl_ctx *Context, unsigned MaxLoopDepth)
    : DT(DT), SE(&ScalarEvolution), R(R), AccFuncMap(AccFuncMap),
      IsOptimized(false), HasSingleExitEdge(R.getExitingBlock()),
      MaxLoopDepth(MaxLoopDepth), IslCtx(Context), Affinator(this) {}

void Scop::init(LoopInfo &LI, ScopDetection &SD, AliasAnalysis &AA) {
  buildContext();

  buildDomains(&R, LI, SD, DT);

  DenseMap<Loop *, std::pair<isl_schedule *, unsigned>> LoopSchedules;

  Loop *L = getLoopSurroundingRegion(R, LI);
  LoopSchedules[L];
  buildSchedule(&R, LI, SD, LoopSchedules);
  Schedule = LoopSchedules[L].first;

  realignParams();
  addParameterBounds();
  addUserContext();
  simplifyAssumedContext();
  buildAliasChecks(AA);
}

Scop::~Scop() {
  isl_set_free(Context);
  isl_set_free(AssumedContext);
  isl_schedule_free(Schedule);

  for (auto It : DomainMap)
    isl_set_free(It.second);

  // Free the alias groups
  for (MinMaxVectorPairTy &MinMaxAccessPair : MinMaxAliasGroups) {
    for (MinMaxAccessTy &MMA : MinMaxAccessPair.first) {
      isl_pw_multi_aff_free(MMA.first);
      isl_pw_multi_aff_free(MMA.second);
    }
    for (MinMaxAccessTy &MMA : MinMaxAccessPair.second) {
      isl_pw_multi_aff_free(MMA.first);
      isl_pw_multi_aff_free(MMA.second);
    }
  }
}

const ScopArrayInfo *
Scop::getOrCreateScopArrayInfo(Value *BasePtr, Type *AccessType,
                               const SmallVector<const SCEV *, 4> &Sizes,
                               bool IsPHI) {
  auto &SAI = ScopArrayInfoMap[std::make_pair(BasePtr, IsPHI)];
  if (!SAI)
    SAI.reset(new ScopArrayInfo(BasePtr, AccessType, getIslCtx(), Sizes, IsPHI,
                                this));
  return SAI.get();
}

const ScopArrayInfo *Scop::getScopArrayInfo(Value *BasePtr, bool IsPHI) {
  auto *SAI = ScopArrayInfoMap[std::make_pair(BasePtr, IsPHI)].get();
  assert(SAI && "No ScopArrayInfo available for this base pointer");
  return SAI;
}

std::string Scop::getContextStr() const { return stringFromIslObj(Context); }
std::string Scop::getAssumedContextStr() const {
  return stringFromIslObj(AssumedContext);
}

std::string Scop::getNameStr() const {
  std::string ExitName, EntryName;
  raw_string_ostream ExitStr(ExitName);
  raw_string_ostream EntryStr(EntryName);

  R.getEntry()->printAsOperand(EntryStr, false);
  EntryStr.str();

  if (R.getExit()) {
    R.getExit()->printAsOperand(ExitStr, false);
    ExitStr.str();
  } else
    ExitName = "FunctionExit";

  return EntryName + "---" + ExitName;
}

__isl_give isl_set *Scop::getContext() const { return isl_set_copy(Context); }
__isl_give isl_space *Scop::getParamSpace() const {
  return isl_set_get_space(Context);
}

__isl_give isl_set *Scop::getAssumedContext() const {
  return isl_set_copy(AssumedContext);
}

__isl_give isl_set *Scop::getRuntimeCheckContext() const {
  isl_set *RuntimeCheckContext = getAssumedContext();
  return RuntimeCheckContext;
}

bool Scop::hasFeasibleRuntimeContext() const {
  isl_set *RuntimeCheckContext = getRuntimeCheckContext();
  RuntimeCheckContext = addNonEmptyDomainConstraints(RuntimeCheckContext);
  bool IsFeasible = !isl_set_is_empty(RuntimeCheckContext);
  isl_set_free(RuntimeCheckContext);
  return IsFeasible;
}

void Scop::addAssumption(__isl_take isl_set *Set) {
  AssumedContext = isl_set_intersect(AssumedContext, Set);
  AssumedContext = isl_set_coalesce(AssumedContext);
}

void Scop::printContext(raw_ostream &OS) const {
  OS << "Context:\n";

  if (!Context) {
    OS.indent(4) << "n/a\n\n";
    return;
  }

  OS.indent(4) << getContextStr() << "\n";

  OS.indent(4) << "Assumed Context:\n";
  if (!AssumedContext) {
    OS.indent(4) << "n/a\n\n";
    return;
  }

  OS.indent(4) << getAssumedContextStr() << "\n";

  for (const SCEV *Parameter : Parameters) {
    int Dim = ParameterIds.find(Parameter)->second;
    OS.indent(4) << "p" << Dim << ": " << *Parameter << "\n";
  }
}

void Scop::printAliasAssumptions(raw_ostream &OS) const {
  int noOfGroups = 0;
  for (const MinMaxVectorPairTy &Pair : MinMaxAliasGroups) {
    if (Pair.second.size() == 0)
      noOfGroups += 1;
    else
      noOfGroups += Pair.second.size();
  }

  OS.indent(4) << "Alias Groups (" << noOfGroups << "):\n";
  if (MinMaxAliasGroups.empty()) {
    OS.indent(8) << "n/a\n";
    return;
  }

  for (const MinMaxVectorPairTy &Pair : MinMaxAliasGroups) {

    // If the group has no read only accesses print the write accesses.
    if (Pair.second.empty()) {
      OS.indent(8) << "[[";
      for (const MinMaxAccessTy &MMANonReadOnly : Pair.first) {
        OS << " <" << MMANonReadOnly.first << ", " << MMANonReadOnly.second
           << ">";
      }
      OS << " ]]\n";
    }

    for (const MinMaxAccessTy &MMAReadOnly : Pair.second) {
      OS.indent(8) << "[[";
      OS << " <" << MMAReadOnly.first << ", " << MMAReadOnly.second << ">";
      for (const MinMaxAccessTy &MMANonReadOnly : Pair.first) {
        OS << " <" << MMANonReadOnly.first << ", " << MMANonReadOnly.second
           << ">";
      }
      OS << " ]]\n";
    }
  }
}

void Scop::printStatements(raw_ostream &OS) const {
  OS << "Statements {\n";

  for (const ScopStmt &Stmt : *this)
    OS.indent(4) << Stmt;

  OS.indent(4) << "}\n";
}

void Scop::printArrayInfo(raw_ostream &OS) const {
  OS << "Arrays {\n";

  for (auto &Array : arrays())
    Array.second->print(OS);

  OS.indent(4) << "}\n";

  OS.indent(4) << "Arrays (Bounds as pw_affs) {\n";

  for (auto &Array : arrays())
    Array.second->print(OS, /* SizeAsPwAff */ true);

  OS.indent(4) << "}\n";
}

void Scop::print(raw_ostream &OS) const {
  OS.indent(4) << "Function: " << getRegion().getEntry()->getParent()->getName()
               << "\n";
  OS.indent(4) << "Region: " << getNameStr() << "\n";
  OS.indent(4) << "Max Loop Depth:  " << getMaxLoopDepth() << "\n";
  printContext(OS.indent(4));
  printArrayInfo(OS.indent(4));
  printAliasAssumptions(OS);
  printStatements(OS.indent(4));
}

void Scop::dump() const { print(dbgs()); }

isl_ctx *Scop::getIslCtx() const { return IslCtx; }

__isl_give isl_pw_aff *Scop::getPwAff(const SCEV *E, isl_set *Domain) {
  return Affinator.getPwAff(E, Domain);
}

__isl_give isl_union_set *Scop::getDomains() const {
  isl_union_set *Domain = isl_union_set_empty(getParamSpace());

  for (const ScopStmt &Stmt : *this)
    Domain = isl_union_set_add_set(Domain, Stmt.getDomain());

  return Domain;
}

__isl_give isl_union_map *Scop::getMustWrites() {
  isl_union_map *Write = isl_union_map_empty(getParamSpace());

  for (ScopStmt &Stmt : *this) {
    for (MemoryAccess *MA : Stmt) {
      if (!MA->isMustWrite())
        continue;

      isl_set *Domain = Stmt.getDomain();
      isl_map *AccessDomain = MA->getAccessRelation();
      AccessDomain = isl_map_intersect_domain(AccessDomain, Domain);
      Write = isl_union_map_add_map(Write, AccessDomain);
    }
  }
  return isl_union_map_coalesce(Write);
}

__isl_give isl_union_map *Scop::getMayWrites() {
  isl_union_map *Write = isl_union_map_empty(getParamSpace());

  for (ScopStmt &Stmt : *this) {
    for (MemoryAccess *MA : Stmt) {
      if (!MA->isMayWrite())
        continue;

      isl_set *Domain = Stmt.getDomain();
      isl_map *AccessDomain = MA->getAccessRelation();
      AccessDomain = isl_map_intersect_domain(AccessDomain, Domain);
      Write = isl_union_map_add_map(Write, AccessDomain);
    }
  }
  return isl_union_map_coalesce(Write);
}

__isl_give isl_union_map *Scop::getWrites() {
  isl_union_map *Write = isl_union_map_empty(getParamSpace());

  for (ScopStmt &Stmt : *this) {
    for (MemoryAccess *MA : Stmt) {
      if (!MA->isWrite())
        continue;

      isl_set *Domain = Stmt.getDomain();
      isl_map *AccessDomain = MA->getAccessRelation();
      AccessDomain = isl_map_intersect_domain(AccessDomain, Domain);
      Write = isl_union_map_add_map(Write, AccessDomain);
    }
  }
  return isl_union_map_coalesce(Write);
}

__isl_give isl_union_map *Scop::getReads() {
  isl_union_map *Read = isl_union_map_empty(getParamSpace());

  for (ScopStmt &Stmt : *this) {
    for (MemoryAccess *MA : Stmt) {
      if (!MA->isRead())
        continue;

      isl_set *Domain = Stmt.getDomain();
      isl_map *AccessDomain = MA->getAccessRelation();

      AccessDomain = isl_map_intersect_domain(AccessDomain, Domain);
      Read = isl_union_map_add_map(Read, AccessDomain);
    }
  }
  return isl_union_map_coalesce(Read);
}

__isl_give isl_union_map *Scop::getSchedule() const {
  auto Tree = getScheduleTree();
  auto S = isl_schedule_get_map(Tree);
  isl_schedule_free(Tree);
  return S;
}

__isl_give isl_schedule *Scop::getScheduleTree() const {
  return isl_schedule_intersect_domain(isl_schedule_copy(Schedule),
                                       getDomains());
}

void Scop::setSchedule(__isl_take isl_union_map *NewSchedule) {
  auto *S = isl_schedule_from_domain(getDomains());
  S = isl_schedule_insert_partial_schedule(
      S, isl_multi_union_pw_aff_from_union_map(NewSchedule));
  isl_schedule_free(Schedule);
  Schedule = S;
}

void Scop::setScheduleTree(__isl_take isl_schedule *NewSchedule) {
  isl_schedule_free(Schedule);
  Schedule = NewSchedule;
}

bool Scop::restrictDomains(__isl_take isl_union_set *Domain) {
  bool Changed = false;
  for (ScopStmt &Stmt : *this) {
    isl_union_set *StmtDomain = isl_union_set_from_set(Stmt.getDomain());
    isl_union_set *NewStmtDomain = isl_union_set_intersect(
        isl_union_set_copy(StmtDomain), isl_union_set_copy(Domain));

    if (isl_union_set_is_subset(StmtDomain, NewStmtDomain)) {
      isl_union_set_free(StmtDomain);
      isl_union_set_free(NewStmtDomain);
      continue;
    }

    Changed = true;

    isl_union_set_free(StmtDomain);
    NewStmtDomain = isl_union_set_coalesce(NewStmtDomain);

    if (isl_union_set_is_empty(NewStmtDomain)) {
      Stmt.restrictDomain(isl_set_empty(Stmt.getDomainSpace()));
      isl_union_set_free(NewStmtDomain);
    } else
      Stmt.restrictDomain(isl_set_from_union_set(NewStmtDomain));
  }
  isl_union_set_free(Domain);
  return Changed;
}

ScalarEvolution *Scop::getSE() const { return SE; }

bool Scop::isTrivialBB(BasicBlock *BB) {
  if (getAccessFunctions(BB) && !isErrorBlock(*BB))
    return false;

  return true;
}

struct MapToDimensionDataTy {
  int N;
  isl_union_pw_multi_aff *Res;
};

// @brief Create a function that maps the elements of 'Set' to its N-th
//        dimension.
//
// The result is added to 'User->Res'.
//
// @param Set The input set.
// @param N   The dimension to map to.
//
// @returns   Zero if no error occurred, non-zero otherwise.
static isl_stat mapToDimension_AddSet(__isl_take isl_set *Set, void *User) {
  struct MapToDimensionDataTy *Data = (struct MapToDimensionDataTy *)User;
  int Dim;
  isl_space *Space;
  isl_pw_multi_aff *PMA;

  Dim = isl_set_dim(Set, isl_dim_set);
  Space = isl_set_get_space(Set);
  PMA = isl_pw_multi_aff_project_out_map(Space, isl_dim_set, Data->N,
                                         Dim - Data->N);
  if (Data->N > 1)
    PMA = isl_pw_multi_aff_drop_dims(PMA, isl_dim_out, 0, Data->N - 1);
  Data->Res = isl_union_pw_multi_aff_add_pw_multi_aff(Data->Res, PMA);

  isl_set_free(Set);

  return isl_stat_ok;
}

// @brief Create a function that maps the elements of Domain to their Nth
//        dimension.
//
// @param Domain The set of elements to map.
// @param N      The dimension to map to.
static __isl_give isl_multi_union_pw_aff *
mapToDimension(__isl_take isl_union_set *Domain, int N) {
  if (N <= 0 || isl_union_set_is_empty(Domain)) {
    isl_union_set_free(Domain);
    return nullptr;
  }

  struct MapToDimensionDataTy Data;
  isl_space *Space;

  Space = isl_union_set_get_space(Domain);
  Data.N = N;
  Data.Res = isl_union_pw_multi_aff_empty(Space);
  if (isl_union_set_foreach_set(Domain, &mapToDimension_AddSet, &Data) < 0)
    Data.Res = isl_union_pw_multi_aff_free(Data.Res);

  isl_union_set_free(Domain);
  return isl_multi_union_pw_aff_from_union_pw_multi_aff(Data.Res);
}

ScopStmt *Scop::addScopStmt(BasicBlock *BB, Region *R) {
  ScopStmt *Stmt;
  if (BB) {
    Stmts.emplace_back(*this, *BB);
    Stmt = &Stmts.back();
    StmtMap[BB] = Stmt;
  } else {
    assert(R && "Either basic block or a region expected.");
    Stmts.emplace_back(*this, *R);
    Stmt = &Stmts.back();
    for (BasicBlock *BB : R->blocks())
      StmtMap[BB] = Stmt;
  }
  return Stmt;
}

void Scop::buildSchedule(
    Region *R, LoopInfo &LI, ScopDetection &SD,
    DenseMap<Loop *, std::pair<isl_schedule *, unsigned>> &LoopSchedules) {

  ReversePostOrderTraversal<Region *> RTraversal(R);
  for (auto *RN : RTraversal) {

    if (RN->isSubRegion()) {
      Region *SubRegion = RN->getNodeAs<Region>();
      if (!SD.isNonAffineSubRegion(SubRegion, &getRegion())) {
        buildSchedule(SubRegion, LI, SD, LoopSchedules);
        continue;
      }
    }

    Loop *L = getRegionNodeLoop(RN, LI);
    int LD = getRelativeLoopDepth(L);
    auto &LSchedulePair = LoopSchedules[L];
    LSchedulePair.second += getNumBlocksInRegionNode(RN);

    BasicBlock *BB = getRegionNodeBasicBlock(RN);
    if (RN->isSubRegion() || !isTrivialBB(BB)) {

      ScopStmt *Stmt;
      if (RN->isSubRegion())
        Stmt = addScopStmt(nullptr, RN->getNodeAs<Region>());
      else
        Stmt = addScopStmt(BB, nullptr);

      auto *UDomain = isl_union_set_from_set(Stmt->getDomain());
      auto *StmtSchedule = isl_schedule_from_domain(UDomain);
      LSchedulePair.first =
          combineInSequence(LSchedulePair.first, StmtSchedule);
    }

    unsigned NumVisited = LSchedulePair.second;
    while (L && NumVisited == L->getNumBlocks()) {
      auto *LDomain = isl_schedule_get_domain(LSchedulePair.first);
      if (auto *MUPA = mapToDimension(LDomain, LD + 1))
        LSchedulePair.first =
            isl_schedule_insert_partial_schedule(LSchedulePair.first, MUPA);

      auto *PL = L->getParentLoop();
      assert(LoopSchedules.count(PL));
      auto &PSchedulePair = LoopSchedules[PL];
      PSchedulePair.first =
          combineInSequence(PSchedulePair.first, LSchedulePair.first);
      PSchedulePair.second += NumVisited;

      L = PL;
      NumVisited = PSchedulePair.second;
    }
  }
}

ScopStmt *Scop::getStmtForBasicBlock(BasicBlock *BB) const {
  auto StmtMapIt = StmtMap.find(BB);
  if (StmtMapIt == StmtMap.end())
    return nullptr;
  return StmtMapIt->second;
}

void Scop::printIRAccesses(raw_ostream &OS, ScalarEvolution *SE,
                           LoopInfo *LI) const {
  OS << "Scop: " << R.getNameStr() << "\n";

  printIRAccessesDetail(OS, SE, LI, &R, 0);
}

void Scop::printIRAccessesDetail(raw_ostream &OS, ScalarEvolution *SE,
                                 LoopInfo *LI, const Region *CurR,
                                 unsigned ind) const {
  // FIXME: Print other details rather than memory accesses.
  for (const auto &CurBlock : CurR->blocks()) {
    AccFuncMapType::const_iterator AccSetIt = AccFuncMap.find(CurBlock);

    // Ignore trivial blocks that do not contain any memory access.
    if (AccSetIt == AccFuncMap.end())
      continue;

    OS.indent(ind) << "BB: " << CurBlock->getName() << '\n';
    typedef AccFuncSetType::const_iterator access_iterator;
    const AccFuncSetType &AccFuncs = AccSetIt->second;

    for (access_iterator AI = AccFuncs.begin(), AE = AccFuncs.end(); AI != AE;
         ++AI)
      AI->first.print(OS.indent(ind + 2));
  }
}

int Scop::getRelativeLoopDepth(const Loop *L) const {
  Loop *OuterLoop =
      L ? R.outermostLoopInRegion(const_cast<Loop *>(L)) : nullptr;
  if (!OuterLoop)
    return -1;
  return L->getLoopDepth() - OuterLoop->getLoopDepth();
}

void ScopInfo::buildPHIAccesses(PHINode *PHI, Region &R,
                                AccFuncSetType &Functions,
                                Region *NonAffineSubRegion, bool IsExitBlock) {

  // PHI nodes that are in the exit block of the region, hence if IsExitBlock is
  // true, are not modeled as ordinary PHI nodes as they are not part of the
  // region. However, we model the operands in the predecessor blocks that are
  // part of the region as regular scalar accesses.

  // If we can synthesize a PHI we can skip it, however only if it is in
  // the region. If it is not it can only be in the exit block of the region.
  // In this case we model the operands but not the PHI itself.
  if (!IsExitBlock && canSynthesize(PHI, LI, SE, &R))
    return;

  // PHI nodes are modeled as if they had been demoted prior to the SCoP
  // detection. Hence, the PHI is a load of a new memory location in which the
  // incoming value was written at the end of the incoming basic block.
  bool OnlyNonAffineSubRegionOperands = true;
  for (unsigned u = 0; u < PHI->getNumIncomingValues(); u++) {
    Value *Op = PHI->getIncomingValue(u);
    BasicBlock *OpBB = PHI->getIncomingBlock(u);

    // Do not build scalar dependences inside a non-affine subregion.
    if (NonAffineSubRegion && NonAffineSubRegion->contains(OpBB))
      continue;

    OnlyNonAffineSubRegionOperands = false;

    if (!R.contains(OpBB))
      continue;

    Instruction *OpI = dyn_cast<Instruction>(Op);
    if (OpI) {
      BasicBlock *OpIBB = OpI->getParent();
      // As we pretend there is a use (or more precise a write) of OpI in OpBB
      // we have to insert a scalar dependence from the definition of OpI to
      // OpBB if the definition is not in OpBB.
      if (OpIBB != OpBB) {
        IRAccess ScalarRead(IRAccess::READ, OpI, ZeroOffset, 1, true, OpI);
        AccFuncMap[OpBB].push_back(std::make_pair(ScalarRead, PHI));
        IRAccess ScalarWrite(IRAccess::MUST_WRITE, OpI, ZeroOffset, 1, true,
                             OpI);
        AccFuncMap[OpIBB].push_back(std::make_pair(ScalarWrite, OpI));
      }
    }

    // Always use the terminator of the incoming basic block as the access
    // instruction.
    OpI = OpBB->getTerminator();

    IRAccess ScalarAccess(IRAccess::MUST_WRITE, PHI, ZeroOffset, 1, true, Op,
                          /* IsPHI */ !IsExitBlock);
    AccFuncMap[OpBB].push_back(std::make_pair(ScalarAccess, OpI));
  }

  if (!OnlyNonAffineSubRegionOperands) {
    IRAccess ScalarAccess(IRAccess::READ, PHI, ZeroOffset, 1, true, PHI,
                          /* IsPHI */ !IsExitBlock);
    Functions.push_back(std::make_pair(ScalarAccess, PHI));
  }
}

bool ScopInfo::buildScalarDependences(Instruction *Inst, Region *R,
                                      Region *NonAffineSubRegion) {
  bool canSynthesizeInst = canSynthesize(Inst, LI, SE, R);
  if (isIgnoredIntrinsic(Inst))
    return false;

  bool AnyCrossStmtUse = false;
  BasicBlock *ParentBB = Inst->getParent();

  for (User *U : Inst->users()) {
    Instruction *UI = dyn_cast<Instruction>(U);

    // Ignore the strange user
    if (UI == 0)
      continue;

    BasicBlock *UseParent = UI->getParent();

    // Ignore the users in the same BB (statement)
    if (UseParent == ParentBB)
      continue;

    // Do not build scalar dependences inside a non-affine subregion.
    if (NonAffineSubRegion && NonAffineSubRegion->contains(UseParent))
      continue;

    // Check whether or not the use is in the SCoP.
    if (!R->contains(UseParent)) {
      AnyCrossStmtUse = true;
      continue;
    }

    // If the instruction can be synthesized and the user is in the region
    // we do not need to add scalar dependences.
    if (canSynthesizeInst)
      continue;

    // No need to translate these scalar dependences into polyhedral form,
    // because synthesizable scalars can be generated by the code generator.
    if (canSynthesize(UI, LI, SE, R))
      continue;

    // Skip PHI nodes in the region as they handle their operands on their own.
    if (isa<PHINode>(UI))
      continue;

    // Now U is used in another statement.
    AnyCrossStmtUse = true;

    // Do not build a read access that is not in the current SCoP
    // Use the def instruction as base address of the IRAccess, so that it will
    // become the name of the scalar access in the polyhedral form.
    IRAccess ScalarAccess(IRAccess::READ, Inst, ZeroOffset, 1, true, Inst);
    AccFuncMap[UseParent].push_back(std::make_pair(ScalarAccess, UI));
  }

  if (ModelReadOnlyScalars) {
    for (Value *Op : Inst->operands()) {
      if (canSynthesize(Op, LI, SE, R))
        continue;

      if (Instruction *OpInst = dyn_cast<Instruction>(Op))
        if (R->contains(OpInst))
          continue;

      if (isa<Constant>(Op))
        continue;

      IRAccess ScalarAccess(IRAccess::READ, Op, ZeroOffset, 1, true, Op);
      AccFuncMap[Inst->getParent()].push_back(
          std::make_pair(ScalarAccess, Inst));
    }
  }

  return AnyCrossStmtUse;
}

extern MapInsnToMemAcc InsnToMemAcc;

IRAccess
ScopInfo::buildIRAccess(Instruction *Inst, Loop *L, Region *R,
                        const ScopDetection::BoxedLoopsSetTy *BoxedLoops) {
  unsigned Size;
  Type *SizeType;
  Value *Val;
  enum IRAccess::TypeKind Type;

  if (LoadInst *Load = dyn_cast<LoadInst>(Inst)) {
    SizeType = Load->getType();
    Size = TD->getTypeStoreSize(SizeType);
    Type = IRAccess::READ;
    Val = Load;
  } else {
    StoreInst *Store = cast<StoreInst>(Inst);
    SizeType = Store->getValueOperand()->getType();
    Size = TD->getTypeStoreSize(SizeType);
    Type = IRAccess::MUST_WRITE;
    Val = Store->getValueOperand();
  }

  const SCEV *AccessFunction = SE->getSCEVAtScope(getPointerOperand(*Inst), L);
  const SCEVUnknown *BasePointer =
      dyn_cast<SCEVUnknown>(SE->getPointerBase(AccessFunction));

  assert(BasePointer && "Could not find base pointer");
  AccessFunction = SE->getMinusSCEV(AccessFunction, BasePointer);

  auto AccItr = InsnToMemAcc.find(Inst);
  if (PollyDelinearize && AccItr != InsnToMemAcc.end())
    return IRAccess(Type, BasePointer->getValue(), AccessFunction, Size, true,
                    AccItr->second.DelinearizedSubscripts,
                    AccItr->second.Shape->DelinearizedSizes, Val);

  // Check if the access depends on a loop contained in a non-affine subregion.
  bool isVariantInNonAffineLoop = false;
  if (BoxedLoops) {
    SetVector<const Loop *> Loops;
    findLoops(AccessFunction, Loops);
    for (const Loop *L : Loops)
      if (BoxedLoops->count(L))
        isVariantInNonAffineLoop = true;
  }

  bool IsAffine = !isVariantInNonAffineLoop &&
                  isAffineExpr(R, AccessFunction, *SE, BasePointer->getValue());

  SmallVector<const SCEV *, 4> Subscripts, Sizes;
  Subscripts.push_back(AccessFunction);
  Sizes.push_back(SE->getConstant(ZeroOffset->getType(), Size));

  if (!IsAffine && Type == IRAccess::MUST_WRITE)
    Type = IRAccess::MAY_WRITE;

  return IRAccess(Type, BasePointer->getValue(), AccessFunction, Size, IsAffine,
                  Subscripts, Sizes, Val);
}

void ScopInfo::buildAccessFunctions(Region &R, Region &SR) {

  if (SD->isNonAffineSubRegion(&SR, &R)) {
    for (BasicBlock *BB : SR.blocks())
      buildAccessFunctions(R, *BB, &SR);
    return;
  }

  for (auto I = SR.element_begin(), E = SR.element_end(); I != E; ++I)
    if (I->isSubRegion())
      buildAccessFunctions(R, *I->getNodeAs<Region>());
    else
      buildAccessFunctions(R, *I->getNodeAs<BasicBlock>());
}

void ScopInfo::buildAccessFunctions(Region &R, BasicBlock &BB,
                                    Region *NonAffineSubRegion,
                                    bool IsExitBlock) {
  AccFuncSetType Functions;
  Loop *L = LI->getLoopFor(&BB);

  // The set of loops contained in non-affine subregions that are part of R.
  const ScopDetection::BoxedLoopsSetTy *BoxedLoops = SD->getBoxedLoops(&R);

  for (BasicBlock::iterator I = BB.begin(), E = --BB.end(); I != E; ++I) {
    Instruction *Inst = I;

    PHINode *PHI = dyn_cast<PHINode>(Inst);
    if (PHI)
      buildPHIAccesses(PHI, R, Functions, NonAffineSubRegion, IsExitBlock);

    // For the exit block we stop modeling after the last PHI node.
    if (!PHI && IsExitBlock)
      break;

    if (isa<LoadInst>(Inst) || isa<StoreInst>(Inst))
      Functions.push_back(
          std::make_pair(buildIRAccess(Inst, L, &R, BoxedLoops), Inst));

    if (isIgnoredIntrinsic(Inst))
      continue;

    if (buildScalarDependences(Inst, &R, NonAffineSubRegion)) {
      // If the Instruction is used outside the statement, we need to build the
      // write access.
      if (!isa<StoreInst>(Inst)) {
        IRAccess ScalarAccess(IRAccess::MUST_WRITE, Inst, ZeroOffset, 1, true,
                              Inst);
        Functions.push_back(std::make_pair(ScalarAccess, Inst));
      }
    }
  }

  if (Functions.empty())
    return;

  AccFuncSetType &Accs = AccFuncMap[&BB];
  Accs.insert(Accs.end(), Functions.begin(), Functions.end());
}

Scop *ScopInfo::buildScop(Region &R, DominatorTree &DT) {
  unsigned MaxLoopDepth = getMaxLoopDepthInRegion(R, *LI, *SD);
  Scop *S = new Scop(R, AccFuncMap, *SE, DT, ctx, MaxLoopDepth);

  buildAccessFunctions(R, R);

  // In case the region does not have an exiting block we will later (during
  // code generation) split the exit block. This will move potential PHI nodes
  // from the current exit block into the new region exiting block. Hence, PHI
  // nodes that are at this point not part of the region will be.
  // To handle these PHI nodes later we will now model their operands as scalar
  // accesses. Note that we do not model anything in the exit block if we have
  // an exiting block in the region, as there will not be any splitting later.
  if (!R.getExitingBlock())
    buildAccessFunctions(R, *R.getExit(), nullptr, /* IsExitBlock */ true);

  S->init(*LI, *SD, *AA);
  return S;
}

void ScopInfo::print(raw_ostream &OS, const Module *) const {
  if (!scop) {
    OS << "Invalid Scop!\n";
    return;
  }

  scop->printIRAccesses(OS, SE, LI);
  scop->print(OS);
}

void ScopInfo::clear() {
  AccFuncMap.clear();
  if (scop) {
    delete scop;
    scop = 0;
  }
}

//===----------------------------------------------------------------------===//
ScopInfo::ScopInfo() : RegionPass(ID), scop(0) {
  ctx = isl_ctx_alloc();
  isl_options_set_on_error(ctx, ISL_ON_ERROR_ABORT);
}

ScopInfo::~ScopInfo() {
  clear();
  isl_ctx_free(ctx);
}

void ScopInfo::getAnalysisUsage(AnalysisUsage &AU) const {
  AU.addRequiredID(IndependentBlocksID);
  AU.addRequired<LoopInfoWrapperPass>();
  AU.addRequired<RegionInfoPass>();
  AU.addRequired<DominatorTreeWrapperPass>();
  AU.addRequiredTransitive<ScalarEvolutionWrapperPass>();
  AU.addRequiredTransitive<ScopDetection>();
  AU.addRequired<AAResultsWrapperPass>();
  AU.setPreservesAll();
}

bool ScopInfo::runOnRegion(Region *R, RGPassManager &RGM) {
  SD = &getAnalysis<ScopDetection>();

  if (!SD->isMaxRegionInScop(*R))
    return false;

  Function *F = R->getEntry()->getParent();
  SE = &getAnalysis<ScalarEvolutionWrapperPass>().getSE();
  LI = &getAnalysis<LoopInfoWrapperPass>().getLoopInfo();
  AA = &getAnalysis<AAResultsWrapperPass>().getAAResults();
  TD = &F->getParent()->getDataLayout();
  DominatorTree &DT = getAnalysis<DominatorTreeWrapperPass>().getDomTree();
  ZeroOffset = SE->getConstant(TD->getIntPtrType(F->getContext()), 0);

  scop = buildScop(*R, DT);

  DEBUG(scop->print(dbgs()));

  if (!scop->hasFeasibleRuntimeContext()) {
    delete scop;
    scop = nullptr;
    return false;
  }

  // Statistics.
  ++ScopFound;
  if (scop->getMaxLoopDepth() > 0)
    ++RichScopFound;
  return false;
}

char ScopInfo::ID = 0;

Pass *polly::createScopInfoPass() { return new ScopInfo(); }

INITIALIZE_PASS_BEGIN(ScopInfo, "polly-scops",
                      "Polly - Create polyhedral description of Scops", false,
                      false);
INITIALIZE_PASS_DEPENDENCY(AAResultsWrapperPass);
INITIALIZE_PASS_DEPENDENCY(LoopInfoWrapperPass);
INITIALIZE_PASS_DEPENDENCY(RegionInfoPass);
INITIALIZE_PASS_DEPENDENCY(ScalarEvolutionWrapperPass);
INITIALIZE_PASS_DEPENDENCY(ScopDetection);
INITIALIZE_PASS_DEPENDENCY(DominatorTreeWrapperPass);
INITIALIZE_PASS_END(ScopInfo, "polly-scops",
                    "Polly - Create polyhedral description of Scops", false,
                    false)