lib/Transforms/Scalar/PRE.cpp

//===- PRE.cpp - Partial Redundancy Elimination ---------------------------===//
//
// This file implements the well known Partial Redundancy Elimination
// optimization, using an SSA formulation based on e-paths.  See this paper for
// more information:
//
//  E-path_PRE: partial redundancy elimination made easy
//  By: Dhananjay M. Dhamdhere   In: ACM SIGPLAN Notices. Vol 37, #8, 2002
//    http://doi.acm.org/10.1145/596992.597004
//
// This file actually implements a sparse version of the algorithm, using SSA
// and CFG properties instead of bit-vectors.
//
//===----------------------------------------------------------------------===//

#include "llvm/Pass.h"
#include "llvm/Function.h"
#include "llvm/Type.h"
#include "llvm/iPHINode.h"
#include "llvm/iMemory.h"
#include "llvm/Support/CFG.h"
#include "llvm/Analysis/Dominators.h"
#include "llvm/Analysis/PostDominators.h"
#include "llvm/Analysis/ValueNumbering.h"
#include "llvm/Transforms/Scalar.h"
#include "Support/Debug.h"
#include "Support/DepthFirstIterator.h"
#include "Support/PostOrderIterator.h"
#include "Support/Statistic.h"
#include "Support/hash_set"

namespace {
  Statistic<> NumExprsEliminated("pre", "Number of expressions constantified");
  Statistic<> NumRedundant      ("pre", "Number of redundant exprs eliminated");
  Statistic<> NumInserted       ("pre", "Number of expressions inserted");

  struct PRE : public FunctionPass {
    virtual void getAnalysisUsage(AnalysisUsage &AU) const {
      AU.addRequiredID(BreakCriticalEdgesID);  // No critical edges for now!
      AU.addRequired<PostDominatorTree>();
      AU.addRequired<PostDominanceFrontier>();
      AU.addRequired<DominatorSet>();
      AU.addRequired<DominatorTree>();
      AU.addRequired<DominanceFrontier>();
      AU.addRequired<ValueNumbering>();
    }
    virtual bool runOnFunction(Function &F);

  private:
    // Block information - Map basic blocks in a function back and forth to
    // unsigned integers.
    std::vector<BasicBlock*> BlockMapping;
    hash_map<BasicBlock*, unsigned> BlockNumbering;

    // ProcessedExpressions - Keep track of which expressions have already been
    // processed.
    hash_set<Instruction*> ProcessedExpressions;

    // Provide access to the various analyses used...
    DominatorSet      *DS;
    DominatorTree     *DT; PostDominatorTree *PDT;
    DominanceFrontier *DF; PostDominanceFrontier *PDF;
    ValueNumbering    *VN;

    // AvailableBlocks - Contain a mapping of blocks with available expression
    // values to the expression value itself.  This can be used as an efficient
    // way to find out if the expression is available in the block, and if so,
    // which version to use.  This map is only used while processing a single
    // expression.
    //
    typedef hash_map<BasicBlock*, Instruction*> AvailableBlocksTy;
    AvailableBlocksTy AvailableBlocks;

    bool ProcessBlock(BasicBlock *BB);
    
    // Anticipatibility calculation...
    void MarkPostDominatingBlocksAnticipatible(PostDominatorTree::Node *N,
                                               std::vector<char> &AntBlocks,
                                               Instruction *Occurance);
    void CalculateAnticipatiblityForOccurance(unsigned BlockNo,
                                              std::vector<char> &AntBlocks,
                                              Instruction *Occurance);
    void CalculateAnticipatibleBlocks(const std::map<unsigned, Instruction*> &D,
                                      std::vector<char> &AnticipatibleBlocks);

    // PRE for an expression
    void MarkOccuranceAvailableInAllDominatedBlocks(Instruction *Occurance,
                                                    BasicBlock *StartBlock);
    void ReplaceDominatedAvailableOccurancesWith(Instruction *NewOcc,
                                                 DominatorTree::Node *N);
    bool ProcessExpression(Instruction *I);
  };

  RegisterOpt<PRE> Z("pre", "Partial Redundancy Elimination");
}


bool PRE::runOnFunction(Function &F) {
  VN  = &getAnalysis<ValueNumbering>();
  DS  = &getAnalysis<DominatorSet>();
  DT  = &getAnalysis<DominatorTree>();
  DF  = &getAnalysis<DominanceFrontier>();
  PDT = &getAnalysis<PostDominatorTree>();
  PDF = &getAnalysis<PostDominanceFrontier>();

  DEBUG(std::cerr << "\n*** Running PRE on func '" << F.getName() << "'...\n");

  // Number the basic blocks based on a reverse post-order traversal of the CFG
  // so that all predecessors of a block (ignoring back edges) are visited
  // before a block is visited.
  //
  BlockMapping.reserve(F.size());
  {
    ReversePostOrderTraversal<Function*> RPOT(&F);
    DEBUG(std::cerr << "Block order: ");
    for (ReversePostOrderTraversal<Function*>::rpo_iterator I = RPOT.begin(),
           E = RPOT.end(); I != E; ++I) {
      // Keep track of mapping...
      BasicBlock *BB = *I;
      BlockNumbering.insert(std::make_pair(BB, BlockMapping.size()));
      BlockMapping.push_back(BB);
      DEBUG(std::cerr << BB->getName() << " ");
    }
    DEBUG(std::cerr << "\n");
  }

  // Traverse the current function depth-first in dominator-tree order.  This
  // ensures that we see all definitions before their uses (except for PHI
  // nodes), allowing us to hoist dependent expressions correctly.
  bool Changed = false;
  for (unsigned i = 0, e = BlockMapping.size(); i != e; ++i)
    Changed |= ProcessBlock(BlockMapping[i]);

  // Free memory
  BlockMapping.clear();
  BlockNumbering.clear();
  ProcessedExpressions.clear();
  return Changed;
}


// ProcessBlock - Process any expressions first seen in this block...
//
bool PRE::ProcessBlock(BasicBlock *BB) {
  bool Changed = false;

  // PRE expressions first defined in this block...
  Instruction *PrevInst = 0;
  for (BasicBlock::iterator I = BB->begin(); I != BB->end(); )
    if (ProcessExpression(I)) {
      // The current instruction may have been deleted, make sure to back up to
      // PrevInst instead.
      if (PrevInst)
        I = PrevInst;
      else
        I = BB->begin();
      Changed = true;
    } else {
      PrevInst = I++;
    }

  return Changed;
}

void PRE::MarkPostDominatingBlocksAnticipatible(PostDominatorTree::Node *N,
                                                std::vector<char> &AntBlocks,
                                                Instruction *Occurance) {
  unsigned BlockNo = BlockNumbering[N->getNode()];

  if (AntBlocks[BlockNo]) return;  // Already known to be anticipatible??

  // Check to see if any of the operands are defined in this block, if so, the
  // entry of this block does not anticipate the expression.  This computes
  // "transparency".
  for (unsigned i = 0, e = Occurance->getNumOperands(); i != e; ++i)
    if (Instruction *I = dyn_cast<Instruction>(Occurance->getOperand(i)))
      if (I->getParent() == N->getNode())  // Operand is defined in this block!
        return;

  if (isa<LoadInst>(Occurance))
    return;        // FIXME: compute transparency for load instructions using AA

  // Insert block into AntBlocks list...
  AntBlocks[BlockNo] = true;

  for (PostDominatorTree::Node::iterator I = N->begin(), E = N->end(); I != E;
       ++I)
    MarkPostDominatingBlocksAnticipatible(*I, AntBlocks, Occurance);
}

void PRE::CalculateAnticipatiblityForOccurance(unsigned BlockNo,
                                               std::vector<char> &AntBlocks,
                                               Instruction *Occurance) {
  if (AntBlocks[BlockNo]) return;  // Block already anticipatible!

  BasicBlock *BB = BlockMapping[BlockNo];

  // For each occurance, mark all post-dominated blocks as anticipatible...
  MarkPostDominatingBlocksAnticipatible(PDT->getNode(BB), AntBlocks,
                                        Occurance);

  // Next, mark any blocks in the post-dominance frontier as anticipatible iff
  // all successors are anticipatible.
  //
  PostDominanceFrontier::iterator PDFI = PDF->find(BB);
  if (PDFI != DF->end())
    for (std::set<BasicBlock*>::iterator DI = PDFI->second.begin();
         DI != PDFI->second.end(); ++DI) {
      BasicBlock *PDFBlock = *DI;
      bool AllSuccessorsAnticipatible = true;
      for (succ_iterator SI = succ_begin(PDFBlock), SE = succ_end(PDFBlock);
           SI != SE; ++SI)
        if (!AntBlocks[BlockNumbering[*SI]]) {
          AllSuccessorsAnticipatible = false;
          break;
        }

      if (AllSuccessorsAnticipatible)
        CalculateAnticipatiblityForOccurance(BlockNumbering[PDFBlock],
                                             AntBlocks, Occurance);
    }
}


void PRE::CalculateAnticipatibleBlocks(const std::map<unsigned,
                                                      Instruction*> &Defs,
                                       std::vector<char> &AntBlocks) {
  // Initialize to zeros...
  AntBlocks.resize(BlockMapping.size());

  // Loop over all of the expressions...
  for (std::map<unsigned, Instruction*>::const_iterator I = Defs.begin(),
         E = Defs.end(); I != E; ++I)
    CalculateAnticipatiblityForOccurance(I->first, AntBlocks, I->second);
}

/// MarkOccuranceAvailableInAllDominatedBlocks - Add entries to AvailableBlocks
/// for all nodes dominated by the occurance to indicate that it is now the
/// available occurance to use in any of these blocks.
///
void PRE::MarkOccuranceAvailableInAllDominatedBlocks(Instruction *Occurance,
                                                     BasicBlock *BB) {
  // FIXME: There are much more efficient ways to get the blocks dominated
  // by a block.  Use them.
  //
  DominatorTree::Node *N = DT->getNode(Occurance->getParent());
  for (df_iterator<DominatorTree::Node*> DI = df_begin(N), E = df_end(N);
       DI != E; ++DI)
    AvailableBlocks[(*DI)->getNode()] = Occurance;
}

/// ReplaceDominatedAvailableOccurancesWith - This loops over the region
/// dominated by N, replacing any available expressions with NewOcc.
void PRE::ReplaceDominatedAvailableOccurancesWith(Instruction *NewOcc,
                                                  DominatorTree::Node *N) {
  BasicBlock *BB = N->getNode();
  Instruction *&ExistingAvailableVal = AvailableBlocks[BB];

  // If there isn't a definition already active in this node, make this the new
  // active definition...
  if (ExistingAvailableVal == 0) {
    ExistingAvailableVal = NewOcc;
    
    for (DominatorTree::Node::iterator I = N->begin(), E = N->end(); I != E;++I)
      ReplaceDominatedAvailableOccurancesWith(NewOcc, *I);
  } else {
    // If there is already an active definition in this block, replace it with
    // NewOcc, and force it into all dominated blocks.
    DEBUG(std::cerr << "  Replacing dominated occ %"
          << ExistingAvailableVal->getName() << " with %" << NewOcc->getName()
          << "\n");
    assert(ExistingAvailableVal != NewOcc && "NewOcc already inserted??");
    ExistingAvailableVal->replaceAllUsesWith(NewOcc);
    ++NumRedundant;

    assert(ExistingAvailableVal->getParent() == BB &&
           "OldOcc not defined in current block?");
    BB->getInstList().erase(ExistingAvailableVal);

    // Mark NewOCC as the Available expression in all blocks dominated by BB
    for (df_iterator<DominatorTree::Node*> DI = df_begin(N), E = df_end(N);
         DI != E; ++DI)
      AvailableBlocks[(*DI)->getNode()] = NewOcc;
  }  
}


/// ProcessExpression - Given an expression (instruction) process the
/// instruction to remove any partial redundancies induced by equivalent
/// computations.  Note that we only need to PRE each expression once, so we
/// keep track of whether an expression has been PRE'd already, and don't PRE an
/// expression again.  Expressions may be seen multiple times because process
/// the entire equivalence class at once, which may leave expressions later in
/// the control path.
///
bool PRE::ProcessExpression(Instruction *Expr) {
  if (Expr->mayWriteToMemory() || Expr->getType() == Type::VoidTy ||
      isa<PHINode>(Expr))
    return false;         // Cannot move expression
  if (ProcessedExpressions.count(Expr)) return false; // Already processed.

  // Ok, this is the first time we have seen the expression.  Build a set of
  // equivalent expressions using SSA def/use information.  We consider
  // expressions to be equivalent if they are the same opcode and have
  // equivalent operands.  As a special case for SSA, values produced by PHI
  // nodes are considered to be equivalent to all of their operands.
  //
  std::vector<Value*> Values;
  VN->getEqualNumberNodes(Expr, Values);

#if 0
  // FIXME: This should handle PHI nodes correctly.  To do this, we need to
  // consider expressions of the following form equivalent to this set of
  // expressions:
  //
  // If an operand is a PHI node, add any occurances of the expression with the
  // PHI operand replaced with the PHI node operands.  This is only valid if the
  // PHI operand occurances exist in blocks post-dominated by the incoming edge
  // of the PHI node.
#endif

  // We have to be careful to handle expression definitions which dominated by
  // other expressions.  These can be directly eliminated in favor of their
  // dominating value.  Keep track of which blocks contain definitions (the key)
  // and if a block contains a definition, which instruction it is.
  //
  std::map<unsigned, Instruction*> Definitions;
  Definitions.insert(std::make_pair(BlockNumbering[Expr->getParent()], Expr));

  bool Changed = false;

  // Look at all of the equal values.  If any of the values is not an
  // instruction, replace all other expressions immediately with it (it must be
  // an argument or a constant or something). Otherwise, convert the list of
  // values into a list of expression (instruction) definitions ordering
  // according to their dominator tree ordering.
  //
  Value *NonInstValue = 0;
  for (unsigned i = 0, e = Values.size(); i != e; ++i)
    if (Instruction *I = dyn_cast<Instruction>(Values[i])) {
      Instruction *&BlockInst = Definitions[BlockNumbering[I->getParent()]];
      if (BlockInst && BlockInst != I) {    // Eliminate direct redundancy
        if (DS->dominates(I, BlockInst)) {  // I dom BlockInst
          BlockInst->replaceAllUsesWith(I);
          BlockInst->getParent()->getInstList().erase(BlockInst);
        } else {                            // BlockInst dom I
          I->replaceAllUsesWith(BlockInst);
          I->getParent()->getInstList().erase(I);
          I = BlockInst;
        }
        ++NumRedundant;
      }
      BlockInst = I;
    } else {
      NonInstValue = Values[i];
    }

  std::vector<Value*>().swap(Values);  // Done with the values list

  if (NonInstValue) {
    // This is the good, though unlikely, case where we find out that this
    // expression is equal to a constant or argument directly.  We can replace
    // this and all of the other equivalent instructions with the value
    // directly.
    //
    for (std::map<unsigned, Instruction*>::iterator I = Definitions.begin(),
           E = Definitions.end(); I != E; ++I) {
      Instruction *Inst = I->second;
      // Replace the value with the specified non-instruction value.
      Inst->replaceAllUsesWith(NonInstValue);       // Fixup any uses
      Inst->getParent()->getInstList().erase(Inst); // Erase the instruction
    }
    NumExprsEliminated += Definitions.size();
    return true;   // Program modified!
  }

  // There are no expressions equal to this one.  Exit early.
  assert(!Definitions.empty() && "no equal expressions??");
#if 0
  if (Definitions.size() == 1) {
    ProcessedExpressions.insert(Definitions.begin()->second);
    return Changed;
  }
#endif
  DEBUG(std::cerr << "\n====--- Expression: " << Expr);
  const Type *ExprType = Expr->getType();

  // AnticipatibleBlocks - Blocks where the current expression is anticipatible.
  // This is logically std::vector<bool> but using 'char' for performance.
  std::vector<char> AnticipatibleBlocks;

  // Calculate all of the blocks which the current expression is anticipatible.
  CalculateAnticipatibleBlocks(Definitions, AnticipatibleBlocks);

  // Print out anticipatible blocks...
  DEBUG(std::cerr << "AntBlocks: ";
        for (unsigned i = 0, e = AnticipatibleBlocks.size(); i != e; ++i)
          if (AnticipatibleBlocks[i])
            std::cerr << BlockMapping[i]->getName() <<" ";
        std::cerr << "\n";);
  


  // AvailabilityFrontier - Calculates the availability frontier for the current
  // expression.  The availability frontier contains the blocks on the dominance
  // frontier of the current available expressions, iff they anticipate a
  // definition of the expression.
  hash_set<unsigned> AvailabilityFrontier;

  Instruction *NonPHIOccurance = 0;

  while (!Definitions.empty() || !AvailabilityFrontier.empty()) {
    if (!Definitions.empty() &&
        (AvailabilityFrontier.empty() ||
         Definitions.begin()->first < *AvailabilityFrontier.begin())) {
      Instruction *Occurance = Definitions.begin()->second;
      BasicBlock *BB = Occurance->getParent();
      Definitions.erase(Definitions.begin());

      DEBUG(std::cerr << "PROCESSING Occurance: " << Occurance);

      // Check to see if there is already an incoming value for this block...
      AvailableBlocksTy::iterator LBI = AvailableBlocks.find(BB);
      if (LBI != AvailableBlocks.end()) {
        // Yes, there is a dominating definition for this block.  Replace this
        // occurance with the incoming value.
        if (LBI->second != Occurance) {
          DEBUG(std::cerr << "  replacing with: " << LBI->second);
          Occurance->replaceAllUsesWith(LBI->second);
          BB->getInstList().erase(Occurance);   // Delete instruction
          ++NumRedundant;
        }

      } else {
        ProcessedExpressions.insert(Occurance);
        if (!isa<PHINode>(Occurance))
          NonPHIOccurance = Occurance;  // Keep an occurance of this expr

        // Okay, there is no incoming value for this block, so this expression
        // is a new definition that is good for this block and all blocks
        // dominated by it.  Add this information to the AvailableBlocks map.
        //
        MarkOccuranceAvailableInAllDominatedBlocks(Occurance, BB);

        // Update the dominance frontier for the definitions so far... if a node
        // in the dominator frontier now has all of its predecessors available,
        // and the block is in an anticipatible region, we can insert a PHI node
        // in that block.
        DominanceFrontier::iterator DFI = DF->find(BB);
        if (DFI != DF->end()) {
          for (std::set<BasicBlock*>::iterator DI = DFI->second.begin();
               DI != DFI->second.end(); ++DI) {
            BasicBlock *DFBlock = *DI;
            unsigned DFBlockID = BlockNumbering[DFBlock];
            if (AnticipatibleBlocks[DFBlockID]) {
              // Check to see if any of the predecessors of this block on the
              // frontier are not available...
              bool AnyNotAvailable = false;
              for (pred_iterator PI = pred_begin(DFBlock),
                     PE = pred_end(DFBlock); PI != PE; ++PI)
                if (!AvailableBlocks.count(*PI)) {
                  AnyNotAvailable = true;
                  break;
                }
            
              // If any predecessor blocks are not available, add the node to
              // the current expression dominance frontier.
              if (AnyNotAvailable) {
                AvailabilityFrontier.insert(DFBlockID);
              } else {
                // This block is no longer in the availability frontier, it IS
                // available.
                AvailabilityFrontier.erase(DFBlockID);

                // If all of the predecessor blocks are available (and the block
                // anticipates a definition along the path to the exit), we need
                // to insert a new PHI node in this block.  This block serves as
                // a new definition for the expression, extending the available
                // region.
                //
                PHINode *PN = new PHINode(ExprType, Expr->getName()+".pre",
                                          DFBlock->begin());
                ProcessedExpressions.insert(PN);

                DEBUG(std::cerr << "  INSERTING PHI on frontier: " << PN);

                // Add the incoming blocks for the PHI node
                for (pred_iterator PI = pred_begin(DFBlock),
                       PE = pred_end(DFBlock); PI != PE; ++PI)
                  if (*PI != DFBlock)
                    PN->addIncoming(AvailableBlocks[*PI], *PI);
                  else                          // edge from the current block
                    PN->addIncoming(PN, DFBlock);

                Instruction *&BlockOcc = Definitions[DFBlockID];
                if (BlockOcc) {
                  DEBUG(std::cerr <<"    PHI superceeds occurance: "<<BlockOcc);
                  BlockOcc->replaceAllUsesWith(PN);
                  BlockOcc->getParent()->getInstList().erase(BlockOcc);
                  ++NumRedundant;
                }
                BlockOcc = PN;
              }
            }
          }
        }
      }

    } else {
      // Otherwise we must be looking at a node in the availability frontier!
      unsigned AFBlockID = *AvailabilityFrontier.begin();
      AvailabilityFrontier.erase(AvailabilityFrontier.begin());
      BasicBlock *AFBlock = BlockMapping[AFBlockID];

      // We eliminate the partial redundancy on this frontier by inserting a PHI
      // node into this block, merging any incoming available versions into the
      // PHI and inserting a new computation into predecessors without an
      // incoming value.  Note that we would have to insert the expression on
      // the edge if the predecessor didn't anticipate the expression and we
      // didn't break critical edges.
      //
      PHINode *PN = new PHINode(ExprType, Expr->getName()+".PRE",
                                AFBlock->begin());
      DEBUG(std::cerr << "INSERTING PHI for PR: " << PN);

      // If there is a pending occurance in this block, make sure to replace it
      // with the PHI node...
      std::map<unsigned, Instruction*>::iterator EDFI =
        Definitions.find(AFBlockID);
      if (EDFI != Definitions.end()) {
        // There is already an occurance in this block.  Replace it with PN and
        // remove it.
        Instruction *OldOcc = EDFI->second;
        DEBUG(std::cerr << "  Replaces occurance: " << OldOcc);
        OldOcc->replaceAllUsesWith(PN);
        AFBlock->getInstList().erase(OldOcc);
        Definitions.erase(EDFI);
        ++NumRedundant;
      }

      for (pred_iterator PI = pred_begin(AFBlock), PE = pred_end(AFBlock);
           PI != PE; ++PI) {
        BasicBlock *Pred = *PI;
        AvailableBlocksTy::iterator LBI = AvailableBlocks.find(Pred);
        if (LBI != AvailableBlocks.end()) {    // If there is a available value
          PN->addIncoming(LBI->second, Pred);  // for this pred, use it.
        } else {                         // No available value yet...
          unsigned PredID = BlockNumbering[Pred];

          // Is the predecessor the same block that we inserted the PHI into?
          // (self loop)
          if (Pred == AFBlock) {
            // Yes, reuse the incoming value here...
            PN->addIncoming(PN, Pred);
          } else {
            // No, we must insert a new computation into this block and add it
            // to the definitions list...
            assert(NonPHIOccurance && "No non-phi occurances seen so far???");
            Instruction *New = NonPHIOccurance->clone();
            New->setName(NonPHIOccurance->getName() + ".PRE-inserted");
            ProcessedExpressions.insert(New);

            DEBUG(std::cerr << "  INSERTING OCCURANCE: " << New);

            // Insert it into the bottom of the predecessor, right before the
            // terminator instruction...
            Pred->getInstList().insert(Pred->getTerminator(), New);

            // Make this block be the available definition for any blocks it
            // dominates.  The ONLY case that this can affect more than just the
            // block itself is when we are moving a computation to a loop
            // header.  In all other cases, because we don't have critical
            // edges, the node is guaranteed to only dominate itself.
            //
            ReplaceDominatedAvailableOccurancesWith(New, DT->getNode(Pred));

            // Add it as an incoming value on this edge to the PHI node
            PN->addIncoming(New, Pred);
            NonPHIOccurance = New;
            NumInserted++;
          }
        }
      }

      // Find out if there is already an available value in this block.  If so,
      // we need to replace the available value with the PHI node.  This can
      // only happen when we just inserted a PHI node on a backedge.
      //
      AvailableBlocksTy::iterator LBBlockAvailableValIt =
        AvailableBlocks.find(AFBlock);
      if (LBBlockAvailableValIt != AvailableBlocks.end()) {
        if (LBBlockAvailableValIt->second->getParent() == AFBlock) {
          Instruction *OldVal = LBBlockAvailableValIt->second;
          OldVal->replaceAllUsesWith(PN);        // Use the new PHI node now
          ++NumRedundant;
          DEBUG(std::cerr << "  PHI replaces available value: %"
                << OldVal->getName() << "\n");
          
          // Loop over all of the blocks dominated by this PHI node, and change
          // the AvailableBlocks entries to be the PHI node instead of the old
          // instruction.
          MarkOccuranceAvailableInAllDominatedBlocks(PN, AFBlock);
          
          AFBlock->getInstList().erase(OldVal);  // Delete old instruction!

          // The resultant PHI node is a new definition of the value!
          Definitions.insert(std::make_pair(AFBlockID, PN));
        } else {
          // If the value is not defined in this block, that means that an
          // inserted occurance in a predecessor is now the live value for the
          // region (occurs when hoisting loop invariants, f.e.).  In this case,
          // the PHI node should actually just be removed.
          assert(PN->use_empty() && "No uses should exist for dead PHI node!");
          PN->getParent()->getInstList().erase(PN);            
        }
      } else {
        // The resultant PHI node is a new definition of the value!
        Definitions.insert(std::make_pair(AFBlockID, PN));
      }
    }
  }

  AvailableBlocks.clear();

  return Changed;
}