llvm/lib/Analysis/LoopInfo.cpp
Yaron Keren 55307987a1 Annotate dump() methods with LLVM_DUMP_METHOD, addressing Richard Smith r259192 post commit comment.
clang part in r259232, this is the LLVM part of the patch.



git-svn-id: https://llvm.org/svn/llvm-project/llvm/trunk@259240 91177308-0d34-0410-b5e6-96231b3b80d8
2016-01-29 20:50:44 +00:00

726 lines
24 KiB
C++

//===- LoopInfo.cpp - Natural Loop Calculator -----------------------------===//
//
// The LLVM Compiler Infrastructure
//
// This file is distributed under the University of Illinois Open Source
// License. See LICENSE.TXT for details.
//
//===----------------------------------------------------------------------===//
//
// This file defines the LoopInfo class that is used to identify natural loops
// and determine the loop depth of various nodes of the CFG. Note that the
// loops identified may actually be several natural loops that share the same
// header node... not just a single natural loop.
//
//===----------------------------------------------------------------------===//
#include "llvm/Analysis/LoopInfo.h"
#include "llvm/ADT/DepthFirstIterator.h"
#include "llvm/ADT/SmallPtrSet.h"
#include "llvm/Analysis/LoopInfoImpl.h"
#include "llvm/Analysis/LoopIterator.h"
#include "llvm/Analysis/ValueTracking.h"
#include "llvm/IR/CFG.h"
#include "llvm/IR/Constants.h"
#include "llvm/IR/Dominators.h"
#include "llvm/IR/Instructions.h"
#include "llvm/IR/LLVMContext.h"
#include "llvm/IR/Metadata.h"
#include "llvm/IR/PassManager.h"
#include "llvm/Support/CommandLine.h"
#include "llvm/Support/Debug.h"
#include "llvm/Support/raw_ostream.h"
#include <algorithm>
using namespace llvm;
// Explicitly instantiate methods in LoopInfoImpl.h for IR-level Loops.
template class llvm::LoopBase<BasicBlock, Loop>;
template class llvm::LoopInfoBase<BasicBlock, Loop>;
// Always verify loopinfo if expensive checking is enabled.
#ifdef XDEBUG
static bool VerifyLoopInfo = true;
#else
static bool VerifyLoopInfo = false;
#endif
static cl::opt<bool,true>
VerifyLoopInfoX("verify-loop-info", cl::location(VerifyLoopInfo),
cl::desc("Verify loop info (time consuming)"));
// Loop identifier metadata name.
static const char *const LoopMDName = "llvm.loop";
//===----------------------------------------------------------------------===//
// Loop implementation
//
bool Loop::isLoopInvariant(const Value *V) const {
if (const Instruction *I = dyn_cast<Instruction>(V))
return !contains(I);
return true; // All non-instructions are loop invariant
}
bool Loop::hasLoopInvariantOperands(const Instruction *I) const {
return all_of(I->operands(), [this](Value *V) { return isLoopInvariant(V); });
}
bool Loop::makeLoopInvariant(Value *V, bool &Changed,
Instruction *InsertPt) const {
if (Instruction *I = dyn_cast<Instruction>(V))
return makeLoopInvariant(I, Changed, InsertPt);
return true; // All non-instructions are loop-invariant.
}
bool Loop::makeLoopInvariant(Instruction *I, bool &Changed,
Instruction *InsertPt) const {
// Test if the value is already loop-invariant.
if (isLoopInvariant(I))
return true;
if (!isSafeToSpeculativelyExecute(I))
return false;
if (I->mayReadFromMemory())
return false;
// EH block instructions are immobile.
if (I->isEHPad())
return false;
// Determine the insertion point, unless one was given.
if (!InsertPt) {
BasicBlock *Preheader = getLoopPreheader();
// Without a preheader, hoisting is not feasible.
if (!Preheader)
return false;
InsertPt = Preheader->getTerminator();
}
// Don't hoist instructions with loop-variant operands.
for (Value *Operand : I->operands())
if (!makeLoopInvariant(Operand, Changed, InsertPt))
return false;
// Hoist.
I->moveBefore(InsertPt);
// There is possibility of hoisting this instruction above some arbitrary
// condition. Any metadata defined on it can be control dependent on this
// condition. Conservatively strip it here so that we don't give any wrong
// information to the optimizer.
I->dropUnknownNonDebugMetadata();
Changed = true;
return true;
}
PHINode *Loop::getCanonicalInductionVariable() const {
BasicBlock *H = getHeader();
BasicBlock *Incoming = nullptr, *Backedge = nullptr;
pred_iterator PI = pred_begin(H);
assert(PI != pred_end(H) &&
"Loop must have at least one backedge!");
Backedge = *PI++;
if (PI == pred_end(H)) return nullptr; // dead loop
Incoming = *PI++;
if (PI != pred_end(H)) return nullptr; // multiple backedges?
if (contains(Incoming)) {
if (contains(Backedge))
return nullptr;
std::swap(Incoming, Backedge);
} else if (!contains(Backedge))
return nullptr;
// Loop over all of the PHI nodes, looking for a canonical indvar.
for (BasicBlock::iterator I = H->begin(); isa<PHINode>(I); ++I) {
PHINode *PN = cast<PHINode>(I);
if (ConstantInt *CI =
dyn_cast<ConstantInt>(PN->getIncomingValueForBlock(Incoming)))
if (CI->isNullValue())
if (Instruction *Inc =
dyn_cast<Instruction>(PN->getIncomingValueForBlock(Backedge)))
if (Inc->getOpcode() == Instruction::Add &&
Inc->getOperand(0) == PN)
if (ConstantInt *CI = dyn_cast<ConstantInt>(Inc->getOperand(1)))
if (CI->equalsInt(1))
return PN;
}
return nullptr;
}
bool Loop::isLCSSAForm(DominatorTree &DT) const {
for (BasicBlock *BB : this->blocks()) {
for (Instruction &I : *BB) {
// Tokens can't be used in PHI nodes and live-out tokens prevent loop
// optimizations, so for the purposes of considered LCSSA form, we
// can ignore them.
if (I.getType()->isTokenTy())
continue;
for (Use &U : I.uses()) {
Instruction *UI = cast<Instruction>(U.getUser());
BasicBlock *UserBB = UI->getParent();
if (PHINode *P = dyn_cast<PHINode>(UI))
UserBB = P->getIncomingBlock(U);
// Check the current block, as a fast-path, before checking whether
// the use is anywhere in the loop. Most values are used in the same
// block they are defined in. Also, blocks not reachable from the
// entry are special; uses in them don't need to go through PHIs.
if (UserBB != BB &&
!contains(UserBB) &&
DT.isReachableFromEntry(UserBB))
return false;
}
}
}
return true;
}
bool Loop::isRecursivelyLCSSAForm(DominatorTree &DT) const {
if (!isLCSSAForm(DT))
return false;
return std::all_of(begin(), end(), [&](const Loop *L) {
return L->isRecursivelyLCSSAForm(DT);
});
}
bool Loop::isLoopSimplifyForm() const {
// Normal-form loops have a preheader, a single backedge, and all of their
// exits have all their predecessors inside the loop.
return getLoopPreheader() && getLoopLatch() && hasDedicatedExits();
}
// Routines that reform the loop CFG and split edges often fail on indirectbr.
bool Loop::isSafeToClone() const {
// Return false if any loop blocks contain indirectbrs, or there are any calls
// to noduplicate functions.
for (BasicBlock *BB : this->blocks()) {
if (isa<IndirectBrInst>(BB->getTerminator()))
return false;
if (const InvokeInst *II = dyn_cast<InvokeInst>(BB->getTerminator())) {
if (II->cannotDuplicate())
return false;
// Return false if any loop blocks contain invokes to EH-pads other than
// landingpads; we don't know how to split those edges yet.
auto *FirstNonPHI = II->getUnwindDest()->getFirstNonPHI();
if (FirstNonPHI->isEHPad() && !isa<LandingPadInst>(FirstNonPHI))
return false;
}
for (Instruction &I : *BB) {
if (const CallInst *CI = dyn_cast<CallInst>(&I)) {
if (CI->cannotDuplicate())
return false;
}
if (I.getType()->isTokenTy() && I.isUsedOutsideOfBlock(BB))
return false;
}
}
return true;
}
MDNode *Loop::getLoopID() const {
MDNode *LoopID = nullptr;
if (isLoopSimplifyForm()) {
LoopID = getLoopLatch()->getTerminator()->getMetadata(LoopMDName);
} else {
// Go through each predecessor of the loop header and check the
// terminator for the metadata.
BasicBlock *H = getHeader();
for (BasicBlock *BB : this->blocks()) {
TerminatorInst *TI = BB->getTerminator();
MDNode *MD = nullptr;
// Check if this terminator branches to the loop header.
for (BasicBlock *Successor : TI->successors()) {
if (Successor == H) {
MD = TI->getMetadata(LoopMDName);
break;
}
}
if (!MD)
return nullptr;
if (!LoopID)
LoopID = MD;
else if (MD != LoopID)
return nullptr;
}
}
if (!LoopID || LoopID->getNumOperands() == 0 ||
LoopID->getOperand(0) != LoopID)
return nullptr;
return LoopID;
}
void Loop::setLoopID(MDNode *LoopID) const {
assert(LoopID && "Loop ID should not be null");
assert(LoopID->getNumOperands() > 0 && "Loop ID needs at least one operand");
assert(LoopID->getOperand(0) == LoopID && "Loop ID should refer to itself");
if (isLoopSimplifyForm()) {
getLoopLatch()->getTerminator()->setMetadata(LoopMDName, LoopID);
return;
}
BasicBlock *H = getHeader();
for (BasicBlock *BB : this->blocks()) {
TerminatorInst *TI = BB->getTerminator();
for (BasicBlock *Successor : TI->successors()) {
if (Successor == H)
TI->setMetadata(LoopMDName, LoopID);
}
}
}
bool Loop::isAnnotatedParallel() const {
MDNode *DesiredLoopIdMetadata = getLoopID();
if (!DesiredLoopIdMetadata)
return false;
// The loop branch contains the parallel loop metadata. In order to ensure
// that any parallel-loop-unaware optimization pass hasn't added loop-carried
// dependencies (thus converted the loop back to a sequential loop), check
// that all the memory instructions in the loop contain parallelism metadata
// that point to the same unique "loop id metadata" the loop branch does.
for (BasicBlock *BB : this->blocks()) {
for (Instruction &I : *BB) {
if (!I.mayReadOrWriteMemory())
continue;
// The memory instruction can refer to the loop identifier metadata
// directly or indirectly through another list metadata (in case of
// nested parallel loops). The loop identifier metadata refers to
// itself so we can check both cases with the same routine.
MDNode *LoopIdMD =
I.getMetadata(LLVMContext::MD_mem_parallel_loop_access);
if (!LoopIdMD)
return false;
bool LoopIdMDFound = false;
for (const MDOperand &MDOp : LoopIdMD->operands()) {
if (MDOp == DesiredLoopIdMetadata) {
LoopIdMDFound = true;
break;
}
}
if (!LoopIdMDFound)
return false;
}
}
return true;
}
bool Loop::hasDedicatedExits() const {
// Each predecessor of each exit block of a normal loop is contained
// within the loop.
SmallVector<BasicBlock *, 4> ExitBlocks;
getExitBlocks(ExitBlocks);
for (BasicBlock *BB : ExitBlocks)
for (BasicBlock *Predecessor : predecessors(BB))
if (!contains(Predecessor))
return false;
// All the requirements are met.
return true;
}
void
Loop::getUniqueExitBlocks(SmallVectorImpl<BasicBlock *> &ExitBlocks) const {
assert(hasDedicatedExits() &&
"getUniqueExitBlocks assumes the loop has canonical form exits!");
SmallVector<BasicBlock *, 32> SwitchExitBlocks;
for (BasicBlock *BB : this->blocks()) {
SwitchExitBlocks.clear();
for (BasicBlock *Successor : successors(BB)) {
// If block is inside the loop then it is not an exit block.
if (contains(Successor))
continue;
pred_iterator PI = pred_begin(Successor);
BasicBlock *FirstPred = *PI;
// If current basic block is this exit block's first predecessor
// then only insert exit block in to the output ExitBlocks vector.
// This ensures that same exit block is not inserted twice into
// ExitBlocks vector.
if (BB != FirstPred)
continue;
// If a terminator has more then two successors, for example SwitchInst,
// then it is possible that there are multiple edges from current block
// to one exit block.
if (std::distance(succ_begin(BB), succ_end(BB)) <= 2) {
ExitBlocks.push_back(Successor);
continue;
}
// In case of multiple edges from current block to exit block, collect
// only one edge in ExitBlocks. Use switchExitBlocks to keep track of
// duplicate edges.
if (std::find(SwitchExitBlocks.begin(), SwitchExitBlocks.end(), Successor)
== SwitchExitBlocks.end()) {
SwitchExitBlocks.push_back(Successor);
ExitBlocks.push_back(Successor);
}
}
}
}
BasicBlock *Loop::getUniqueExitBlock() const {
SmallVector<BasicBlock *, 8> UniqueExitBlocks;
getUniqueExitBlocks(UniqueExitBlocks);
if (UniqueExitBlocks.size() == 1)
return UniqueExitBlocks[0];
return nullptr;
}
#if !defined(NDEBUG) || defined(LLVM_ENABLE_DUMP)
LLVM_DUMP_METHOD void Loop::dump() const {
print(dbgs());
}
#endif
//===----------------------------------------------------------------------===//
// UnloopUpdater implementation
//
namespace {
/// Find the new parent loop for all blocks within the "unloop" whose last
/// backedges has just been removed.
class UnloopUpdater {
Loop *Unloop;
LoopInfo *LI;
LoopBlocksDFS DFS;
// Map unloop's immediate subloops to their nearest reachable parents. Nested
// loops within these subloops will not change parents. However, an immediate
// subloop's new parent will be the nearest loop reachable from either its own
// exits *or* any of its nested loop's exits.
DenseMap<Loop*, Loop*> SubloopParents;
// Flag the presence of an irreducible backedge whose destination is a block
// directly contained by the original unloop.
bool FoundIB;
public:
UnloopUpdater(Loop *UL, LoopInfo *LInfo) :
Unloop(UL), LI(LInfo), DFS(UL), FoundIB(false) {}
void updateBlockParents();
void removeBlocksFromAncestors();
void updateSubloopParents();
protected:
Loop *getNearestLoop(BasicBlock *BB, Loop *BBLoop);
};
} // end anonymous namespace
/// Update the parent loop for all blocks that are directly contained within the
/// original "unloop".
void UnloopUpdater::updateBlockParents() {
if (Unloop->getNumBlocks()) {
// Perform a post order CFG traversal of all blocks within this loop,
// propagating the nearest loop from sucessors to predecessors.
LoopBlocksTraversal Traversal(DFS, LI);
for (LoopBlocksTraversal::POTIterator POI = Traversal.begin(),
POE = Traversal.end(); POI != POE; ++POI) {
Loop *L = LI->getLoopFor(*POI);
Loop *NL = getNearestLoop(*POI, L);
if (NL != L) {
// For reducible loops, NL is now an ancestor of Unloop.
assert((NL != Unloop && (!NL || NL->contains(Unloop))) &&
"uninitialized successor");
LI->changeLoopFor(*POI, NL);
}
else {
// Or the current block is part of a subloop, in which case its parent
// is unchanged.
assert((FoundIB || Unloop->contains(L)) && "uninitialized successor");
}
}
}
// Each irreducible loop within the unloop induces a round of iteration using
// the DFS result cached by Traversal.
bool Changed = FoundIB;
for (unsigned NIters = 0; Changed; ++NIters) {
assert(NIters < Unloop->getNumBlocks() && "runaway iterative algorithm");
// Iterate over the postorder list of blocks, propagating the nearest loop
// from successors to predecessors as before.
Changed = false;
for (LoopBlocksDFS::POIterator POI = DFS.beginPostorder(),
POE = DFS.endPostorder(); POI != POE; ++POI) {
Loop *L = LI->getLoopFor(*POI);
Loop *NL = getNearestLoop(*POI, L);
if (NL != L) {
assert(NL != Unloop && (!NL || NL->contains(Unloop)) &&
"uninitialized successor");
LI->changeLoopFor(*POI, NL);
Changed = true;
}
}
}
}
/// Remove unloop's blocks from all ancestors below their new parents.
void UnloopUpdater::removeBlocksFromAncestors() {
// Remove all unloop's blocks (including those in nested subloops) from
// ancestors below the new parent loop.
for (Loop::block_iterator BI = Unloop->block_begin(),
BE = Unloop->block_end(); BI != BE; ++BI) {
Loop *OuterParent = LI->getLoopFor(*BI);
if (Unloop->contains(OuterParent)) {
while (OuterParent->getParentLoop() != Unloop)
OuterParent = OuterParent->getParentLoop();
OuterParent = SubloopParents[OuterParent];
}
// Remove blocks from former Ancestors except Unloop itself which will be
// deleted.
for (Loop *OldParent = Unloop->getParentLoop(); OldParent != OuterParent;
OldParent = OldParent->getParentLoop()) {
assert(OldParent && "new loop is not an ancestor of the original");
OldParent->removeBlockFromLoop(*BI);
}
}
}
/// Update the parent loop for all subloops directly nested within unloop.
void UnloopUpdater::updateSubloopParents() {
while (!Unloop->empty()) {
Loop *Subloop = *std::prev(Unloop->end());
Unloop->removeChildLoop(std::prev(Unloop->end()));
assert(SubloopParents.count(Subloop) && "DFS failed to visit subloop");
if (Loop *Parent = SubloopParents[Subloop])
Parent->addChildLoop(Subloop);
else
LI->addTopLevelLoop(Subloop);
}
}
/// Return the nearest parent loop among this block's successors. If a successor
/// is a subloop header, consider its parent to be the nearest parent of the
/// subloop's exits.
///
/// For subloop blocks, simply update SubloopParents and return NULL.
Loop *UnloopUpdater::getNearestLoop(BasicBlock *BB, Loop *BBLoop) {
// Initially for blocks directly contained by Unloop, NearLoop == Unloop and
// is considered uninitialized.
Loop *NearLoop = BBLoop;
Loop *Subloop = nullptr;
if (NearLoop != Unloop && Unloop->contains(NearLoop)) {
Subloop = NearLoop;
// Find the subloop ancestor that is directly contained within Unloop.
while (Subloop->getParentLoop() != Unloop) {
Subloop = Subloop->getParentLoop();
assert(Subloop && "subloop is not an ancestor of the original loop");
}
// Get the current nearest parent of the Subloop exits, initially Unloop.
NearLoop =
SubloopParents.insert(std::make_pair(Subloop, Unloop)).first->second;
}
succ_iterator I = succ_begin(BB), E = succ_end(BB);
if (I == E) {
assert(!Subloop && "subloop blocks must have a successor");
NearLoop = nullptr; // unloop blocks may now exit the function.
}
for (; I != E; ++I) {
if (*I == BB)
continue; // self loops are uninteresting
Loop *L = LI->getLoopFor(*I);
if (L == Unloop) {
// This successor has not been processed. This path must lead to an
// irreducible backedge.
assert((FoundIB || !DFS.hasPostorder(*I)) && "should have seen IB");
FoundIB = true;
}
if (L != Unloop && Unloop->contains(L)) {
// Successor is in a subloop.
if (Subloop)
continue; // Branching within subloops. Ignore it.
// BB branches from the original into a subloop header.
assert(L->getParentLoop() == Unloop && "cannot skip into nested loops");
// Get the current nearest parent of the Subloop's exits.
L = SubloopParents[L];
// L could be Unloop if the only exit was an irreducible backedge.
}
if (L == Unloop) {
continue;
}
// Handle critical edges from Unloop into a sibling loop.
if (L && !L->contains(Unloop)) {
L = L->getParentLoop();
}
// Remember the nearest parent loop among successors or subloop exits.
if (NearLoop == Unloop || !NearLoop || NearLoop->contains(L))
NearLoop = L;
}
if (Subloop) {
SubloopParents[Subloop] = NearLoop;
return BBLoop;
}
return NearLoop;
}
LoopInfo::LoopInfo(const DominatorTreeBase<BasicBlock> &DomTree) {
analyze(DomTree);
}
void LoopInfo::markAsRemoved(Loop *Unloop) {
assert(!Unloop->isInvalid() && "Loop has already been removed");
Unloop->invalidate();
RemovedLoops.push_back(Unloop);
// First handle the special case of no parent loop to simplify the algorithm.
if (!Unloop->getParentLoop()) {
// Since BBLoop had no parent, Unloop blocks are no longer in a loop.
for (Loop::block_iterator I = Unloop->block_begin(),
E = Unloop->block_end();
I != E; ++I) {
// Don't reparent blocks in subloops.
if (getLoopFor(*I) != Unloop)
continue;
// Blocks no longer have a parent but are still referenced by Unloop until
// the Unloop object is deleted.
changeLoopFor(*I, nullptr);
}
// Remove the loop from the top-level LoopInfo object.
for (iterator I = begin();; ++I) {
assert(I != end() && "Couldn't find loop");
if (*I == Unloop) {
removeLoop(I);
break;
}
}
// Move all of the subloops to the top-level.
while (!Unloop->empty())
addTopLevelLoop(Unloop->removeChildLoop(std::prev(Unloop->end())));
return;
}
// Update the parent loop for all blocks within the loop. Blocks within
// subloops will not change parents.
UnloopUpdater Updater(Unloop, this);
Updater.updateBlockParents();
// Remove blocks from former ancestor loops.
Updater.removeBlocksFromAncestors();
// Add direct subloops as children in their new parent loop.
Updater.updateSubloopParents();
// Remove unloop from its parent loop.
Loop *ParentLoop = Unloop->getParentLoop();
for (Loop::iterator I = ParentLoop->begin();; ++I) {
assert(I != ParentLoop->end() && "Couldn't find loop");
if (*I == Unloop) {
ParentLoop->removeChildLoop(I);
break;
}
}
}
char LoopAnalysis::PassID;
LoopInfo LoopAnalysis::run(Function &F, AnalysisManager<Function> *AM) {
// FIXME: Currently we create a LoopInfo from scratch for every function.
// This may prove to be too wasteful due to deallocating and re-allocating
// memory each time for the underlying map and vector datastructures. At some
// point it may prove worthwhile to use a freelist and recycle LoopInfo
// objects. I don't want to add that kind of complexity until the scope of
// the problem is better understood.
LoopInfo LI;
LI.analyze(AM->getResult<DominatorTreeAnalysis>(F));
return LI;
}
PreservedAnalyses LoopPrinterPass::run(Function &F,
AnalysisManager<Function> *AM) {
AM->getResult<LoopAnalysis>(F).print(OS);
return PreservedAnalyses::all();
}
PrintLoopPass::PrintLoopPass() : OS(dbgs()) {}
PrintLoopPass::PrintLoopPass(raw_ostream &OS, const std::string &Banner)
: OS(OS), Banner(Banner) {}
PreservedAnalyses PrintLoopPass::run(Loop &L) {
OS << Banner;
for (auto *Block : L.blocks())
if (Block)
Block->print(OS);
else
OS << "Printing <null> block";
return PreservedAnalyses::all();
}
//===----------------------------------------------------------------------===//
// LoopInfo implementation
//
char LoopInfoWrapperPass::ID = 0;
INITIALIZE_PASS_BEGIN(LoopInfoWrapperPass, "loops", "Natural Loop Information",
true, true)
INITIALIZE_PASS_DEPENDENCY(DominatorTreeWrapperPass)
INITIALIZE_PASS_END(LoopInfoWrapperPass, "loops", "Natural Loop Information",
true, true)
bool LoopInfoWrapperPass::runOnFunction(Function &) {
releaseMemory();
LI.analyze(getAnalysis<DominatorTreeWrapperPass>().getDomTree());
return false;
}
void LoopInfoWrapperPass::verifyAnalysis() const {
// LoopInfoWrapperPass is a FunctionPass, but verifying every loop in the
// function each time verifyAnalysis is called is very expensive. The
// -verify-loop-info option can enable this. In order to perform some
// checking by default, LoopPass has been taught to call verifyLoop manually
// during loop pass sequences.
if (VerifyLoopInfo)
LI.verify();
}
void LoopInfoWrapperPass::getAnalysisUsage(AnalysisUsage &AU) const {
AU.setPreservesAll();
AU.addRequired<DominatorTreeWrapperPass>();
}
void LoopInfoWrapperPass::print(raw_ostream &OS, const Module *) const {
LI.print(OS);
}
//===----------------------------------------------------------------------===//
// LoopBlocksDFS implementation
//
/// Traverse the loop blocks and store the DFS result.
/// Useful for clients that just want the final DFS result and don't need to
/// visit blocks during the initial traversal.
void LoopBlocksDFS::perform(LoopInfo *LI) {
LoopBlocksTraversal Traversal(*this, LI);
for (LoopBlocksTraversal::POTIterator POI = Traversal.begin(),
POE = Traversal.end(); POI != POE; ++POI) ;
}