#include "p-spin.hpp"

template <class Scalar>
double segmentCost(const Vector<Scalar>& z, const Vector<Scalar>& dz, const Vector<Scalar>& dH) {
  Vector<Scalar> zD = zDot(z, dH);
  return 1.0 - pow(real(zD.dot(dz)), 2) / zD.squaredNorm() / dz.squaredNorm();
}

class ropeRelaxationStallException: public std::exception {
  virtual const char* what() const throw() {
    return "Gradient descent stalled.";
  }
};

template <class Scalar>
Vector<Scalar> variation(const Vector<Scalar>& z, const Vector<Scalar>& z´, const Vector<Scalar>& z´´, const Vector<Scalar>& dH, const Matrix<Scalar>& ddH) {
  double z² = z.squaredNorm();
  double z´² = z´.squaredNorm();

  Vector<Scalar> ż = zDot(z, dH);
  double ż² = ż.squaredNorm();

  double Reż·z´ = real(ż.dot(z´));

  Matrix<Scalar> dż = (dH.conjugate() - (dH.dot(z) / z²) * z.conjugate()) * z.adjoint() / z²;
  Matrix<Scalar> dżc = -ddH + (ddH * z.conjugate()) * z.transpose() / z²
    + (z.dot(dH) / z²) * (
        Matrix<Scalar>::Identity(ddH.rows(), ddH.cols()) - z.conjugate() * z.transpose() / z²
      );

  Vector<Scalar> dLdz = - Reż·z´ * (
        dżc * z´ + dż * z´.conjugate() - (dż * ż.conjugate() + dżc * ż) * Reż·z´ / ż²
      ) / (ż² * z´²);

  Vector<Scalar> ż´ = -(ddH * z´).conjugate() + ((ddH * z´).dot(z) / z²) * z.conjugate() + (
        dH.dot(z) * z´.conjugate() + dH.dot(z´) * z.conjugate() - (
          dH.dot(z) * (z´.dot(z) + z.dot(z´)) / z²
        ) * z.conjugate()
      ) / z²;

  double dReż·z´ = real(ż.dot(z´´) + ż´.dot(z´));

  Vector<Scalar> ddtdLdz´ = - (
      Reż·z´ * (
        ż´.conjugate() - (
          Reż·z´ * z´´.conjugate() + dReż·z´ * z´.conjugate()
          - (Reż·z´ / z´²) * (z´´.dot(z´) + z´.dot(z´´)) * z´.conjugate() 
          ) / z´²
        )
      + dReż·z´ * (ż.conjugate() - Reż·z´ / z´² * z´.conjugate())
      - Reż·z´ * (
        (ż.dot(ż´) + ż´.dot(ż)) / ż² + (z´´.dot(z´) + z´.dot(z´´)) / z´²
        ) * (ż.conjugate() - Reż·z´ / z´² * z´.conjugate())
      ) / (ż² * z´²);

  return dLdz - ddtdLdz´;
}

template <class Scalar>
class Rope {
  public:
    std::vector<Vector<Scalar>> z;

    Rope(unsigned N, const Vector<Scalar>& z1, const Vector<Scalar>& z2) : z(N + 2) {
      for (unsigned i = 0; i < N + 2; i++) {
        z[i] = normalize(z1 + (z2 - z1) * ((double)i / (N + 1.0)));
      }
    }

    Vector<Scalar> dz(unsigned i) const {
      return z[i + 1] - z[i - 1];
    }

    Vector<Scalar> ddz(unsigned i) const {
      return 4.0 * (z[i + 1] + z[i - 1] - 2.0 * z[i]);
    }

    template <int p>
    std::vector<Vector<Scalar>> δz(const Tensor<Scalar, p>& J) const {
      std::vector<Vector<Scalar>> δz(z.size());

#pragma omp parallel for
      for (unsigned i = 1; i < z.size() - 1; i++) {
        Vector<Scalar> dH;
        Matrix<Scalar> ddH;
        std::tie(std::ignore, dH, ddH) = hamGradHess(J, z[i]);

        δz[i] = variation(z[i], this->dz(i), this->ddz(i), dH, ddH);
      }

      return δz;
    }

    template <int p>
    double perturb(const Tensor<Scalar, p>& J, double δ0) {
      std::vector<Vector<Scalar>> Δz = this->δz(J);

      Rope rNew = *this;
      double δ = δ0;

      while (rNew.cost(J) >= this->cost(J)) {
        for (unsigned i = 1; i < z.size() - 1; i++) {
          rNew.z[i] = normalize(z[i] - δ * Δz[i].conjugate());
        }

        δ /= 2;

        if (δ < 1e-30) {
          throw ropeRelaxationStallException();
        }
      }

      z = rNew.z;

      return δ;
    }

    void spread() {
      double l = 0;

      for (unsigned i = 0; i < z.size() - 1; i++) {
        l += (z[i + 1] - z[i]).norm();
      }

      double a = 0;
      unsigned pos = 0;

      std::vector<Vector<Scalar>> zNew = z;

      for (unsigned i = 1; i < z.size() - 1; i++) {
        double b = i * l / (z.size() - 1);

        while (b > a) {
          pos++;
          a += (z[pos] - z[pos - 1]).norm();
        } 

        Vector<Scalar> δz = z[pos] - z[pos - 1];

        zNew[i] = normalize(z[pos] - (a - b) / δz.norm() * δz);
      }

      z = zNew;
    }

    template <int p>
    double step(const Tensor<Scalar, p>& J, double δ) {
      double δNew = this->perturb(J, δ);
      this->spread();

      return δNew;
    }

    template <int p>
    double cost(const Tensor<Scalar, p>& J) const {
      double c = 0;

      for (unsigned i = 1; i < z.size() - 1; i++) {
        Vector<Scalar> dH;
        std::tie(std::ignore, dH, std::ignore) = hamGradHess(J, z[i]);

        c += segmentCost(z[i], this->dz(i), dH);
      }

      return c;
    }

    template <int p>
    void relax(const Tensor<Scalar, p>& J, unsigned N, double δ0) {
      double δ = δ0;
      try {
        for (unsigned i = 0; i < N; i++) {
          δ = 2 * this->step(J, δ);
        }
      } catch (std::exception& e) {
      }
    }

    Rope<Scalar> interpolate() const {
      Rope<Scalar> r(2 * (z.size() - 2) + 1, z[0], z[z.size() - 1]);

      for (unsigned i = 0; i < z.size(); i++) {
        r.z[2 * i] = z[i];
      }

      for (unsigned i = 0; i < z.size() - 1; i++) {
        r.z[2 * i + 1] = normalize(((z[i] + z[i + 1]) / 2.0));
      }

      return r;
    }
};