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namespace Intrepid {

template<class Scalar, class UserVector>
bool AdaptiveSparseGrid<Scalar,UserVector>::isAdmissible(
               std::vector<int> index, 
               int direction, 
               std::set<std::vector<int> > inOldIndex,
               AdaptiveSparseGridInterface<Scalar,UserVector> & problem_data) {
  /*
    Check if inOldIndex remains admissible if index is added, i.e.
       index-ek in inOldIndex for all k=1,...,dim.
  */
  int dimension = problem_data.getDimension();
  for (int i=0; i<dimension; i++) {
    if (index[i]>1 && i!=direction) {
      index[i]--;
      if (!inOldIndex.count(index)) {
        return false;
      }
      index[i]++;
    }
  }
  return true;
}

template<class Scalar, class UserVector>
void AdaptiveSparseGrid<Scalar,UserVector>::build_diffRule(
               CubatureTensorSorted<Scalar> & outRule, 
               std::vector<int> index,
               AdaptiveSparseGridInterface<Scalar,UserVector> & problem_data) {
               
  int numPoints     = 0;
  int dimension     = problem_data.getDimension();
  bool isNormalized = problem_data.isNormalized();
  std::vector<EIntrepidBurkardt> rule1D; problem_data.getRule(rule1D);
  std::vector<EIntrepidGrowth> growth1D; problem_data.getGrowth(growth1D);

  for (int i=0; i<dimension; i++) {
    // Compute 1D rules
    numPoints = growthRule1D(index[i],growth1D[i],rule1D[i]);
    CubatureLineSorted<Scalar> diffRule1(rule1D[i],numPoints,isNormalized);
 
    if (numPoints!=1) { // Compute differential rule
      numPoints = growthRule1D(index[i]-1,growth1D[i],rule1D[i]);
      CubatureLineSorted<Scalar> rule1(rule1D[i],numPoints,isNormalized);
      diffRule1.update(-1.0,rule1,1.0);
    }
    // Build Tensor Rule
    outRule = kron_prod<Scalar>(outRule,diffRule1); 
  }
}

template<class Scalar, class UserVector>
void AdaptiveSparseGrid<Scalar,UserVector>::build_diffRule(
               CubatureTensorSorted<Scalar> & outRule, 
               std::vector<int> index, int dimension,
               std::vector<EIntrepidBurkardt> rule1D, 
               std::vector<EIntrepidGrowth> growth1D,
               bool isNormalized) {
               
  int numPoints = 0;
  for (int i=0; i<dimension; i++) {
    // Compute 1D rules
    numPoints = growthRule1D(index[i],growth1D[i],rule1D[i]);
    CubatureLineSorted<Scalar> diffRule1(rule1D[i],numPoints,isNormalized);
 
    if (numPoints!=1) { // Differential Rule
      numPoints = growthRule1D(index[i]-1,growth1D[i],rule1D[i]);
      CubatureLineSorted<Scalar> rule1(rule1D[i],numPoints,isNormalized);
      diffRule1.update(-1.0,rule1,1.0);
    }
    // Build Tensor Rule
    outRule = kron_prod<Scalar>(outRule,diffRule1); 
  }
}
 
// Update Index Set - no knowledge of active or old indices
template<class Scalar, class UserVector>
Scalar AdaptiveSparseGrid<Scalar,UserVector>::refine_grid(
                 typename std::multimap<Scalar,std::vector<int> >  & indexSet, 
                 UserVector & integralValue,
                 AdaptiveSparseGridInterface<Scalar,UserVector> & problem_data) {
  
  int dimension = problem_data.getDimension();
  std::vector<EIntrepidBurkardt> rule1D; problem_data.getRule(rule1D);
  std::vector<EIntrepidGrowth> growth1D; problem_data.getGrowth(growth1D);
  
  // Copy Multimap into a Set for ease of use
  typename std::multimap<Scalar,std::vector<int> >::iterator it;
  std::set<std::vector<int> > oldSet;
  std::set<std::vector<int> >::iterator it1(oldSet.begin());
  for (it=indexSet.begin(); it!=indexSet.end(); it++) {
    oldSet.insert(it1,it->second);
    it1++;
  }
  indexSet.clear();
  
  // Find Possible Active Points
  int flag = 1;
  std::vector<int> index(dimension,0);
  typename std::multimap<Scalar,std::vector<int> > activeSet;
  for (it1=oldSet.begin(); it1!=oldSet.end(); it1++) {
    index = *it1;
    for (int i=0; i<dimension; i++) {
      index[i]++;
      flag = (int)(!oldSet.count(index));
      index[i]--;
      if (flag) {
        activeSet.insert(std::pair<Scalar,std::vector<int> >(1.0,index));
        oldSet.erase(it1);
        break;
      }
    }
  }

  // Compute local and global error indicators for active set
  typename std::multimap<Scalar,std::vector<int> >::iterator it2;
  Scalar eta = 0.0;
  Scalar G   = 0.0;
  Teuchos::RCP<UserVector> s = integralValue.Create();
  for (it2=activeSet.begin(); it2!=activeSet.end(); it2++) {
    // Build Differential Quarature Rule
    index = it2->second;
    CubatureTensorSorted<Scalar> diffRule(0,dimension);
    build_diffRule(diffRule,index,problem_data);
    
    // Apply Rule to function
    problem_data.eval_cubature(*s,diffRule);
    
    // Update local error indicator and index set
    G  = problem_data.error_indicator(*s);
    activeSet.erase(it2);
    activeSet.insert(it2,std::pair<Scalar,std::vector<int> >(G,index));
    eta += G;
  }

  // Refine Sparse Grid
  eta = refine_grid(activeSet,oldSet,integralValue,eta,
                    dimension,rule1D,growth1D);

  // Insert New Active and Old Index sets into indexSet
  indexSet.insert(activeSet.begin(),activeSet.end());
  for (it1=oldSet.begin(); it1!=oldSet.end(); it1++) {
    index = *it1;
    indexSet.insert(std::pair<Scalar,std::vector<int> >(-1.0,index));
  }
    
  return eta;
}

// Update index set and output integral
template<class Scalar, class UserVector> 
Scalar AdaptiveSparseGrid<Scalar,UserVector>::refine_grid(
          typename std::multimap<Scalar,std::vector<int> > & activeIndex, 
          std::set<std::vector<int> > & oldIndex, 
          UserVector & integralValue,
          Scalar globalErrorIndicator,
          AdaptiveSparseGridInterface<Scalar,UserVector> & problem_data) {
 
 TEUCHOS_TEST_FOR_EXCEPTION((activeIndex.empty()),std::out_of_range,
              ">>> ERROR (AdaptiveSparseGrid): Active Index set is empty.");  

  int dimension = problem_data.getDimension();
  std::vector<EIntrepidBurkardt> rule1D; problem_data.getRule(rule1D);
  std::vector<EIntrepidGrowth> growth1D; problem_data.getGrowth(growth1D);

  // Initialize Flags
  bool maxLevelFlag     = true;
  bool isAdmissibleFlag = true;

  // Initialize Cubature Rule
  Teuchos::RCP<UserVector> s = integralValue.Create();
  // Initialize iterator at end of inOldIndex
  std::set<std::vector<int> >::iterator it1(oldIndex.end()); 

  // Initialize iterator at end of inActiveIndex
  typename std::multimap<Scalar,std::vector<int> >::iterator it;

  // Obtain Global Error Indicator as sum of key values of inActiveIndex
  Scalar eta = globalErrorIndicator;

  // Select Index to refine
  it = activeIndex.end(); it--;        // Decrement to position of final value
  Scalar G               = it->first;  // Largest Error Indicator is at End 
  eta                   -= G;          // Update global error indicator
  std::vector<int> index = it->second; // Get Corresponding index
  activeIndex.erase(it);               // Erase Index from active index set
  // Insert Index into old index set
  oldIndex.insert(it1,index); it1 = oldIndex.end();   

  // Refinement process
  for (int k=0; k<dimension; k++) {
    index[k]++; // index + ek
    // Check Max Level
    maxLevelFlag = problem_data.max_level(index);
    if (maxLevelFlag) {
      // Check Admissibility
      isAdmissibleFlag = isAdmissible(index,k,oldIndex,problem_data);
      if (isAdmissibleFlag) { // If admissible
        // Build Differential Quarature Rule
        CubatureTensorSorted<Scalar> diffRule(0,dimension);
        build_diffRule(diffRule,index,problem_data);    

        // Apply Rule to function
        problem_data.eval_cubature(*s,diffRule);
        
        // Update integral value
        integralValue.Update(*s);

        // Update local error indicator and index set
        G  = problem_data.error_indicator(*s); 
        if (activeIndex.end()!=activeIndex.begin()) 
          activeIndex.insert(activeIndex.end()--,
                           std::pair<Scalar,std::vector<int> >(G,index));
        else
          activeIndex.insert(std::pair<Scalar,std::vector<int> >(G,index));

        // Update global error indicators
        eta += G;
      }
    }
    else { // Max Level Exceeded 
      //std::cout << "Maximum Level Exceeded" << std::endl;
    }
    index[k]--;
  }
  return eta;
}

// Update index set and output integral/sparse grid
template<class Scalar, class UserVector> 
Scalar AdaptiveSparseGrid<Scalar,UserVector>::refine_grid(
        typename std::multimap<Scalar,std::vector<int> > & activeIndex, 
        std::set<std::vector<int> > & oldIndex, 
        UserVector & integralValue,
        CubatureTensorSorted<Scalar> & cubRule,
        Scalar globalErrorIndicator,
        AdaptiveSparseGridInterface<Scalar,UserVector> & problem_data) {

  TEUCHOS_TEST_FOR_EXCEPTION((activeIndex.empty()),std::out_of_range,
              ">>> ERROR (AdaptiveSparseGrid): Active Index set is empty.");  

  int dimension = problem_data.getDimension();
  std::vector<EIntrepidBurkardt> rule1D; problem_data.getRule(rule1D);
  std::vector<EIntrepidGrowth> growth1D; problem_data.getGrowth(growth1D);

  // Initialize Flags
  bool maxLevelFlag     = true;
  bool isAdmissibleFlag = true;

  // Initialize Cubature Rule
  Teuchos::RCP<UserVector> s = integralValue.Create();

  // Initialize iterator at end of inOldIndex
  std::set<std::vector<int> >::iterator it1(oldIndex.end());  

  // Initialize iterator at end of inActiveIndex
  typename std::multimap<Scalar,std::vector<int> >::iterator it;

  // Obtain Global Error Indicator as sum of key values of inActiveIndex
  Scalar eta = globalErrorIndicator;

  // Select Index to refine
  it = activeIndex.end(); it--;        // Decrement to position of final value 
  Scalar G               = it->first;  // Largest Error Indicator is at End
  eta                   -= G;          // Update global error indicator
  std::vector<int> index = it->second; // Get Corresponding index
  activeIndex.erase(it);               // Erase Index from active index set
  // Insert Index into old index set
  oldIndex.insert(it1,index); it1 = oldIndex.end(); 
  
  // Refinement process
  for (int k=0; k<dimension; k++) {
    index[k]++; // index + ek
    // Check Max Level
    maxLevelFlag = problem_data.max_level(index);
    if (maxLevelFlag) {
      // Check Admissibility
      isAdmissibleFlag = isAdmissible(index,k,oldIndex,problem_data);
      if (isAdmissibleFlag) { // If admissible
        // Build Differential Quarature Rule
        CubatureTensorSorted<Scalar> diffRule(0,dimension);
        build_diffRule(diffRule,index,problem_data);
        
        // Apply Rule to function
        problem_data.eval_cubature(*s,diffRule);
        
        // Update integral value
        integralValue.Update(*s);
        
        // Update local error indicator and index set
        G  = problem_data.error_indicator(*s);  
        if (activeIndex.end()!=activeIndex.begin()) 
          activeIndex.insert(activeIndex.end()--,
                           std::pair<Scalar,std::vector<int> >(G,index));
        else
          activeIndex.insert(std::pair<Scalar,std::vector<int> >(G,index));
                
        // Update global error indicators
        eta += G;

        // Update adapted quadrature rule nodes and weights
        cubRule.update(1.0,diffRule,1.0);
      }
    }
    else { // Max Level Exceeded 
      //std::cout << "Maximum Level Exceeded" << std::endl;
    }
    index[k]--;
  }
  return eta;
}

template<class Scalar, class UserVector>
void AdaptiveSparseGrid<Scalar,UserVector>::buildSparseGrid(
              CubatureTensorSorted<Scalar> & output,
              int dimension, int maxlevel,
              std::vector<EIntrepidBurkardt> rule1D, 
              std::vector<EIntrepidGrowth> growth1D,
              bool isNormalized) {

  if (dimension == 2) {
    std::vector<int> index(dimension,0);
    for (int i=0; i<maxlevel; i++) {
      for (int j=0; j<maxlevel; j++) {
        if(i+j+dimension <= maxlevel+dimension-1) {
          index[0] = i+1; index[1] = j+1; 
          CubatureTensorSorted<Scalar> diffRule(0,dimension);
          build_diffRule(diffRule,index,dimension,rule1D,growth1D,isNormalized);
          output.update(1.0,diffRule,1.0);
        }
      }
    } 
  }
  else if (dimension == 3) {    
    std::vector<int> index(dimension,0);
    for (int i=0; i<maxlevel; i++) {
      for (int j=0; j<maxlevel; j++) {
        for (int k=0; k<maxlevel; k++) {
          if(i+j+k+dimension <= maxlevel+dimension-1) {
            index[0] = i+1; index[1] = j+1; index[2] = k+1;
            CubatureTensorSorted<Scalar> diffRule(0,dimension);
            build_diffRule(diffRule,index,dimension,rule1D,
                          growth1D,isNormalized);
            output.update(1.0,diffRule,1.0);
          }
        }
      }
    } 
  }
  else if (dimension == 4) {
    std::vector<int> index(dimension,0);
    for (int i=0; i<maxlevel; i++) {
      for (int j=0; j<maxlevel; j++) {
        for (int k=0; k<maxlevel; k++) {
          for (int l=0; l<maxlevel; l++) {
            if(i+j+k+l+dimension <= maxlevel+dimension-1) {
              index[0] = i+1; index[1] = j+1; index[2] = k+1; index[3] = l+1;
              CubatureTensorSorted<Scalar> diffRule(0,dimension);
              build_diffRule(diffRule,index,dimension,rule1D,
                            growth1D,isNormalized);
              output.update(1.0,diffRule,1.0);
            }
          }
        }
      } 
    }
  }
  else if (dimension == 5) {
    std::vector<int> index(dimension,0);
    for (int i=0; i<maxlevel; i++) {
      for (int j=0; j<maxlevel; j++) {
        for (int k=0; k<maxlevel; k++) {
          for (int l=0; l<maxlevel; l++) {
            for (int m=0; m<maxlevel; m++) {
              if(i+j+k+l+m+dimension <= maxlevel+dimension-1) {
                index[0] = i+1; index[1] = j+1; index[2] = k+1; 
                index[3] = l+1; index[4] = m+1;
                CubatureTensorSorted<Scalar> diffRule(0,dimension);
                build_diffRule(diffRule,index,dimension,rule1D,
                              growth1D,isNormalized);
                output.update(1.0,diffRule,1.0);
              }
            }
          }
        } 
      }
    }
  }
  else 
    std::cout << "Dimension Must Be Less Than 5\n";
}

} // end Intrepid namespace