polly/lib/Transform/DeLICM.cpp

//===------ DeLICM.cpp -----------------------------------------*- C++ -*-===//
//
//                     The LLVM Compiler Infrastructure
//
// This file is distributed under the University of Illinois Open Source
// License. See LICENSE.TXT for details.
//
//===----------------------------------------------------------------------===//
//
// Undo the effect of Loop Invariant Code Motion (LICM) and
// GVN Partial Redundancy Elimination (PRE) on SCoP-level.
//
// Namely, remove register/scalar dependencies by mapping them back to array
// elements.
//
// The algorithms here work on the scatter space - the image space of the
// schedule returned by Scop::getSchedule(). We call an element in that space a
// "timepoint". Timepoints are lexicographically ordered such that we can
// defined ranges in the scatter space. We use two flavors of such ranges:
// Timepoint sets and zones. A timepoint set is simply a subset of the scatter
// space and is directly stored as isl_set.
//
// Zones are used to describe the space between timepoints as open sets, i.e.
// they do not contain the extrema. Using isl rational sets to express these
// would be overkill. We also cannot store them as the integer timepoints they
// contain; the (nonempty) zone between 1 and 2 would be empty and
// indistinguishable from e.g. the zone between 3 and 4. Also, we cannot store
// the integer set including the extrema; the set ]1,2[ + ]3,4[ could be
// coalesced to ]1,3[, although we defined the range [2,3] to be not in the set.
// Instead, we store the "half-open" integer extrema, including the lower bound,
// but excluding the upper bound. Examples:
//
// * The set { [i] : 1 <= i <= 3 } represents the zone ]0,3[ (which contains the
//   integer points 1 and 2, but not 0 or 3)
//
// * { [1] } represents the zone ]0,1[
//
// * { [i] : i = 1 or i = 3 } represents the zone ]0,1[ + ]2,3[
//
// Therefore, an integer i in the set represents the zone ]i-1,i[, i.e. strictly
// speaking the integer points never belong to the zone. However, depending an
// the interpretation, one might want to include them. Part of the
// interpretation may not be known when the zone is constructed.
//
// Reads are assumed to always take place before writes, hence we can think of
// reads taking place at the beginning of a timepoint and writes at the end.
//
// Let's assume that the zone represents the lifetime of a variable. That is,
// the zone begins with a write that defines the value during its lifetime and
// ends with the last read of that value. In the following we consider whether a
// read/write at the beginning/ending of the lifetime zone should be within the
// zone or outside of it.
//
// * A read at the timepoint that starts the live-range loads the previous
//   value. Hence, exclude the timepoint starting the zone.
//
// * A write at the timepoint that starts the live-range is not defined whether
//   it occurs before or after the write that starts the lifetime. We do not
//   allow this situation to occur. Hence, we include the timepoint starting the
//   zone to determine whether they are conflicting.
//
// * A read at the timepoint that ends the live-range reads the same variable.
//   We include the timepoint at the end of the zone to include that read into
//   the live-range. Doing otherwise would mean that the two reads access
//   different values, which would mean that the value they read are both alive
//   at the same time but occupy the same variable.
//
// * A write at the timepoint that ends the live-range starts a new live-range.
//   It must not be included in the live-range of the previous definition.
//
// All combinations of reads and writes at the endpoints are possible, but most
// of the time only the write->read (for instance, a live-range from definition
// to last use) and read->write (for instance, an unused range from last use to
// overwrite) and combinations are interesting (half-open ranges). write->write
// zones might be useful as well in some context to represent
// output-dependencies.
//
// @see convertZoneToTimepoints
//
//
// The code makes use of maps and sets in many different spaces. To not loose
// track in which space a set or map is expected to be in, variables holding an
// isl reference are usually annotated in the comments. They roughly follow isl
// syntax for spaces, but only the tuples, not the dimensions. The tuples have a
// meaning as follows:
//
// * Space[] - An unspecified tuple. Used for function parameters such that the
//             function caller can use it for anything they like.
//
// * Domain[] - A statement instance as returned by ScopStmt::getDomain()
//     isl_id_get_name: Stmt_<NameOfBasicBlock>
//     isl_id_get_user: Pointer to ScopStmt
//
// * Element[] - An array element as in the range part of
//               MemoryAccess::getAccessRelation()
//     isl_id_get_name: MemRef_<NameOfArrayVariable>
//     isl_id_get_user: Pointer to ScopArrayInfo
//
// * Scatter[] - Scatter space or space of timepoints
//     Has no tuple id
//
// * Zone[] - Range between timepoints as described above
//     Has no tuple id
//
// An annotation "{ Domain[] -> Scatter[] }" therefore means: A map from a
// statement instance to a timepoint, aka a schedule. There is only one scatter
// space, but most of the time multiple statements are processed in one set.
// This is why most of the time isl_union_map has to be used.
//
// The basic algorithm works as follows:
// At first we verify that the SCoP is compatible with this technique. For
// instance, two writes cannot write to the same location at the same statement
// instance because we cannot determine within the polyhedral model which one
// comes first. Once this was verified, we compute zones at which an array
// element is unused. This computation can fail if it takes too long. Then the
// main algorithm is executed. Because every store potentially trails an unused
// zone, we start at stores. We search for a scalar (MemoryKind::Value or
// MemoryKind::PHI) that we can map to the array element overwritten by the
// store, preferably one that is used by the store or at least the ScopStmt.
// When it does not conflict with the lifetime of the values in the array
// element, the map is applied and the unused zone updated as it is now used. We
// continue to try to map scalars to the array element until there are no more
// candidates to map. The algorithm is greedy in the sense that the first scalar
// not conflicting will be mapped. Other scalars processed later that could have
// fit the same unused zone will be rejected. As such the result depends on the
// processing order.
//
//===----------------------------------------------------------------------===//

#include "polly/DeLICM.h"
#include "polly/Options.h"
#include "polly/ScopInfo.h"
#include "polly/ScopPass.h"
#include "polly/Support/ISLTools.h"
#include "llvm/ADT/Statistic.h"
#define DEBUG_TYPE "polly-delicm"

using namespace polly;
using namespace llvm;

namespace {

cl::opt<int>
    DelicmMaxOps("polly-delicm-max-ops",
                 cl::desc("Maximum number of isl operations to invest for "
                          "lifetime analysis; 0=no limit"),
                 cl::init(1000000), cl::cat(PollyCategory));

cl::opt<bool> DelicmOverapproximateWrites(
    "polly-delicm-overapproximate-writes",
    cl::desc(
        "Do more PHI writes than necessary in order to avoid partial accesses"),
    cl::init(false), cl::Hidden, cl::cat(PollyCategory));

STATISTIC(DeLICMAnalyzed, "Number of successfully analyzed SCoPs");
STATISTIC(DeLICMOutOfQuota,
          "Analyses aborted because max_operations was reached");
STATISTIC(DeLICMIncompatible, "Number of SCoPs incompatible for analysis");
STATISTIC(MappedValueScalars, "Number of mapped Value scalars");
STATISTIC(MappedPHIScalars, "Number of mapped PHI scalars");
STATISTIC(TargetsMapped, "Number of stores used for at least one mapping");
STATISTIC(DeLICMScopsModified, "Number of SCoPs optimized");

/// Class for keeping track of scalar def-use chains in the polyhedral
/// representation.
///
/// MemoryKind::Value:
/// There is one definition per llvm::Value or zero (read-only values defined
/// before the SCoP) and an arbitrary number of reads.
///
/// MemoryKind::PHI, MemoryKind::ExitPHI:
/// There is at least one write (the incoming blocks/stmts) and one
/// (MemoryKind::PHI) or zero (MemoryKind::ExitPHI) reads per llvm::PHINode.
class ScalarDefUseChains {
private:
  /// The definitions (i.e. write MemoryAccess) of a MemoryKind::Value scalar.
  DenseMap<const ScopArrayInfo *, MemoryAccess *> ValueDefAccs;

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

  /// The receiving part (i.e. read MemoryAccess) of a MemoryKind::PHI scalar.
  DenseMap<const ScopArrayInfo *, MemoryAccess *> PHIReadAccs;

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

public:
  /// Find the MemoryAccesses that access the ScopArrayInfo-represented memory.
  ///
  /// @param S The SCoP to analyze.
  void compute(Scop *S) {
    // Purge any previous result.
    reset();

    for (auto &Stmt : *S) {
      for (auto *MA : Stmt) {
        if (MA->isOriginalValueKind() && MA->isWrite()) {
          auto *SAI = MA->getScopArrayInfo();
          assert(!ValueDefAccs.count(SAI) &&
                 "There can be at most one "
                 "definition per MemoryKind::Value scalar");
          ValueDefAccs[SAI] = MA;
        }

        if (MA->isOriginalValueKind() && MA->isRead())
          ValueUseAccs[MA->getScopArrayInfo()].push_back(MA);

        if (MA->isOriginalAnyPHIKind() && MA->isRead()) {
          auto *SAI = MA->getScopArrayInfo();
          assert(!PHIReadAccs.count(SAI) &&
                 "There must be exactly one read "
                 "per PHI (that's where the PHINode is)");
          PHIReadAccs[SAI] = MA;
        }

        if (MA->isOriginalAnyPHIKind() && MA->isWrite())
          PHIIncomingAccs[MA->getScopArrayInfo()].push_back(MA);
      }
    }
  }

  /// Free all memory used by the analysis.
  void reset() {
    ValueDefAccs.clear();
    ValueUseAccs.clear();
    PHIReadAccs.clear();
    PHIIncomingAccs.clear();
  }

  MemoryAccess *getValueDef(const ScopArrayInfo *SAI) const {
    return ValueDefAccs.lookup(SAI);
  }

  ArrayRef<MemoryAccess *> getValueUses(const ScopArrayInfo *SAI) const {
    auto It = ValueUseAccs.find(SAI);
    if (It == ValueUseAccs.end())
      return {};
    return It->second;
  }

  MemoryAccess *getPHIRead(const ScopArrayInfo *SAI) const {
    return PHIReadAccs.lookup(SAI);
  }

  ArrayRef<MemoryAccess *> getPHIIncomings(const ScopArrayInfo *SAI) const {
    auto It = PHIIncomingAccs.find(SAI);
    if (It == PHIIncomingAccs.end())
      return {};
    return It->second;
  }
};

isl::union_map computeReachingDefinition(isl::union_map Schedule,
                                         isl::union_map Writes, bool InclDef,
                                         bool InclRedef) {
  return computeReachingWrite(Schedule, Writes, false, InclDef, InclRedef);
}

isl::union_map computeReachingOverwrite(isl::union_map Schedule,
                                        isl::union_map Writes,
                                        bool InclPrevWrite,
                                        bool InclOverwrite) {
  return computeReachingWrite(Schedule, Writes, true, InclPrevWrite,
                              InclOverwrite);
}

/// Compute the next overwrite for a scalar.
///
/// @param Schedule      { DomainWrite[] -> Scatter[] }
///                      Schedule of (at least) all writes. Instances not in @p
///                      Writes are ignored.
/// @param Writes        { DomainWrite[] }
///                      The element instances that write to the scalar.
/// @param InclPrevWrite Whether to extend the timepoints to include
///                      the timepoint where the previous write happens.
/// @param InclOverwrite Whether the reaching overwrite includes the timepoint
///                      of the overwrite itself.
///
/// @return { Scatter[] -> DomainDef[] }
isl::union_map computeScalarReachingOverwrite(isl::union_map Schedule,
                                              isl::union_set Writes,
                                              bool InclPrevWrite,
                                              bool InclOverwrite) {

  // { DomainWrite[] }
  auto WritesMap = give(isl_union_map_from_domain(Writes.take()));

  // { [Element[] -> Scatter[]] -> DomainWrite[] }
  auto Result = computeReachingOverwrite(
      std::move(Schedule), std::move(WritesMap), InclPrevWrite, InclOverwrite);

  return give(isl_union_map_domain_factor_range(Result.take()));
}

/// Overload of computeScalarReachingOverwrite, with only one writing statement.
/// Consequently, the result consists of only one map space.
///
/// @param Schedule      { DomainWrite[] -> Scatter[] }
/// @param Writes        { DomainWrite[] }
/// @param InclPrevWrite Include the previous write to result.
/// @param InclOverwrite Include the overwrite to the result.
///
/// @return { Scatter[] -> DomainWrite[] }
isl::map computeScalarReachingOverwrite(isl::union_map Schedule,
                                        isl::set Writes, bool InclPrevWrite,
                                        bool InclOverwrite) {
  auto ScatterSpace = getScatterSpace(Schedule);
  auto DomSpace = give(isl_set_get_space(Writes.keep()));

  auto ReachOverwrite = computeScalarReachingOverwrite(
      Schedule, give(isl_union_set_from_set(Writes.take())), InclPrevWrite,
      InclOverwrite);

  auto ResultSpace = give(isl_space_map_from_domain_and_range(
      ScatterSpace.take(), DomSpace.take()));
  return singleton(std::move(ReachOverwrite), ResultSpace);
}

/// Compute the reaching definition of a scalar.
///
/// Compared to computeReachingDefinition, there is just one element which is
/// accessed and therefore only a set if instances that accesses that element is
/// required.
///
/// @param Schedule  { DomainWrite[] -> Scatter[] }
/// @param Writes    { DomainWrite[] }
/// @param InclDef   Include the timepoint of the definition to the result.
/// @param InclRedef Include the timepoint of the overwrite into the result.
///
/// @return { Scatter[] -> DomainWrite[] }
isl::union_map computeScalarReachingDefinition(isl::union_map Schedule,
                                               isl::union_set Writes,
                                               bool InclDef, bool InclRedef) {

  // { DomainWrite[] -> Element[] }
  auto Defs = give(isl_union_map_from_domain(Writes.take()));

  // { [Element[] -> Scatter[]] -> DomainWrite[] }
  auto ReachDefs =
      computeReachingDefinition(Schedule, Defs, InclDef, InclRedef);

  // { Scatter[] -> DomainWrite[] }
  return give(isl_union_set_unwrap(
      isl_union_map_range(isl_union_map_curry(ReachDefs.take()))));
}

/// Compute the reaching definition of a scalar.
///
/// This overload accepts only a single writing statement as an isl_map,
/// consequently the result also is only a single isl_map.
///
/// @param Schedule  { DomainWrite[] -> Scatter[] }
/// @param Writes    { DomainWrite[] }
/// @param InclDef   Include the timepoint of the definition to the result.
/// @param InclRedef Include the timepoint of the overwrite into the result.
///
/// @return { Scatter[] -> DomainWrite[] }
isl::map computeScalarReachingDefinition( // { Domain[] -> Zone[] }
    isl::union_map Schedule, isl::set Writes, bool InclDef, bool InclRedef) {
  auto DomainSpace = give(isl_set_get_space(Writes.keep()));
  auto ScatterSpace = getScatterSpace(Schedule);

  //  { Scatter[] -> DomainWrite[] }
  auto UMap = computeScalarReachingDefinition(
      Schedule, give(isl_union_set_from_set(Writes.take())), InclDef,
      InclRedef);

  auto ResultSpace = give(isl_space_map_from_domain_and_range(
      ScatterSpace.take(), DomainSpace.take()));
  return singleton(UMap, ResultSpace);
}

/// If InputVal is not defined in the stmt itself, return the MemoryAccess that
/// reads the scalar. Return nullptr otherwise (if the value is defined in the
/// scop, or is synthesizable).
MemoryAccess *getInputAccessOf(Value *InputVal, ScopStmt *Stmt) {
  for (auto *MA : *Stmt) {
    if (!MA->isRead())
      continue;
    if (!MA->isLatestScalarKind())
      continue;

    assert(MA->getAccessValue() == MA->getBaseAddr());
    if (MA->getAccessValue() == InputVal)
      return MA;
  }
  return nullptr;
}

/// Return whether @p Map maps to an unknown value.
///
/// @param { [] -> ValInst[] }
bool isMapToUnknown(const isl::map &Map) {
  auto Space = give(isl_space_range(isl_map_get_space(Map.keep())));
  return !isl_map_has_tuple_id(Map.keep(), isl_dim_set) &&
         !isl_space_is_wrapping(Space.keep()) &&
         isl_map_dim(Map.keep(), isl_dim_out) == 0;
}

/// Return only the mappings that map to known values.
///
/// @param UMap { [] -> ValInst[] }
///
/// @return { [] -> ValInst[] }
isl::union_map filterKnownValInst(const isl::union_map &UMap) {
  auto Result = give(isl_union_map_empty(isl_union_map_get_space(UMap.keep())));
  UMap.foreach_map([=, &Result](isl::map Map) -> isl::stat {
    if (!isMapToUnknown(Map))
      Result = give(isl_union_map_add_map(Result.take(), Map.take()));
    return isl::stat::ok;
  });
  return Result;
}

/// Try to find a 'natural' extension of a mapped to elements outside its
/// domain.
///
/// @param Relevant The map with mapping that may not be modified.
/// @param Universe The domain to which @p Relevant needs to be extended.
///
/// @return A map with that associates the domain elements of @p Relevant to the
///         same elements and in addition the elements of @p Universe to some
///         undefined elements. The function prefers to return simple maps.
isl::union_map expandMapping(isl::union_map Relevant, isl::union_set Universe) {
  Relevant = give(isl_union_map_coalesce(Relevant.take()));
  auto RelevantDomain = give(isl_union_map_domain(Relevant.copy()));
  auto Simplified =
      give(isl_union_map_gist_domain(Relevant.take(), RelevantDomain.take()));
  Simplified = give(isl_union_map_coalesce(Simplified.take()));
  return give(
      isl_union_map_intersect_domain(Simplified.take(), Universe.take()));
}

/// Represent the knowledge of the contents of any array elements in any zone or
/// the knowledge we would add when mapping a scalar to an array element.
///
/// Every array element at every zone unit has one of two states:
///
/// - Unused: Not occupied by any value so a transformation can change it to
///   other values.
///
/// - Occupied: The element contains a value that is still needed.
///
/// The union of Unused and Unknown zones forms the universe, the set of all
/// elements at every timepoint. The universe can easily be derived from the
/// array elements that are accessed someway. Arrays that are never accessed
/// also never play a role in any computation and can hence be ignored. With a
/// given universe, only one of the sets needs to stored implicitly. Computing
/// the complement is also an expensive operation, hence this class has been
/// designed that only one of sets is needed while the other is assumed to be
/// implicit. It can still be given, but is mostly ignored.
///
/// There are two use cases for the Knowledge class:
///
/// 1) To represent the knowledge of the current state of ScopInfo. The unused
///    state means that an element is currently unused: there is no read of it
///    before the next overwrite. Also called 'Existing'.
///
/// 2) To represent the requirements for mapping a scalar to array elements. The
///    unused state means that there is no change/requirement. Also called
///    'Proposed'.
///
/// In addition to these states at unit zones, Knowledge needs to know when
/// values are written. This is because written values may have no lifetime (one
/// reason is that the value is never read). Such writes would therefore never
/// conflict, but overwrite values that might still be required. Another source
/// of problems are multiple writes to the same element at the same timepoint,
/// because their order is undefined.
class Knowledge {
private:
  /// { [Element[] -> Zone[]] }
  /// Set of array elements and when they are alive.
  /// Can contain a nullptr; in this case the set is implicitly defined as the
  /// complement of #Unused.
  ///
  /// The set of alive array elements is represented as zone, as the set of live
  /// values can differ depending on how the elements are interpreted.
  /// Assuming a value X is written at timestep [0] and read at timestep [1]
  /// without being used at any later point, then the value is alive in the
  /// interval ]0,1[. This interval cannot be represented by an integer set, as
  /// it does not contain any integer point. Zones allow us to represent this
  /// interval and can be converted to sets of timepoints when needed (e.g., in
  /// isConflicting when comparing to the write sets).
  /// @see convertZoneToTimepoints and this file's comment for more details.
  isl::union_set Occupied;

  /// { [Element[] -> Zone[]] }
  /// Set of array elements when they are not alive, i.e. their memory can be
  /// used for other purposed. Can contain a nullptr; in this case the set is
  /// implicitly defined as the complement of #Occupied.
  isl::union_set Unused;

  /// { [Element[] -> Zone[]] -> ValInst[] }
  /// Maps to the known content for each array element at any interval.
  ///
  /// Any element/interval can map to multiple known elements. This is due to
  /// multiple llvm::Value referring to the same content. Examples are
  ///
  /// - A value stored and loaded again. The LoadInst represents the same value
  /// as the StoreInst's value operand.
  ///
  /// - A PHINode is equal to any one of the incoming values. In case of
  /// LCSSA-form, it is always equal to its single incoming value.
  ///
  /// Two Knowledges are considered not conflicting if at least one of the known
  /// values match. Not known values are not stored as an unnamed tuple (as
  /// #Written does), but maps to nothing.
  ///
  ///  Known values are usually just defined for #Occupied elements. Knowing
  ///  #Unused contents has no advantage as it can be overwritten.
  isl::union_map Known;

  /// { [Element[] -> Scatter[]] -> ValInst[] }
  /// The write actions currently in the scop or that would be added when
  /// mapping a scalar. Maps to the value that is written.
  ///
  /// Written values that cannot be identified are represented by an unknown
  /// ValInst[] (an unnamed tuple of 0 dimension). It conflicts with itself.
  isl::union_map Written;

  /// Check whether this Knowledge object is well-formed.
  void checkConsistency() const {
#ifndef NDEBUG
    // Default-initialized object
    if (!Occupied && !Unused && !Known && !Written)
      return;

    assert(Occupied || Unused);
    assert(Known);
    assert(Written);

    // If not all fields are defined, we cannot derived the universe.
    if (!Occupied || !Unused)
      return;

    assert(isl_union_set_is_disjoint(Occupied.keep(), Unused.keep()) ==
           isl_bool_true);
    auto Universe = give(isl_union_set_union(Occupied.copy(), Unused.copy()));

    assert(!Known.domain().is_subset(Universe).is_false());
    assert(!Written.domain().is_subset(Universe).is_false());
#endif
  }

public:
  /// Initialize a nullptr-Knowledge. This is only provided for convenience; do
  /// not use such an object.
  Knowledge() {}

  /// Create a new object with the given members.
  Knowledge(isl::union_set Occupied, isl::union_set Unused,
            isl::union_set Written)
      : Occupied(std::move(Occupied)), Unused(std::move(Unused)),
        Known(isl::manage(
            isl_union_map_empty(isl_union_set_get_space(Written.keep())))),
        Written(isl::manage(isl_union_map_from_domain(Written.take()))) {
    checkConsistency();
  }

  /// Create a new object with the given members.
  Knowledge(isl::union_set Occupied, isl::union_set Unused,
            isl::union_map Known, isl::union_map Written)
      : Occupied(std::move(Occupied)), Unused(std::move(Unused)),
        Known(std::move(Known)), Written(std::move(Written)) {
    checkConsistency();
  }

  /// Return whether this object was not default-constructed.
  bool isUsable() const { return (Occupied || Unused) && Known && Written; }

  /// Print the content of this object to @p OS.
  void print(llvm::raw_ostream &OS, unsigned Indent = 0) const {
    if (isUsable()) {
      if (Occupied)
        OS.indent(Indent) << "Occupied: " << Occupied << "\n";
      else
        OS.indent(Indent) << "Occupied: <Everything else not in Unused>\n";
      if (Unused)
        OS.indent(Indent) << "Unused:   " << Unused << "\n";
      else
        OS.indent(Indent) << "Unused:   <Everything else not in Occupied>\n";
      OS.indent(Indent) << "Known:    " << Known << "\n";
      OS.indent(Indent) << "Written : " << Written << '\n';
    } else {
      OS.indent(Indent) << "Invalid knowledge\n";
    }
  }

  /// Combine two knowledges, this and @p That.
  void learnFrom(Knowledge That) {
    assert(!isConflicting(*this, That));
    assert(Unused && That.Occupied);
    assert(
        !That.Unused &&
        "This function is only prepared to learn occupied elements from That");
    assert(!Occupied && "This function does not implement "
                        "`this->Occupied = "
                        "give(isl_union_set_union(this->Occupied.take(), "
                        "That.Occupied.copy()));`");

    Unused = give(isl_union_set_subtract(Unused.take(), That.Occupied.copy()));
    Known = give(isl_union_map_union(Known.take(), That.Known.copy()));
    Written = give(isl_union_map_union(Written.take(), That.Written.take()));

    checkConsistency();
  }

  /// Determine whether two Knowledges conflict with each other.
  ///
  /// In theory @p Existing and @p Proposed are symmetric, but the
  /// implementation is constrained by the implicit interpretation. That is, @p
  /// Existing must have #Unused defined (use case 1) and @p Proposed must have
  /// #Occupied defined (use case 1).
  ///
  /// A conflict is defined as non-preserved semantics when they are merged. For
  /// instance, when for the same array and zone they assume different
  /// llvm::Values.
  ///
  /// @param Existing One of the knowledges with #Unused defined.
  /// @param Proposed One of the knowledges with #Occupied defined.
  /// @param OS       Dump the conflict reason to this output stream; use
  ///                 nullptr to not output anything.
  /// @param Indent   Indention for the conflict reason.
  ///
  /// @return True, iff the two knowledges are conflicting.
  static bool isConflicting(const Knowledge &Existing,
                            const Knowledge &Proposed,
                            llvm::raw_ostream *OS = nullptr,
                            unsigned Indent = 0) {
    assert(Existing.Unused);
    assert(Proposed.Occupied);

#ifndef NDEBUG
    if (Existing.Occupied && Proposed.Unused) {
      auto ExistingUniverse = give(isl_union_set_union(Existing.Occupied.copy(),
                                                       Existing.Unused.copy()));
      auto ProposedUniverse = give(isl_union_set_union(Proposed.Occupied.copy(),
                                                       Proposed.Unused.copy()));
      assert(isl_union_set_is_equal(ExistingUniverse.keep(),
                                    ProposedUniverse.keep()) == isl_bool_true &&
             "Both inputs' Knowledges must be over the same universe");
    }
#endif

    // Are the new lifetimes required for Proposed unused in Existing?
    if (isl_union_set_is_subset(Proposed.Occupied.keep(),
                                Existing.Unused.keep()) != isl_bool_true) {
      if (OS) {
        auto ConflictingLifetimes = give(isl_union_set_subtract(
            Proposed.Occupied.copy(), Existing.Unused.copy()));
        OS->indent(Indent) << "Proposed lifetimes are not unused in existing\n";
        OS->indent(Indent) << "Conflicting lifetimes: " << ConflictingLifetimes
                           << "\n";
      }
      return true;
    }

    // Do the writes in Existing only overwrite unused values in Proposed?
    // We convert here the set of lifetimes to actual timepoints. A lifetime is
    // in conflict with a set of write timepoints, if either a live timepoint is
    // clearly within the lifetime or if a write happens at the beginning of the
    // lifetime (where it would conflict with the value that actually writes the
    // value alive). There is no conflict at the end of a lifetime, as the alive
    // value will always be read, before it is overwritten again. The last
    // property holds in Polly for all scalar values and we expect all users of
    // Knowledge to check this property also for accesses to MemoryKind::Array.
    auto ProposedFixedDefs =
        convertZoneToTimepoints(Proposed.Occupied, true, false);
    auto ExistingWrittenDomain =
        isl::manage(isl_union_map_domain(Existing.Written.copy()));
    if (isl_union_set_is_disjoint(ExistingWrittenDomain.keep(),
                                  ProposedFixedDefs.keep()) != isl_bool_true) {
      if (OS) {
        auto ConflictingWrites = give(isl_union_set_intersect(
            ExistingWrittenDomain.copy(), ProposedFixedDefs.copy()));
        OS->indent(Indent) << "Proposed writes into range used by existing\n";
        OS->indent(Indent) << "Conflicting writes: " << ConflictingWrites
                           << "\n";
      }
      return true;
    }

    // Do the new writes in Proposed only overwrite unused values in Existing?
    auto ExistingAvailableDefs =
        convertZoneToTimepoints(Existing.Unused, true, false);
    auto ProposedWrittenDomain =
        isl::manage(isl_union_map_domain(Proposed.Written.copy()));
    if (isl_union_set_is_subset(ProposedWrittenDomain.keep(),
                                ExistingAvailableDefs.keep()) !=
        isl_bool_true) {
      if (OS) {
        auto ConflictingWrites = give(isl_union_set_subtract(
            ProposedWrittenDomain.copy(), ExistingAvailableDefs.copy()));
        OS->indent(Indent)
            << "Proposed a lifetime where there is an Existing write into it\n";
        OS->indent(Indent) << "Conflicting writes: " << ConflictingWrites
                           << "\n";
      }
      return true;
    }

    // Does Proposed write at the same time as Existing already does (order of
    // writes is undefined)? Writing the same value is permitted.
    auto BothWritten =
        Existing.Written.domain().intersect(Proposed.Written.domain());
    auto ExistingKnownWritten = filterKnownValInst(Existing.Written);
    auto ProposedKnownWritten = filterKnownValInst(Proposed.Written);
    auto CommonWritten =
        ExistingKnownWritten.intersect(ProposedKnownWritten).domain();

    if (!BothWritten.is_subset(CommonWritten)) {
      if (OS) {
        auto Conflicting = BothWritten.subtract(CommonWritten);
        auto ExistingConflictingWritten =
            Existing.Written.intersect_domain(Conflicting);
        auto ProposedConflictingWritten =
            Proposed.Written.intersect_domain(Conflicting);

        OS->indent(Indent) << "Proposed writes at the same time as an already "
                              "Existing write\n";
        OS->indent(Indent) << "Conflicting writes: " << Conflicting << "\n";
        if (!ExistingConflictingWritten.is_empty())
          OS->indent(Indent)
              << "Exiting write:      " << ExistingConflictingWritten << "\n";
        if (!ProposedConflictingWritten.is_empty())
          OS->indent(Indent)
              << "Proposed write:     " << ProposedConflictingWritten << "\n";
      }
      return true;
    }

    return false;
  }
};

std::string printIntruction(Instruction *Instr, bool IsForDebug = false) {
  std::string Result;
  raw_string_ostream OS(Result);
  Instr->print(OS, IsForDebug);
  OS.flush();
  size_t i = 0;
  while (i < Result.size() && Result[i] == ' ')
    i += 1;
  return Result.substr(i);
}

/// Base class for algorithms based on zones, like DeLICM.
class ZoneAlgorithm {
protected:
  /// Hold a reference to the isl_ctx to avoid it being freed before we released
  /// all of the isl objects.
  ///
  /// This must be declared before any other member that holds an isl object.
  /// This guarantees that the shared_ptr and its isl_ctx is destructed last,
  /// after all other members free'd the isl objects they were holding.
  std::shared_ptr<isl_ctx> IslCtx;

  /// Cached reaching definitions for each ScopStmt.
  ///
  /// Use getScalarReachingDefinition() to get its contents.
  DenseMap<ScopStmt *, isl::map> ScalarReachDefZone;

  /// The analyzed Scop.
  Scop *S;

  /// Parameter space that does not need realignment.
  isl::space ParamSpace;

  /// Space the schedule maps to.
  isl::space ScatterSpace;

  /// Cached version of the schedule and domains.
  isl::union_map Schedule;

  /// Combined access relations of all MemoryKind::Array READ accesses.
  /// { DomainRead[] -> Element[] }
  isl::union_map AllReads;

  /// Combined access relations of all MemoryKind::Array, MAY_WRITE accesses.
  /// { DomainMayWrite[] -> Element[] }
  isl::union_map AllMayWrites;

  /// Combined access relations of all MemoryKind::Array, MUST_WRITE accesses.
  /// { DomainMustWrite[] -> Element[] }
  isl::union_map AllMustWrites;

  /// Prepare the object before computing the zones of @p S.
  ZoneAlgorithm(Scop *S)
      : IslCtx(S->getSharedIslCtx()), S(S), Schedule(give(S->getSchedule())) {

    auto Domains = give(S->getDomains());

    Schedule =
        give(isl_union_map_intersect_domain(Schedule.take(), Domains.take()));
    ParamSpace = give(isl_union_map_get_space(Schedule.keep()));
    ScatterSpace = getScatterSpace(Schedule);
  }

private:
  /// Check whether @p Stmt can be accurately analyzed by zones.
  ///
  /// What violates our assumptions:
  /// - A load after a write of the same location; we assume that all reads
  ///   occur before the writes.
  /// - Two writes to the same location; we cannot model the order in which
  ///   these occur.
  ///
  /// Scalar reads implicitly always occur before other accesses therefore never
  /// violate the first condition. There is also at most one write to a scalar,
  /// satisfying the second condition.
  bool isCompatibleStmt(ScopStmt *Stmt) {
    auto Stores = makeEmptyUnionMap();
    auto Loads = makeEmptyUnionMap();

    // This assumes that the MemoryKind::Array MemoryAccesses are iterated in
    // order.
    for (auto *MA : *Stmt) {
      if (!MA->isLatestArrayKind())
        continue;

      auto AccRel =
          give(isl_union_map_from_map(getAccessRelationFor(MA).take()));

      if (MA->isRead()) {
        // Reject load after store to same location.
        if (!isl_union_map_is_disjoint(Stores.keep(), AccRel.keep())) {
          OptimizationRemarkMissed R(DEBUG_TYPE, "LoadAfterStore",
                                     MA->getAccessInstruction());
          R << "load after store of same element in same statement";
          R << " (previous stores: " << Stores;
          R << ", loading: " << AccRel << ")";
          S->getFunction().getContext().diagnose(R);
          return false;
        }

        Loads = give(isl_union_map_union(Loads.take(), AccRel.take()));

        continue;
      }

      if (!isa<StoreInst>(MA->getAccessInstruction())) {
        DEBUG(dbgs() << "WRITE that is not a StoreInst not supported\n");
        OptimizationRemarkMissed R(DEBUG_TYPE, "UnusualStore",
                                   MA->getAccessInstruction());
        R << "encountered write that is not a StoreInst: "
          << printIntruction(MA->getAccessInstruction());
        S->getFunction().getContext().diagnose(R);
        return false;
      }

      // In region statements the order is less clear, eg. the load and store
      // might be in a boxed loop.
      if (Stmt->isRegionStmt() &&
          !isl_union_map_is_disjoint(Loads.keep(), AccRel.keep())) {
        OptimizationRemarkMissed R(DEBUG_TYPE, "StoreInSubregion",
                                   MA->getAccessInstruction());
        R << "store is in a non-affine subregion";
        S->getFunction().getContext().diagnose(R);
        return false;
      }

      // Do not allow more than one store to the same location.
      if (!isl_union_map_is_disjoint(Stores.keep(), AccRel.keep())) {
        OptimizationRemarkMissed R(DEBUG_TYPE, "StoreAfterStore",
                                   MA->getAccessInstruction());
        R << "store after store of same element in same statement";
        R << " (previous stores: " << Stores;
        R << ", storing: " << AccRel << ")";
        S->getFunction().getContext().diagnose(R);
        return false;
      }

      Stores = give(isl_union_map_union(Stores.take(), AccRel.take()));
    }

    return true;
  }

  void addArrayReadAccess(MemoryAccess *MA) {
    assert(MA->isLatestArrayKind());
    assert(MA->isRead());

    // { DomainRead[] -> Element[] }
    auto AccRel = getAccessRelationFor(MA);
    AllReads = give(isl_union_map_add_map(AllReads.take(), AccRel.copy()));
  }

  void addArrayWriteAccess(MemoryAccess *MA) {
    assert(MA->isLatestArrayKind());
    assert(MA->isWrite());

    // { Domain[] -> Element[] }
    auto AccRel = getAccessRelationFor(MA);

    if (MA->isMustWrite())
      AllMustWrites =
          give(isl_union_map_add_map(AllMustWrites.take(), AccRel.copy()));

    if (MA->isMayWrite())
      AllMayWrites =
          give(isl_union_map_add_map(AllMayWrites.take(), AccRel.copy()));
  }

protected:
  isl::union_set makeEmptyUnionSet() const {
    return give(isl_union_set_empty(ParamSpace.copy()));
  }

  isl::union_map makeEmptyUnionMap() const {
    return give(isl_union_map_empty(ParamSpace.copy()));
  }

  /// Check whether @p S can be accurately analyzed by zones.
  bool isCompatibleScop() {
    for (auto &Stmt : *S) {
      if (!isCompatibleStmt(&Stmt))
        return false;
    }
    return true;
  }

  /// Get the schedule for @p Stmt.
  ///
  /// The domain of the result is as narrow as possible.
  isl::map getScatterFor(ScopStmt *Stmt) const {
    auto ResultSpace = give(isl_space_map_from_domain_and_range(
        Stmt->getDomainSpace(), ScatterSpace.copy()));
    return give(isl_union_map_extract_map(Schedule.keep(), ResultSpace.take()));
  }

  /// Get the schedule of @p MA's parent statement.
  isl::map getScatterFor(MemoryAccess *MA) const {
    return getScatterFor(MA->getStatement());
  }

  /// Get the schedule for the statement instances of @p Domain.
  isl::union_map getScatterFor(isl::union_set Domain) const {
    return give(isl_union_map_intersect_domain(Schedule.copy(), Domain.take()));
  }

  /// Get the schedule for the statement instances of @p Domain.
  isl::map getScatterFor(isl::set Domain) const {
    auto ResultSpace = give(isl_space_map_from_domain_and_range(
        isl_set_get_space(Domain.keep()), ScatterSpace.copy()));
    auto UDomain = give(isl_union_set_from_set(Domain.copy()));
    auto UResult = getScatterFor(std::move(UDomain));
    auto Result = singleton(std::move(UResult), std::move(ResultSpace));
    assert(isl_set_is_equal(give(isl_map_domain(Result.copy())).keep(),
                            Domain.keep()) == isl_bool_true);
    return Result;
  }

  /// Get the domain of @p Stmt.
  isl::set getDomainFor(ScopStmt *Stmt) const {
    return give(Stmt->getDomain());
  }

  /// Get the domain @p MA's parent statement.
  isl::set getDomainFor(MemoryAccess *MA) const {
    return getDomainFor(MA->getStatement());
  }

  /// Get the access relation of @p MA.
  ///
  /// The domain of the result is as narrow as possible.
  isl::map getAccessRelationFor(MemoryAccess *MA) const {
    auto Domain = getDomainFor(MA);
    auto AccRel = give(MA->getLatestAccessRelation());
    return give(isl_map_intersect_domain(AccRel.take(), Domain.take()));
  }

  /// Get the reaching definition of a scalar defined in @p Stmt.
  ///
  /// Note that this does not depend on the llvm::Instruction, only on the
  /// statement it is defined in. Therefore the same computation can be reused.
  ///
  /// @param Stmt The statement in which a scalar is defined.
  ///
  /// @return { Scatter[] -> DomainDef[] }
  isl::map getScalarReachingDefinition(ScopStmt *Stmt) {
    auto &Result = ScalarReachDefZone[Stmt];
    if (Result)
      return Result;

    auto Domain = getDomainFor(Stmt);
    Result = computeScalarReachingDefinition(Schedule, Domain, false, true);
    simplify(Result);

    assert(Result);
    return Result;
  }

  /// Compute the different zones.
  void computeCommon() {
    AllReads = makeEmptyUnionMap();
    AllMayWrites = makeEmptyUnionMap();
    AllMustWrites = makeEmptyUnionMap();

    for (auto &Stmt : *S) {
      for (auto *MA : Stmt) {
        if (!MA->isLatestArrayKind())
          continue;

        if (MA->isRead())
          addArrayReadAccess(MA);

        if (MA->isWrite())
          addArrayWriteAccess(MA);
      }
    }

    // { DomainWrite[] -> Element[] }
    auto AllWrites =
        give(isl_union_map_union(AllMustWrites.copy(), AllMayWrites.copy()));
  }

  /// Print the current state of all MemoryAccesses to @p.
  void printAccesses(llvm::raw_ostream &OS, int Indent = 0) const {
    OS.indent(Indent) << "After accesses {\n";
    for (auto &Stmt : *S) {
      OS.indent(Indent + 4) << Stmt.getBaseName() << "\n";
      for (auto *MA : Stmt)
        MA->print(OS);
    }
    OS.indent(Indent) << "}\n";
  }

public:
  /// Return the SCoP this object is analyzing.
  Scop *getScop() const { return S; }
};

/// Implementation of the DeLICM/DePRE transformation.
class DeLICMImpl : public ZoneAlgorithm {
private:
  /// Knowledge before any transformation took place.
  Knowledge OriginalZone;

  /// Current knowledge of the SCoP including all already applied
  /// transformations.
  Knowledge Zone;

  /// For getting the MemoryAccesses that write or read a given scalar.
  ScalarDefUseChains DefUse;

  /// Number of StoreInsts something can be mapped to.
  int NumberOfCompatibleTargets = 0;

  /// The number of StoreInsts to which at least one value or PHI has been
  /// mapped to.
  int NumberOfTargetsMapped = 0;

  /// The number of llvm::Value mapped to some array element.
  int NumberOfMappedValueScalars = 0;

  /// The number of PHIs mapped to some array element.
  int NumberOfMappedPHIScalars = 0;

  /// Determine whether two knowledges are conflicting with each other.
  ///
  /// @see Knowledge::isConflicting
  bool isConflicting(const Knowledge &Proposed) {
    raw_ostream *OS = nullptr;
    DEBUG(OS = &llvm::dbgs());
    return Knowledge::isConflicting(Zone, Proposed, OS, 4);
  }

  /// Determine whether @p SAI is a scalar that can be mapped to an array
  /// element.
  bool isMappable(const ScopArrayInfo *SAI) {
    assert(SAI);

    if (SAI->isValueKind()) {
      auto *MA = DefUse.getValueDef(SAI);
      if (!MA) {
        DEBUG(dbgs()
              << "    Reject because value is read-only within the scop\n");
        return false;
      }

      // Mapping if value is used after scop is not supported. The code
      // generator would need to reload the scalar after the scop, but it
      // does not have the information to where it is mapped to. Only the
      // MemoryAccesses have that information, not the ScopArrayInfo.
      auto Inst = MA->getAccessInstruction();
      for (auto User : Inst->users()) {
        if (!isa<Instruction>(User))
          return false;
        auto UserInst = cast<Instruction>(User);

        if (!S->contains(UserInst)) {
          DEBUG(dbgs() << "    Reject because value is escaping\n");
          return false;
        }
      }

      return true;
    }

    if (SAI->isPHIKind()) {
      auto *MA = DefUse.getPHIRead(SAI);
      assert(MA);

      // Mapping of an incoming block from before the SCoP is not supported by
      // the code generator.
      auto PHI = cast<PHINode>(MA->getAccessInstruction());
      for (auto Incoming : PHI->blocks()) {
        if (!S->contains(Incoming)) {
          DEBUG(dbgs() << "    Reject because at least one incoming block is "
                          "not in the scop region\n");
          return false;
        }
      }

      return true;
    }

    DEBUG(dbgs() << "    Reject ExitPHI or other non-value\n");
    return false;
  }

  /// Compute the uses of a MemoryKind::Value and its lifetime (from its
  /// definition to the last use).
  ///
  /// @param SAI The ScopArrayInfo representing the value's storage.
  ///
  /// @return { DomainDef[] -> DomainUse[] }, { DomainDef[] -> Zone[] }
  ///         First element is the set of uses for each definition.
  ///         The second is the lifetime of each definition.
  std::tuple<isl::union_map, isl::map>
  computeValueUses(const ScopArrayInfo *SAI) {
    assert(SAI->isValueKind());

    // { DomainRead[] }
    auto Reads = makeEmptyUnionSet();

    // Find all uses.
    for (auto *MA : DefUse.getValueUses(SAI))
      Reads =
          give(isl_union_set_add_set(Reads.take(), getDomainFor(MA).take()));

    // { DomainRead[] -> Scatter[] }
    auto ReadSchedule = getScatterFor(Reads);

    auto *DefMA = DefUse.getValueDef(SAI);
    assert(DefMA);

    // { DomainDef[] }
    auto Writes = getDomainFor(DefMA);

    // { DomainDef[] -> Scatter[] }
    auto WriteScatter = getScatterFor(Writes);

    // { Scatter[] -> DomainDef[] }
    auto ReachDef = getScalarReachingDefinition(DefMA->getStatement());

    // { [DomainDef[] -> Scatter[]] -> DomainUse[] }
    auto Uses = give(
        isl_union_map_apply_range(isl_union_map_from_map(isl_map_range_map(
                                      isl_map_reverse(ReachDef.take()))),
                                  isl_union_map_reverse(ReadSchedule.take())));

    // { DomainDef[] -> Scatter[] }
    auto UseScatter =
        singleton(give(isl_union_set_unwrap(isl_union_map_domain(Uses.copy()))),
                  give(isl_space_map_from_domain_and_range(
                      isl_set_get_space(Writes.keep()), ScatterSpace.copy())));

    // { DomainDef[] -> Zone[] }
    auto Lifetime = betweenScatter(WriteScatter, UseScatter, false, true);

    // { DomainDef[] -> DomainRead[] }
    auto DefUses = give(isl_union_map_domain_factor_domain(Uses.take()));

    return std::make_pair(DefUses, Lifetime);
  }

  /// For each 'execution' of a PHINode, get the incoming block that was
  /// executed before.
  ///
  /// For each PHI instance we can directly determine which was the incoming
  /// block, and hence derive which value the PHI has.
  ///
  /// @param SAI The ScopArrayInfo representing the PHI's storage.
  ///
  /// @return { DomainPHIRead[] -> DomainPHIWrite[] }
  isl::union_map computePerPHI(const ScopArrayInfo *SAI) {
    assert(SAI->isPHIKind());

    // { DomainPHIWrite[] -> Scatter[] }
    auto PHIWriteScatter = makeEmptyUnionMap();

    // Collect all incoming block timepoint.
    for (auto *MA : DefUse.getPHIIncomings(SAI)) {
      auto Scatter = getScatterFor(MA);
      PHIWriteScatter =
          give(isl_union_map_add_map(PHIWriteScatter.take(), Scatter.take()));
    }

    // { DomainPHIRead[] -> Scatter[] }
    auto PHIReadScatter = getScatterFor(DefUse.getPHIRead(SAI));

    // { DomainPHIRead[] -> Scatter[] }
    auto BeforeRead = beforeScatter(PHIReadScatter, true);

    // { Scatter[] }
    auto WriteTimes = singleton(
        give(isl_union_map_range(PHIWriteScatter.copy())), ScatterSpace);

    // { DomainPHIRead[] -> Scatter[] }
    auto PHIWriteTimes =
        give(isl_map_intersect_range(BeforeRead.take(), WriteTimes.take()));
    auto LastPerPHIWrites = give(isl_map_lexmax(PHIWriteTimes.take()));

    // { DomainPHIRead[] -> DomainPHIWrite[] }
    auto Result = give(isl_union_map_apply_range(
        isl_union_map_from_map(LastPerPHIWrites.take()),
        isl_union_map_reverse(PHIWriteScatter.take())));
    assert(isl_union_map_is_single_valued(Result.keep()) == isl_bool_true);
    assert(isl_union_map_is_injective(Result.keep()) == isl_bool_true);
    return Result;
  }

  /// Try to map a MemoryKind::Value to a given array element.
  ///
  /// @param SAI       Representation of the scalar's memory to map.
  /// @param TargetElt { Scatter[] -> Element[] }
  ///                  Suggestion where to map a scalar to when at a timepoint.
  ///
  /// @return true if the scalar was successfully mapped.
  bool tryMapValue(const ScopArrayInfo *SAI, isl::map TargetElt) {
    assert(SAI->isValueKind());

    auto *DefMA = DefUse.getValueDef(SAI);
    assert(DefMA->isValueKind());
    assert(DefMA->isMustWrite());

    // Stop if the scalar has already been mapped.
    if (!DefMA->getLatestScopArrayInfo()->isValueKind())
      return false;

    // { DomainDef[] -> Scatter[] }
    auto DefSched = getScatterFor(DefMA);

    // Where each write is mapped to, according to the suggestion.
    // { DomainDef[] -> Element[] }
    auto DefTarget = give(isl_map_apply_domain(
        TargetElt.copy(), isl_map_reverse(DefSched.copy())));
    simplify(DefTarget);
    DEBUG(dbgs() << "    Def Mapping: " << DefTarget << '\n');

    auto OrigDomain = getDomainFor(DefMA);
    auto MappedDomain = give(isl_map_domain(DefTarget.copy()));
    if (!isl_set_is_subset(OrigDomain.keep(), MappedDomain.keep())) {
      DEBUG(dbgs()
            << "    Reject because mapping does not encompass all instances\n");
      return false;
    }

    // { DomainDef[] -> Zone[] }
    isl::map Lifetime;

    // { DomainDef[] -> DomainUse[] }
    isl::union_map DefUses;

    std::tie(DefUses, Lifetime) = computeValueUses(SAI);
    DEBUG(dbgs() << "    Lifetime: " << Lifetime << '\n');

    /// { [Element[] -> Zone[]] }
    auto EltZone = give(
        isl_map_wrap(isl_map_apply_domain(Lifetime.copy(), DefTarget.copy())));
    simplify(EltZone);

    // { [Element[] -> Scatter[]] }
    auto DefEltSched = give(isl_map_wrap(isl_map_reverse(
        isl_map_apply_domain(DefTarget.copy(), DefSched.copy()))));
    simplify(DefEltSched);

    Knowledge Proposed(EltZone, nullptr, DefEltSched);
    if (isConflicting(Proposed))
      return false;

    // { DomainUse[] -> Element[] }
    auto UseTarget = give(
        isl_union_map_apply_range(isl_union_map_reverse(DefUses.take()),
                                  isl_union_map_from_map(DefTarget.copy())));

    mapValue(SAI, std::move(DefTarget), std::move(UseTarget),
             std::move(Lifetime), std::move(Proposed));
    return true;
  }

  /// After a scalar has been mapped, update the global knowledge.
  void applyLifetime(Knowledge Proposed) {
    Zone.learnFrom(std::move(Proposed));
  }

  /// Map a MemoryKind::Value scalar to an array element.
  ///
  /// Callers must have ensured that the mapping is valid and not conflicting.
  ///
  /// @param SAI       The ScopArrayInfo representing the scalar's memory to
  ///                  map.
  /// @param DefTarget { DomainDef[] -> Element[] }
  ///                  The array element to map the scalar to.
  /// @param UseTarget { DomainUse[] -> Element[] }
  ///                  The array elements the uses are mapped to.
  /// @param Lifetime  { DomainDef[] -> Zone[] }
  ///                  The lifetime of each llvm::Value definition for
  ///                  reporting.
  /// @param Proposed  Mapping constraints for reporting.
  void mapValue(const ScopArrayInfo *SAI, isl::map DefTarget,
                isl::union_map UseTarget, isl::map Lifetime,
                Knowledge Proposed) {
    // Redirect the read accesses.
    for (auto *MA : DefUse.getValueUses(SAI)) {
      // { DomainUse[] }
      auto Domain = getDomainFor(MA);

      // { DomainUse[] -> Element[] }
      auto NewAccRel = give(isl_union_map_intersect_domain(
          UseTarget.copy(), isl_union_set_from_set(Domain.take())));
      simplify(NewAccRel);

      assert(isl_union_map_n_map(NewAccRel.keep()) == 1);
      MA->setNewAccessRelation(isl_map_from_union_map(NewAccRel.take()));
    }

    auto *WA = DefUse.getValueDef(SAI);
    WA->setNewAccessRelation(DefTarget.copy());
    applyLifetime(Proposed);

    MappedValueScalars++;
    NumberOfMappedValueScalars += 1;
  }

  /// Try to map a MemoryKind::PHI scalar to a given array element.
  ///
  /// @param SAI       Representation of the scalar's memory to map.
  /// @param TargetElt { Scatter[] -> Element[] }
  ///                  Suggestion where to map the scalar to when at a
  ///                  timepoint.
  ///
  /// @return true if the PHI scalar has been mapped.
  bool tryMapPHI(const ScopArrayInfo *SAI, isl::map TargetElt) {
    auto *PHIRead = DefUse.getPHIRead(SAI);
    assert(PHIRead->isPHIKind());
    assert(PHIRead->isRead());

    // Skip if already been mapped.
    if (!PHIRead->getLatestScopArrayInfo()->isPHIKind())
      return false;

    // { DomainRead[] -> Scatter[] }
    auto PHISched = getScatterFor(PHIRead);

    // { DomainRead[] -> Element[] }
    auto PHITarget =
        give(isl_map_apply_range(PHISched.copy(), TargetElt.copy()));
    simplify(PHITarget);
    DEBUG(dbgs() << "    Mapping: " << PHITarget << '\n');

    auto OrigDomain = getDomainFor(PHIRead);
    auto MappedDomain = give(isl_map_domain(PHITarget.copy()));
    if (!isl_set_is_subset(OrigDomain.keep(), MappedDomain.keep())) {
      DEBUG(dbgs()
            << "    Reject because mapping does not encompass all instances\n");
      return false;
    }

    // { DomainRead[] -> DomainWrite[] }
    auto PerPHIWrites = computePerPHI(SAI);

    // { DomainWrite[] -> Element[] }
    auto WritesTarget = give(isl_union_map_reverse(isl_union_map_apply_domain(
        PerPHIWrites.copy(), isl_union_map_from_map(PHITarget.copy()))));
    simplify(WritesTarget);

    // { DomainWrite[] }
    auto UniverseWritesDom = give(isl_union_set_empty(ParamSpace.copy()));

    for (auto *MA : DefUse.getPHIIncomings(SAI))
      UniverseWritesDom = give(isl_union_set_add_set(UniverseWritesDom.take(),
                                                     getDomainFor(MA).take()));

    auto RelevantWritesTarget = WritesTarget;
    if (DelicmOverapproximateWrites)
      WritesTarget = expandMapping(WritesTarget, UniverseWritesDom);

    auto ExpandedWritesDom = give(isl_union_map_domain(WritesTarget.copy()));
    if (!isl_union_set_is_subset(UniverseWritesDom.keep(),
                                 ExpandedWritesDom.keep())) {
      DEBUG(dbgs() << "    Reject because did not find PHI write mapping for "
                      "all instances\n");
      if (DelicmOverapproximateWrites)
        DEBUG(dbgs() << "      Relevant Mapping:    " << RelevantWritesTarget
                     << '\n');
      DEBUG(dbgs() << "      Deduced Mapping:     " << WritesTarget << '\n');
      DEBUG(dbgs() << "      Missing instances:    "
                   << give(isl_union_set_subtract(UniverseWritesDom.copy(),
                                                  ExpandedWritesDom.copy()))
                   << '\n');
      return false;
    }

    //  { DomainRead[] -> Scatter[] }
    auto PerPHIWriteScatter = give(isl_map_from_union_map(
        isl_union_map_apply_range(PerPHIWrites.copy(), Schedule.copy())));

    // { DomainRead[] -> Zone[] }
    auto Lifetime = betweenScatter(PerPHIWriteScatter, PHISched, false, true);
    simplify(Lifetime);
    DEBUG(dbgs() << "    Lifetime: " << Lifetime << "\n");

    // { DomainWrite[] -> Zone[] }
    auto WriteLifetime = give(isl_union_map_apply_domain(
        isl_union_map_from_map(Lifetime.copy()), PerPHIWrites.copy()));

    // { DomainWrite[] -> [Element[] -> Scatter[]] }
    auto WrittenTranslator =
        give(isl_union_map_range_product(WritesTarget.copy(), Schedule.copy()));

    // { [Element[] -> Scatter[]] }
    auto Written = give(isl_union_map_range(WrittenTranslator.copy()));
    simplify(Written);

    // { DomainWrite[] -> [Element[] -> Zone[]] }
    auto LifetimeTranslator = give(
        isl_union_map_range_product(WritesTarget.copy(), WriteLifetime.take()));

    // { [Element[] -> Zone[] }
    auto Occupied = give(isl_union_map_range(LifetimeTranslator.copy()));
    simplify(Occupied);

    Knowledge Proposed(Occupied, nullptr, Written);
    if (isConflicting(Proposed))
      return false;

    mapPHI(SAI, std::move(PHITarget), std::move(WritesTarget),
           std::move(Lifetime), std::move(Proposed));
    return true;
  }

  /// Map a MemoryKind::PHI scalar to an array element.
  ///
  /// Callers must have ensured that the mapping is valid and not conflicting
  /// with the common knowledge.
  ///
  /// @param SAI         The ScopArrayInfo representing the scalar's memory to
  ///                    map.
  /// @param ReadTarget  { DomainRead[] -> Element[] }
  ///                    The array element to map the scalar to.
  /// @param WriteTarget { DomainWrite[] -> Element[] }
  ///                    New access target for each PHI incoming write.
  /// @param Lifetime    { DomainRead[] -> Zone[] }
  ///                    The lifetime of each PHI for reporting.
  /// @param Proposed    Mapping constraints for reporting.
  void mapPHI(const ScopArrayInfo *SAI, isl::map ReadTarget,
              isl::union_map WriteTarget, isl::map Lifetime,
              Knowledge Proposed) {
    // Redirect the PHI incoming writes.
    for (auto *MA : DefUse.getPHIIncomings(SAI)) {
      // { DomainWrite[] }
      auto Domain = getDomainFor(MA);

      // { DomainWrite[] -> Element[] }
      auto NewAccRel = give(isl_union_map_intersect_domain(
          WriteTarget.copy(), isl_union_set_from_set(Domain.take())));
      simplify(NewAccRel);

      assert(isl_union_map_n_map(NewAccRel.keep()) == 1);
      MA->setNewAccessRelation(isl_map_from_union_map(NewAccRel.take()));
    }

    // Redirect the PHI read.
    auto *PHIRead = DefUse.getPHIRead(SAI);
    PHIRead->setNewAccessRelation(ReadTarget.copy());
    applyLifetime(Proposed);

    MappedPHIScalars++;
    NumberOfMappedPHIScalars++;
  }

  /// Search and map scalars to memory overwritten by @p TargetStoreMA.
  ///
  /// Start trying to map scalars that are used in the same statement as the
  /// store. For every successful mapping, try to also map scalars of the
  /// statements where those are written. Repeat, until no more mapping
  /// opportunity is found.
  ///
  /// There is currently no preference in which order scalars are tried.
  /// Ideally, we would direct it towards a load instruction of the same array
  /// element.
  bool collapseScalarsToStore(MemoryAccess *TargetStoreMA) {
    assert(TargetStoreMA->isLatestArrayKind());
    assert(TargetStoreMA->isMustWrite());

    auto TargetStmt = TargetStoreMA->getStatement();

    // { DomTarget[] }
    auto TargetDom = getDomainFor(TargetStmt);

    // { DomTarget[] -> Element[] }
    auto TargetAccRel = getAccessRelationFor(TargetStoreMA);

    // { Zone[] -> DomTarget[] }
    // For each point in time, find the next target store instance.
    auto Target =
        computeScalarReachingOverwrite(Schedule, TargetDom, false, true);

    // { Zone[] -> Element[] }
    // Use the target store's write location as a suggestion to map scalars to.
    auto EltTarget =
        give(isl_map_apply_range(Target.take(), TargetAccRel.take()));
    simplify(EltTarget);
    DEBUG(dbgs() << "    Target mapping is " << EltTarget << '\n');

    // Stack of elements not yet processed.
    SmallVector<MemoryAccess *, 16> Worklist;

    // Set of scalars already tested.
    SmallPtrSet<const ScopArrayInfo *, 16> Closed;

    // Lambda to add all scalar reads to the work list.
    auto ProcessAllIncoming = [&](ScopStmt *Stmt) {
      for (auto *MA : *Stmt) {
        if (!MA->isLatestScalarKind())
          continue;
        if (!MA->isRead())
          continue;

        Worklist.push_back(MA);
      }
    };

    // Add initial scalar. Either the value written by the store, or all inputs
    // of its statement.
    auto WrittenVal = TargetStoreMA->getAccessValue();
    if (auto InputAcc = getInputAccessOf(WrittenVal, TargetStmt))
      Worklist.push_back(InputAcc);
    else
      ProcessAllIncoming(TargetStmt);

    auto AnyMapped = false;
    auto &DL = S->getRegion().getEntry()->getModule()->getDataLayout();
    auto StoreSize =
        DL.getTypeAllocSize(TargetStoreMA->getAccessValue()->getType());

    while (!Worklist.empty()) {
      auto *MA = Worklist.pop_back_val();

      auto *SAI = MA->getScopArrayInfo();
      if (Closed.count(SAI))
        continue;
      Closed.insert(SAI);
      DEBUG(dbgs() << "\n    Trying to map " << MA << " (SAI: " << SAI
                   << ")\n");

      // Skip non-mappable scalars.
      if (!isMappable(SAI))
        continue;

      auto MASize = DL.getTypeAllocSize(MA->getAccessValue()->getType());
      if (MASize > StoreSize) {
        DEBUG(dbgs() << "    Reject because storage size is insufficient\n");
        continue;
      }

      // Try to map MemoryKind::Value scalars.
      if (SAI->isValueKind()) {
        if (!tryMapValue(SAI, EltTarget))
          continue;

        auto *DefAcc = DefUse.getValueDef(SAI);
        ProcessAllIncoming(DefAcc->getStatement());

        AnyMapped = true;
        continue;
      }

      // Try to map MemoryKind::PHI scalars.
      if (SAI->isPHIKind()) {
        if (!tryMapPHI(SAI, EltTarget))
          continue;
        // Add inputs of all incoming statements to the worklist.
        for (auto *PHIWrite : DefUse.getPHIIncomings(SAI))
          ProcessAllIncoming(PHIWrite->getStatement());

        AnyMapped = true;
        continue;
      }
    }

    if (AnyMapped) {
      TargetsMapped++;
      NumberOfTargetsMapped++;
    }
    return AnyMapped;
  }

  /// Compute when an array element is unused.
  ///
  /// @return { [Element[] -> Zone[]] }
  isl::union_set computeLifetime() const {
    // { Element[] -> Zone[] }
    auto ArrayUnused = computeArrayUnused(Schedule, AllMustWrites, AllReads,
                                          false, false, true);

    auto Result = give(isl_union_map_wrap(ArrayUnused.copy()));

    simplify(Result);
    return Result;
  }

  /// Determine when an array element is written to.
  ///
  /// @return { [Element[] -> Scatter[]] }
  isl::union_set computeWritten() const {
    // { WriteDomain[] -> Element[] }
    auto AllWrites =
        give(isl_union_map_union(AllMustWrites.copy(), AllMayWrites.copy()));

    // { Scatter[] -> Element[] }
    auto WriteTimepoints =
        give(isl_union_map_apply_domain(AllWrites.copy(), Schedule.copy()));

    auto Result =
        give(isl_union_map_wrap(isl_union_map_reverse(WriteTimepoints.copy())));

    simplify(Result);
    return Result;
  }

  /// Determine whether an access touches at most one element.
  ///
  /// The accessed element could be a scalar or accessing an array with constant
  /// subscript, such that all instances access only that element.
  ///
  /// @param MA The access to test.
  ///
  /// @return True, if zero or one elements are accessed; False if at least two
  ///         different elements are accessed.
  bool isScalarAccess(MemoryAccess *MA) {
    auto Map = getAccessRelationFor(MA);
    auto Set = give(isl_map_range(Map.take()));
    return isl_set_is_singleton(Set.keep()) == isl_bool_true;
  }

  /// Print mapping statistics to @p OS.
  void printStatistics(llvm::raw_ostream &OS, int Indent = 0) const {
    OS.indent(Indent) << "Statistics {\n";
    OS.indent(Indent + 4) << "Compatible overwrites: "
                          << NumberOfCompatibleTargets << "\n";
    OS.indent(Indent + 4) << "Overwrites mapped to:  " << NumberOfTargetsMapped
                          << '\n';
    OS.indent(Indent + 4) << "Value scalars mapped:  "
                          << NumberOfMappedValueScalars << '\n';
    OS.indent(Indent + 4) << "PHI scalars mapped:    "
                          << NumberOfMappedPHIScalars << '\n';
    OS.indent(Indent) << "}\n";
  }

  /// Return whether at least one transformation been applied.
  bool isModified() const { return NumberOfTargetsMapped > 0; }

public:
  DeLICMImpl(Scop *S) : ZoneAlgorithm(S) {}

  /// Calculate the lifetime (definition to last use) of every array element.
  ///
  /// @return True if the computed lifetimes (#Zone) is usable.
  bool computeZone() {
    // Check that nothing strange occurs.
    if (!isCompatibleScop()) {
      DeLICMIncompatible++;
      return false;
    }

    DefUse.compute(S);
    isl::union_set EltUnused, EltWritten;

    {
      IslMaxOperationsGuard MaxOpGuard(IslCtx.get(), DelicmMaxOps);

      computeCommon();

      EltUnused = computeLifetime();
      EltWritten = computeWritten();
    }
    DeLICMAnalyzed++;

    if (!EltUnused || !EltWritten) {
      assert(isl_ctx_last_error(IslCtx.get()) == isl_error_quota &&
             "The only reason that these things have not been computed should "
             "be if the max-operations limit hit");
      DeLICMOutOfQuota++;
      DEBUG(dbgs() << "DeLICM analysis exceeded max_operations\n");
      DebugLoc Begin, End;
      getDebugLocations(getBBPairForRegion(&S->getRegion()), Begin, End);
      OptimizationRemarkAnalysis R(DEBUG_TYPE, "OutOfQuota", Begin,
                                   S->getEntry());
      R << "maximal number of operations exceeded during zone analysis";
      S->getFunction().getContext().diagnose(R);
      return false;
    }

    Zone = OriginalZone = Knowledge(nullptr, EltUnused, EltWritten);
    DEBUG(dbgs() << "Computed Zone:\n"; OriginalZone.print(dbgs(), 4));

    assert(Zone.isUsable() && OriginalZone.isUsable());
    return true;
  }

  /// Try to map as many scalars to unused array elements as possible.
  ///
  /// Multiple scalars might be mappable to intersecting unused array element
  /// zones, but we can only chose one. This is a greedy algorithm, therefore
  /// the first processed element claims it.
  void greedyCollapse() {
    bool Modified = false;

    for (auto &Stmt : *S) {
      for (auto *MA : Stmt) {
        if (!MA->isLatestArrayKind())
          continue;
        if (!MA->isWrite())
          continue;

        if (MA->isMayWrite()) {
          DEBUG(dbgs() << "Access " << MA
                       << " pruned because it is a MAY_WRITE\n");
          OptimizationRemarkMissed R(DEBUG_TYPE, "TargetMayWrite",
                                     MA->getAccessInstruction());
          R << "Skipped possible mapping target because it is not an "
               "unconditional overwrite";
          S->getFunction().getContext().diagnose(R);
          continue;
        }

        if (Stmt.getNumIterators() == 0) {
          DEBUG(dbgs() << "Access " << MA
                       << " pruned because it is not in a loop\n");
          OptimizationRemarkMissed R(DEBUG_TYPE, "WriteNotInLoop",
                                     MA->getAccessInstruction());
          R << "skipped possible mapping target because it is not in a loop";
          S->getFunction().getContext().diagnose(R);
          continue;
        }

        if (isScalarAccess(MA)) {
          DEBUG(dbgs() << "Access " << MA
                       << " pruned because it writes only a single element\n");
          OptimizationRemarkMissed R(DEBUG_TYPE, "ScalarWrite",
                                     MA->getAccessInstruction());
          R << "skipped possible mapping target because the memory location "
               "written to does not depend on its outer loop";
          S->getFunction().getContext().diagnose(R);
          continue;
        }

        NumberOfCompatibleTargets++;
        DEBUG(dbgs() << "Analyzing target access " << MA << "\n");
        if (collapseScalarsToStore(MA))
          Modified = true;
      }
    }

    if (Modified)
      DeLICMScopsModified++;
  }

  /// Dump the internal information about a performed DeLICM to @p OS.
  void print(llvm::raw_ostream &OS, int Indent = 0) {
    if (!Zone.isUsable()) {
      OS.indent(Indent) << "Zone not computed\n";
      return;
    }

    printStatistics(OS, Indent);
    if (!isModified()) {
      OS.indent(Indent) << "No modification has been made\n";
      return;
    }
    printAccesses(OS, Indent);
  }
};

class DeLICM : public ScopPass {
private:
  DeLICM(const DeLICM &) = delete;
  const DeLICM &operator=(const DeLICM &) = delete;

  /// The pass implementation, also holding per-scop data.
  std::unique_ptr<DeLICMImpl> Impl;

  void collapseToUnused(Scop &S) {
    Impl = make_unique<DeLICMImpl>(&S);

    if (!Impl->computeZone()) {
      DEBUG(dbgs() << "Abort because cannot reliably compute lifetimes\n");
      return;
    }

    DEBUG(dbgs() << "Collapsing scalars to unused array elements...\n");
    Impl->greedyCollapse();

    DEBUG(dbgs() << "\nFinal Scop:\n");
    DEBUG(S.print(dbgs()));
  }

public:
  static char ID;
  explicit DeLICM() : ScopPass(ID) {}

  virtual void getAnalysisUsage(AnalysisUsage &AU) const override {
    AU.addRequiredTransitive<ScopInfoRegionPass>();
    AU.setPreservesAll();
  }

  virtual bool runOnScop(Scop &S) override {
    // Free resources for previous scop's computation, if not yet done.
    releaseMemory();

    collapseToUnused(S);

    return false;
  }

  virtual void printScop(raw_ostream &OS, Scop &S) const override {
    if (!Impl)
      return;
    assert(Impl->getScop() == &S);

    OS << "DeLICM result:\n";
    Impl->print(OS);
  }

  virtual void releaseMemory() override { Impl.reset(); }
};

char DeLICM::ID;
} // anonymous namespace

Pass *polly::createDeLICMPass() { return new DeLICM(); }

INITIALIZE_PASS_BEGIN(DeLICM, "polly-delicm", "Polly - DeLICM/DePRE", false,
                      false)
INITIALIZE_PASS_DEPENDENCY(ScopInfoWrapperPass)
INITIALIZE_PASS_END(DeLICM, "polly-delicm", "Polly - DeLICM/DePRE", false,
                    false)

bool polly::isConflicting(
    isl::union_set ExistingOccupied, isl::union_set ExistingUnused,
    isl::union_map ExistingKnown, isl::union_map ExistingWrites,
    isl::union_set ProposedOccupied, isl::union_set ProposedUnused,
    isl::union_map ProposedKnown, isl::union_map ProposedWrites,
    llvm::raw_ostream *OS, unsigned Indent) {
  Knowledge Existing(std::move(ExistingOccupied), std::move(ExistingUnused),
                     std::move(ExistingKnown), std::move(ExistingWrites));
  Knowledge Proposed(std::move(ProposedOccupied), std::move(ProposedUnused),
                     std::move(ProposedKnown), std::move(ProposedWrites));

  return Knowledge::isConflicting(Existing, Proposed, OS, Indent);
}