/* ----------------------------------------------------------------------------

 * GTSAM Copyright 2010, Georgia Tech Research Corporation, 
 * Atlanta, Georgia 30332-0415
 * All Rights Reserved
 * Authors: Frank Dellaert, et al. (see THANKS for the full author list)

 * See LICENSE for the license information

 * -------------------------------------------------------------------------- */

/**
 * @file Expression-inl.h
 * @date September 18, 2014
 * @author Frank Dellaert
 * @author Paul Furgale
 * @brief Internals for Expression.h, not for general consumption
 */

#pragma once

#include <gtsam/nonlinear/Values.h>
#include <gtsam/base/Matrix.h>
#include <boost/foreach.hpp>
#include <boost/tuple/tuple.hpp>

// template meta-programming headers
#include <boost/mpl/vector.hpp>
#include <boost/mpl/plus.hpp>
#include <boost/mpl/front.hpp>
#include <boost/mpl/pop_front.hpp>
#include <boost/mpl/fold.hpp>
#include<boost/mpl/empty_base.hpp>
#include<boost/mpl/placeholders.hpp>
namespace MPL = boost::mpl::placeholders;

namespace gtsam {

template<typename T>
class Expression;

typedef std::map<Key, Matrix> JacobianMap;

//-----------------------------------------------------------------------------
/// The JacobinaTrace class records a tree-structured expression's execution
template<class T>
struct JacobianTrace {

  // Some fixed matrix sizes we need
  typedef Eigen::Matrix<double, 2, T::dimension> Jacobian2T;

  /**
   *  The Pointer class is a tagged union that obviates the need to create
   *  a JacobianTrace subclass for Constants and Leaf Expressions. Instead
   *  the key for the leaf is stored in the space normally used to store a
   *  JacobianTrace*. Nothing is stored for a Constant.
   */
  ///
  class Pointer {
    enum {
      Constant, Leaf, Function
    } type;
    union {
      Key key;
      JacobianTrace* ptr;
    } content;
  public:
    /// Pointer always starts out as a Constant
    Pointer() :
        type(Constant) {
    }
    /// Destructor cleans up pointer if Function
    ~Pointer() {
      if (type == Function)
        delete content.ptr;
    }
    /// Change pointer to a Leaf Trace
    void setLeaf(Key key) {
      type = Leaf;
      content.key = key;
    }
    /// Take ownership of pointer to a Function Trace
    void setFunction(JacobianTrace* trace) {
      type = Function;
      content.ptr = trace;
    }
    // *** This is the main entry point for reverseAD, called from Expression::augmented ***
    // Called only once, either inserts identity into Jacobians (Leaf) or starts AD (Function)
    void startReverseAD(JacobianMap& jacobians) const {
      if (type == Leaf) {
        // This branch will only be called on trivial Leaf expressions, i.e. Priors
        size_t n = T::Dim();
        jacobians[content.key] = Eigen::MatrixXd::Identity(n, n);
      } else if (type == Function)
        // This is the more typical entry point, starting the AD pipeline
        // It is inside the startReverseAD that the correctly dimensioned pipeline is chosen.
        content.ptr->startReverseAD(jacobians);
    }
    // Either add to Jacobians (Leaf) or propagate (Function)
    void reverseAD(const Matrix& dTdA, JacobianMap& jacobians) const {
      if (type == Leaf) {
        JacobianMap::iterator it = jacobians.find(content.key);
        if (it != jacobians.end())
          it->second += dTdA;
        else
          jacobians[content.key] = dTdA;
      } else if (type == Function)
        content.ptr->reverseAD(dTdA, jacobians);
    }
    // Either add to Jacobians (Leaf) or propagate (Function)
    void reverseAD2(const Jacobian2T& dTdA, JacobianMap& jacobians) const {
      if (type == Leaf) {
        JacobianMap::iterator it = jacobians.find(content.key);
        if (it != jacobians.end())
          it->second += dTdA;
        else
          jacobians[content.key] = dTdA;
      } else if (type == Function)
        content.ptr->reverseAD2(dTdA, jacobians);
    }
  };

  /// Make sure destructor is virtual
  virtual ~JacobianTrace() {
  }
  virtual void startReverseAD(JacobianMap& jacobians) const {
  }
  virtual void reverseAD(const Matrix& dFdT, JacobianMap& jacobians) const {
  }
  virtual void reverseAD2(const Jacobian2T& dFdT,
      JacobianMap& jacobians) const {
  }
};

/// Primary template calls the generic Matrix reverseAD pipeline
template<size_t M, class A>
struct Select {
  typedef Eigen::Matrix<double, M, A::dimension> Jacobian;
  static void reverseAD(const typename JacobianTrace<A>::Pointer& trace,
      const Jacobian& dTdA, JacobianMap& jacobians) {
    trace.reverseAD(dTdA, jacobians);
  }
};

/// Partially specialized template calls the 2-dimensional output version
template<class A>
struct Select<2, A> {
  typedef Eigen::Matrix<double, 2, A::dimension> Jacobian;
  static void reverseAD(const typename JacobianTrace<A>::Pointer& trace,
      const Jacobian& dTdA, JacobianMap& jacobians) {
    trace.reverseAD2(dTdA, jacobians);
  }
};

/**
 * Recursive Trace Class for Functional Expressions
 * Abrahams, David; Gurtovoy, Aleksey (2004-12-10).
 * C++ Template Metaprogramming: Concepts, Tools, and Techniques from Boost
 * and Beyond (Kindle Location 1244). Pearson Education.
 */
template<class T, class A, class More>
struct Trace: More {
  // define dimensions
  static const size_t m = T::dimension;
  static const size_t n = A::dimension;

  // define fixed size Jacobian matrix types
  typedef Eigen::Matrix<double, m, n> JacobianTA;
  typedef Eigen::Matrix<double, 2, m> Jacobian2T;

  // declare trace that produces value A, and corresponding Jacobian
  typename JacobianTrace<A>::Pointer trace;
  JacobianTA dTdA;

  /// Start the reverse AD process
  virtual void startReverseAD(JacobianMap& jacobians) const {
    More::startReverseAD(jacobians);
    Select<T::dimension, A>::reverseAD(trace, dTdA, jacobians);
  }
  /// Given df/dT, multiply in dT/dA and continue reverse AD process
  virtual void reverseAD(const Matrix& dFdT, JacobianMap& jacobians) const {
    More::reverseAD(dFdT, jacobians);
    trace.reverseAD(dFdT * dTdA, jacobians);
  }
  /// Version specialized to 2-dimensional output
  virtual void reverseAD2(const Jacobian2T& dFdT,
      JacobianMap& jacobians) const {
    More::reverseAD2(dFdT, jacobians);
    trace.reverseAD2(dFdT * dTdA, jacobians);
  }
};

/// Recursive Trace Class Generator
template<class T, class TYPES>
struct GenerateTrace {
  typedef typename boost::mpl::fold<TYPES, JacobianTrace<T>,
      Trace<T, MPL::_2, MPL::_1> >::type type;
};

//-----------------------------------------------------------------------------
/**
 * Value and Jacobians
 */
template<class T>
class Augmented {

private:

  T value_;
  JacobianMap jacobians_;

  typedef std::pair<Key, Matrix> Pair;

  /// Insert terms into jacobians_, adding if already exists
  void add(const JacobianMap& terms) {
    BOOST_FOREACH(const Pair& term, terms) {
      JacobianMap::iterator it = jacobians_.find(term.first);
      if (it != jacobians_.end())
        it->second += term.second;
      else
        jacobians_[term.first] = term.second;
    }
  }

  /// Insert terms into jacobians_, premultiplying by H, adding if already exists
  void add(const Matrix& H, const JacobianMap& terms) {
    BOOST_FOREACH(const Pair& term, terms) {
      JacobianMap::iterator it = jacobians_.find(term.first);
      if (it != jacobians_.end())
        it->second += H * term.second;
      else
        jacobians_[term.first] = H * term.second;
    }
  }

public:

  /// Construct value that does not depend on anything
  Augmented(const T& t) :
      value_(t) {
  }

  /// Construct value dependent on a single key
  Augmented(const T& t, Key key) :
      value_(t) {
    size_t n = t.dim();
    jacobians_[key] = Eigen::MatrixXd::Identity(n, n);
  }

  /// Construct value, pre-multiply jacobians by dTdA
  Augmented(const T& t, const Matrix& dTdA, const JacobianMap& jacobians) :
      value_(t) {
    add(dTdA, jacobians);
  }

  /// Construct value, pre-multiply jacobians
  Augmented(const T& t, const Matrix& dTdA1, const JacobianMap& jacobians1,
      const Matrix& dTdA2, const JacobianMap& jacobians2) :
      value_(t) {
    add(dTdA1, jacobians1);
    add(dTdA2, jacobians2);
  }

  /// Construct value, pre-multiply jacobians
  Augmented(const T& t, const Matrix& dTdA1, const JacobianMap& jacobians1,
      const Matrix& dTdA2, const JacobianMap& jacobians2, const Matrix& dTdA3,
      const JacobianMap& jacobians3) :
      value_(t) {
    add(dTdA1, jacobians1);
    add(dTdA2, jacobians2);
    add(dTdA3, jacobians3);
  }

  /// Return value
  const T& value() const {
    return value_;
  }

  /// Return jacobians
  const JacobianMap& jacobians() const {
    return jacobians_;
  }

  /// Return jacobians
  JacobianMap& jacobians() {
    return jacobians_;
  }

  /// Not dependent on any key
  bool constant() const {
    return jacobians_.empty();
  }

  /// debugging
  void print(const KeyFormatter& keyFormatter = DefaultKeyFormatter) {
    BOOST_FOREACH(const Pair& term, jacobians_)
      std::cout << "(" << keyFormatter(term.first) << ", " << term.second.rows()
          << "x" << term.second.cols() << ") ";
    std::cout << std::endl;
  }

  /// Move terms to array, destroys content
  void move(std::vector<Matrix>& H) {
    assert(H.size()==jacobains.size());
    size_t j = 0;
    JacobianMap::iterator it = jacobians_.begin();
    for (; it != jacobians_.end(); ++it)
      it->second.swap(H[j++]);
  }

};

//-----------------------------------------------------------------------------
/**
 * Expression node. The superclass for objects that do the heavy lifting
 * An Expression<T> has a pointer to an ExpressionNode<T> underneath
 * allowing Expressions to have polymorphic behaviour even though they
 * are passed by value. This is the same way boost::function works.
 * http://loki-lib.sourceforge.net/html/a00652.html
 */
template<class T>
class ExpressionNode {

protected:
  ExpressionNode() {
  }

public:

  /// Destructor
  virtual ~ExpressionNode() {
  }

  /// Return keys that play in this expression as a set
  virtual std::set<Key> keys() const = 0;

  /// Return value
  virtual T value(const Values& values) const = 0;

  /// Return value and derivatives
  virtual Augmented<T> forward(const Values& values) const = 0;

  /// Construct an execution trace for reverse AD
  virtual T traceExecution(const Values& values,
      typename JacobianTrace<T>::Pointer& p) const = 0;
};

//-----------------------------------------------------------------------------
/// Constant Expression
template<class T>
class ConstantExpression: public ExpressionNode<T> {

  /// The constant value
  T constant_;

  /// Constructor with a value, yielding a constant
  ConstantExpression(const T& value) :
      constant_(value) {
  }

  friend class Expression<T> ;

public:

  /// Return keys that play in this expression, i.e., the empty set
  virtual std::set<Key> keys() const {
    std::set<Key> keys;
    return keys;
  }

  /// Return value
  virtual T value(const Values& values) const {
    return constant_;
  }

  /// Return value and derivatives
  virtual Augmented<T> forward(const Values& values) const {
    return Augmented<T>(constant_);
  }

  /// Construct an execution trace for reverse AD
  virtual T traceExecution(const Values& values,
      typename JacobianTrace<T>::Pointer& p) const {
    return constant_;
  }
};

//-----------------------------------------------------------------------------
/// Leaf Expression
template<class T>
class LeafExpression: public ExpressionNode<T> {

  /// The key into values
  Key key_;

  /// Constructor with a single key
  LeafExpression(Key key) :
      key_(key) {
  }

  friend class Expression<T> ;

public:

  /// Return keys that play in this expression
  virtual std::set<Key> keys() const {
    std::set<Key> keys;
    keys.insert(key_);
    return keys;
  }

  /// Return value
  virtual T value(const Values& values) const {
    return values.at<T>(key_);
  }

  /// Return value and derivatives
  virtual Augmented<T> forward(const Values& values) const {
    return Augmented<T>(values.at<T>(key_), key_);
  }

  /// Construct an execution trace for reverse AD
  virtual T traceExecution(const Values& values,
      typename JacobianTrace<T>::Pointer& p) const {
    p.setLeaf(key_);
    return values.at<T>(key_);
  }

};

//-----------------------------------------------------------------------------
/// Unary Function Expression
template<class T, class A1>
class UnaryExpression: public ExpressionNode<T> {

public:

  typedef Eigen::Matrix<double, T::dimension, A1::dimension> JacobianTA;
  typedef boost::function<T(const A1&, boost::optional<JacobianTA&>)> Function;

private:

  Function function_;
  boost::shared_ptr<ExpressionNode<A1> > expressionA1_;

  /// Constructor with a unary function f, and input argument e
  UnaryExpression(Function f, const Expression<A1>& e) :
      function_(f), expressionA1_(e.root()) {
  }

  friend class Expression<T> ;

public:

  /// Return keys that play in this expression
  virtual std::set<Key> keys() const {
    return expressionA1_->keys();
  }

  /// Return value
  virtual T value(const Values& values) const {
    return function_(this->expressionA1_->value(values), boost::none);
  }

  /// Return value and derivatives
  virtual Augmented<T> forward(const Values& values) const {
    using boost::none;
    Augmented<A1> argument = this->expressionA1_->forward(values);
    JacobianTA dTdA;
    T t = function_(argument.value(),
        argument.constant() ? none : boost::optional<JacobianTA&>(dTdA));
    return Augmented<T>(t, dTdA, argument.jacobians());
  }

  /// Trace structure for reverse AD
  typedef boost::mpl::vector<A1> Arguments;
  typedef typename GenerateTrace<T, Arguments>::type Trace;

  /// Construct an execution trace for reverse AD
  virtual T traceExecution(const Values& values,
      typename JacobianTrace<T>::Pointer& p) const {
    Trace* trace = new Trace();
    p.setFunction(trace);
    A1 a = this->expressionA1_->traceExecution(values, trace->trace);
    return function_(a, trace->dTdA);
  }
};

//-----------------------------------------------------------------------------
/// Binary Expression

template<class T, class A1, class A2>
class BinaryExpression: public ExpressionNode<T> {

public:

  typedef Eigen::Matrix<double, T::dimension, A1::dimension> JacobianTA1;
  typedef Eigen::Matrix<double, T::dimension, A2::dimension> JacobianTA2;
  typedef boost::function<
      T(const A1&, const A2&, boost::optional<JacobianTA1&>,
          boost::optional<JacobianTA2&>)> Function;

private:

  Function function_;
  boost::shared_ptr<ExpressionNode<A1> > expressionA1_;
  boost::shared_ptr<ExpressionNode<A2> > expressionA2_;

  /// Constructor with a binary function f, and two input arguments
  BinaryExpression(Function f, //
      const Expression<A1>& e1, const Expression<A2>& e2) :
      function_(f), expressionA1_(e1.root()), expressionA2_(e2.root()) {
  }

  friend class Expression<T> ;

public:

  /// Return keys that play in this expression
  virtual std::set<Key> keys() const {
    std::set<Key> keys1 = expressionA1_->keys();
    std::set<Key> keys2 = expressionA2_->keys();
    keys1.insert(keys2.begin(), keys2.end());
    return keys1;
  }

  /// Return value
  virtual T value(const Values& values) const {
    using boost::none;
    return function_(this->expressionA1_->value(values),
        this->expressionA2_->value(values), none, none);
  }

  /// Return value and derivatives
  virtual Augmented<T> forward(const Values& values) const {
    using boost::none;
    Augmented<A1> a1 = this->expressionA1_->forward(values);
    Augmented<A2> a2 = this->expressionA2_->forward(values);
    JacobianTA1 dTdA1;
    JacobianTA2 dTdA2;
    T t = function_(a1.value(), a2.value(),
        a1.constant() ? none : boost::optional<JacobianTA1&>(dTdA1),
        a2.constant() ? none : boost::optional<JacobianTA2&>(dTdA2));
    return Augmented<T>(t, dTdA1, a1.jacobians(), dTdA2, a2.jacobians());
  }

  /// Trace structure for reverse AD
  struct Trace: public JacobianTrace<T> {
    typename JacobianTrace<A1>::Pointer trace1;
    typename JacobianTrace<A2>::Pointer trace2;
    JacobianTA1 dTdA1;
    JacobianTA2 dTdA2;

    /// Start the reverse AD process
    virtual void startReverseAD(JacobianMap& jacobians) const {
      Select<T::dimension, A1>::reverseAD(trace1, dTdA1, jacobians);
      Select<T::dimension, A2>::reverseAD(trace2, dTdA2, jacobians);
    }
    /// Given df/dT, multiply in dT/dA and continue reverse AD process
    virtual void reverseAD(const Matrix& dFdT, JacobianMap& jacobians) const {
      trace1.reverseAD(dFdT * dTdA1, jacobians);
      trace2.reverseAD(dFdT * dTdA2, jacobians);
    }
    /// Version specialized to 2-dimensional output
    typedef Eigen::Matrix<double, 2, T::dimension> Jacobian2T;
    virtual void reverseAD2(const Jacobian2T& dFdT,
        JacobianMap& jacobians) const {
      trace1.reverseAD2(dFdT * dTdA1, jacobians);
      trace2.reverseAD2(dFdT * dTdA2, jacobians);
    }
  };

  /// Construct an execution trace for reverse AD
  virtual T traceExecution(const Values& values,
      typename JacobianTrace<T>::Pointer& p) const {
    Trace* trace = new Trace();
    p.setFunction(trace);
    A1 a1 = this->expressionA1_->traceExecution(values, trace->trace1);
    A2 a2 = this->expressionA2_->traceExecution(values, trace->trace2);
    return function_(a1, a2, trace->dTdA1, trace->dTdA2);
  }

};

//-----------------------------------------------------------------------------
/// Ternary Expression

template<class T, class A1, class A2, class A3>
class TernaryExpression: public ExpressionNode<T> {

public:

  typedef Eigen::Matrix<double, T::dimension, A1::dimension> JacobianTA1;
  typedef Eigen::Matrix<double, T::dimension, A2::dimension> JacobianTA2;
  typedef Eigen::Matrix<double, T::dimension, A3::dimension> JacobianTA3;
  typedef boost::function<
      T(const A1&, const A2&, const A3&, boost::optional<JacobianTA1&>,
          boost::optional<JacobianTA2&>, boost::optional<JacobianTA3&>)> Function;

private:

  Function function_;
  boost::shared_ptr<ExpressionNode<A1> > expressionA1_;
  boost::shared_ptr<ExpressionNode<A2> > expressionA2_;
  boost::shared_ptr<ExpressionNode<A3> > expressionA3_;

  /// Constructor with a ternary function f, and three input arguments
  TernaryExpression(
      Function f, //
      const Expression<A1>& e1, const Expression<A2>& e2,
      const Expression<A3>& e3) :
      function_(f), expressionA1_(e1.root()), expressionA2_(e2.root()), expressionA3_(
          e3.root()) {
  }

  friend class Expression<T> ;

public:

  /// Return keys that play in this expression
  virtual std::set<Key> keys() const {
    std::set<Key> keys1 = expressionA1_->keys();
    std::set<Key> keys2 = expressionA2_->keys();
    std::set<Key> keys3 = expressionA3_->keys();
    keys2.insert(keys3.begin(), keys3.end());
    keys1.insert(keys2.begin(), keys2.end());
    return keys1;
  }

  /// Return value
  virtual T value(const Values& values) const {
    using boost::none;
    return function_(this->expressionA1_->value(values),
        this->expressionA2_->value(values), this->expressionA3_->value(values),
        none, none, none);
  }

  /// Return value and derivatives
  virtual Augmented<T> forward(const Values& values) const {
    using boost::none;
    Augmented<A1> a1 = this->expressionA1_->forward(values);
    Augmented<A2> a2 = this->expressionA2_->forward(values);
    Augmented<A3> a3 = this->expressionA3_->forward(values);
    JacobianTA1 dTdA1;
    JacobianTA2 dTdA2;
    JacobianTA3 dTdA3;
    T t = function_(a1.value(), a2.value(), a3.value(),
        a1.constant() ? none : boost::optional<JacobianTA1&>(dTdA1),
        a2.constant() ? none : boost::optional<JacobianTA2&>(dTdA2),
        a3.constant() ? none : boost::optional<JacobianTA3&>(dTdA3));
    return Augmented<T>(t, dTdA1, a1.jacobians(), dTdA2, a2.jacobians(), dTdA3,
        a3.jacobians());
  }

  /// Trace structure for reverse AD
  struct Trace: public JacobianTrace<T> {
    typename JacobianTrace<A1>::Pointer trace1;
    typename JacobianTrace<A2>::Pointer trace2;
    typename JacobianTrace<A3>::Pointer trace3;
    JacobianTA1 dTdA1;
    JacobianTA2 dTdA2;
    JacobianTA3 dTdA3;

    /// Start the reverse AD process
    virtual void startReverseAD(JacobianMap& jacobians) const {
      Select<T::dimension, A1>::reverseAD(trace1, dTdA1, jacobians);
      Select<T::dimension, A2>::reverseAD(trace2, dTdA2, jacobians);
      Select<T::dimension, A3>::reverseAD(trace3, dTdA3, jacobians);
    }
    /// Given df/dT, multiply in dT/dA and continue reverse AD process
    virtual void reverseAD(const Matrix& dFdT, JacobianMap& jacobians) const {
      trace1.reverseAD(dFdT * dTdA1, jacobians);
      trace2.reverseAD(dFdT * dTdA2, jacobians);
      trace3.reverseAD(dFdT * dTdA3, jacobians);
    }
    /// Version specialized to 2-dimensional output
    typedef Eigen::Matrix<double, 2, T::dimension> Jacobian2T;
    virtual void reverseAD2(const Jacobian2T& dFdT,
        JacobianMap& jacobians) const {
      trace1.reverseAD2(dFdT * dTdA1, jacobians);
      trace2.reverseAD2(dFdT * dTdA2, jacobians);
      trace3.reverseAD2(dFdT * dTdA3, jacobians);
    }
  };

  /// Construct an execution trace for reverse AD
  virtual T traceExecution(const Values& values,
      typename JacobianTrace<T>::Pointer& p) const {
    Trace* trace = new Trace();
    p.setFunction(trace);
    A1 a1 = this->expressionA1_->traceExecution(values, trace->trace1);
    A2 a2 = this->expressionA2_->traceExecution(values, trace->trace2);
    A3 a3 = this->expressionA3_->traceExecution(values, trace->trace3);
    return function_(a1, a2, a3, trace->dTdA1, trace->dTdA2, trace->dTdA3);
  }

};
//-----------------------------------------------------------------------------
}