lib/Support/ISLTools.cpp

//===------ ISLTools.cpp ----------------------------------------*- C++ -*-===//
//
//                     The LLVM Compiler Infrastructure
//
// This file is distributed under the University of Illinois Open Source
// License. See LICENSE.TXT for details.
//
//===----------------------------------------------------------------------===//
//
// Tools, utilities, helpers and extensions useful in conjunction with the
// Integer Set Library (isl).
//
//===----------------------------------------------------------------------===//

#include "polly/Support/ISLTools.h"
#include "llvm/ADT/StringRef.h"

using namespace polly;

namespace {
/// Create a map that shifts one dimension by an offset.
///
/// Example:
/// makeShiftDimAff({ [i0, i1] -> [o0, o1] }, 1, -2)
///   = { [i0, i1] -> [i0, i1 - 1] }
///
/// @param Space  The map space of the result. Must have equal number of in- and
///               out-dimensions.
/// @param Pos    Position to shift.
/// @param Amount Value added to the shifted dimension.
///
/// @return An isl_multi_aff for the map with this shifted dimension.
isl::multi_aff makeShiftDimAff(isl::space Space, int Pos, int Amount) {
  auto Identity = give(isl_multi_aff_identity(Space.take()));
  if (Amount == 0)
    return Identity;
  auto ShiftAff = give(isl_multi_aff_get_aff(Identity.keep(), Pos));
  ShiftAff = give(isl_aff_set_constant_si(ShiftAff.take(), Amount));
  return give(isl_multi_aff_set_aff(Identity.take(), Pos, ShiftAff.take()));
}

/// Construct a map that swaps two nested tuples.
///
/// @param FromSpace1 { Space1[] }
/// @param FromSpace2 { Space2[] }
///
/// @return { [Space1[] -> Space2[]] -> [Space2[] -> Space1[]] }
isl::basic_map makeTupleSwapBasicMap(isl::space FromSpace1,
                                     isl::space FromSpace2) {
  assert(isl_space_is_set(FromSpace1.keep()) != isl_bool_false);
  assert(isl_space_is_set(FromSpace2.keep()) != isl_bool_false);

  auto Dims1 = isl_space_dim(FromSpace1.keep(), isl_dim_set);
  auto Dims2 = isl_space_dim(FromSpace2.keep(), isl_dim_set);
  auto FromSpace = give(isl_space_wrap(isl_space_map_from_domain_and_range(
      FromSpace1.copy(), FromSpace2.copy())));
  auto ToSpace = give(isl_space_wrap(isl_space_map_from_domain_and_range(
      FromSpace2.take(), FromSpace1.take())));
  auto MapSpace = give(
      isl_space_map_from_domain_and_range(FromSpace.take(), ToSpace.take()));

  auto Result = give(isl_basic_map_universe(MapSpace.take()));
  for (auto i = Dims1 - Dims1; i < Dims1; i += 1) {
    Result = give(isl_basic_map_equate(Result.take(), isl_dim_in, i,
                                       isl_dim_out, Dims2 + i));
  }
  for (auto i = Dims2 - Dims2; i < Dims2; i += 1) {
    Result = give(isl_basic_map_equate(Result.take(), isl_dim_in, Dims1 + i,
                                       isl_dim_out, i));
  }

  return Result;
}

/// Like makeTupleSwapBasicMap(isl::space,isl::space), but returns
/// an isl_map.
isl::map makeTupleSwapMap(isl::space FromSpace1, isl::space FromSpace2) {
  auto BMapResult =
      makeTupleSwapBasicMap(std::move(FromSpace1), std::move(FromSpace2));
  return give(isl_map_from_basic_map(BMapResult.take()));
}
} // anonymous namespace

isl::map polly::beforeScatter(isl::map Map, bool Strict) {
  auto RangeSpace = give(isl_space_range(isl_map_get_space(Map.keep())));
  auto ScatterRel = give(Strict ? isl_map_lex_gt(RangeSpace.take())
                                : isl_map_lex_ge(RangeSpace.take()));
  return give(isl_map_apply_range(Map.take(), ScatterRel.take()));
}

isl::union_map polly::beforeScatter(isl::union_map UMap, bool Strict) {
  auto Result = give(isl_union_map_empty(isl_union_map_get_space(UMap.keep())));
  UMap.foreach_map([=, &Result](isl::map Map) -> isl::stat {
    auto After = beforeScatter(Map, Strict);
    Result = give(isl_union_map_add_map(Result.take(), After.take()));
    return isl::stat::ok;
  });
  return Result;
}

isl::map polly::afterScatter(isl::map Map, bool Strict) {
  auto RangeSpace = give(isl_space_range(isl_map_get_space(Map.keep())));
  auto ScatterRel = give(Strict ? isl_map_lex_lt(RangeSpace.take())
                                : isl_map_lex_le(RangeSpace.take()));
  return give(isl_map_apply_range(Map.take(), ScatterRel.take()));
}

isl::union_map polly::afterScatter(const isl::union_map &UMap, bool Strict) {
  auto Result = give(isl_union_map_empty(isl_union_map_get_space(UMap.keep())));
  UMap.foreach_map([=, &Result](isl::map Map) -> isl::stat {
    auto After = afterScatter(Map, Strict);
    Result = give(isl_union_map_add_map(Result.take(), After.take()));
    return isl::stat::ok;
  });
  return Result;
}

isl::map polly::betweenScatter(isl::map From, isl::map To, bool InclFrom,
                               bool InclTo) {
  auto AfterFrom = afterScatter(From, !InclFrom);
  auto BeforeTo = beforeScatter(To, !InclTo);

  return give(isl_map_intersect(AfterFrom.take(), BeforeTo.take()));
}

isl::union_map polly::betweenScatter(isl::union_map From, isl::union_map To,
                                     bool InclFrom, bool InclTo) {
  auto AfterFrom = afterScatter(From, !InclFrom);
  auto BeforeTo = beforeScatter(To, !InclTo);

  return give(isl_union_map_intersect(AfterFrom.take(), BeforeTo.take()));
}

isl::map polly::singleton(isl::union_map UMap, isl::space ExpectedSpace) {
  if (!UMap)
    return nullptr;

  if (isl_union_map_n_map(UMap.keep()) == 0)
    return isl::map::empty(ExpectedSpace);

  isl::map Result = isl::map::from_union_map(UMap);
  assert(!Result || Result.get_space().has_equal_tuples(ExpectedSpace));

  return Result;
}

isl::set polly::singleton(isl::union_set USet, isl::space ExpectedSpace) {
  if (!USet)
    return nullptr;

  if (isl_union_set_n_set(USet.keep()) == 0)
    return isl::set::empty(ExpectedSpace);

  isl::set Result(USet);
  assert(!Result || Result.get_space().has_equal_tuples(ExpectedSpace));

  return Result;
}

unsigned polly::getNumScatterDims(const isl::union_map &Schedule) {
  unsigned Dims = 0;
  Schedule.foreach_map([&Dims](isl::map Map) -> isl::stat {
    Dims = std::max(Dims, isl_map_dim(Map.keep(), isl_dim_out));
    return isl::stat::ok;
  });
  return Dims;
}

isl::space polly::getScatterSpace(const isl::union_map &Schedule) {
  if (!Schedule)
    return nullptr;
  auto Dims = getNumScatterDims(Schedule);
  auto ScatterSpace =
      give(isl_space_set_from_params(isl_union_map_get_space(Schedule.keep())));
  return give(isl_space_add_dims(ScatterSpace.take(), isl_dim_set, Dims));
}

isl::union_map polly::makeIdentityMap(const isl::union_set &USet,
                                      bool RestrictDomain) {
  auto Result = give(isl_union_map_empty(isl_union_set_get_space(USet.keep())));
  USet.foreach_set([=, &Result](isl::set Set) -> isl::stat {
    auto IdentityMap = give(isl_map_identity(
        isl_space_map_from_set(isl_set_get_space(Set.keep()))));
    if (RestrictDomain)
      IdentityMap =
          give(isl_map_intersect_domain(IdentityMap.take(), Set.take()));
    Result = give(isl_union_map_add_map(Result.take(), IdentityMap.take()));
    return isl::stat::ok;
  });
  return Result;
}

isl::map polly::reverseDomain(isl::map Map) {
  auto DomSpace =
      give(isl_space_unwrap(isl_space_domain(isl_map_get_space(Map.keep()))));
  auto Space1 = give(isl_space_domain(DomSpace.copy()));
  auto Space2 = give(isl_space_range(DomSpace.take()));
  auto Swap = makeTupleSwapMap(std::move(Space1), std::move(Space2));
  return give(isl_map_apply_domain(Map.take(), Swap.take()));
}

isl::union_map polly::reverseDomain(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 {
    auto Reversed = reverseDomain(std::move(Map));
    Result = give(isl_union_map_add_map(Result.take(), Reversed.take()));
    return isl::stat::ok;
  });
  return Result;
}

isl::set polly::shiftDim(isl::set Set, int Pos, int Amount) {
  int NumDims = isl_set_dim(Set.keep(), isl_dim_set);
  if (Pos < 0)
    Pos = NumDims + Pos;
  assert(Pos < NumDims && "Dimension index must be in range");
  auto Space = give(isl_set_get_space(Set.keep()));
  Space = give(isl_space_map_from_domain_and_range(Space.copy(), Space.copy()));
  auto Translator = makeShiftDimAff(std::move(Space), Pos, Amount);
  auto TranslatorMap = give(isl_map_from_multi_aff(Translator.take()));
  return give(isl_set_apply(Set.take(), TranslatorMap.take()));
}

isl::union_set polly::shiftDim(isl::union_set USet, int Pos, int Amount) {
  auto Result = give(isl_union_set_empty(isl_union_set_get_space(USet.keep())));
  USet.foreach_set([=, &Result](isl::set Set) -> isl::stat {
    auto Shifted = shiftDim(Set, Pos, Amount);
    Result = give(isl_union_set_add_set(Result.take(), Shifted.take()));
    return isl::stat::ok;
  });
  return Result;
}

isl::map polly::shiftDim(isl::map Map, isl::dim Dim, int Pos, int Amount) {
  int NumDims = Map.dim(Dim);
  if (Pos < 0)
    Pos = NumDims + Pos;
  assert(Pos < NumDims && "Dimension index must be in range");
  auto Space = give(isl_map_get_space(Map.keep()));
  switch (Dim) {
  case isl::dim::in:
    Space = std::move(Space).domain();
    break;
  case isl::dim::out:
    Space = give(isl_space_range(Space.take()));
    break;
  default:
    llvm_unreachable("Unsupported value for 'dim'");
  }
  Space = give(isl_space_map_from_domain_and_range(Space.copy(), Space.copy()));
  auto Translator = makeShiftDimAff(std::move(Space), Pos, Amount);
  auto TranslatorMap = give(isl_map_from_multi_aff(Translator.take()));
  switch (Dim) {
  case isl::dim::in:
    return Map.apply_domain(TranslatorMap);
  case isl::dim::out:
    return Map.apply_range(TranslatorMap);
  default:
    llvm_unreachable("Unsupported value for 'dim'");
  }
}

isl::union_map polly::shiftDim(isl::union_map UMap, isl::dim Dim, int Pos,
                               int Amount) {
  auto Result = isl::union_map::empty(UMap.get_space());

  UMap.foreach_map([=, &Result](isl::map Map) -> isl::stat {
    auto Shifted = shiftDim(Map, Dim, Pos, Amount);
    Result = std::move(Result).add_map(Shifted);
    return isl::stat::ok;
  });
  return Result;
}

void polly::simplify(isl::set &Set) {
  Set = give(isl_set_compute_divs(Set.take()));
  Set = give(isl_set_detect_equalities(Set.take()));
  Set = give(isl_set_coalesce(Set.take()));
}

void polly::simplify(isl::union_set &USet) {
  USet = give(isl_union_set_compute_divs(USet.take()));
  USet = give(isl_union_set_detect_equalities(USet.take()));
  USet = give(isl_union_set_coalesce(USet.take()));
}

void polly::simplify(isl::map &Map) {
  Map = give(isl_map_compute_divs(Map.take()));
  Map = give(isl_map_detect_equalities(Map.take()));
  Map = give(isl_map_coalesce(Map.take()));
}

void polly::simplify(isl::union_map &UMap) {
  UMap = give(isl_union_map_compute_divs(UMap.take()));
  UMap = give(isl_union_map_detect_equalities(UMap.take()));
  UMap = give(isl_union_map_coalesce(UMap.take()));
}

isl::union_map polly::computeReachingWrite(isl::union_map Schedule,
                                           isl::union_map Writes, bool Reverse,
                                           bool InclPrevDef, bool InclNextDef) {

  // { Scatter[] }
  isl::space ScatterSpace = getScatterSpace(Schedule);

  // { ScatterRead[] -> ScatterWrite[] }
  isl::map Relation;
  if (Reverse)
    Relation = InclPrevDef ? isl::map::lex_lt(ScatterSpace)
                           : isl::map::lex_le(ScatterSpace);
  else
    Relation = InclNextDef ? isl::map::lex_gt(ScatterSpace)
                           : isl::map::lex_ge(ScatterSpace);

  // { ScatterWrite[] -> [ScatterRead[] -> ScatterWrite[]] }
  isl::map RelationMap = Relation.range_map().reverse();

  // { Element[] -> ScatterWrite[] }
  isl::union_map WriteAction = Schedule.apply_domain(Writes);

  // { ScatterWrite[] -> Element[] }
  isl::union_map WriteActionRev = WriteAction.reverse();

  // { Element[] -> [ScatterUse[] -> ScatterWrite[]] }
  isl::union_map DefSchedRelation =
      isl::union_map(RelationMap).apply_domain(WriteActionRev);

  // For each element, at every point in time, map to the times of previous
  // definitions. { [Element[] -> ScatterRead[]] -> ScatterWrite[] }
  isl::union_map ReachableWrites = DefSchedRelation.uncurry();
  if (Reverse)
    ReachableWrites = ReachableWrites.lexmin();
  else
    ReachableWrites = ReachableWrites.lexmax();

  // { [Element[] -> ScatterWrite[]] -> ScatterWrite[] }
  isl::union_map SelfUse = WriteAction.range_map();

  if (InclPrevDef && InclNextDef) {
    // Add the Def itself to the solution.
    ReachableWrites = ReachableWrites.unite(SelfUse).coalesce();
  } else if (!InclPrevDef && !InclNextDef) {
    // Remove Def itself from the solution.
    ReachableWrites = ReachableWrites.subtract(SelfUse);
  }

  // { [Element[] -> ScatterRead[]] -> Domain[] }
  return ReachableWrites.apply_range(Schedule.reverse());
}

isl::union_map
polly::computeArrayUnused(isl::union_map Schedule, isl::union_map Writes,
                          isl::union_map Reads, bool ReadEltInSameInst,
                          bool IncludeLastRead, bool IncludeWrite) {
  // { Element[] -> Scatter[] }
  auto ReadActions =
      give(isl_union_map_apply_domain(Schedule.copy(), Reads.take()));
  auto WriteActions =
      give(isl_union_map_apply_domain(Schedule.copy(), Writes.copy()));

  // { [Element[] -> DomainWrite[]] -> Scatter[] }
  auto EltDomWrites = give(isl_union_map_apply_range(
      isl_union_map_range_map(isl_union_map_reverse(Writes.copy())),
      Schedule.copy()));

  // { [Element[] -> Scatter[]] -> DomainWrite[] }
  auto ReachingOverwrite = computeReachingWrite(
      Schedule, Writes, true, ReadEltInSameInst, !ReadEltInSameInst);

  // { [Element[] -> Scatter[]] -> DomainWrite[] }
  auto ReadsOverwritten = give(isl_union_map_intersect_domain(
      ReachingOverwrite.take(), isl_union_map_wrap(ReadActions.take())));

  // { [Element[] -> DomainWrite[]] -> Scatter[] }
  auto ReadsOverwrittenRotated = give(isl_union_map_reverse(
      isl_union_map_curry(reverseDomain(ReadsOverwritten).take())));
  auto LastOverwrittenRead =
      give(isl_union_map_lexmax(ReadsOverwrittenRotated.copy()));

  // { [Element[] -> DomainWrite[]] -> Scatter[] }
  auto BetweenLastReadOverwrite = betweenScatter(
      LastOverwrittenRead, EltDomWrites, IncludeLastRead, IncludeWrite);

  // { [Element[] -> Scatter[]] -> DomainWrite[] }
  isl::union_map ReachingOverwriteZone = computeReachingWrite(
      Schedule, Writes, true, IncludeLastRead, IncludeWrite);

  // { [Element[] -> DomainWrite[]] -> Scatter[] }
  isl::union_map ReachingOverwriteRotated =
      reverseDomain(ReachingOverwriteZone).curry().reverse();

  // { [Element[] -> DomainWrite[]] -> Scatter[] }
  isl::union_map WritesWithoutReads = ReachingOverwriteRotated.subtract_domain(
      ReadsOverwrittenRotated.domain());

  return BetweenLastReadOverwrite.unite(WritesWithoutReads)
      .domain_factor_domain();
}

isl::union_set polly::convertZoneToTimepoints(isl::union_set Zone,
                                              bool InclStart, bool InclEnd) {
  if (!InclStart && InclEnd)
    return Zone;

  auto ShiftedZone = shiftDim(Zone, -1, -1);
  if (InclStart && !InclEnd)
    return ShiftedZone;
  else if (!InclStart && !InclEnd)
    return give(isl_union_set_intersect(Zone.take(), ShiftedZone.take()));

  assert(InclStart && InclEnd);
  return give(isl_union_set_union(Zone.take(), ShiftedZone.take()));
}

isl::union_map polly::convertZoneToTimepoints(isl::union_map Zone, isl::dim Dim,
                                              bool InclStart, bool InclEnd) {
  if (!InclStart && InclEnd)
    return Zone;

  auto ShiftedZone = shiftDim(Zone, Dim, -1, -1);
  if (InclStart && !InclEnd)
    return ShiftedZone;
  else if (!InclStart && !InclEnd)
    return give(isl_union_map_intersect(Zone.take(), ShiftedZone.take()));

  assert(InclStart && InclEnd);
  return give(isl_union_map_union(Zone.take(), ShiftedZone.take()));
}

isl::map polly::convertZoneToTimepoints(isl::map Zone, isl::dim Dim,
                                        bool InclStart, bool InclEnd) {
  if (!InclStart && InclEnd)
    return Zone;

  auto ShiftedZone = shiftDim(Zone, Dim, -1, -1);
  if (InclStart && !InclEnd)
    return ShiftedZone;
  else if (!InclStart && !InclEnd)
    return give(isl_map_intersect(Zone.take(), ShiftedZone.take()));

  assert(InclStart && InclEnd);
  return give(isl_map_union(Zone.take(), ShiftedZone.take()));
}

isl::map polly::distributeDomain(isl::map Map) {
  // Note that we cannot take Map apart into { Domain[] -> Range1[] } and {
  // Domain[] -> Range2[] } and combine again. We would loose any relation
  // between Range1[] and Range2[] that is not also a constraint to Domain[].

  auto Space = give(isl_map_get_space(Map.keep()));
  auto DomainSpace = give(isl_space_domain(Space.copy()));
  auto DomainDims = isl_space_dim(DomainSpace.keep(), isl_dim_set);
  auto RangeSpace = give(isl_space_unwrap(isl_space_range(Space.copy())));
  auto Range1Space = give(isl_space_domain(RangeSpace.copy()));
  auto Range1Dims = isl_space_dim(Range1Space.keep(), isl_dim_set);
  auto Range2Space = give(isl_space_range(RangeSpace.copy()));
  auto Range2Dims = isl_space_dim(Range2Space.keep(), isl_dim_set);

  auto OutputSpace = give(isl_space_map_from_domain_and_range(
      isl_space_wrap(isl_space_map_from_domain_and_range(DomainSpace.copy(),
                                                         Range1Space.copy())),
      isl_space_wrap(isl_space_map_from_domain_and_range(DomainSpace.copy(),
                                                         Range2Space.copy()))));

  auto Translator =
      give(isl_basic_map_universe(isl_space_map_from_domain_and_range(
          isl_space_wrap(Space.copy()), isl_space_wrap(OutputSpace.copy()))));

  for (unsigned i = 0; i < DomainDims; i += 1) {
    Translator = give(
        isl_basic_map_equate(Translator.take(), isl_dim_in, i, isl_dim_out, i));
    Translator =
        give(isl_basic_map_equate(Translator.take(), isl_dim_in, i, isl_dim_out,
                                  DomainDims + Range1Dims + i));
  }
  for (unsigned i = 0; i < Range1Dims; i += 1) {
    Translator =
        give(isl_basic_map_equate(Translator.take(), isl_dim_in, DomainDims + i,
                                  isl_dim_out, DomainDims + i));
  }
  for (unsigned i = 0; i < Range2Dims; i += 1) {
    Translator = give(isl_basic_map_equate(
        Translator.take(), isl_dim_in, DomainDims + Range1Dims + i, isl_dim_out,
        DomainDims + Range1Dims + DomainDims + i));
  }

  return give(isl_set_unwrap(isl_set_apply(
      isl_map_wrap(Map.copy()), isl_map_from_basic_map(Translator.copy()))));
}

isl::union_map polly::distributeDomain(isl::union_map UMap) {
  auto Result = give(isl_union_map_empty(isl_union_map_get_space(UMap.keep())));
  isl::stat Success = UMap.foreach_map([=, &Result](isl::map Map) {
    auto Distributed = distributeDomain(Map);
    Result = give(isl_union_map_add_map(Result.take(), Distributed.copy()));
    return isl::stat::ok;
  });
  if (Success != isl::stat::ok)
    return {};
  return Result;
}

isl::union_map polly::liftDomains(isl::union_map UMap, isl::union_set Factor) {

  // { Factor[] -> Factor[] }
  auto Factors = makeIdentityMap(std::move(Factor), true);

  return std::move(Factors).product(std::move(UMap));
}

isl::union_map polly::applyDomainRange(isl::union_map UMap,
                                       isl::union_map Func) {
  // This implementation creates unnecessary cross products of the
  // DomainDomain[] and Func. An alternative implementation could reverse
  // domain+uncurry,apply Func to what now is the domain, then undo the
  // preparing transformation. Another alternative implementation could create a
  // translator map for each piece.

  // { DomainDomain[] }
  auto DomainDomain = UMap.domain().unwrap().domain();

  // { [DomainDomain[] -> DomainRange[]] -> [DomainDomain[] -> NewDomainRange[]]
  // }
  auto LifetedFunc = liftDomains(std::move(Func), DomainDomain);

  return std::move(UMap).apply_domain(std::move(LifetedFunc));
}

isl::map polly::intersectRange(isl::map Map, isl::union_set Range) {
  isl::set RangeSet = Range.extract_set(Map.get_space().range());
  return Map.intersect_range(RangeSet);
}

isl::val polly::getConstant(isl::pw_aff PwAff, bool Max, bool Min) {
  assert(!Max || !Min); // Cannot return min and max at the same time.
  isl::val Result;
  PwAff.foreach_piece([=, &Result](isl::set Set, isl::aff Aff) -> isl::stat {
    if (Result && Result.is_nan())
      return isl::stat::ok;

    // TODO: If Min/Max, we can also determine a minimum/maximum value if
    // Set is constant-bounded.
    if (!Aff.is_cst()) {
      Result = isl::val::nan(Aff.get_ctx());
      return isl::stat::error;
    }

    isl::val ThisVal = Aff.get_constant_val();
    if (!Result) {
      Result = ThisVal;
      return isl::stat::ok;
    }

    if (Result.eq(ThisVal))
      return isl::stat::ok;

    if (Max && ThisVal.gt(Result)) {
      Result = ThisVal;
      return isl::stat::ok;
    }

    if (Min && ThisVal.lt(Result)) {
      Result = ThisVal;
      return isl::stat::ok;
    }

    // Not compatible
    Result = isl::val::nan(Aff.get_ctx());
    return isl::stat::error;
  });
  return Result;
}

#if !defined(NDEBUG) || defined(LLVM_ENABLE_DUMP)
static void foreachPoint(const isl::set &Set,
                         const std::function<void(isl::point P)> &F) {
  isl_set_foreach_point(
      Set.keep(),
      [](__isl_take isl_point *p, void *User) -> isl_stat {
        auto &F = *static_cast<const std::function<void(isl::point)> *>(User);
        F(give(p));
        return isl_stat_ok;
      },
      const_cast<void *>(static_cast<const void *>(&F)));
}

static void foreachPoint(isl::basic_set BSet,
                         const std::function<void(isl::point P)> &F) {
  foreachPoint(give(isl_set_from_basic_set(BSet.take())), F);
}

/// Determine the sorting order of the sets @p A and @p B without considering
/// the space structure.
///
/// Ordering is based on the lower bounds of the set's dimensions. First
/// dimensions are considered first.
static int flatCompare(const isl::basic_set &A, const isl::basic_set &B) {
  int ALen = A.dim(isl::dim::set);
  int BLen = B.dim(isl::dim::set);
  int Len = std::min(ALen, BLen);

  for (int i = 0; i < Len; i += 1) {
    isl::basic_set ADim =
        A.project_out(isl::dim::param, 0, A.dim(isl::dim::param))
            .project_out(isl::dim::set, i + 1, ALen - i - 1)
            .project_out(isl::dim::set, 0, i);
    isl::basic_set BDim =
        B.project_out(isl::dim::param, 0, B.dim(isl::dim::param))
            .project_out(isl::dim::set, i + 1, BLen - i - 1)
            .project_out(isl::dim::set, 0, i);

    isl::basic_set AHull = isl::set(ADim).convex_hull();
    isl::basic_set BHull = isl::set(BDim).convex_hull();

    bool ALowerBounded =
        bool(isl::set(AHull).dim_has_any_lower_bound(isl::dim::set, 0));
    bool BLowerBounded =
        bool(isl::set(BHull).dim_has_any_lower_bound(isl::dim::set, 0));

    int BoundedCompare = BLowerBounded - ALowerBounded;
    if (BoundedCompare != 0)
      return BoundedCompare;

    if (!ALowerBounded || !BLowerBounded)
      continue;

    isl::pw_aff AMin = isl::set(ADim).dim_min(0);
    isl::pw_aff BMin = isl::set(BDim).dim_min(0);

    isl::val AMinVal = polly::getConstant(AMin, false, true);
    isl::val BMinVal = polly::getConstant(BMin, false, true);

    int MinCompare = AMinVal.sub(BMinVal).sgn();
    if (MinCompare != 0)
      return MinCompare;
  }

  // If all the dimensions' lower bounds are equal or incomparable, sort based
  // on the number of dimensions.
  return ALen - BLen;
}

/// Compare the sets @p A and @p B according to their nested space structure. If
/// the structure is the same, sort using the dimension lower bounds.
static int recursiveCompare(const isl::basic_set &A, const isl::basic_set &B) {
  isl::space ASpace = A.get_space();
  isl::space BSpace = B.get_space();

  int WrappingCompare = bool(ASpace.is_wrapping()) - bool(BSpace.is_wrapping());
  if (WrappingCompare != 0)
    return WrappingCompare;

  if (ASpace.is_wrapping() && B.is_wrapping()) {
    isl::basic_map AMap = A.unwrap();
    isl::basic_map BMap = B.unwrap();

    int FirstResult = recursiveCompare(AMap.domain(), BMap.domain());
    if (FirstResult != 0)
      return FirstResult;

    return recursiveCompare(AMap.range(), BMap.range());
  }

  std::string AName = ASpace.has_tuple_name(isl::dim::set)
                          ? ASpace.get_tuple_name(isl::dim::set)
                          : std::string();
  std::string BName = BSpace.has_tuple_name(isl::dim::set)
                          ? BSpace.get_tuple_name(isl::dim::set)
                          : std::string();

  int NameCompare = AName.compare(BName);
  if (NameCompare != 0)
    return NameCompare;

  return flatCompare(A, B);
}

/// Wrapper for recursiveCompare, convert a {-1,0,1} compare result to what
/// std::sort expects.
static bool orderComparer(const isl::basic_set &A, const isl::basic_set &B) {
  return recursiveCompare(A, B) < 0;
}

/// Print a string representation of @p USet to @p OS.
///
/// The pieces of @p USet are printed in a sorted order. Spaces with equal or
/// similar nesting structure are printed together. Compared to isl's own
/// printing function the uses the structure itself as base of the sorting, not
/// a hash of it. It ensures that e.g. maps spaces with same domain structure
/// are printed together. Set pieces with same structure are printed in order of
/// their lower bounds.
///
/// @param USet     Polyhedra to print.
/// @param OS       Target stream.
/// @param Simplify Whether to simplify the polyhedron before printing.
/// @param IsMap    Whether @p USet is a wrapped map. If true, sets are
///                 unwrapped before printing to again appear as a map.
static void printSortedPolyhedra(isl::union_set USet, llvm::raw_ostream &OS,
                                 bool Simplify, bool IsMap) {
  if (!USet) {
    OS << "<null>\n";
    return;
  }

  if (Simplify)
    simplify(USet);

  // Get all the polyhedra.
  std::vector<isl::basic_set> BSets;
  USet.foreach_set([&BSets](isl::set Set) -> isl::stat {
    Set.foreach_basic_set([&BSets](isl::basic_set BSet) -> isl::stat {
      BSets.push_back(BSet);
      return isl::stat::ok;
    });
    return isl::stat::ok;
  });

  if (BSets.empty()) {
    OS << "{\n}\n";
    return;
  }

  // Sort the polyhedra.
  std::sort(BSets.begin(), BSets.end(), orderComparer);

  // Print the polyhedra.
  bool First = true;
  for (const isl::basic_set &BSet : BSets) {
    std::string Str;
    if (IsMap)
      Str = isl::map(BSet.unwrap()).to_str();
    else
      Str = isl::set(BSet).to_str();
    size_t OpenPos = Str.find_first_of('{');
    assert(OpenPos != std::string::npos);
    size_t ClosePos = Str.find_last_of('}');
    assert(ClosePos != std::string::npos);

    if (First)
      OS << llvm::StringRef(Str).substr(0, OpenPos + 1) << "\n ";
    else
      OS << ";\n ";

    OS << llvm::StringRef(Str).substr(OpenPos + 1, ClosePos - OpenPos - 2);
    First = false;
  }
  assert(!First);
  OS << "\n}\n";
}

static void recursiveExpand(isl::basic_set BSet, int Dim, isl::set &Expanded) {
  int Dims = BSet.dim(isl::dim::set);
  if (Dim >= Dims) {
    Expanded = Expanded.unite(BSet);
    return;
  }

  isl::basic_set DimOnly =
      BSet.project_out(isl::dim::param, 0, BSet.dim(isl::dim::param))
          .project_out(isl::dim::set, Dim + 1, Dims - Dim - 1)
          .project_out(isl::dim::set, 0, Dim);
  if (!DimOnly.is_bounded()) {
    recursiveExpand(BSet, Dim + 1, Expanded);
    return;
  }

  foreachPoint(DimOnly, [&, Dim](isl::point P) {
    isl::val Val = P.get_coordinate_val(isl::dim::set, 0);
    isl::basic_set FixBSet = BSet.fix_val(isl::dim::set, Dim, Val);
    recursiveExpand(FixBSet, Dim + 1, Expanded);
  });
}

/// Make each point of a set explicit.
///
/// "Expanding" makes each point a set contains explicit. That is, the result is
/// a set of singleton polyhedra. Unbounded dimensions are not expanded.
///
/// Example:
///   { [i] : 0 <= i < 2 }
/// is expanded to:
///   { [0]; [1] }
static isl::set expand(const isl::set &Set) {
  isl::set Expanded = isl::set::empty(Set.get_space());
  Set.foreach_basic_set([&](isl::basic_set BSet) -> isl::stat {
    recursiveExpand(BSet, 0, Expanded);
    return isl::stat::ok;
  });
  return Expanded;
}

/// Expand all points of a union set explicit.
///
/// @see expand(const isl::set)
static isl::union_set expand(const isl::union_set &USet) {
  isl::union_set Expanded =
      give(isl_union_set_empty(isl_union_set_get_space(USet.keep())));
  USet.foreach_set([&](isl::set Set) -> isl::stat {
    isl::set SetExpanded = expand(Set);
    Expanded = Expanded.add_set(SetExpanded);
    return isl::stat::ok;
  });
  return Expanded;
}

LLVM_DUMP_METHOD void polly::dumpPw(const isl::set &Set) {
  printSortedPolyhedra(Set, llvm::errs(), true, false);
}

LLVM_DUMP_METHOD void polly::dumpPw(const isl::map &Map) {
  printSortedPolyhedra(Map.wrap(), llvm::errs(), true, true);
}

LLVM_DUMP_METHOD void polly::dumpPw(const isl::union_set &USet) {
  printSortedPolyhedra(USet, llvm::errs(), true, false);
}

LLVM_DUMP_METHOD void polly::dumpPw(const isl::union_map &UMap) {
  printSortedPolyhedra(UMap.wrap(), llvm::errs(), true, true);
}

LLVM_DUMP_METHOD void polly::dumpPw(__isl_keep isl_set *Set) {
  dumpPw(isl::manage(isl_set_copy(Set)));
}

LLVM_DUMP_METHOD void polly::dumpPw(__isl_keep isl_map *Map) {
  dumpPw(isl::manage(isl_map_copy(Map)));
}

LLVM_DUMP_METHOD void polly::dumpPw(__isl_keep isl_union_set *USet) {
  dumpPw(isl::manage(isl_union_set_copy(USet)));
}

LLVM_DUMP_METHOD void polly::dumpPw(__isl_keep isl_union_map *UMap) {
  dumpPw(isl::manage(isl_union_map_copy(UMap)));
}

LLVM_DUMP_METHOD void polly::dumpExpanded(const isl::set &Set) {
  printSortedPolyhedra(expand(Set), llvm::errs(), false, false);
}

LLVM_DUMP_METHOD void polly::dumpExpanded(const isl::map &Map) {
  printSortedPolyhedra(expand(Map.wrap()), llvm::errs(), false, true);
}

LLVM_DUMP_METHOD void polly::dumpExpanded(const isl::union_set &USet) {
  printSortedPolyhedra(expand(USet), llvm::errs(), false, false);
}

LLVM_DUMP_METHOD void polly::dumpExpanded(const isl::union_map &UMap) {
  printSortedPolyhedra(expand(UMap.wrap()), llvm::errs(), false, true);
}

LLVM_DUMP_METHOD void polly::dumpExpanded(__isl_keep isl_set *Set) {
  dumpExpanded(isl::manage(isl_set_copy(Set)));
}

LLVM_DUMP_METHOD void polly::dumpExpanded(__isl_keep isl_map *Map) {
  dumpExpanded(isl::manage(isl_map_copy(Map)));
}

LLVM_DUMP_METHOD void polly::dumpExpanded(__isl_keep isl_union_set *USet) {
  dumpExpanded(isl::manage(isl_union_set_copy(USet)));
}

LLVM_DUMP_METHOD void polly::dumpExpanded(__isl_keep isl_union_map *UMap) {
  dumpExpanded(isl::manage(isl_union_map_copy(UMap)));
}
#endif