[DeLICM] Refactor ZoneAlgorithm into ZoneAlgo.cpp. NFC.

Extract ZoneAlgorithm from DeLICM.cpp into its own file.
It will gain a second use by the load forwarding part of
-polly-optree.

llvm-svn: 310146
diff --git a/polly/lib/Transform/ZoneAlgo.cpp b/polly/lib/Transform/ZoneAlgo.cpp
new file mode 100644
index 0000000..917f55e
--- /dev/null
+++ b/polly/lib/Transform/ZoneAlgo.cpp
@@ -0,0 +1,613 @@
+//===------ ZoneAlgo.cpp ----------------------------------------*- C++ -*-===//
+//
+//                     The LLVM Compiler Infrastructure
+//
+// This file is distributed under the University of Illinois Open Source
+// License. See LICENSE.TXT for details.
+//
+//===----------------------------------------------------------------------===//
+//
+// Derive information about array elements between statements ("Zones").
+//
+// 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
+//
+// * ValInst[] - An llvm::Value as defined at a specific timepoint.
+//
+//     A ValInst[] itself can be structured as one of:
+//
+//     * [] - An unknown value.
+//         Always zero dimensions
+//         Has no tuple id
+//
+//     * Value[] - An llvm::Value that is read-only in the SCoP, i.e. its
+//                 runtime content does not depend on the timepoint.
+//         Always zero dimensions
+//         isl_id_get_name: Val_<NameOfValue>
+//         isl_id_get_user: A pointer to an llvm::Value
+//
+//     * SCEV[...] - A synthesizable llvm::SCEV Expression.
+//         In contrast to a Value[] is has at least one dimension per
+//         SCEVAddRecExpr in the SCEV.
+//
+//     * [Domain[] -> Value[]] - An llvm::Value that may change during the
+//                               Scop's execution.
+//         The tuple itself has no id, but it wraps a map space holding a
+//         statement instance which defines the llvm::Value as the map's domain
+//         and llvm::Value itself as range.
+//
+// @see makeValInst()
+//
+// 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/ZoneAlgo.h"
+#include "polly/ScopInfo.h"
+#include "polly/Support/GICHelper.h"
+#include "polly/Support/ISLTools.h"
+#include "polly/Support/VirtualInstruction.h"
+
+#define DEBUG_TYPE "polly-zone"
+
+using namespace polly;
+using namespace llvm;
+
+static isl::union_map computeReachingDefinition(isl::union_map Schedule,
+                                                isl::union_map Writes,
+                                                bool InclDef, bool InclRedef) {
+  return computeReachingWrite(Schedule, Writes, false, InclDef, InclRedef);
+}
+
+/// 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[] }
+static 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[] }
+static isl::map computeScalarReachingDefinition(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);
+}
+
+isl::union_map polly::makeUnknownForDomain(isl::union_set Domain) {
+  return give(isl_union_map_from_domain(Domain.take()));
+}
+
+/// Create a domain-to-unknown value mapping.
+///
+/// @see makeUnknownForDomain(isl::union_set)
+///
+/// @param Domain { Domain[] }
+///
+/// @return { Domain[] -> ValInst[] }
+static isl::map makeUnknownForDomain(isl::set Domain) {
+  return give(isl_map_from_domain(Domain.take()));
+}
+
+static std::string printInstruction(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);
+}
+
+ZoneAlgorithm::ZoneAlgorithm(const char *PassName, Scop *S, LoopInfo *LI)
+    : PassName(PassName), IslCtx(S->getSharedIslCtx()), S(S), LI(LI),
+      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);
+}
+
+bool ZoneAlgorithm::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(PassName, "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(PassName, "UnusualStore",
+                                 MA->getAccessInstruction());
+      R << "encountered write that is not a StoreInst: "
+        << printInstruction(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(PassName, "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(PassName, "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 ZoneAlgorithm::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 ZoneAlgorithm::addArrayWriteAccess(MemoryAccess *MA) {
+  assert(MA->isLatestArrayKind());
+  assert(MA->isWrite());
+  auto *Stmt = MA->getStatement();
+
+  // { 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()));
+
+  // { Domain[] -> ValInst[] }
+  auto WriteValInstance =
+      makeValInst(MA->getAccessValue(), Stmt,
+                  LI->getLoopFor(MA->getAccessInstruction()->getParent()),
+                  MA->isMustWrite());
+
+  // { Domain[] -> [Element[] -> Domain[]] }
+  auto IncludeElement = give(isl_map_curry(isl_map_domain_map(AccRel.copy())));
+
+  // { [Element[] -> DomainWrite[]] -> ValInst[] }
+  auto EltWriteValInst = give(
+      isl_map_apply_domain(WriteValInstance.take(), IncludeElement.take()));
+
+  AllWriteValInst = give(
+      isl_union_map_add_map(AllWriteValInst.take(), EltWriteValInst.take()));
+}
+
+isl::union_set ZoneAlgorithm::makeEmptyUnionSet() const {
+  return give(isl_union_set_empty(ParamSpace.copy()));
+}
+
+isl::union_map ZoneAlgorithm::makeEmptyUnionMap() const {
+  return give(isl_union_map_empty(ParamSpace.copy()));
+}
+
+bool ZoneAlgorithm::isCompatibleScop() {
+  for (auto &Stmt : *S) {
+    if (!isCompatibleStmt(&Stmt))
+      return false;
+  }
+  return true;
+}
+
+isl::map ZoneAlgorithm::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()));
+}
+
+isl::map ZoneAlgorithm::getScatterFor(MemoryAccess *MA) const {
+  return getScatterFor(MA->getStatement());
+}
+
+isl::union_map ZoneAlgorithm::getScatterFor(isl::union_set Domain) const {
+  return give(isl_union_map_intersect_domain(Schedule.copy(), Domain.take()));
+}
+
+isl::map ZoneAlgorithm::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(!Result || isl_set_is_equal(give(isl_map_domain(Result.copy())).keep(),
+                                     Domain.keep()) == isl_bool_true);
+  return Result;
+}
+
+isl::set ZoneAlgorithm::getDomainFor(ScopStmt *Stmt) const {
+  return give(isl_set_remove_redundancies(Stmt->getDomain()));
+}
+
+isl::set ZoneAlgorithm::getDomainFor(MemoryAccess *MA) const {
+  return getDomainFor(MA->getStatement());
+}
+
+isl::map ZoneAlgorithm::getAccessRelationFor(MemoryAccess *MA) const {
+  auto Domain = getDomainFor(MA);
+  auto AccRel = MA->getLatestAccessRelation();
+  return give(isl_map_intersect_domain(AccRel.take(), Domain.take()));
+}
+
+isl::map ZoneAlgorithm::getScalarReachingDefinition(ScopStmt *Stmt) {
+  auto &Result = ScalarReachDefZone[Stmt];
+  if (Result)
+    return Result;
+
+  auto Domain = getDomainFor(Stmt);
+  Result = computeScalarReachingDefinition(Schedule, Domain, false, true);
+  simplify(Result);
+
+  return Result;
+}
+
+isl::map ZoneAlgorithm::getScalarReachingDefinition(isl::set DomainDef) {
+  auto DomId = give(isl_set_get_tuple_id(DomainDef.keep()));
+  auto *Stmt = static_cast<ScopStmt *>(isl_id_get_user(DomId.keep()));
+
+  auto StmtResult = getScalarReachingDefinition(Stmt);
+
+  return give(isl_map_intersect_range(StmtResult.take(), DomainDef.take()));
+}
+
+isl::map ZoneAlgorithm::makeUnknownForDomain(ScopStmt *Stmt) const {
+  return ::makeUnknownForDomain(getDomainFor(Stmt));
+}
+
+isl::id ZoneAlgorithm::makeValueId(Value *V) {
+  if (!V)
+    return nullptr;
+
+  auto &Id = ValueIds[V];
+  if (Id.is_null()) {
+    auto Name = getIslCompatibleName("Val_", V, ValueIds.size() - 1,
+                                     std::string(), UseInstructionNames);
+    Id = give(isl_id_alloc(IslCtx.get(), Name.c_str(), V));
+  }
+  return Id;
+}
+
+isl::space ZoneAlgorithm::makeValueSpace(Value *V) {
+  auto Result = give(isl_space_set_from_params(ParamSpace.copy()));
+  return give(isl_space_set_tuple_id(Result.take(), isl_dim_set,
+                                     makeValueId(V).take()));
+}
+
+isl::set ZoneAlgorithm::makeValueSet(Value *V) {
+  auto Space = makeValueSpace(V);
+  return give(isl_set_universe(Space.take()));
+}
+
+isl::map ZoneAlgorithm::makeValInst(Value *Val, ScopStmt *UserStmt, Loop *Scope,
+                                    bool IsCertain) {
+  // If the definition/write is conditional, the value at the location could
+  // be either the written value or the old value. Since we cannot know which
+  // one, consider the value to be unknown.
+  if (!IsCertain)
+    return makeUnknownForDomain(UserStmt);
+
+  auto DomainUse = getDomainFor(UserStmt);
+  auto VUse = VirtualUse::create(S, UserStmt, Scope, Val, true);
+  switch (VUse.getKind()) {
+  case VirtualUse::Constant:
+  case VirtualUse::Block:
+  case VirtualUse::Hoisted:
+  case VirtualUse::ReadOnly: {
+    // The definition does not depend on the statement which uses it.
+    auto ValSet = makeValueSet(Val);
+    return give(isl_map_from_domain_and_range(DomainUse.take(), ValSet.take()));
+  }
+
+  case VirtualUse::Synthesizable: {
+    auto *ScevExpr = VUse.getScevExpr();
+    auto UseDomainSpace = give(isl_set_get_space(DomainUse.keep()));
+
+    // Construct the SCEV space.
+    // TODO: Add only the induction variables referenced in SCEVAddRecExpr
+    // expressions, not just all of them.
+    auto ScevId = give(isl_id_alloc(UseDomainSpace.get_ctx().get(), nullptr,
+                                    const_cast<SCEV *>(ScevExpr)));
+    auto ScevSpace =
+        give(isl_space_drop_dims(UseDomainSpace.copy(), isl_dim_set, 0, 0));
+    ScevSpace = give(
+        isl_space_set_tuple_id(ScevSpace.take(), isl_dim_set, ScevId.copy()));
+
+    // { DomainUse[] -> ScevExpr[] }
+    auto ValInst = give(isl_map_identity(isl_space_map_from_domain_and_range(
+        UseDomainSpace.copy(), ScevSpace.copy())));
+    return ValInst;
+  }
+
+  case VirtualUse::Intra: {
+    // Definition and use is in the same statement. We do not need to compute
+    // a reaching definition.
+
+    // { llvm::Value }
+    auto ValSet = makeValueSet(Val);
+
+    // {  UserDomain[] -> llvm::Value }
+    auto ValInstSet =
+        give(isl_map_from_domain_and_range(DomainUse.take(), ValSet.take()));
+
+    // { UserDomain[] -> [UserDomain[] - >llvm::Value] }
+    auto Result = give(isl_map_reverse(isl_map_domain_map(ValInstSet.take())));
+    simplify(Result);
+    return Result;
+  }
+
+  case VirtualUse::Inter: {
+    // The value is defined in a different statement.
+
+    auto *Inst = cast<Instruction>(Val);
+    auto *ValStmt = S->getStmtFor(Inst);
+
+    // If the llvm::Value is defined in a removed Stmt, we cannot derive its
+    // domain. We could use an arbitrary statement, but this could result in
+    // different ValInst[] for the same llvm::Value.
+    if (!ValStmt)
+      return ::makeUnknownForDomain(DomainUse);
+
+    // { DomainDef[] }
+    auto DomainDef = getDomainFor(ValStmt);
+
+    // { Scatter[] -> DomainDef[] }
+    auto ReachDef = getScalarReachingDefinition(DomainDef);
+
+    // { DomainUse[] -> Scatter[] }
+    auto UserSched = getScatterFor(DomainUse);
+
+    // { DomainUse[] -> DomainDef[] }
+    auto UsedInstance =
+        give(isl_map_apply_range(UserSched.take(), ReachDef.take()));
+
+    // { llvm::Value }
+    auto ValSet = makeValueSet(Val);
+
+    // { DomainUse[] -> llvm::Value[] }
+    auto ValInstSet =
+        give(isl_map_from_domain_and_range(DomainUse.take(), ValSet.take()));
+
+    // { DomainUse[] -> [DomainDef[] -> llvm::Value]  }
+    auto Result =
+        give(isl_map_range_product(UsedInstance.take(), ValInstSet.take()));
+
+    simplify(Result);
+    return Result;
+  }
+  }
+  llvm_unreachable("Unhandled use type");
+}
+
+void ZoneAlgorithm::computeCommon() {
+  AllReads = makeEmptyUnionMap();
+  AllMayWrites = makeEmptyUnionMap();
+  AllMustWrites = makeEmptyUnionMap();
+  AllWriteValInst = 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()));
+
+  // { [Element[] -> Zone[]] -> DomainWrite[] }
+  WriteReachDefZone =
+      computeReachingDefinition(Schedule, AllWrites, false, true);
+  simplify(WriteReachDefZone);
+}
+
+void ZoneAlgorithm::printAccesses(llvm::raw_ostream &OS, int Indent) 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";
+}