path: root/src/Persistence_representations/include/gudhi/Persistence_landscape_on_grid.h

/*    This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
 *    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
 *    Author(s):       Pawel Dlotko
 *
 *    Copyright (C) 2016 Inria
 *
 *    Modification(s):
 *      - YYYY/MM Author: Description of the modification
 */

#ifndef PERSISTENCE_LANDSCAPE_ON_GRID_H_
#define PERSISTENCE_LANDSCAPE_ON_GRID_H_

// gudhi include
#include <gudhi/read_persistence_from_file.h>
#include <gudhi/common_persistence_representations.h>

// standard include
#include <iostream>
#include <vector>
#include <limits>
#include <fstream>
#include <sstream>
#include <algorithm>
#include <cmath>
#include <functional>
#include <utility>
#include <string>
#include <cstdint>

namespace Gudhi {
namespace Persistence_representations {

// pre declaration
class Persistence_landscape_on_grid;
template <typename operation>
Persistence_landscape_on_grid operation_on_pair_of_landscapes_on_grid(const Persistence_landscape_on_grid& land1,
                                                                      const Persistence_landscape_on_grid& land2);

/**
 * \class Persistence_landscape_on_grid Persistence_landscape_on_grid.h gudhi/Persistence_landscape_on_grid.h
 * \brief A class implementing persistence landscapes by approximating them on a collection of grid points.
 *
 * \ingroup Persistence_representations
 *
 * \details
 * Persistence landscapes on grid allows vectorization, computations of distances, computations of projections to Real,
 * computations of averages and scalar products. Therefore they implement suitable interfaces.
 * It implements the following concepts: Vectorized_topological_data, Topological_data_with_distances,
 * Real_valued_topological_data, Topological_data_with_averages, Topological_data_with_scalar_product
 *
 * Note that at the moment, due to rounding errors during the construction of persistence landscapes on a grid,
 * elements which are different by 0.000005 are considered the same. If the scale in your persistence diagrams
 * is comparable to this value, please rescale them before use this code.
**/

// this class implements the following concepts: Vectorized_topological_data, Topological_data_with_distances,
// Real_valued_topological_data, Topological_data_with_averages, Topological_data_with_scalar_product
class Persistence_landscape_on_grid {
 public:
  /**
   * Default constructor.
  **/
  Persistence_landscape_on_grid() {
    this->set_up_numbers_of_functions_for_vectorization_and_projections_to_reals();
    this->grid_min = this->grid_max = 0;
  }

  /**
       * Constructor that takes as an input a vector of birth-death pairs.
      **/
  Persistence_landscape_on_grid(const std::vector<std::pair<double, double> >& p, double grid_min_, double grid_max_,
                                size_t number_of_points_);

  /**
       * Constructor that takes as an input a vector of birth-death pairs, parameters of the grid and number of
    *landscape function to be created.
      **/
  Persistence_landscape_on_grid(const std::vector<std::pair<double, double> >& p, double grid_min_, double grid_max_,
                                size_t number_of_points_, unsigned number_of_levels_of_landscape);

  /**
       * Constructor that reads persistence intervals from file and creates persistence landscape. The format of the
    *input file is the following: in each line we put birth-death pair. Last line is assumed
       * to be empty. Even if the points within a line are not ordered, they will be ordered while the input is read.
    *The additional parameters of this procedure are: ranges of grid, resolution of a grid
       * number of landscape functions to be created and the dimension of intervals that are need to be read from a file
    *(in case of Gudhi format files).
      **/
  Persistence_landscape_on_grid(const char* filename, double grid_min_, double grid_max_, size_t number_of_points_,
                                unsigned number_of_levels_of_landscape,
                                uint16_t dimension_ = std::numeric_limits<uint16_t>::max());

  /**
      * Constructor that reads persistence intervals from file and creates persistence landscape. The format of the
    *input file is the following: in each line we put birth-death pair. Last line is assumed
      * to be empty. Even if the points within a line are not ordered, they will be ordered while the input is read. The
    *additional parameters of this procedure are: ranges of grid, resolution of a grid
      * and the dimension of intervals that are need to be read from a file (in case of Gudhi format files).
     **/
  Persistence_landscape_on_grid(const char* filename, double grid_min_, double grid_max_, size_t number_of_points_,
                                uint16_t dimension_ = std::numeric_limits<uint16_t>::max());

  /**
       * Constructor that reads persistence intervals from file and creates persistence landscape. The format of the
    *input file is the following: in each line we put birth-death pair. Last line is assumed
       * to be empty. Even if the points within a line are not ordered, they will be ordered while the input is read.
    *The additional parameter is the resolution of a grid and the number of landscape
       * functions to be created. The remaining parameters are calculated based on data.
      **/
  Persistence_landscape_on_grid(const char* filename, size_t number_of_points, unsigned number_of_levels_of_landscape,
                                uint16_t dimension = std::numeric_limits<uint16_t>::max());

  /**
     * Constructor that reads persistence intervals from file and creates persistence landscape. The format of the input
    *file is the following: in each line we put birth-death pair. Last line is assumed
     * to be empty. Even if the points within a line are not ordered, they will be ordered while the input is read. The
    *additional parameter is the resolution of a grid. The last parameter is the dimension
     * of a persistence to read from the file. If your file contains only persistence pair in a single dimension, please
    *set it up to std::numeric_limits<unsigned>::max().
     * The remaining parameters are calculated based on data.
    **/
  Persistence_landscape_on_grid(const char* filename, size_t number_of_points,
                                uint16_t dimension = std::numeric_limits<uint16_t>::max());

  /**
   * This procedure loads a landscape from file. It erase all the data that was previously stored in this landscape.
  **/
  void load_landscape_from_file(const char* filename);

  /**
   * The procedure stores a landscape to a file. The file can be later used by a procedure load_landscape_from_file.
  **/
  void print_to_file(const char* filename) const;

  /**
   * This function compute integral of the landscape (defined formally as sum of integrals on R of all landscape
   *functions)
  **/
  double compute_integral_of_landscape() const {
    size_t maximal_level = this->number_of_nonzero_levels();
    double result = 0;
    for (size_t i = 0; i != maximal_level; ++i) {
      result += this->compute_integral_of_landscape(i);
    }
    return result;
  }

  /**
       * This function compute integral of the 'level'-level of a landscape.
      **/
  double compute_integral_of_landscape(size_t level) const {
    bool dbg = false;
    double result = 0;
    double dx = (this->grid_max - this->grid_min) / static_cast<double>(this->values_of_landscapes.size() - 1);

    if (dbg) {
      std::clog << "this->grid_max : " << this->grid_max << std::endl;
      std::clog << "this->grid_min : " << this->grid_min << std::endl;
      std::clog << "this->values_of_landscapes.size() : " << this->values_of_landscapes.size() << std::endl;
      getchar();
    }

    double previous_x = this->grid_min - dx;
    double previous_y = 0;
    for (size_t i = 0; i != this->values_of_landscapes.size(); ++i) {
      double current_x = previous_x + dx;
      double current_y = 0;
      if (this->values_of_landscapes[i].size() > level) current_y = this->values_of_landscapes[i][level];

      if (dbg) {
        std::clog << "this->values_of_landscapes[i].size() : " << this->values_of_landscapes[i].size()
                  << " , level : " << level << std::endl;
        if (this->values_of_landscapes[i].size() > level)
          std::clog << "this->values_of_landscapes[i][level] : " << this->values_of_landscapes[i][level] << std::endl;
        std::clog << "previous_y : " << previous_y << std::endl;
        std::clog << "current_y : " << current_y << std::endl;
        std::clog << "dx : " << dx << std::endl;
        std::clog << "0.5*dx*( previous_y + current_y ); " << 0.5 * dx * (previous_y + current_y) << std::endl;
      }

      result += 0.5 * dx * (previous_y + current_y);
      previous_x = current_x;
      previous_y = current_y;
    }
    return result;
  }

  /**
   * This function compute integral of the landscape p-th power of a landscape (defined formally as sum of integrals on
   *R of p-th powers of all landscape functions)
  **/
  double compute_integral_of_landscape(double p) const {
    size_t maximal_level = this->number_of_nonzero_levels();
    double result = 0;
    for (size_t i = 0; i != maximal_level; ++i) {
      result += this->compute_integral_of_landscape(p, i);
    }
    return result;
  }

  /**
       * This function compute integral of the landscape p-th power of a level of a landscape (defined formally as sum
    *of integrals on R of p-th powers of all landscape functions)
      **/
  double compute_integral_of_landscape(double p, size_t level) const {
    bool dbg = false;

    double result = 0;
    double dx = (this->grid_max - this->grid_min) / static_cast<double>(this->values_of_landscapes.size() - 1);
    double previous_x = this->grid_min;
    double previous_y = 0;
    if (this->values_of_landscapes[0].size() > level) previous_y = this->values_of_landscapes[0][level];

    if (dbg) {
      std::clog << "dx : " << dx << std::endl;
      std::clog << "previous_x : " << previous_x << std::endl;
      std::clog << "previous_y : " << previous_y << std::endl;
      std::clog << "power : " << p << std::endl;
      getchar();
    }

    for (size_t i = 0; i != this->values_of_landscapes.size(); ++i) {
      double current_x = previous_x + dx;
      double current_y = 0;
      if (this->values_of_landscapes[i].size() > level) current_y = this->values_of_landscapes[i][level];

      if (dbg) std::clog << "current_y : " << current_y << std::endl;

      if (current_y == previous_y) continue;

      std::pair<double, double> coef =
          compute_parameters_of_a_line(std::make_pair(previous_x, previous_y), std::make_pair(current_x, current_y));
      double a = coef.first;
      double b = coef.second;

      if (dbg) {
        std::clog << "A line passing through points : (" << previous_x << "," << previous_y << ") and (" << current_x
                  << "," << current_y << ") is : " << a << "x+" << b << std::endl;
      }

      // In this interval, the landscape has a form f(x) = ax+b. We want to compute integral of (ax+b)^p = 1/a *
      // (ax+b)^{p+1}/(p+1)
      double value_to_add = 0;
      if (a != 0) {
        value_to_add = 1 / (a * (p + 1)) * (pow((a * current_x + b), p + 1) - pow((a * previous_x + b), p + 1));
      } else {
        value_to_add = (current_x - previous_x) * (pow(b, p));
      }
      result += value_to_add;
      if (dbg) {
        std::clog << "Increasing result by : " << value_to_add << std::endl;
        std::clog << "result : " << result << std::endl;
        getchar();
      }
      previous_x = current_x;
      previous_y = current_y;
    }
    if (dbg) std::clog << "The total result is : " << result << std::endl;
    return result;
  }

  /**
* Writing landscape into a stream. A i-th level landscape starts with a string "lambda_i". Then the discontinuity points
*of the landscapes follows.
* Shall those points be joined with lines, we will obtain the i-th landscape function.
**/
  friend std::ostream& operator<<(std::ostream& out, const Persistence_landscape_on_grid& land) {
    double dx = (land.grid_max - land.grid_min) / static_cast<double>(land.values_of_landscapes.size() - 1);
    double x = land.grid_min;
    for (size_t i = 0; i != land.values_of_landscapes.size(); ++i) {
      out << x << " : ";
      for (size_t j = 0; j != land.values_of_landscapes[i].size(); ++j) {
        out << land.values_of_landscapes[i][j] << " ";
      }
      out << std::endl;
      x += dx;
    }
    return out;
  }

  template <typename oper>
  friend Persistence_landscape_on_grid operation_on_pair_of_landscapes_on_grid(
      const Persistence_landscape_on_grid& land1, const Persistence_landscape_on_grid& land2);

  /**
   * A function that computes the value of a landscape at a given point. The parameters of the function are: unsigned
   *level and double x.
   * The procedure will compute the value of the level-landscape at the point x.
  **/
  double compute_value_at_a_given_point(unsigned level, double x) const {
    bool dbg = false;
    if ((x < this->grid_min) || (x > this->grid_max)) return 0;

    // find a position of a vector closest to x:
    double dx = (this->grid_max - this->grid_min) / static_cast<double>(this->values_of_landscapes.size() - 1);
    size_t position = size_t((x - this->grid_min) / dx);

    if (dbg) {
      std::clog << "This is a procedure compute_value_at_a_given_point \n";
      std::clog << "level : " << level << std::endl;
      std::clog << "x : " << x << std::endl;
      std::clog << "position : " << position << std::endl;
    }
    // check if we are not exactly in the grid point:
    if (almost_equal(position * dx + this->grid_min, x)) {
      if (this->values_of_landscapes[position].size() < level) {
        return this->values_of_landscapes[position][level];
      } else {
        return 0;
      }
    }
    // in the other case, approximate with a line:
    std::pair<double, double> line;
    if ((this->values_of_landscapes[position].size() > level) &&
        (this->values_of_landscapes[position + 1].size() > level)) {
      line = compute_parameters_of_a_line(
          std::make_pair(position * dx + this->grid_min, this->values_of_landscapes[position][level]),
          std::make_pair((position + 1) * dx + this->grid_min, this->values_of_landscapes[position + 1][level]));
    } else {
      if ((this->values_of_landscapes[position].size() > level) ||
          (this->values_of_landscapes[position + 1].size() > level)) {
        if ((this->values_of_landscapes[position].size() > level)) {
          line = compute_parameters_of_a_line(
              std::make_pair(position * dx + this->grid_min, this->values_of_landscapes[position][level]),
              std::make_pair((position + 1) * dx + this->grid_min, 0));
        } else {
          line = compute_parameters_of_a_line(
              std::make_pair(position * dx + this->grid_min, 0),
              std::make_pair((position + 1) * dx + this->grid_min, this->values_of_landscapes[position + 1][level]));
        }
      } else {
        return 0;
      }
    }
    // compute the value of the linear function parametrized by line on a point x:
    return line.first * x + line.second;
  }

 public:
  /**
   *\private A function that compute sum of two landscapes.
  **/
  friend Persistence_landscape_on_grid add_two_landscapes(const Persistence_landscape_on_grid& land1,
                                                          const Persistence_landscape_on_grid& land2) {
    return operation_on_pair_of_landscapes_on_grid<std::plus<double> >(land1, land2);
  }

  /**
       *\private A function that compute difference of two landscapes.
      **/
  friend Persistence_landscape_on_grid subtract_two_landscapes(const Persistence_landscape_on_grid& land1,
                                                               const Persistence_landscape_on_grid& land2) {
    return operation_on_pair_of_landscapes_on_grid<std::minus<double> >(land1, land2);
  }

  /**
   * An operator +, that compute sum of two landscapes.
  **/
  friend Persistence_landscape_on_grid operator+(const Persistence_landscape_on_grid& first,
                                                 const Persistence_landscape_on_grid& second) {
    return add_two_landscapes(first, second);
  }

  /**
   * An operator -, that compute difference of two landscapes.
  **/
  friend Persistence_landscape_on_grid operator-(const Persistence_landscape_on_grid& first,
                                                 const Persistence_landscape_on_grid& second) {
    return subtract_two_landscapes(first, second);
  }

  /**
   * An operator * that allows multiplication of a landscape by a real number.
  **/
  friend Persistence_landscape_on_grid operator*(const Persistence_landscape_on_grid& first, double con) {
    return first.multiply_lanscape_by_real_number_not_overwrite(con);
  }

  /**
   * An operator * that allows multiplication of a landscape by a real number (order of parameters swapped).
  **/
  friend Persistence_landscape_on_grid operator*(double con, const Persistence_landscape_on_grid& first) {
    return first.multiply_lanscape_by_real_number_not_overwrite(con);
  }

  friend bool check_if_defined_on_the_same_domain(const Persistence_landscape_on_grid& land1,
                                                  const Persistence_landscape_on_grid& land2) {
    if (land1.values_of_landscapes.size() != land2.values_of_landscapes.size()) return false;
    if (land1.grid_min != land2.grid_min) return false;
    if (land1.grid_max != land2.grid_max) return false;
    return true;
  }

  /**
   * Operator +=. The second parameter is persistence landscape.
  **/
  Persistence_landscape_on_grid operator+=(const Persistence_landscape_on_grid& rhs) {
    *this = *this + rhs;
    return *this;
  }

  /**
   * Operator -=. The second parameter is persistence landscape.
  **/
  Persistence_landscape_on_grid operator-=(const Persistence_landscape_on_grid& rhs) {
    *this = *this - rhs;
    return *this;
  }

  /**
   * Operator *=. The second parameter is a real number by which the y values of all landscape functions are multiplied.
   *The x-values remain unchanged.
  **/
  Persistence_landscape_on_grid operator*=(double x) {
    *this = *this * x;
    return *this;
  }

  /**
   * Operator /=. The second parameter is a real number.
  **/
  Persistence_landscape_on_grid operator/=(double x) {
    if (x == 0) throw("In operator /=, division by 0. Program terminated.");
    *this = *this * (1 / x);
    return *this;
  }

  /**
   * An operator to compare two persistence landscapes.
  **/
  bool operator==(const Persistence_landscape_on_grid& rhs) const {
    bool dbg = true;
    if (this->values_of_landscapes.size() != rhs.values_of_landscapes.size()) {
      if (dbg) std::clog << "values_of_landscapes of incompatible sizes\n";
      return false;
    }
    if (!almost_equal(this->grid_min, rhs.grid_min)) {
      if (dbg) std::clog << "grid_min not equal\n";
      return false;
    }
    if (!almost_equal(this->grid_max, rhs.grid_max)) {
      if (dbg) std::clog << "grid_max not equal\n";
      return false;
    }
    for (size_t i = 0; i != this->values_of_landscapes.size(); ++i) {
      for (size_t aa = 0; aa != this->values_of_landscapes[i].size(); ++aa) {
        if (!almost_equal(this->values_of_landscapes[i][aa], rhs.values_of_landscapes[i][aa])) {
          if (dbg) {
            std::clog << "Problem in the position : " << i << " of values_of_landscapes. \n";
            std::clog << this->values_of_landscapes[i][aa] << " " << rhs.values_of_landscapes[i][aa] << std::endl;
          }
          return false;
        }
      }
    }
    return true;
  }

  /**
       * An operator to compare two persistence landscapes.
      **/
  bool operator!=(const Persistence_landscape_on_grid& rhs) const { return !((*this) == rhs); }

  /**
   * Computations of maximum (y) value of landscape.
  **/
  double compute_maximum() const {
    // since the function can only be entirely positive or negative, the maximal value will be an extremal value in the
    // arrays:
    double max_value = -std::numeric_limits<double>::max();
    for (size_t i = 0; i != this->values_of_landscapes.size(); ++i) {
      if (this->values_of_landscapes[i].size()) {
        if (this->values_of_landscapes[i][0] > max_value) max_value = this->values_of_landscapes[i][0];
        if (this->values_of_landscapes[i][this->values_of_landscapes[i].size() - 1] > max_value)
          max_value = this->values_of_landscapes[i][this->values_of_landscapes[i].size() - 1];
      }
    }
    return max_value;
  }

  /**
       * Computations of minimum and maximum value of landscape.
      **/
  std::pair<double, double> compute_minimum_maximum() const {
    // since the function can only be entirely positive or negative, the maximal value will be an extremal value in the
    // arrays:
    double max_value = -std::numeric_limits<double>::max();
    double min_value = 0;
    for (size_t i = 0; i != this->values_of_landscapes.size(); ++i) {
      if (this->values_of_landscapes[i].size()) {
        if (this->values_of_landscapes[i][0] > max_value) max_value = this->values_of_landscapes[i][0];
        if (this->values_of_landscapes[i][this->values_of_landscapes[i].size() - 1] > max_value)
          max_value = this->values_of_landscapes[i][this->values_of_landscapes[i].size() - 1];

        if (this->values_of_landscapes[i][0] < min_value) min_value = this->values_of_landscapes[i][0];
        if (this->values_of_landscapes[i][this->values_of_landscapes[i].size() - 1] < min_value)
          min_value = this->values_of_landscapes[i][this->values_of_landscapes[i].size() - 1];
      }
    }
    return std::make_pair(min_value, max_value);
  }

  /**
       * This procedure returns x-range of a given level persistence landscape. If a default value is used, the x-range
       * of 0th level landscape is given (and this range contains the ranges of all other landscapes).
      **/
  std::pair<double, double> get_x_range(size_t level = 0) const {
    return std::make_pair(this->grid_min, this->grid_max);
  }

  /**
   * This procedure returns y-range of a persistence landscape. If a default value is used, the y-range
   * of 0th level landscape is given (and this range contains the ranges of all other landscapes).
  **/
  std::pair<double, double> get_y_range(size_t level = 0) const { return this->compute_minimum_maximum(); }

  /**
   * This function computes maximal lambda for which lambda-level landscape is nonzero.
  **/
  size_t number_of_nonzero_levels() const {
    size_t result = 0;
    for (size_t i = 0; i != this->values_of_landscapes.size(); ++i) {
      if (this->values_of_landscapes[i].size() > result) result = this->values_of_landscapes[i].size();
    }
    return result;
  }

  /**
   * Computations of a \f$L^i\f$ norm of landscape, where i is the input parameter.
  **/
  double compute_norm_of_landscape(double i) const {
    std::vector<std::pair<double, double> > p;
    Persistence_landscape_on_grid l(p, this->grid_min, this->grid_max, this->values_of_landscapes.size() - 1);

    if (i < std::numeric_limits<double>::max()) {
      return compute_distance_of_landscapes_on_grid(*this, l, i);
    } else {
      return compute_max_norm_distance_of_landscapes(*this, l);
    }
  }

  /**
   * An operator to compute the value of a landscape in the level 'level' at the argument 'x'.
  **/
  double operator()(unsigned level, double x) const { return this->compute_value_at_a_given_point(level, x); }

  /**
   * Computations of \f$L^{\infty}\f$ distance between two landscapes.
  **/
  friend double compute_max_norm_distance_of_landscapes(const Persistence_landscape_on_grid& first,
                                                        const Persistence_landscape_on_grid& second);

  /**
   * Function to compute absolute value of a PL function. The representation of persistence landscapes allow to store
   *general PL-function. When computing distance between two landscapes, we compute difference between
   * them. In this case, a general PL-function with negative value can appear as a result. Then in order to compute
   *distance, we need to take its absolute value. This is the purpose of this procedure.
  **/
  void abs() {
    for (size_t i = 0; i != this->values_of_landscapes.size(); ++i) {
      for (size_t j = 0; j != this->values_of_landscapes[i].size(); ++j) {
        this->values_of_landscapes[i][j] = std::abs(this->values_of_landscapes[i][j]);
      }
    }
  }

  /**
   * Computes the number of landscape functions.
  **/
  size_t size() const { return this->number_of_nonzero_levels(); }

  /**
   *  Compute maximal value of lambda-level landscape.
  **/
  double find_max(unsigned lambda) const {
    double max_value = -std::numeric_limits<double>::max();
    for (size_t i = 0; i != this->values_of_landscapes.size(); ++i) {
      if (this->values_of_landscapes[i].size() > lambda) {
        if (this->values_of_landscapes[i][lambda] > max_value) max_value = this->values_of_landscapes[i][lambda];
      }
    }
    return max_value;
  }

  /**
   * Function to compute inner (scalar) product of two landscapes.
  **/
  friend double compute_inner_product(const Persistence_landscape_on_grid& l1,
                                      const Persistence_landscape_on_grid& l2) {
    if (!check_if_defined_on_the_same_domain(l1, l2))
      throw "Landscapes are not defined on the same grid, the program will now terminate";
    size_t maximal_level = l1.number_of_nonzero_levels();
    double result = 0;
    for (size_t i = 0; i != maximal_level; ++i) {
      result += compute_inner_product(l1, l2, i);
    }
    return result;
  }

  /**
   * Function to compute inner (scalar) product of given levels of two landscapes.
  **/
  friend double compute_inner_product(const Persistence_landscape_on_grid& l1, const Persistence_landscape_on_grid& l2,
                                      size_t level) {
    bool dbg = false;

    if (!check_if_defined_on_the_same_domain(l1, l2))
      throw "Landscapes are not defined on the same grid, the program will now terminate";
    double result = 0;

    double dx = (l1.grid_max - l1.grid_min) / static_cast<double>(l1.values_of_landscapes.size() - 1);

    double previous_x = l1.grid_min - dx;
    double previous_y_l1 = 0;
    double previous_y_l2 = 0;
    for (size_t i = 0; i != l1.values_of_landscapes.size(); ++i) {
      if (dbg) std::clog << "i : " << i << std::endl;

      double current_x = previous_x + dx;
      double current_y_l1 = 0;
      if (l1.values_of_landscapes[i].size() > level) current_y_l1 = l1.values_of_landscapes[i][level];

      double current_y_l2 = 0;
      if (l2.values_of_landscapes[i].size() > level) current_y_l2 = l2.values_of_landscapes[i][level];

      if (dbg) {
        std::clog << "previous_x  : " << previous_x << std::endl;
        std::clog << "previous_y_l1 : " << previous_y_l1 << std::endl;
        std::clog << "current_y_l1 : " << current_y_l1 << std::endl;
        std::clog << "previous_y_l2 : " << previous_y_l2 << std::endl;
        std::clog << "current_y_l2 : " << current_y_l2 << std::endl;
      }

      std::pair<double, double> l1_coords = compute_parameters_of_a_line(std::make_pair(previous_x, previous_y_l1),
                                                                         std::make_pair(current_x, current_y_l1));
      std::pair<double, double> l2_coords = compute_parameters_of_a_line(std::make_pair(previous_x, previous_y_l2),
                                                                         std::make_pair(current_x, current_y_l2));

      // let us assume that the first line is of a form y = ax+b, and the second one is of a form y = cx + d. Then here
      // are a,b,c,d:
      double a = l1_coords.first;
      double b = l1_coords.second;

      double c = l2_coords.first;
      double d = l2_coords.second;

      if (dbg) {
        std::clog << "Here are the formulas for a line: \n";
        std::clog << "a : " << a << std::endl;
        std::clog << "b : " << b << std::endl;
        std::clog << "c : " << c << std::endl;
        std::clog << "d : " << d << std::endl;
      }

      // now, to compute the inner product in this interval we need to compute the integral of (ax+b)(cx+d) = acx^2 +
      // (ad+bc)x + bd in the interval from previous_x to current_x:
      // The integral is ac/3*x^3 + (ac+bd)/2*x^2 + bd*x

      double added_value = (a * c / 3 * current_x * current_x * current_x +
                            (a * d + b * c) / 2 * current_x * current_x + b * d * current_x) -
                           (a * c / 3 * previous_x * previous_x * previous_x +
                            (a * d + b * c) / 2 * previous_x * previous_x + b * d * previous_x);

      if (dbg) {
        std::clog << "Value of the integral on the left end i.e. : " << previous_x << " is : "
                  << a * c / 3 * previous_x * previous_x * previous_x + (a * d + b * c) / 2 * previous_x * previous_x +
                         b * d * previous_x
                  << std::endl;
        std::clog << "Value of the integral on the right end i.e. : " << current_x << " is "
                  << a * c / 3 * current_x * current_x * current_x + (a * d + b * c) / 2 * current_x * current_x +
                         b * d * current_x
                  << std::endl;
      }

      result += added_value;

      if (dbg) {
        std::clog << "added_value : " << added_value << std::endl;
        std::clog << "result : " << result << std::endl;
        getchar();
      }

      previous_x = current_x;
      previous_y_l1 = current_y_l1;
      previous_y_l2 = current_y_l2;
    }
    return result;
  }

  /**
   * Computations of \f$L^{p}\f$ distance between two landscapes on a grid. p is the parameter of the procedure.
   * FIXME: Note that, due to the grid representation, the method below may give non--accurate results in case when the
   *landscape P and Q the difference of which we want to compute
   * are intersecting. This is a consequence of a general way they are computed. In the future, an integral of absolute
   *value of a difference of P and Q will be given as a separated
   * function to fix that inaccuracy.
  **/
  friend double compute_distance_of_landscapes_on_grid(const Persistence_landscape_on_grid& first,
                                                       const Persistence_landscape_on_grid& second, double p) {
    bool dbg = false;
    // This is what we want to compute: (\int_{- \infty}^{+\infty}| first-second |^p)^(1/p). We will do it one step at a
    // time:

    if (dbg) {
      std::clog << "first : " << first << std::endl;
      std::clog << "second : " << second << std::endl;
      getchar();
    }

    // first-second :
    Persistence_landscape_on_grid lan = first - second;

    if (dbg) {
      std::clog << "Difference : " << lan << std::endl;
    }

    //| first-second |:
    lan.abs();

    if (dbg) {
      std::clog << "Abs : " << lan << std::endl;
    }

    if (p < std::numeric_limits<double>::max()) {
      // \int_{- \infty}^{+\infty}| first-second |^p
      double result;
      if (p != 1) {
        if (dbg) {
          std::clog << "p : " << p << std::endl;
          getchar();
        }
        result = lan.compute_integral_of_landscape(p);
        if (dbg) {
          std::clog << "integral : " << result << std::endl;
          getchar();
        }
      } else {
        result = lan.compute_integral_of_landscape();
        if (dbg) {
          std::clog << "integral, without power : " << result << std::endl;
          getchar();
        }
      }
      // (\int_{- \infty}^{+\infty}| first-second |^p)^(1/p)
      return pow(result, 1.0 / p);
    } else {
      // p == infty
      return lan.compute_maximum();
    }
  }

  // Functions that are needed for that class to implement the concept.

  /**
   * The number of projections to R is defined to the number of nonzero landscape functions. I-th projection is an
   *integral of i-th landscape function over whole R.
   * This function is required by the Real_valued_topological_data concept.
   * At the moment this function is not tested, since it is quite likely to be changed in the future. Given this, when
   *using it, keep in mind that it
   * will be most likely changed in the next versions.
  **/
  double project_to_R(int number_of_function) const {
    return this->compute_integral_of_landscape((size_t)number_of_function);
  }

  /**
   * The function gives the number of possible projections to R. This function is required by the
   *Real_valued_topological_data concept.
  **/
  size_t number_of_projections_to_R() const { return number_of_functions_for_projections_to_reals; }

  /**
   * This function produce a vector of doubles based on a landscape. It is required in a concept
   * Vectorized_topological_data
  */
  std::vector<double> vectorize(int number_of_function) const {
    // TODO(PD) think of something smarter over here
    if ((number_of_function < 0) || ((size_t)number_of_function >= this->values_of_landscapes.size())) {
      throw "Wrong number of function\n";
    }
    std::vector<double> v(this->values_of_landscapes.size());
    for (size_t i = 0; i != this->values_of_landscapes.size(); ++i) {
      v[i] = 0;
      if (this->values_of_landscapes[i].size() > (size_t)number_of_function) {
        v[i] = this->values_of_landscapes[i][number_of_function];
      }
    }
    return v;
  }

  /**
   * This function return the number of functions that allows vectorization of persistence landscape. It is required in
   *a concept Vectorized_topological_data.
   **/
  size_t number_of_vectorize_functions() const { return number_of_functions_for_vectorization; }

  /**
   * A function to compute averaged persistence landscape on a grid, based on vector of persistence landscapes on grid.
   * This function is required by Topological_data_with_averages concept.
  **/
  void compute_average(const std::vector<Persistence_landscape_on_grid*>& to_average) {
    bool dbg = false;
    // After execution of this procedure, the average is supposed to be in the current object. To make sure that this is
    // the case, we need to do some cleaning first.
    this->values_of_landscapes.clear();
    this->grid_min = this->grid_max = 0;

    // if there is nothing to average, then the average is a zero landscape.
    if (to_average.size() == 0) return;

    // now we need to check if the grids in all objects of to_average are the same:
    for (size_t i = 0; i != to_average.size(); ++i) {
      if (!check_if_defined_on_the_same_domain(*(to_average[0]), *(to_average[i])))
        throw "Two grids are not compatible";
    }

    this->values_of_landscapes = std::vector<std::vector<double> >((to_average[0])->values_of_landscapes.size());
    this->grid_min = (to_average[0])->grid_min;
    this->grid_max = (to_average[0])->grid_max;

    if (dbg) {
      std::clog << "Computations of average. The data from the current landscape have been cleared. We are ready to do "
                   "the computations. \n";
    }

    // for every point in the grid:
    for (size_t grid_point = 0; grid_point != (to_average[0])->values_of_landscapes.size(); ++grid_point) {
      // set up a vector of the correct size:
      size_t maximal_size_of_vector = 0;
      for (size_t land_no = 0; land_no != to_average.size(); ++land_no) {
        if ((to_average[land_no])->values_of_landscapes[grid_point].size() > maximal_size_of_vector)
          maximal_size_of_vector = (to_average[land_no])->values_of_landscapes[grid_point].size();
      }
      this->values_of_landscapes[grid_point] = std::vector<double>(maximal_size_of_vector);

      if (dbg) {
        std::clog << "We are considering the point : " << grid_point
                  << " of the grid. In this point, there are at most : " << maximal_size_of_vector
                  << " nonzero landscape functions \n";
      }

      // and compute an arithmetic average:
      for (size_t land_no = 0; land_no != to_average.size(); ++land_no) {
        // summing:
        for (size_t i = 0; i != (to_average[land_no])->values_of_landscapes[grid_point].size(); ++i) {
          // compute the average in a smarter way.
          this->values_of_landscapes[grid_point][i] += (to_average[land_no])->values_of_landscapes[grid_point][i];
        }
      }
      // normalizing:
      for (size_t i = 0; i != this->values_of_landscapes[grid_point].size(); ++i) {
        this->values_of_landscapes[grid_point][i] /= static_cast<double>(to_average.size());
      }
    }
  }  // compute_average

  /**
  * A function to compute distance between persistence landscape on a grid.
  * The parameter of this function is a Persistence_landscape_on_grid.
  * This function is required in Topological_data_with_distances concept.
  * For max norm distance, set power to std::numeric_limits<double>::max()
  **/
  double distance(const Persistence_landscape_on_grid& second, double power = 1) const {
    if (power < std::numeric_limits<double>::max()) {
      return compute_distance_of_landscapes_on_grid(*this, second, power);
    } else {
      return compute_max_norm_distance_of_landscapes(*this, second);
    }
  }

  /**
  * A function to compute scalar product of persistence landscape on a grid.
  * The parameter of this function is a Persistence_landscape_on_grid.
  * This function is required in Topological_data_with_scalar_product concept.
  **/
  double compute_scalar_product(const Persistence_landscape_on_grid& second) {
    return compute_inner_product((*this), second);
  }

  // end of implementation of functions needed for concepts.

  /**
  * A function that returns values of landscapes. It can be used for visualization
  **/
  std::vector<std::vector<double> > output_for_visualization() const { return this->values_of_landscapes; }

  /**
  * function used to create a gnuplot script for visualization of landscapes. Over here we need to specify which
  *landscapes do we want to plot.
  * In addition, the user may specify the range (min and max) where landscape is plot. The default values for min and
  *max are std::numeric_limits<double>::max(). If the procedure detect those
  * values, it will determine the range so that the whole landscape is supported there. If at least one min or max value
  *is different from std::numeric_limits<double>::max(), then the values
  * provided by the user will be used.
  **/
  void plot(const char* filename, size_t from_, size_t to_) const {
    this->plot(filename, std::numeric_limits<double>::max(), std::numeric_limits<double>::max(),
               std::numeric_limits<double>::max(), std::numeric_limits<double>::max(), from_, to_);
  }

  /**
  * function used to create a gnuplot script for visualization of landscapes. Over here we can restrict also x and y
  *range of the landscape.
  **/
  void plot(const char* filename, double min_x = std::numeric_limits<double>::max(),
            double max_x = std::numeric_limits<double>::max(), double min_y = std::numeric_limits<double>::max(),
            double max_y = std::numeric_limits<double>::max(), size_t from_ = std::numeric_limits<size_t>::max(),
            size_t to_ = std::numeric_limits<size_t>::max()) const;

 protected:
  double grid_min;
  double grid_max;
  std::vector<std::vector<double> > values_of_landscapes;
  size_t number_of_functions_for_vectorization;
  size_t number_of_functions_for_projections_to_reals;

  void set_up_numbers_of_functions_for_vectorization_and_projections_to_reals() {
    // warning, this function can be only called after filling in the values_of_landscapes vector.
    this->number_of_functions_for_vectorization = this->values_of_landscapes.size();
    this->number_of_functions_for_projections_to_reals = this->values_of_landscapes.size();
  }
  void set_up_values_of_landscapes(const std::vector<std::pair<double, double> >& p, double grid_min_, double grid_max_,
                                   size_t number_of_points_,
                                   unsigned number_of_levels = std::numeric_limits<unsigned>::max());
  Persistence_landscape_on_grid multiply_lanscape_by_real_number_not_overwrite(double x) const;
};

void Persistence_landscape_on_grid::set_up_values_of_landscapes(const std::vector<std::pair<double, double> >& p,
                                                                double grid_min_, double grid_max_,
                                                                size_t number_of_points_, unsigned number_of_levels) {
  bool dbg = false;
  if (dbg) {
    std::clog << "Here is the procedure : set_up_values_of_landscapes. The parameters are : grid_min_ : " << grid_min_
              << ", grid_max_ : " << grid_max_ << ", number_of_points_ : " << number_of_points_
              << ", number_of_levels: " << number_of_levels << std::endl;
    std::clog << "Here are the intervals at our disposal : \n";
    for (size_t i = 0; i != p.size(); ++i) {
      std::clog << p[i].first << " , " << p[i].second << std::endl;
    }
  }

  if ((grid_min_ == std::numeric_limits<double>::max()) || (grid_max_ == std::numeric_limits<double>::max())) {
    // in this case, we need to find grid_min_ and grid_min_ based on the data.
    double min = std::numeric_limits<double>::max();
    double max = std::numeric_limits<double>::min();
    for (size_t i = 0; i != p.size(); ++i) {
      if (p[i].first < min) min = p[i].first;
      if (p[i].second > max) max = p[i].second;
    }
    if (grid_min_ == std::numeric_limits<double>::max()) {
      grid_min_ = min;
    } else {
      // in this case grid_max_ == std::numeric_limits<double>::max()
      grid_max_ = max;
    }
  }

  // if number_of_levels == std::numeric_limits<size_t>::max(), then we will have all the nonzero values of landscapes,
  // and will store them in a vector
  // if number_of_levels != std::numeric_limits<size_t>::max(), then we will use those vectors as heaps.
  this->values_of_landscapes = std::vector<std::vector<double> >(number_of_points_ + 1);

  this->grid_min = grid_min_;
  this->grid_max = grid_max_;

  if (grid_max_ <= grid_min_) {
    throw "Wrong parameters of grid_min and grid_max given to the procedure. The program will now terminate.\n";
  }

  double dx = (grid_max_ - grid_min_) / static_cast<double>(number_of_points_);
  // for every interval in the diagram:
  for (size_t int_no = 0; int_no != p.size(); ++int_no) {
    size_t grid_interval_begin = (p[int_no].first - grid_min_) / dx;
    size_t grid_interval_end = (p[int_no].second - grid_min_) / dx;
    size_t grid_interval_midpoint = (size_t)(0.5 * (grid_interval_begin + grid_interval_end));

    if (dbg) {
      std::clog << "Considering an interval : " << p[int_no].first << "," << p[int_no].second << std::endl;

      std::clog << "grid_interval_begin : " << grid_interval_begin << std::endl;
      std::clog << "grid_interval_end : " << grid_interval_end << std::endl;
      std::clog << "grid_interval_midpoint : " << grid_interval_midpoint << std::endl;
    }

    double landscape_value = dx;
    for (size_t i = grid_interval_begin + 1; i < grid_interval_midpoint; ++i) {
      if (dbg) {
        std::clog << "Adding landscape value (going up) for a point : " << i << " equal : " << landscape_value
                  << std::endl;
      }
      if (number_of_levels != std::numeric_limits<unsigned>::max()) {
        // we have a heap of no more that number_of_levels values.
        // Note that if we are using heaps, we want to know the shortest distance in the heap.
        // This is achieved by putting -distance to the heap.
        if (this->values_of_landscapes[i].size() >= number_of_levels) {
          // in this case, the full heap is build, and we need to check if the landscape_value is not larger than the
          // smallest element in the heap.
          if (-landscape_value < this->values_of_landscapes[i].front()) {
            // if it is, we remove the largest value in the heap, and move on.
            std::pop_heap(this->values_of_landscapes[i].begin(), this->values_of_landscapes[i].end());
            this->values_of_landscapes[i][this->values_of_landscapes[i].size() - 1] = -landscape_value;
            std::push_heap(this->values_of_landscapes[i].begin(), this->values_of_landscapes[i].end());
          }
        } else {
          // in this case we are still filling in the array.
          this->values_of_landscapes[i].push_back(-landscape_value);
          if (this->values_of_landscapes[i].size() == number_of_levels - 1) {
            // this->values_of_landscapes[i].size() == number_of_levels
            // in this case we need to create the heap.
            std::make_heap(this->values_of_landscapes[i].begin(), this->values_of_landscapes[i].end());
          }
        }
      } else {
        // we have vector of all values
        this->values_of_landscapes[i].push_back(landscape_value);
      }
      landscape_value += dx;
    }
    for (size_t i = grid_interval_midpoint; i <= grid_interval_end; ++i) {
      if (landscape_value > 0) {
        if (number_of_levels != std::numeric_limits<unsigned>::max()) {
          // we have a heap of no more that number_of_levels values
          if (this->values_of_landscapes[i].size() >= number_of_levels) {
            // in this case, the full heap is build, and we need to check if the landscape_value is not larger than the
            // smallest element in the heap.
            if (-landscape_value < this->values_of_landscapes[i].front()) {
              // if it is, we remove the largest value in the heap, and move on.
              std::pop_heap(this->values_of_landscapes[i].begin(), this->values_of_landscapes[i].end());
              this->values_of_landscapes[i][this->values_of_landscapes[i].size() - 1] = -landscape_value;
              std::push_heap(this->values_of_landscapes[i].begin(), this->values_of_landscapes[i].end());
            }
          } else {
            // in this case we are still filling in the array.
            this->values_of_landscapes[i].push_back(-landscape_value);
            if (this->values_of_landscapes[i].size() == number_of_levels - 1) {
              // this->values_of_landscapes[i].size() == number_of_levels
              // in this case we need to create the heap.
              std::make_heap(this->values_of_landscapes[i].begin(), this->values_of_landscapes[i].end());
            }
          }
        } else {
          this->values_of_landscapes[i].push_back(landscape_value);
        }

        if (dbg) {
          std::clog << "Adding landscape value (going down) for a point : " << i << " equal : " << landscape_value
                    << std::endl;
        }
      }
      landscape_value -= dx;
    }
  }

  if (number_of_levels != std::numeric_limits<unsigned>::max()) {
    // in this case, vectors are used as heaps. And, since we want to have the smallest element at the top of
    // each heap, we store minus distances. To get if right at the end, we need to multiply each value
    // in the heap by -1 to get real vector of distances.
    for (size_t pt = 0; pt != this->values_of_landscapes.size(); ++pt) {
      for (size_t j = 0; j != this->values_of_landscapes[pt].size(); ++j) {
        this->values_of_landscapes[pt][j] *= -1;
      }
    }
  }

  // and now we need to sort the values:
  for (size_t pt = 0; pt != this->values_of_landscapes.size(); ++pt) {
    std::sort(this->values_of_landscapes[pt].begin(), this->values_of_landscapes[pt].end(), std::greater<double>());
  }
}  // set_up_values_of_landscapes

Persistence_landscape_on_grid::Persistence_landscape_on_grid(const std::vector<std::pair<double, double> >& p,
                                                             double grid_min_, double grid_max_,
                                                             size_t number_of_points_) {
  this->set_up_values_of_landscapes(p, grid_min_, grid_max_, number_of_points_);
}  // Persistence_landscape_on_grid

Persistence_landscape_on_grid::Persistence_landscape_on_grid(const std::vector<std::pair<double, double> >& p,
                                                             double grid_min_, double grid_max_,
                                                             size_t number_of_points_,
                                                             unsigned number_of_levels_of_landscape) {
  this->set_up_values_of_landscapes(p, grid_min_, grid_max_, number_of_points_, number_of_levels_of_landscape);
}

Persistence_landscape_on_grid::Persistence_landscape_on_grid(const char* filename, double grid_min_, double grid_max_,
                                                             size_t number_of_points_, uint16_t dimension) {
  std::vector<std::pair<double, double> > p;
  if (dimension == std::numeric_limits<uint16_t>::max()) {
    p = read_persistence_intervals_in_one_dimension_from_file(filename);
  } else {
    p = read_persistence_intervals_in_one_dimension_from_file(filename, dimension);
  }
  this->set_up_values_of_landscapes(p, grid_min_, grid_max_, number_of_points_);
}

Persistence_landscape_on_grid::Persistence_landscape_on_grid(const char* filename, double grid_min_, double grid_max_,
                                                             size_t number_of_points_,
                                                             unsigned number_of_levels_of_landscape,
                                                             uint16_t dimension) {
  std::vector<std::pair<double, double> > p;
  if (dimension == std::numeric_limits<uint16_t>::max()) {
    p = read_persistence_intervals_in_one_dimension_from_file(filename);
  } else {
    p = read_persistence_intervals_in_one_dimension_from_file(filename, dimension);
  }
  this->set_up_values_of_landscapes(p, grid_min_, grid_max_, number_of_points_, number_of_levels_of_landscape);
}

Persistence_landscape_on_grid::Persistence_landscape_on_grid(const char* filename, size_t number_of_points_,
                                                             uint16_t dimension) {
  std::vector<std::pair<double, double> > p;
  if (dimension == std::numeric_limits<uint16_t>::max()) {
    p = read_persistence_intervals_in_one_dimension_from_file(filename);
  } else {
    p = read_persistence_intervals_in_one_dimension_from_file(filename, dimension);
  }
  double grid_min_ = std::numeric_limits<double>::max();
  double grid_max_ = -std::numeric_limits<double>::max();
  for (size_t i = 0; i != p.size(); ++i) {
    if (p[i].first < grid_min_) grid_min_ = p[i].first;
    if (p[i].second > grid_max_) grid_max_ = p[i].second;
  }
  this->set_up_values_of_landscapes(p, grid_min_, grid_max_, number_of_points_);
}

Persistence_landscape_on_grid::Persistence_landscape_on_grid(const char* filename, size_t number_of_points_,
                                                             unsigned number_of_levels_of_landscape,
                                                             uint16_t dimension) {
  std::vector<std::pair<double, double> > p;
  if (dimension == std::numeric_limits<uint16_t>::max()) {
    p = read_persistence_intervals_in_one_dimension_from_file(filename);
  } else {
    p = read_persistence_intervals_in_one_dimension_from_file(filename, dimension);
  }
  double grid_min_ = std::numeric_limits<double>::max();
  double grid_max_ = -std::numeric_limits<double>::max();
  for (size_t i = 0; i != p.size(); ++i) {
    if (p[i].first < grid_min_) grid_min_ = p[i].first;
    if (p[i].second > grid_max_) grid_max_ = p[i].second;
  }
  this->set_up_values_of_landscapes(p, grid_min_, grid_max_, number_of_points_, number_of_levels_of_landscape);
}

void Persistence_landscape_on_grid::load_landscape_from_file(const char* filename) {
  std::ifstream in;
  in.open(filename);
  // check if the file exist.
  if (!in.good()) {
    std::cerr << "The file : " << filename << " do not exist. The program will now terminate \n";
    throw "The persistence landscape file do not exist. The program will now terminate \n";
  }

  size_t number_of_points_in_the_grid = 0;
  in >> this->grid_min >> this->grid_max >> number_of_points_in_the_grid;

  std::vector<std::vector<double> > v(number_of_points_in_the_grid);
  std::string line;
  std::getline(in, line);
  double number;
  for (size_t i = 0; i != number_of_points_in_the_grid; ++i) {
    // read a line of a file and convert it to a vector.
    std::vector<double> vv;
    std::getline(in, line);
    std::istringstream stream(line);
    while (stream >> number) {
      vv.push_back(number);
    }
    v[i] = vv;
  }
  this->values_of_landscapes = v;
  in.close();
}

void Persistence_landscape_on_grid::print_to_file(const char* filename) const {
  std::ofstream out;
  out.open(filename);

  // first we store the parameters of the grid:
  out << grid_min << std::endl << grid_max << std::endl << this->values_of_landscapes.size() << std::endl;

  // and now in the following lines, the values of this->values_of_landscapes for the following arguments:
  for (size_t i = 0; i != this->values_of_landscapes.size(); ++i) {
    for (size_t j = 0; j != this->values_of_landscapes[i].size(); ++j) {
      out << this->values_of_landscapes[i][j] << " ";
    }
    out << std::endl;
  }

  out.close();
}

void Persistence_landscape_on_grid::plot(const char* filename, double min_x, double max_x, double min_y, double max_y,
                                         size_t from_, size_t to_) const {
  // this program create a gnuplot script file that allows to plot persistence diagram.
  std::ofstream out;

  std::ostringstream gnuplot_script;
  gnuplot_script << filename << "_GnuplotScript";
  out.open(gnuplot_script.str().c_str());

  if (min_x == max_x) {
    std::pair<double, double> min_max = compute_minimum_maximum();
    out << "set xrange [" << this->grid_min << " : " << this->grid_max << "]" << std::endl;
    out << "set yrange [" << min_max.first << " : " << min_max.second << "]" << std::endl;
  } else {
    out << "set xrange [" << min_x << " : " << max_x << "]" << std::endl;
    out << "set yrange [" << min_y << " : " << max_y << "]" << std::endl;
  }

  size_t number_of_nonzero_levels = this->number_of_nonzero_levels();
  double dx = (this->grid_max - this->grid_min) / static_cast<double>(this->values_of_landscapes.size() - 1);

  size_t from = 0;
  if (from_ != std::numeric_limits<size_t>::max()) {
    if (from_ < number_of_nonzero_levels) {
      from = from_;
    } else {
      return;
    }
  }
  size_t to = number_of_nonzero_levels;
  if (to_ != std::numeric_limits<size_t>::max()) {
    if (to_ < number_of_nonzero_levels) {
      to = to_;
    }
  }

  out << "plot ";
  for (size_t lambda = from; lambda != to; ++lambda) {
    out << "     '-' using 1:2 notitle with lp";
    if (lambda + 1 != to) {
      out << ", \\";
    }
    out << std::endl;
  }

  for (size_t lambda = from; lambda != to; ++lambda) {
    double point = this->grid_min;
    for (size_t i = 0; i != this->values_of_landscapes.size(); ++i) {
      double value = 0;
      if (this->values_of_landscapes[i].size() > lambda) {
        value = this->values_of_landscapes[i][lambda];
      }
      out << point << " " << value << std::endl;
      point += dx;
    }
    out << "EOF" << std::endl;
  }
  std::clog << "To visualize, install gnuplot and type the command: gnuplot -persist -e \"load \'"
            << gnuplot_script.str().c_str() << "\'\"" << std::endl;
}

template <typename T>
Persistence_landscape_on_grid operation_on_pair_of_landscapes_on_grid(const Persistence_landscape_on_grid& land1,
                                                                      const Persistence_landscape_on_grid& land2) {
  // first we need to check if the domains are the same:
  if (!check_if_defined_on_the_same_domain(land1, land2)) throw "Two grids are not compatible";

  T oper;
  Persistence_landscape_on_grid result;
  result.values_of_landscapes = std::vector<std::vector<double> >(land1.values_of_landscapes.size());
  result.grid_min = land1.grid_min;
  result.grid_max = land1.grid_max;

  // now we perform the operations:
  for (size_t grid_point = 0; grid_point != land1.values_of_landscapes.size(); ++grid_point) {
    result.values_of_landscapes[grid_point] = std::vector<double>(
        std::max(land1.values_of_landscapes[grid_point].size(), land2.values_of_landscapes[grid_point].size()));
    for (size_t lambda = 0; lambda != std::max(land1.values_of_landscapes[grid_point].size(),
                                               land2.values_of_landscapes[grid_point].size());
         ++lambda) {
      double value1 = 0;
      double value2 = 0;
      if (lambda < land1.values_of_landscapes[grid_point].size())
        value1 = land1.values_of_landscapes[grid_point][lambda];
      if (lambda < land2.values_of_landscapes[grid_point].size())
        value2 = land2.values_of_landscapes[grid_point][lambda];
      result.values_of_landscapes[grid_point][lambda] = oper(value1, value2);
    }
  }

  return result;
}

Persistence_landscape_on_grid Persistence_landscape_on_grid::multiply_lanscape_by_real_number_not_overwrite(
    double x) const {
  Persistence_landscape_on_grid result;
  result.values_of_landscapes = std::vector<std::vector<double> >(this->values_of_landscapes.size());
  result.grid_min = this->grid_min;
  result.grid_max = this->grid_max;

  for (size_t grid_point = 0; grid_point != this->values_of_landscapes.size(); ++grid_point) {
    result.values_of_landscapes[grid_point] = std::vector<double>(this->values_of_landscapes[grid_point].size());
    for (size_t i = 0; i != this->values_of_landscapes[grid_point].size(); ++i) {
      result.values_of_landscapes[grid_point][i] = x * this->values_of_landscapes[grid_point][i];
    }
  }

  return result;
}

double compute_max_norm_distance_of_landscapes(const Persistence_landscape_on_grid& first,
                                               const Persistence_landscape_on_grid& second) {
  double result = 0;

  // first we need to check if first and second is defined on the same domain"
  if (!check_if_defined_on_the_same_domain(first, second)) throw "Two grids are not compatible";

  for (size_t i = 0; i != first.values_of_landscapes.size(); ++i) {
    for (size_t j = 0; j != std::min(first.values_of_landscapes[i].size(), second.values_of_landscapes[i].size());
         ++j) {
      if (result < abs(first.values_of_landscapes[i][j] - second.values_of_landscapes[i][j])) {
        result = abs(first.values_of_landscapes[i][j] - second.values_of_landscapes[i][j]);
      }
    }
    if (first.values_of_landscapes[i].size() ==
        std::min(first.values_of_landscapes[i].size(), second.values_of_landscapes[i].size())) {
      for (size_t j = first.values_of_landscapes[i].size(); j != second.values_of_landscapes[i].size(); ++j) {
        if (result < second.values_of_landscapes[i][j]) result = second.values_of_landscapes[i][j];
      }
    }
    if (second.values_of_landscapes[i].size() ==
        std::min(first.values_of_landscapes[i].size(), second.values_of_landscapes[i].size())) {
      for (size_t j = second.values_of_landscapes[i].size(); j != first.values_of_landscapes[i].size(); ++j) {
        if (result < first.values_of_landscapes[i][j]) result = first.values_of_landscapes[i][j];
      }
    }
  }
  return result;
}

}  // namespace Persistence_representations
}  // namespace Gudhi

#endif  // PERSISTENCE_LANDSCAPE_ON_GRID_H_