// Copyright (c) 2017-2018 INRIA Sophia-Antipolis (France).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
//
// $URL: https://github.com/CGAL/cgal/blob/v6.2-beta1/Surface_mesh_approximation/include/CGAL/Variational_shape_approximation.h $
// $Id: include/CGAL/Variational_shape_approximation.h 431cfb5357d $
// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
//
//
// Author(s)     : Pierre Alliez and Lingjie Zhu


#ifndef CGAL_VARIATIONAL_SHAPE_APPROXIMATION_H
#define CGAL_VARIATIONAL_SHAPE_APPROXIMATION_H

#include <CGAL/license/Surface_mesh_approximation.h>


#include <CGAL/boost/graph/helpers.h>
#include <CGAL/Kernel/global_functions.h>
#include <CGAL/squared_distance_3.h>
#include <CGAL/linear_least_squares_fitting_3.h>
#include <CGAL/array.h>

#include <CGAL/Surface_mesh_approximation/L21_metric_plane_proxy.h>
#include <CGAL/Default.h>
#include <CGAL/tags.h>

#include <CGAL/Polygon_mesh_processing/polygon_soup_to_polygon_mesh.h>

#include <boost/graph/graph_traits.hpp>
#include <boost/graph/adjacency_list.hpp>
#include <boost/graph/dijkstra_shortest_paths.hpp>
#include <boost/graph/subgraph.hpp>
#include <optional>

#include <CGAL/Named_function_parameters.h>
#include <CGAL/boost/graph/named_params_helper.h>

#include <vector>
#include <stack>
#include <queue>
#include <iterator>
#include <cmath>
#include <cstdlib>

#ifdef CGAL_LINKED_WITH_TBB
#include <tbb/parallel_for.h>
#include <tbb/blocked_range.h>
#endif // CGAL_LINKED_WITH_TBB

#ifdef CGAL_SURFACE_MESH_APPROXIMATION_DEBUG
#include <iostream>
#endif

#define CGAL_VSA_INVALID_TAG (std::numeric_limits<std::size_t>::max)()

namespace CGAL {

namespace Surface_mesh_approximation {
/// \ingroup PkgTSMARef
/// @brief Seeding method enumeration for Variational Shape Approximation algorithm.
enum Seeding_method {
  /// Random seeding
  RANDOM,
  /// Incremental seeding
  INCREMENTAL,
  /// Hierarchical seeding
  HIERARCHICAL
};

} // namespace Surface_mesh_approximation

/// \ingroup PkgTSMARef
/// @brief Main class for Variational Shape Approximation algorithm.
/// It is based on \cgalCite{cgal:cad-vsa-04}. For simple use cases, the function `CGAL::Surface_mesh_approximation::approximate_mesh()` might be sufficient.
/// @tparam TriangleMesh a model of `FaceListGraph`
/// @tparam VertexPointMap a `ReadablePropertyMap` with `boost::graph_traits<TriangleMesh>::%vertex_descriptor` as key and `GeomTraits::Point_3` as value type
/// @tparam ErrorMetricProxy a model of `ErrorMetricProxy`
/// @tparam GeomTraits a model of Kernel
/// @tparam Concurrency_tag concurrency tag.
template <typename TriangleMesh,
  typename VertexPointMap,
  typename ErrorMetricProxy = CGAL::Default,
  typename GeomTraits = CGAL::Default,
  typename Concurrency_tag = CGAL::Sequential_tag>
class Variational_shape_approximation {
// public typedefs
public:

  /// \name Types
  /// @{
#ifndef DOXYGEN_RUNNING
  // GeomTraits type
  typedef typename CGAL::Default::Get<
    GeomTraits,
    typename Kernel_traits<
      typename boost::property_traits<VertexPointMap>::value_type
    >::Kernel >::type Geom_traits;
#else
  /// Geometric traits type
  typedef GeomTraits Geom_traits;
#endif
  // ErrorMetricProxy type
#ifndef DOXYGEN_RUNNING
  typedef typename CGAL::Default::Get<ErrorMetricProxy,
    Surface_mesh_approximation::L21_metric_plane_proxy<TriangleMesh, VertexPointMap, Geom_traits> >::type Error_metric;
#else
  /// Error metric for proxy fitting type
  typedef ErrorMetricProxy Error_metric;
#endif
  /// Proxy type
  typedef typename Error_metric::Proxy Proxy;

  /// Indexed triangle type
  typedef std::array<std::size_t, 3> Indexed_triangle;
  /// @}

// private typedefs and data member
private:
  // Geom_traits typedefs
  typedef typename Geom_traits::FT FT;
  typedef typename Geom_traits::Point_3 Point_3;
  typedef typename Geom_traits::Vector_3 Vector_3;
  typedef typename Geom_traits::Segment_3 Segment_3;
  typedef typename Geom_traits::Plane_3 Plane_3;
  typedef typename Geom_traits::Construct_vector_3 Construct_vector_3;
  typedef typename Geom_traits::Construct_point_3 Construct_point_3;
  typedef typename Geom_traits::Construct_scaled_vector_3 Construct_scaled_vector_3;
  typedef typename Geom_traits::Construct_sum_of_vectors_3 Construct_sum_of_vectors_3;
  typedef typename Geom_traits::Construct_translated_point_3 Construct_translated_point_3;
  typedef typename Geom_traits::Construct_cross_product_vector_3 Construct_cross_product_vector_3;
  typedef typename Geom_traits::Collinear_3 Collinear_3;

  // graph_traits typedefs
  typedef typename boost::graph_traits<TriangleMesh>::vertex_descriptor vertex_descriptor;
  typedef typename boost::graph_traits<TriangleMesh>::halfedge_descriptor halfedge_descriptor;
  typedef typename boost::graph_traits<TriangleMesh>::edge_descriptor edge_descriptor;
  typedef typename boost::graph_traits<TriangleMesh>::face_descriptor face_descriptor;

  // internal typedefs
  typedef CGAL::dynamic_vertex_property_t<std::size_t> Vertex_anchor_tag;
  typedef typename boost::property_map<TriangleMesh, Vertex_anchor_tag>::type Vertex_anchor_map;

  typedef CGAL::dynamic_face_property_t<std::size_t> Face_proxy_tag;
  typedef typename boost::property_map<TriangleMesh, Face_proxy_tag>::type Face_proxy_map;

  typedef std::vector<halfedge_descriptor> Boundary_chord;
  typedef typename Boundary_chord::iterator Boundary_chord_iterator;

  /// \cond SKIP_IN_MANUAL
public:
  // The proxy wrapper for approximation.
  struct Proxy_wrapper {
    Proxy_wrapper(const Proxy &p, const std::size_t &i, const face_descriptor s, const FT &e)
      : px(p), idx(i), seed(s), err(e) {}

    Proxy px; // parameterized proxy
    std::size_t idx; // proxy index, maintained to be the same as its position in proxies vector
    face_descriptor seed; // proxy seed
    FT err; // proxy fitting error
  };
  /// \endcond

private:
  // The proxy fitting plane for meshing.
  struct Proxy_plane {
    Proxy_plane(const Plane_3 &p, const Vector_3 &n, const FT &a)
      : plane(p), normal(n), area(a) {}

    Plane_3 plane;
    Vector_3 normal;
    FT area;
  };

  // The face candidate to be queued.
  struct Face_to_integrate {
    Face_to_integrate(const face_descriptor f_, const std::size_t &px_, const FT &err_)
      : f(f_), px(px_), err(err_) {}

    bool operator<(const Face_to_integrate &rhs) const {
      return err > rhs.err;
    }

    face_descriptor f; // face
    std::size_t px; // proxy index
    FT err; // fitting error
  };

  // Proxy error with its index.
  struct Proxy_error {
    Proxy_error(const std::size_t &px_, const FT &err_)
      : px(px_), err(err_) {}

    // in ascending order
    bool operator<(const Proxy_error &rhs) const {
      return err < rhs.err;
    }

    std::size_t px;
    FT err;
  };

  // The anchor attached to a vertex.
  struct Anchor {
    Anchor(const vertex_descriptor vtx_, const Point_3 pos_)
      : vtx(vtx_), pos(pos_) {}

    vertex_descriptor vtx; // The associated vertex.
    Point_3 pos; // The position of the anchor.
  };

  // The boundary cycle of a region.
  // One region may have multiple boundary cycles.
  struct Boundary_cycle {
    Boundary_cycle(const halfedge_descriptor h)
      : he_head(h), num_anchors(0) {}

    halfedge_descriptor he_head; // Heading halfedge of the boundary cycle.
    std::size_t num_anchors; // Number of anchors on the boundary cycle.
  };

  // member variables
  // The triangle mesh.
  const TriangleMesh *m_ptm;
  // The exact number of faces
  const std::size_t m_nb_of_faces;
  // The mesh vertex point map.
  VertexPointMap m_vpoint_map;
  // The approximation object.
  const Error_metric *m_metric;

  Construct_vector_3 vector_functor;
  Construct_scaled_vector_3 scale_functor;
  Construct_sum_of_vectors_3 sum_functor;
  Construct_translated_point_3 translate_point_functor;
  Construct_cross_product_vector_3 cross_product_functor;
  Collinear_3 collinear_functor;

  // Proxies.
  std::vector<Proxy_wrapper> m_proxies;
  // Proxy planes
  std::vector<Proxy_plane> m_px_planes;

  // All anchors.
  std::vector<Anchor> m_anchors;
  // All boundary cycles.
  std::vector<Boundary_cycle> m_bcycles;
  // The indexed triangle approximation.
  std::vector<Indexed_triangle> m_tris;

  // meshing parameters
  FT m_average_edge_length;

  // The face proxy index map.
  Face_proxy_map m_fproxy_map;
  // The attached anchor index of a vertex.
  Vertex_anchor_map m_vanchor_map;

//member functions
public:
  /// \name Construction
  /// @{
  /*!
   * @brief initializes internal data for the approximation.
   * @param tm `CGAL TriangleMesh` on which approximation operates
   * @param vpoint_map vertex point map of the mesh
   * @param error_metric an `ErrorMetricProxy` object
   */
  Variational_shape_approximation(const TriangleMesh &tm,
    const VertexPointMap &vpoint_map,
    const Error_metric &error_metric) :
    m_ptm(&tm),
    m_nb_of_faces(std::distance(faces(tm).first, faces(tm).second)),
    m_vpoint_map(vpoint_map),
    m_metric(&error_metric),
    m_average_edge_length(0.0),
    m_fproxy_map( get(Face_proxy_tag(), *(const_cast<TriangleMesh *>(m_ptm))) ),
    m_vanchor_map( get( Vertex_anchor_tag(), *(const_cast<TriangleMesh *>(m_ptm))) )
  {

    Geom_traits traits;
    vector_functor = traits.construct_vector_3_object();
    scale_functor = traits.construct_scaled_vector_3_object();
    sum_functor = traits.construct_sum_of_vectors_3_object();
    translate_point_functor = traits.construct_translated_point_3_object();
    cross_product_functor = traits.construct_cross_product_vector_3_object();
    collinear_functor = traits.collinear_3_object();
  }
  /// @}

  /// \name Approximation
  /// @{
  /*!
   * @brief initializes the seeds with both maximum number of proxies and minimum error drop stop criteria.
   * The first criterion met stops the seeding.
   * Parameters out of range are ignored.
   * @tparam NamedParameters a sequence of \ref bgl_namedparameters

   * @param np an optional sequence of \ref bgl_namedparameters "Named Parameters" among the ones listed below
   * @return number of proxies initialized

   * \cgalNamedParamsBegin{Seeding Named Parameters}
   *   \cgalParamNBegin{seeding_method}
   *     \cgalParamDescription{the selection of seeding method}
   *     \cgalParamType{`CGAL::Surface_mesh_approximation::Seeding_method`}
   *     \cgalParamDefault{`CGAL::Surface_mesh_approximation::HIERARCHICAL`}
   *   \cgalParamNEnd
   *
   *   \cgalParamNBegin{max_number_of_proxies}
   *     \cgalParamDescription{the maximum number of proxies used to approximate the input mesh}
   *     \cgalParamType{`std::size_t`}
   *     \cgalParamDefault{`num_faces(tm) / 3`, used when `min_error_drop` is also not provided}
   *   \cgalParamNEnd
   *
   *   \cgalParamNBegin{min_error_drop}
   *     \cgalParamDescription{the minimum error drop of the approximation, expressed as
   *                           the ratio between two iterations of proxy addition}
   *     \cgalParamType{`geom_traits::FT`}
   *     \cgalParamDefault{`0.1`, used when `max_number_of_proxies` is also not provided}
   *   \cgalParamNEnd
   *
   *   \cgalParamNBegin{number_of_relaxations}
   *     \cgalParamDescription{the number of relaxation iterations interleaved within seeding}
   *     \cgalParamType{`std::size_t`}
   *     \cgalParamDefault{`5`}
   *   \cgalParamNEnd
   * \cgalNamedParamsEnd
   */
  template <typename NamedParameters>
  std::size_t initialize_seeds(const NamedParameters &np) {
    using parameters::get_parameter;
    using parameters::choose_parameter;

    const Surface_mesh_approximation::Seeding_method method = choose_parameter(
      get_parameter(np, internal_np::seeding_method), Surface_mesh_approximation::HIERARCHICAL);
    std::size_t max_nb_proxies = choose_parameter(
      get_parameter(np, internal_np::max_number_of_proxies), 0);
    FT min_error_drop = choose_parameter(
      get_parameter(np, internal_np::min_error_drop), FT(0.0));
    const std::size_t nb_relaxations = choose_parameter(
      get_parameter(np, internal_np::number_of_relaxations), 5);

    // adjust parameters
    if (max_nb_proxies < (m_nb_of_faces / 3) && max_nb_proxies > 0) {
      if(!(min_error_drop < FT(1.0)) || !(min_error_drop > FT(0.0)))
        min_error_drop = FT(-1.0);
    }
    else {
      max_nb_proxies = m_nb_of_faces / 3;
      if (!(min_error_drop < FT(1.0)) || !(min_error_drop > FT(0.0)))
        min_error_drop = FT(0.1);
    }

    // initialize proxies and the proxy map to prepare for insertion
    bootstrap_from_connected_components();
    if (max_nb_proxies <= m_proxies.size())
      return m_proxies.size();
    switch (method) {
      case Surface_mesh_approximation::RANDOM:
        return init_random(max_nb_proxies, min_error_drop, nb_relaxations);
      case Surface_mesh_approximation::INCREMENTAL:
        return init_incremental(max_nb_proxies, min_error_drop, nb_relaxations);
      case Surface_mesh_approximation::HIERARCHICAL:
        return init_hierarchical(max_nb_proxies, min_error_drop, nb_relaxations);
      default:
        return 0;
    }
  }

  /*!
   * @brief runs the partitioning and fitting processes on the whole surface.
   * @param nb_iterations number of iterations.
   * @return total fitting error
   */
  FT run(std::size_t nb_iterations = 1) {
    for (std::size_t i = 0; i < nb_iterations; ++i) {
      // tag the whole surface
      for(face_descriptor f : faces(*m_ptm))
        put(m_fproxy_map, f, CGAL_VSA_INVALID_TAG);

      partition(m_proxies.begin(), m_proxies.end());
      fit(m_proxies.begin(), m_proxies.end(), Concurrency_tag());
    }

    return compute_total_error();
  }

  /*!
   * @brief calls `run` while error decrease is greater than `cvg_threshold`.
   * @param cvg_threshold the percentage of error change between two successive runs,
   * should be in range `(0, 1)`.
   * @param max_iterations maximum number of iterations allowed
   * @param avg_interval size of error average interval to have smoother convergence curve,
   * if 0 is assigned, 1 is used instead.
   * @return `true` if converged before hitting the maximum iterations.
   */
  bool run_to_convergence(const FT cvg_threshold,
    const std::size_t max_iterations = 100,
    std::size_t avg_interval = 3) {
    if (avg_interval == 0)
      avg_interval = 1;
    FT drop_pct(0.0);
    FT pre_err = compute_total_error();
    for (std::size_t itr_count = 0; itr_count < max_iterations; itr_count += avg_interval) {
      if (pre_err == FT(0.0))
        return true;

      FT avg_err(0.0);
      for (std::size_t i = 0; i < avg_interval; ++i)
        avg_err += run();
      avg_err /= static_cast<FT>(avg_interval);

      drop_pct = (pre_err - avg_err) / pre_err;
      // the error may fluctuate
      if (drop_pct < FT(0.0))
        drop_pct = -drop_pct;
      if (drop_pct < cvg_threshold)
        return true;

      pre_err = avg_err;
    }

    return false;
  }

  /*!
   * @brief computes fitting error of current partition to the proxies.
   * @return total fitting error
   */
  FT compute_total_error() {
    FT sum_error(0.0);
    for(const Proxy_wrapper& pxw : m_proxies)
      sum_error += pxw.err;

#ifdef CGAL_SURFACE_MESH_APPROXIMATION_DEBUG
    static std::size_t count = 0;
    std::cerr << '#' << count++ << ": " << sum_error << std::endl;
#endif

    return sum_error;
  }

  /*!
   * @brief adds proxies to the worst regions one by one.
   * The re-fitting is performed after each proxy is inserted.
   * @param nb_proxies number of proxies to be added
   * @param nb_iterations number of re-fitting iterations
   * @return number of proxies added
   */
  std::size_t add_to_furthest_proxies(const std::size_t nb_proxies,
    const std::size_t nb_iterations = 5) {
    std::size_t num_added = 0;
    while (num_added < nb_proxies) {
      if (!add_to_furthest_proxy())
        break;
      ++num_added;
      run(nb_iterations);
    }
    return num_added;
  }

  /*!
   * @brief adds proxies by diffusing fitting error into current partition.
   * Each partition is added with the number of proxies in proportion to its fitting error.
   * @param nb_proxies number of proxies to be added
   * @return number of proxies successfully added
   */
  std::size_t add_proxies_error_diffusion(const std::size_t nb_proxies) {
#ifdef CGAL_SURFACE_MESH_APPROXIMATION_DEBUG
    std::cerr << "#px " << m_proxies.size() << std::endl;
#endif

    const double sum_error = CGAL::to_double(compute_total_error());
    const double avg_error = sum_error / static_cast<double>(nb_proxies);

    // number of proxies to be added to each region
    std::vector<std::size_t> num_to_add(m_proxies.size(), 0);
    if (avg_error <= 0.0) {
      // rare case on extremely regular geometry like a cube
#ifdef CGAL_SURFACE_MESH_APPROXIMATION_DEBUG
      std::cerr << "zero error, diffuse w.r.t. number of faces" << std::endl;
#endif
      const double avg_face =
        static_cast<double>(m_nb_of_faces) / static_cast<double>(nb_proxies);
      std::vector<double> px_size(m_proxies.size(), 0.0);
      for(face_descriptor f : faces(*m_ptm))
        px_size[get(m_fproxy_map, f)] += 1.0;
      double residual = 0.0;
      for (std::size_t i = 0; i < m_proxies.size(); ++i) {
        const double to_add = (residual + px_size[i]) / avg_face;
        const double to_add_round_up = std::floor(to_add + 0.5);
        residual = (to_add - to_add_round_up) * avg_face;
        num_to_add[i] = static_cast<std::size_t>(to_add_round_up);
      }
    }
    else {
      std::vector<Proxy_error> px_error;
      for (std::size_t i = 0; i < m_proxies.size(); ++i)
        px_error.push_back(Proxy_error(i, m_proxies[i].err));
      // sort partition by error
      std::sort(px_error.begin(), px_error.end());

      // residual from previous proxy in range (-0.5, 0.5] * avg_error
      double residual = 0.0;
      for (std::size_t i = 0; i < m_proxies.size(); ++i) {
        // add error residual from previous proxy
        // to_add maybe negative but greater than -0.5
        const double to_add = (residual + CGAL::to_double(px_error[i].err)) / avg_error;
        const double to_add_round_up = std::floor(to_add + 0.5);
        residual = (to_add - to_add_round_up) * avg_error;
        num_to_add[px_error[i].px] = static_cast<std::size_t>(to_add_round_up);
      }

#ifdef CGAL_SURFACE_MESH_APPROXIMATION_DEBUG
      for (std::size_t i = 0; i < px_error.size(); ++i)
        std::cerr << "#px " << px_error[i].px
          << ", #error " << px_error[i].err
          << ", #num_to_add " << num_to_add[px_error[i].px] << std::endl;
#endif
    }

    std::size_t num_added = 0;
    for(face_descriptor f : faces(*m_ptm)) {
      const std::size_t px_id = get(m_fproxy_map, f);
      if (m_proxies[px_id].seed == f)
        continue;

      if (num_to_add[px_id] > 0) {
        add_one_proxy_at(f);
        --num_to_add[px_id];
        ++num_added;
      }
    }

#ifdef CGAL_SURFACE_MESH_APPROXIMATION_DEBUG
    std::cerr << "#requested/added "
      << nb_proxies << '/' << num_added << std::endl;
#endif

    return num_added;
  }
  /// @}

  /// \name Refinement Operations
  /// @{
  /*!
   * @brief teleports the local minimum to the worst region by combining the merging and adding processes.
   * The re-fitting is performed after each teleportation.
   * Here if we specify more than one proxy this means we teleport in a naive iterative fashion.
   * @param nb_proxies number of proxies requested to teleport.
   * @param nb_iterations number of re-fitting iterations.
   * @param no_threshold_test if `true`, no check on the approximation error before merging a pair of proxies is done. In other words, `find_best_merge(!no_threshold_test)` is called.
   * @return number of proxies teleported.
   */
  std::size_t teleport_proxies(const std::size_t nb_proxies,
    const std::size_t nb_iterations = 5,
    const bool no_threshold_test = false) {
    std::size_t num_teleported = 0;
    while (num_teleported < nb_proxies) {
      // find worst proxy
      std::size_t px_worst = 0;
      FT max_error = m_proxies.front().err;
      for (std::size_t i = 0; i < m_proxies.size(); ++i) {
        if (max_error < m_proxies[i].err) {
          max_error = m_proxies[i].err;
          px_worst = i;
        }
      }
      bool found = false;
      face_descriptor tele_to;
      for(face_descriptor f : faces(*m_ptm)) {
        if (get(m_fproxy_map, f) == px_worst && f != m_proxies[px_worst].seed) {
          // teleport to anywhere but the seed
          tele_to = f;
          found = true;
          break;
        }
      }
      if (!found)
        return num_teleported;

      // find the best merge pair
      std::optional< std::pair<std::size_t, std::size_t> > best_proxies =
        find_best_merge(!no_threshold_test);
      if (best_proxies==std::nullopt)
        return num_teleported;
      if (px_worst == best_proxies->first || px_worst == best_proxies->second)
        return num_teleported;

      // teleport to a face of the worst region
      // update merged proxies
      std::list<face_descriptor> merged_patch;
      for(face_descriptor f : faces(*m_ptm)) {
        std::size_t px_idx = get(m_fproxy_map, f);
        if (px_idx == best_proxies->first || px_idx == best_proxies->second) {
          put(m_fproxy_map, f, best_proxies->first);
          merged_patch.push_back(f);
        }
      }
      m_proxies[best_proxies->first] = fit_proxy_from_patch(merged_patch, best_proxies->first);
      // replace the merged proxy position to the newly teleported proxy
      m_proxies[best_proxies->second] = fit_proxy_from_face(tele_to, best_proxies->second);

      num_teleported++;
      // coarse re-fitting
      run(nb_iterations);

#ifdef CGAL_SURFACE_MESH_APPROXIMATION_DEBUG
      std::cerr << "teleported" << std::endl;
#endif
    }

    return num_teleported;
  }

  /*!
   * @brief merges two specified adjacent regions.
   * The overall re-fitting is not performed and the proxy index map is maintained.
   * @pre two proxies must be adjacent, and 0 <= px0 < px1 < proxies.size()
   * @param px0 the kept proxy index
   * @param px1 the merged and erased proxy index,
   * proxies with greater indeies are decreased by 1 to fill the gap
   * @return change of error
   */
  FT merge(const std::size_t px0, const std::size_t px1) {
    // ensure px0 < px1
    if (px0 >= px1 || px1 >= m_proxies.size())
      return FT(0.0);

    const FT pre_err = m_proxies[px0].err + m_proxies[px1].err;
    // merge px1 to px0
    std::list<face_descriptor> merged_patch;
    for(face_descriptor f : faces(*m_ptm)) {
      std::size_t px_idx = get(m_fproxy_map, f);
      if (px_idx == px1 || px_idx == px0) {
        put(m_fproxy_map, f, px0);
        merged_patch.push_back(f);
      }
    }
    m_proxies[px0] = fit_proxy_from_patch(merged_patch, px0);

    // erase px1 and maintain proxy index
    m_proxies.erase(m_proxies.begin() + px1);
    for (std::size_t i = 0; i < m_proxies.size(); ++i)
      m_proxies[i].idx = i;
    // keep face proxy map valid
    for(face_descriptor f : faces(*m_ptm)) {
      if (get(m_fproxy_map, f) > px1)
        put(m_fproxy_map, f, get(m_fproxy_map, f) - 1);
    }

    return m_proxies[px0].err - pre_err;
  }

  /*!
   * @brief simulates merging and local re-fitting of all pairs of adjacent proxies
   * and finds the best pair to merge.
   * @note The <b>best</b> is defined as the minimum merged sum error
   * <b>change</b> (increase or decrease) among all pairs.
   * @param use_threshold_test if `true` and a best pair of proxies is found,
   * it is returned only if the error change after the merge is lower than the half of the maximum proxy error.
   * @return if the best merge pair is found the optional returned contains the proxy indices, and is empty otherwise.
   */
  std::optional< std::pair<std::size_t, std::size_t> >
  find_best_merge(const bool use_threshold_test) {
    typedef std::pair<std::size_t, std::size_t> Proxy_pair;
    typedef std::set<Proxy_pair> Pair_set;

    std::size_t px0 = 0, px1 = 0;
    std::vector<std::list<face_descriptor> > px_faces(m_proxies.size());
    for(face_descriptor f : faces(*m_ptm))
      px_faces[get(m_fproxy_map, f)].push_back(f);

    // find best merge
    Pair_set merged_set;
    FT min_error_change = FT(0.0);
    bool first_merge = true;
    for(edge_descriptor e : edges(*m_ptm)) {
      if (CGAL::is_border(e, *m_ptm))
        continue;
      std::size_t pxi = get(m_fproxy_map, face(halfedge(e, *m_ptm), *m_ptm));
      std::size_t pxj = get(m_fproxy_map, face(opposite(halfedge(e, *m_ptm), *m_ptm), *m_ptm));
      if (pxi == pxj)
        continue;
      if (pxi > pxj)
        std::swap(pxi, pxj);
      if (merged_set.find(Proxy_pair(pxi, pxj)) != merged_set.end())
        continue;

      merged_set.insert(Proxy_pair(pxi, pxj));
      // simulated merge
      std::list<face_descriptor> merged_patch(px_faces[pxi]);
      for(face_descriptor f : px_faces[pxj])
        merged_patch.push_back(f);
      const Proxy_wrapper pxw_tmp = fit_proxy_from_patch(merged_patch, CGAL_VSA_INVALID_TAG);

      const FT error_change = pxw_tmp.err - (m_proxies[pxi].err + m_proxies[pxj].err);
      if (first_merge || error_change < min_error_change) {
        first_merge = false;
        min_error_change = error_change;
        px0 = pxi;
        px1 = pxj;
      }
    }

    if (merged_set.empty())
      return std::nullopt;

    // test if merge worth it
    if (use_threshold_test) {
      FT max_error = m_proxies.front().err;
      for (std::size_t i = 0; i < m_proxies.size(); ++i) {
        if (max_error < m_proxies[i].err)
          max_error = m_proxies[i].err;
      }
      if (min_error_change > max_error / FT(2.0))
        return std::nullopt;
    }

    return std::make_pair(px0, px1);
  }

  /*!
   * @brief splits within a specified proxy area via N-section (by default bisection),
   * other regions are not affected.
   * @param px_idx proxy index.
   * @param n number of split sections.
   * @param nb_relaxations number of relaxations within the proxy area <em>px_idx</em> after the split
   * @return `true` if split succeeds, and `false` otherwise.
   */
  bool split(const std::size_t px_idx,
    const std::size_t n = 2,
    const std::size_t nb_relaxations = 10) {
    if (px_idx >= m_proxies.size())
      return false;

    // collect confined proxy area
    std::list<face_descriptor> confined_area;
    for(face_descriptor f : faces(*m_ptm))
      if (get(m_fproxy_map, f) == px_idx)
        confined_area.push_back(f);
    // not enough faces to split
    if (n > confined_area.size())
      return false;

    // a copy of confined proxies
    std::vector<Proxy_wrapper> confined_proxies;
    confined_proxies.push_back(m_proxies[px_idx]);

    // select seed faces in the confined area
    std::size_t count = 1;
    for(face_descriptor f : confined_area) {
      if (count >= n)
        break;

      if (f != m_proxies[px_idx].seed) {
        add_one_proxy_at(f);
        ++count;
        // copy
        confined_proxies.push_back(m_proxies.back());
      }
    }

    // relaxation on confined area and proxies
    for (std::size_t i = 0; i < nb_relaxations; ++i) {
      for(face_descriptor f : confined_area)
        put(m_fproxy_map, f, CGAL_VSA_INVALID_TAG);

      partition(confined_proxies.begin(), confined_proxies.end());
      fit(confined_proxies.begin(), confined_proxies.end(), Concurrency_tag());
    }

    // copy back
    for(const Proxy_wrapper& pxw : confined_proxies)
      m_proxies[pxw.idx] = pxw;

    return true;
  }
  /// @}

  /// \name Meshing
  /// @{
  /*!
   * @brief extracts the output mesh in the form of an indexed triangle set.
   * @tparam NamedParameters a sequence of \ref bgl_namedparameters
   *
   * @param np an optional sequence of \ref bgl_namedparameters "Named Parameters" among the ones listed below
   * @return `true` if the extracted surface mesh is manifold, and `false` otherwise.
   *
   * \cgalNamedParamsBegin{Meshing Named Parameters}
   *   \cgalParamNBegin{subdivision_ratio}
   *     \cgalParamDescription{the chord subdivision ratio threshold to the chord length or average edge length}
   *     \cgalParamType{`geom_traits::FT`}
   *     \cgalParamDefault{`5.0`}
   *   \cgalParamNEnd
   *
   *   \cgalParamNBegin{boundary_subdivision_ratio}
   *     \cgalParamDescription{the chord subdivision ratio threshold to the chord length or average edge length for boundary edges}
   *     \cgalParamType{`geom_traits::FT`}
   *     \cgalParamDefault{`subdivision_ratio`}
   *   \cgalParamNEnd
   *
   *   \cgalParamNBegin{relative_to_chord}
   *     \cgalParamDescription{If `true`, the `subdivision_ratio` is the ratio of the furthest vertex distance
   *                           to the chord length, otherwise is the average edge length}
   *     \cgalParamType{`Boolean`}
   *     \cgalParamDefault{`false`}
   *   \cgalParamNEnd
   *
   *   \cgalParamNBegin{with_dihedral_angle}
   *     \cgalParamDescription{If `true`, the `subdivision_ratio` is weighted by dihedral angle}
   *     \cgalParamType{`Boolean`}
   *     \cgalParamDefault{`false`}
   *   \cgalParamNEnd
   *
   *   \cgalParamNBegin{optimize_anchor_location}
   *     \cgalParamDescription{If `true`, optimize the anchor locations}
   *     \cgalParamType{`Boolean`}
   *     \cgalParamDefault{`true`}
   *   \cgalParamNEnd
   *
   *   \cgalParamNBegin{optimize_boundary_anchor_location}
   *     \cgalParamDescription{If `true`, optimize the anchor locations of boundary vertices}
   *     \cgalParamType{`Boolean`}
   *     \cgalParamDefault{`optimize_anchor_location`}
   *   \cgalParamNEnd
   *
   *   \cgalParamNBegin{pca_plane}
   *     \cgalParamDescription{If `true`, use PCA plane fitting, otherwise use the default area averaged plane parameters}
   *     \cgalParamType{`Boolean`}
   *     \cgalParamDefault{`false`}
   *   \cgalParamNEnd
   * \cgalNamedParamsEnd
   */
  template <typename NamedParameters>
  bool extract_mesh(const NamedParameters &np) {
    using parameters::get_parameter;
    using parameters::choose_parameter;

    const FT subdivision_ratio = choose_parameter(get_parameter(np, internal_np::subdivision_ratio), FT(5.0));
    const FT boundary_subdivision_ratio = choose_parameter(get_parameter(np, internal_np::boundary_subdivision_ratio), subdivision_ratio);
    const bool relative_to_chord = choose_parameter(get_parameter(np, internal_np::relative_to_chord), false);
    const bool with_dihedral_angle = choose_parameter(get_parameter(np, internal_np::with_dihedral_angle), false);
    const bool optimize_anchor_location = choose_parameter(get_parameter(np, internal_np::optimize_anchor_location), true);
    const bool optimize_boundary_anchor_location = choose_parameter(get_parameter(np, internal_np::optimize_boundary_anchor_location), optimize_anchor_location);
    const bool pca_plane = choose_parameter(get_parameter(np, internal_np::pca_plane), false);

    // compute averaged edge length, used in chord subdivision
    m_average_edge_length = compute_averaged_edge_length(*m_ptm, m_vpoint_map);

    // initialize all vertex anchor status
    for(vertex_descriptor v : vertices(*m_ptm))
      put(m_vanchor_map, v, CGAL_VSA_INVALID_TAG);
    m_anchors.clear();
    m_bcycles.clear();
    m_tris.clear();
    m_px_planes.clear();

    // compute proxy planes, used for subdivision and anchor location
    compute_proxy_planes(pca_plane);

    // generate anchors
    find_anchors();
    find_edges(subdivision_ratio, boundary_subdivision_ratio, relative_to_chord, with_dihedral_angle);
    add_anchors();

    // discrete constrained Delaunay triangulation
    pseudo_cdt();

    if (optimize_anchor_location)
      this->optimize_anchor_location(optimize_boundary_anchor_location);

    // check manifold-oriented
    return Polygon_mesh_processing::is_polygon_soup_a_polygon_mesh(m_tris);
  }
  /// @}

  /// \name Output
  /// @{
  /*!
   * @brief outputs approximation results.
   * @tparam NamedParameters a sequence of \ref bgl_namedparameters

   * @param np an optional sequence of \ref bgl_namedparameters "Named Parameters" among the ones listed below

   * \cgalNamedParamsBegin{Output Named Parameters}
   *   \cgalParamNBegin{face_proxy_map}
   *     \cgalParamDescription{a property map to output the proxy index of each face of the input polygon mesh}
   *     \cgalParamType{a model of `WritablePropertyMap` with `boost::graph_traits<TriangleMesh>::%face_descriptor`
   *                    as key and `std::size_t` as value type}
   *     \cgalParamDefault{no output operation is performed}
   *     \cgalParamExtra{A proxy is a set of connected faces which are placed under the same proxy patch (see \cgalFigureRef{iterations})}
   *     \cgalParamExtra{The proxy-ids are contiguous in range `[0, number_of_proxies - 1]`}
   *   \cgalParamNEnd
   *
   *   \cgalParamNBegin{proxies}
   *     \cgalParamDescription{an `OutputIterator` to put proxies in}
   *     \cgalParamType{a class model of `OutputIterator` with
   *                    `CGAL::Surface_mesh_approximation::L21_metric_vector_proxy_no_area_weighting::Proxy` value type}
   *     \cgalParamDefault{no output operation is performed}
   *   \cgalParamNEnd
   *
   *   \cgalParamNBegin{anchors}
   *     \cgalParamDescription{an `OutputIterator` to put anchor points in}
   *     \cgalParamType{a class model of `OutputIterator` with `geom_traits::%Point_3` value type}
   *     \cgalParamDefault{no output operation is performed}
   *   \cgalParamNEnd
   *
   *   \cgalParamNBegin{triangles}
   *     \cgalParamDescription{an `OutputIterator` to put indexed triangles in}
   *     \cgalParamType{a class model of `OutputIterator` with `std::array<std::size_t, 3>` value type}
   *     \cgalParamDefault{no output operation is performed}
   *   \cgalParamNEnd
   * \cgalNamedParamsEnd
   */
  template <typename NamedParameters>
  void output(const NamedParameters &np) const {
    using parameters::get_parameter;
    using parameters::choose_parameter;

    // get proxy map
    proxy_map( get_parameter(np, internal_np::face_proxy_map) );
    // get proxies
    proxies( get_parameter(np, internal_np::proxies) );

    // get anchor points
    anchor_points( get_parameter(np, internal_np::anchors) );

    // get indexed triangles
    indexed_triangles( get_parameter(np, internal_np::triangles) );
  }

  /*!
   * @brief returns the number of proxies.
   */
  std::size_t number_of_proxies() const { return m_proxies.size(); }

  /// @cond CGAL_DOCUMENT_INTERNAL
  /*!
   * @brief gets the face-proxy index map.
   * @tparam FaceProxyMap `WritablePropertyMap` with
   * `boost::graph_traits<TriangleMesh>::%face_descriptor` as key and `std::size_t` as value type
   * @param[out] face_proxy_map face proxy index map
   */
  template <typename FaceProxyMap>
  void proxy_map(FaceProxyMap face_proxy_map) const {
    for(face_descriptor f : faces(*m_ptm))
      put(face_proxy_map, f, get(m_fproxy_map, f));
  }

  /*!
   * @brief dummy function for named parameters.
   */
  void proxy_map(internal_np::Param_not_found) const {}

  /*!
   * @brief gets the face region of the specified proxy.
   * @tparam OutputIterator output iterator with `boost::graph_traits<TriangleMesh>::%face_descriptor` as value type
   * @param px_idx proxy index
   * @param out output iterator
   */
  template <typename OutputIterator>
  void proxy_region(const std::size_t px_idx, OutputIterator out) const {
    if (px_idx >= m_proxies.size())
      return;

    for(face_descriptor f : faces(*m_ptm))
      if (get(m_fproxy_map, f) == px_idx)
        *out++ = f;
  }

  /*!
   * @brief gets the proxies.
   * @tparam OutputIterator output iterator with Proxy as value type
   * @param out output iterator
   */
  template <typename OutputIterator>
  void proxies(OutputIterator out) const {
    for(const Proxy_wrapper& pxw : m_proxies)
      *out++ = pxw.px;
  }

  /*!
   * @brief dummy function for named parameters.
   */
  void proxies(internal_np::Param_not_found) const {}

  /*!
   * @brief gets the wrapped proxies.
   * @tparam OutputIterator output iterator with Proxy_wrapper as value type
   * @param out output iterator
   */
  template <typename OutputIterator>
  void wrapped_proxies(OutputIterator out) const {
    for(const Proxy_wrapper& pxw : m_proxies)
      *out++ = pxw;
  }

  /*!
   * @brief gets the anchor points, which have the area-averaged position of the projected anchor vertex points on the incident proxies.
   * @tparam OutputIterator output iterator with Point_3 as value type
   * @param out output iterator
   */
  template <typename OutputIterator>
  void anchor_points(OutputIterator out) const {
    for(const Anchor& a : m_anchors)
      *out++ = a.pos;
  }

  /*!
   * @brief dummy function for named parameters.
   */
  void anchor_points(internal_np::Param_not_found) const {}

  /*!
   * @brief gets the anchor vertices.
   * @tparam OutputIterator output iterator with vertex_descriptor as value type
   * @param out output iterator
   */
  template <typename OutputIterator>
  void anchor_vertices(OutputIterator out) const {
    for(const Anchor& a : m_anchors)
      *out++ = a.vtx;
  }

  /*!
   * @brief gets the indexed triangles, with
   * one triplet of integers per triangles, which refers to the anchor point indices.
   * @tparam OutputIterator output iterator with Indexed_triangle as value type
   * @param out output iterator
   */
  template <typename OutputIterator>
  void indexed_triangles(OutputIterator out) const {
    for(const Indexed_triangle& t : m_tris)
      *out++ = t;
  }

  /*!
   * @brief dummy function for named parameters.
   */
  void indexed_triangles(internal_np::Param_not_found) const {}

  /*!
   * @brief gets the indexed boundary polygon approximation.
   * @tparam OutputIterator output iterator with std::vector<std::size_t> as value type
   * @param out output iterator
   */
  template <typename OutputIterator>
  void indexed_boundary_polygons(OutputIterator out) const {
    for(const Boundary_cycle& bcycle : m_bcycles) {
      std::vector<std::size_t> plg;
      halfedge_descriptor he = bcycle.he_head;
      do {
        Boundary_chord chord;
        walk_to_next_anchor(he, chord);
        plg.push_back(get(m_vanchor_map, target(he, *m_ptm)));
      } while (he != bcycle.he_head);
      *out++ = plg;
    }
  }
  /// @endcond
  /// @}

// private member functions
private:
  /*!
   * @brief randomly initializes proxies
   * with both maximum number of proxies and minimum error drop stop criteria,
   * where the first criterion met stops the seeding.
   * @note To ensure the randomness, call `std::srand()` beforehand.
   * @param max_nb_proxies maximum number of proxies, should be in range `(nb_connected_components, nb_faces / 3)`
   * @param min_error_drop minimum error drop, should be in range `(0.0, 1.0)`,
   * negative value is ignored
   * @param nb_relaxations number of re-fitting iterations
   * @return number of proxies initialized
   */
  std::size_t init_random(const std::size_t max_nb_proxies,
    const FT min_error_drop,
    const std::size_t nb_relaxations) {

    if (!(min_error_drop > FT(0.0))) {
      // pick from current non seed faces randomly
      std::vector<face_descriptor> picked_seeds;
      if (random_pick_non_seed_faces(max_nb_proxies - m_proxies.size(), picked_seeds)) {
        for(face_descriptor f : picked_seeds)
          add_one_proxy_at(f);
        run(nb_relaxations);
      }
      return m_proxies.size();
    }

    const FT initial_err = compute_total_error();
    FT error_drop = min_error_drop * FT(2.0);
    while (m_proxies.size() < max_nb_proxies && error_drop > min_error_drop) {
      // try to double current number of proxies each time
      const std::size_t nb_px = m_proxies.size();
      const std::size_t nb_to_add =
        (nb_px * 2 > max_nb_proxies) ? max_nb_proxies - nb_px : nb_px;

      // pick from current non seed faces randomly
      std::vector<face_descriptor> picked_seeds;
      if (!random_pick_non_seed_faces(nb_to_add, picked_seeds))
        return m_proxies.size();

      for(face_descriptor f : picked_seeds)
        add_one_proxy_at(f);
      const FT err = run(nb_relaxations);
      error_drop = err / initial_err;
    }

    return m_proxies.size();
  }

  /*!
   * @brief incrementally initializes proxies
   * with both maximum number of proxies and minimum error drop stop criteria,
   * The first criterion met stops the seeding.
   * @param max_nb_proxies maximum number of proxies, should be in range `(nb_connected_components, nb_faces / 3)`
   * @param min_error_drop minimum error drop, should be in range `(0.0, 1.0)`,
   * negative value is ignored
   * @param nb_relaxations number of re-fitting iterations
   * @return number of proxies initialized
   */
  std::size_t init_incremental(const std::size_t max_nb_proxies,
    const FT min_error_drop,
    const std::size_t nb_relaxations) {

    if (!(min_error_drop > FT(0.0))) {
      if (m_proxies.size() < max_nb_proxies)
        add_to_furthest_proxies(max_nb_proxies - m_proxies.size(), nb_relaxations);
      return m_proxies.size();
    }

    const FT initial_err = compute_total_error();
    FT error_drop = min_error_drop * FT(2.0);
    while (m_proxies.size() < max_nb_proxies && error_drop > min_error_drop) {
      add_to_furthest_proxy();
      const FT err = run(nb_relaxations);
      error_drop = err / initial_err;
    }

    return m_proxies.size();
  }

  /*!
   * @brief hierarchically initializes proxies
   * with both maximum number of proxies and minimum error drop stop criteria,
   * where the first criterion met stops the seeding.
   * @param max_nb_proxies maximum number of proxies, should be in range `(nb_connected_components, nb_faces / 3)`
   * @param min_error_drop minimum error drop, should be in range `(0.0, 1.0)`,
   * negative value is ignored
   * @param nb_relaxations number of re-fitting iterations
   * @return number of proxies initialized
   */
  std::size_t init_hierarchical(const std::size_t max_nb_proxies,
    const FT min_error_drop,
    const std::size_t nb_relaxations) {

    const FT initial_err = compute_total_error();
    FT error_drop = !(min_error_drop > FT(0.0)) ? FT(1.0) : min_error_drop * FT(2.0);
    while (m_proxies.size() < max_nb_proxies && error_drop > min_error_drop) {
      // try to double current number of proxies each time
      std::size_t target_px = m_proxies.size();
      if (target_px * 2 > max_nb_proxies)
        target_px = max_nb_proxies;
      else
        target_px *= 2;
      // if no proxies could be added, stop
      if( add_proxies_error_diffusion(target_px - m_proxies.size()) == 0)
        break;
      const FT err = run(nb_relaxations);
      error_drop = err / initial_err;
    }

    return m_proxies.size();
  }

  /*!
   * @brief partitions the area tagged with CGAL_VSA_INVALID_TAG with proxies, global face proxy map is updated.
   * Propagates the proxy seed faces and floods the tagged area to minimize the fitting error.
   * @tparam ProxyWrapperIterator forward iterator with Proxy_wrapper as value type
   * @param beg iterator point to the first element
   * @param end iterator point to the one past the last element
   */
  template<typename ProxyWrapperIterator>
  void partition(const ProxyWrapperIterator beg, const ProxyWrapperIterator end) {
    std::priority_queue<Face_to_integrate> face_pqueue;
    for (ProxyWrapperIterator pxw_itr = beg; pxw_itr != end; ++pxw_itr) {
      face_descriptor f = pxw_itr->seed;
      put(m_fproxy_map, f, pxw_itr->idx);

      for(face_descriptor fadj : faces_around_face(halfedge(f, *m_ptm), *m_ptm)) {
        if (fadj != boost::graph_traits<TriangleMesh>::null_face()
            && get(m_fproxy_map, fadj) == CGAL_VSA_INVALID_TAG) {
          face_pqueue.push(Face_to_integrate(
            fadj, pxw_itr->idx, m_metric->compute_error(fadj, *m_ptm, pxw_itr->px)));
        }
      }
    }

    while (!face_pqueue.empty()) {
      const Face_to_integrate c = face_pqueue.top();
      face_pqueue.pop();
      if (get(m_fproxy_map, c.f) == CGAL_VSA_INVALID_TAG) {
        put(m_fproxy_map, c.f, c.px);
        for(face_descriptor fadj : faces_around_face(halfedge(c.f, *m_ptm), *m_ptm)) {
          if (fadj != boost::graph_traits<TriangleMesh>::null_face()
            && get(m_fproxy_map, fadj) == CGAL_VSA_INVALID_TAG) {
            face_pqueue.push(Face_to_integrate(
              fadj, c.px, m_metric->compute_error(fadj, *m_ptm, m_proxies[c.px].px)));
          }
        }
      }
    }
  }

  /*!
   * @brief refits and updates input range of proxies, sequential.
   * @tparam ProxyWrapperIterator forward iterator with Proxy_wrapper as value type
   * @param beg iterator point to the first element
   * @param end iterator point to the one past the last element
   * @param t concurrency tag
   */
  template<typename ProxyWrapperIterator>
  void fit(const ProxyWrapperIterator beg, const ProxyWrapperIterator end, const CGAL::Sequential_tag & t) {
    CGAL_USE(t);
    std::vector<std::list<face_descriptor> > px_faces(m_proxies.size());
    for(face_descriptor f : faces(*m_ptm))
      px_faces[get(m_fproxy_map, f)].push_back(f);

    // update proxy parameters and seed
    for (ProxyWrapperIterator pxw_itr = beg; pxw_itr != end; ++pxw_itr) {
      const std::size_t px_idx = pxw_itr->idx;
      *pxw_itr = fit_proxy_from_patch(px_faces[px_idx], px_idx);
    }
  }

#ifdef CGAL_LINKED_WITH_TBB
  /*!
   * @brief refits and updates input range of proxies, parallel.
   * @tparam ProxyWrapperIterator forward iterator with Proxy_wrapper as value type
   * @param beg iterator point to the first element
   * @param end iterator point to the one past the last element
   * @param t concurrency tag
   */
  template<typename ProxyWrapperIterator>
  void fit(const ProxyWrapperIterator beg, const ProxyWrapperIterator end, const CGAL::Parallel_tag & t) {
    CGAL_USE(t);
    std::vector<std::list<face_descriptor> > px_faces(m_proxies.size());
    for(face_descriptor f : faces(*m_ptm))
      px_faces[get(m_fproxy_map, f)].push_back(f);

    // update proxy parameters and seed
    tbb::parallel_for(tbb::blocked_range<ProxyWrapperIterator>(beg, end),
      [&](tbb::blocked_range<ProxyWrapperIterator> &r) {
        for (ProxyWrapperIterator pxw_itr = r.begin(); pxw_itr != r.end(); ++pxw_itr) {
          const std::size_t px_idx = pxw_itr->idx;
          *pxw_itr = fit_proxy_from_patch(px_faces[px_idx], px_idx);
        }
      });
  }
#endif // CGAL_LINKED_WITH_TBB

  /*!
   * @brief adds a proxy seed at the face with the maximum fitting error.
   * @return `true` if add is successfully, and `false` otherwise
   */
  bool add_to_furthest_proxy() {
#ifdef CGAL_SURFACE_MESH_APPROXIMATION_DEBUG
    std::cerr << "add furthest " << m_proxies.size() << std::endl;
#endif
    FT max_error = m_proxies.front().err;
    std::size_t px_worst = 0;
    for (std::size_t i = 0; i < m_proxies.size(); ++i) {
      if (max_error < m_proxies[i].err) {
        max_error = m_proxies[i].err;
        px_worst = i;
      }
    }

    face_descriptor fworst;
    bool first = true;
    for(face_descriptor f : faces(*m_ptm)) {
      std::size_t px_idx = get(m_fproxy_map, f);
      if (px_idx != px_worst || f == m_proxies[px_idx].seed)
        continue;

      FT err = m_metric->compute_error(f, *m_ptm, m_proxies[px_idx].px);
      if (first || max_error < err) {
        first = false;
        max_error = err;
        fworst = f;
      }
    }

    if (first)
      return false;

    add_one_proxy_at(fworst);

    return true;
  }

  /*!
   * @brief fits a new (wrapped) proxy from a region patch.
   * 1. Compute proxy parameters from a list of faces.
   * 2. Find proxy seed face.
   * 3. Sum the proxy error.
   * @tparam FacePatch container with `face_descriptor` as data type
   * @param px_patch proxy patch container
   * @param px_idx the assigned proxy index
   * @return fitted wrapped proxy
   */
  template<typename FacePatch>
  Proxy_wrapper fit_proxy_from_patch(const FacePatch &px_patch, const std::size_t px_idx) {
    CGAL_assertion(!px_patch.empty());

    // use Proxy_fitting functor to fit proxy parameters
    const Proxy px = m_metric->fit_proxy(px_patch, *m_ptm);

    // find proxy seed and sum error
    face_descriptor seed = *px_patch.begin();
    FT err_min = m_metric->compute_error(seed, *m_ptm, px);
    FT sum_error(0.0);
    for(face_descriptor f : px_patch) {
      const FT err = m_metric->compute_error(f, *m_ptm, px);
      sum_error += err;
      if (err < err_min) {
        err_min = err;
        seed = f;
      }
    }

    return Proxy_wrapper(px, px_idx, seed, sum_error);
  }

  /*!
   * @brief adds a proxy at face f.
   * @param f where to the proxy is initialized from
   */
  void add_one_proxy_at(const face_descriptor f) {
    m_proxies.push_back(fit_proxy_from_face(f, m_proxies.size()));
  }

  /*!
   * @brief fits a new (wrapped) proxy from a face.
   * 1. Compute proxy parameters from the face.
   * 2. Set seed to this face.
   * 3. Update the proxy error.
   * 4. Update proxy map.
   * @pre current face proxy map is valid
   * @param f face
   * @param px_idx proxy index
   * @return fitted wrapped proxy
   */
  Proxy_wrapper fit_proxy_from_face(const face_descriptor f, const std::size_t px_idx) {
    // fit proxy parameters
    std::vector<face_descriptor> fvec(1, f);
    const Proxy px = m_metric->fit_proxy(fvec, *m_ptm);
    const FT err = m_metric->compute_error(f, *m_ptm, px);

    // original proxy map should always be valid
    const std::size_t prev_px_idx = get(m_fproxy_map, f);
    CGAL_assertion(prev_px_idx != CGAL_VSA_INVALID_TAG);
    // update the proxy error and proxy map
    m_proxies[prev_px_idx].err -= m_metric->compute_error(f, *m_ptm, m_proxies[prev_px_idx].px);
    put(m_fproxy_map, f, px_idx);

    return Proxy_wrapper(px, px_idx, f, err);
  }

  /*!
   * @brief picks a number of non-seed faces into an empty vector randomly.
   * @param nb_requested requested number of faces
   * @param[out] picked_faces shuffled faces vector
   * @return `true` if the requested number of faces are selected, and `false` otherwise
   */
  bool random_pick_non_seed_faces(const std::size_t nb_requested,
    std::vector<face_descriptor> &picked_faces) {
    if (nb_requested + m_proxies.size() >= m_nb_of_faces)
      return false;

    std::set<face_descriptor> seed_faces_set;
    for(const Proxy_wrapper& pxw : m_proxies)
      seed_faces_set.insert(pxw.seed);

    const std::size_t nb_nsf = m_nb_of_faces - m_proxies.size();
    std::vector<face_descriptor> non_seed_faces;
    non_seed_faces.reserve(nb_nsf);
    for(face_descriptor f : faces(*m_ptm)) {
      if (seed_faces_set.find(f) != seed_faces_set.end())
        continue;
      non_seed_faces.push_back(f);
    }

    // random shuffle first few faces
    for (std::size_t i = 0; i < nb_requested; ++i) {
      // swap ith element with a random one
      std::size_t r = static_cast<std::size_t>(
        static_cast<double>(std::rand()) / static_cast<double>(RAND_MAX) *
        static_cast<double>(nb_nsf - 1));
      std::swap(non_seed_faces[i], non_seed_faces[r]);
    }

    for (std::size_t i = 0; i < nb_requested; ++i)
      picked_faces.push_back(non_seed_faces[i]);

    return true;
  }

  /*!
   * @brief initializes proxies from each connected component of the input mesh.
   * @note This function clears proxy vector and sets face proxy map to initial state,
   * intended only for bootstrapping initialization.
   * Coarse approximation iteration is not performed, because it is inaccurate anyway
   * and may yield degenerate cases (e.g. a standard cube model).
   */
  void bootstrap_from_connected_components() {
    // set all faces invalid to mark as unvisited / untagged
    for(face_descriptor f : faces(*m_ptm))
      put(m_fproxy_map, f, CGAL_VSA_INVALID_TAG);

    // prepare for connected components visiting
    std::vector<std::list<face_descriptor> > cc_patches;
    bool if_all_visited = false;
    std::size_t cc_idx = 0;
    face_descriptor seed_face = *(faces(*m_ptm).first);
    while (!if_all_visited) {
      // use current seed face to traverse the conneceted componnets
      std::list<face_descriptor> cc_patch;
      cc_patch.push_back(seed_face);
      std::stack<face_descriptor> fstack;
      fstack.push(seed_face);
      put(m_fproxy_map, seed_face, cc_idx);
      while (!fstack.empty()) {
        face_descriptor active_face = fstack.top();
        fstack.pop();
        for(face_descriptor fadj :
          faces_around_face(halfedge(active_face, *m_ptm), *m_ptm)) {
          if (fadj != boost::graph_traits<TriangleMesh>::null_face()
            && get(m_fproxy_map, fadj) == CGAL_VSA_INVALID_TAG) {
            cc_patch.push_back(fadj);
            fstack.push(fadj);
            put(m_fproxy_map, fadj, cc_idx);
          }
        }
      }
      cc_patches.push_back(cc_patch);
      // check if all visited
      if_all_visited = true;
      for(face_descriptor f : faces(*m_ptm)) {
        if (get(m_fproxy_map, f) == CGAL_VSA_INVALID_TAG) {
          if_all_visited = false;
          ++cc_idx;
          seed_face = f;
          break;
        }
      }
    }

    m_proxies.clear();
    for(const std::list<face_descriptor>& cc_patch : cc_patches)
      m_proxies.push_back(fit_proxy_from_patch(cc_patch, m_proxies.size()));
#ifdef CGAL_SURFACE_MESH_APPROXIMATION_DEBUG
    std::cerr << "#cc " << m_proxies.size() << std::endl;
#endif
  }

  /*!
   * @brief computes proxy planes.
   * The proxy may not contain the plane related properties, so we need these internal planes,
   * used in the chord subdivision and anchor location.
   * @param if_pca_plane set `true` to use the PCA plane fitting
   */
  void compute_proxy_planes(const bool if_pca_plane) {
    // fit proxy planes, areas, normals
    std::vector<std::list<face_descriptor> > px_faces(m_proxies.size());
    for(face_descriptor f : faces(*m_ptm))
      px_faces[get(m_fproxy_map, f)].push_back(f);

    for(const std::list<face_descriptor>& px_patch : px_faces) {
      Plane_3 fit_plane = if_pca_plane ?
        fit_plane_pca(px_patch.begin(), px_patch.end()) :
          fit_plane_area_averaged(px_patch.begin(), px_patch.end());

      Vector_3 norm = CGAL::NULL_VECTOR;
      FT area(0.0);
      for(face_descriptor f : px_patch) {
        halfedge_descriptor he = halfedge(f, *m_ptm);
        const Point_3 &p0 = m_vpoint_map[source(he, *m_ptm)];
        const Point_3 &p1 = m_vpoint_map[target(he, *m_ptm)];
        const Point_3 &p2 = m_vpoint_map[target(next(he, *m_ptm), *m_ptm)];
        const FT farea = CGAL::approximate_sqrt(CGAL::squared_area(p0, p1, p2));
        const Vector_3 fnorm =
          collinear_functor(p0, p1, p2) ? CGAL::NULL_VECTOR : CGAL::unit_normal(p0, p1, p2);

        norm = sum_functor(norm, scale_functor(fnorm, farea));
        area += farea;
      }
      if (norm.squared_length() > FT(0.0))
        norm = scale_functor(norm, FT(1.0) / CGAL::approximate_sqrt(norm.squared_length()));
      else
        norm = Vector_3(FT(0.0), FT(0.0), FT(1.0));

      m_px_planes.push_back(Proxy_plane(fit_plane, norm, area));
    }
  }

  /*!
   * @brief finds the anchors.
   */
  void find_anchors() {
    for(vertex_descriptor vtx : vertices(*m_ptm)) {
      std::size_t border_count = 0;

      for(halfedge_descriptor h : halfedges_around_target(vtx, *m_ptm)) {
        if (CGAL::is_border_edge(h, *m_ptm))
          ++border_count;
        else if (get(m_fproxy_map, face(h, *m_ptm)) != get(m_fproxy_map, face(opposite(h, *m_ptm), *m_ptm)))
          ++border_count;
      }
      if (border_count >= 3)
        attach_anchor(vtx);
    }
  }

  /*!
   * @brief finds and approximates the chord connecting the anchors.
   * @param boundary_subdivision_ratio the chord subdivision ratio threshold to the chord length or average edge length for boundary edges
   * @param subdivision_ratio boundary chord approximation recursive split criterion
   * @param relative_to_chord set `true` if the subdivision_ratio is relative to the chord length (relative sense),
   * otherwise it's relative to the average edge length (absolute sense).
   * @param with_dihedral_angle if set to `true`, add dihedral angle weight to the distance.
   */
  void find_edges(const FT subdivision_ratio,
    const FT boundary_subdivision_ratio,
    const bool relative_to_chord,
    const bool with_dihedral_angle) {
    // collect candidate halfedges in a set
    std::set<halfedge_descriptor> he_candidates;
    for(halfedge_descriptor h : halfedges(*m_ptm)) {
      if (!CGAL::is_border(h, *m_ptm)
        && (CGAL::is_border(opposite(h, *m_ptm), *m_ptm)
          || get(m_fproxy_map, face(h, *m_ptm)) != get(m_fproxy_map, face(opposite(h, *m_ptm), *m_ptm))))
        he_candidates.insert(h);
    }

    // pick up one candidate halfedge each time and traverse the connected boundary cycle
    while (!he_candidates.empty()) {
      halfedge_descriptor he_start = *he_candidates.begin();
      walk_to_first_anchor(he_start);
      // no anchor in this connected boundary cycle, make a new anchor
      if (!is_anchor_attached(he_start))
        attach_anchor(he_start);

      // a new connected boundary cycle
      m_bcycles.push_back(Boundary_cycle(he_start));

#ifdef CGAL_SURFACE_MESH_APPROXIMATION_DEBUG
      std::cerr << "#bcycle " << m_bcycles.size() << std::endl;
#endif

      const halfedge_descriptor he_mark = he_start;
      do {
        Boundary_chord chord;
        walk_to_next_anchor(he_start, chord);
        m_bcycles.back().num_anchors += subdivide_chord(chord.begin(), chord.end(),
            subdivision_ratio, boundary_subdivision_ratio, relative_to_chord, with_dihedral_angle);

#ifdef CGAL_SURFACE_MESH_APPROXIMATION_DEBUG
        std::cerr << "#chord_anchor " << m_bcycles.back().num_anchors << std::endl;
#endif

        for(const halfedge_descriptor& he : chord)
          he_candidates.erase(he);
      } while (he_start != he_mark);
    }
  }

  /*!
   * @brief adds anchors to the boundary cycles with only 2 anchors.
   */
  void add_anchors() {
    for(const Boundary_cycle& bcycle : m_bcycles) {
      if (bcycle.num_anchors > 2)
        continue;

      // 2 initial anchors at least
      CGAL_assertion(bcycle.num_anchors == 2);
      // borders with only 2 initial anchors
      Point_3 pt_begin = m_vpoint_map[target(bcycle.he_head, *m_ptm)];
      Point_3 pt_end = pt_begin;

      halfedge_descriptor he = bcycle.he_head;
      Boundary_chord chord;
      std::size_t count = 0;
      do {
        walk_to_next_border_halfedge(he);
        if (!is_anchor_attached(he))
          chord.push_back(he);
        else {
          if (count == 0)
            pt_end = m_vpoint_map[target(he, *m_ptm)];
          ++count;
        }
      } while (he != bcycle.he_head);

      // anchor count may be increased to more than 2 afterwards
      // due to the new anchors added by the neighboring boundary cycle (< 2 anchors)
      if (count > 2) {
        const_cast<Boundary_cycle &>(bcycle).num_anchors = count;
        continue;
      }

      CGAL_assertion(!chord.empty());
      halfedge_descriptor he_max = *chord.begin();
      Vector_3 chord_vec = vector_functor(pt_begin, pt_end);
      if (chord_vec.squared_length() > FT(0.0)) {
        FT dist_max(0.0);
        chord_vec = scale_functor(chord_vec,
          FT(1.0) / CGAL::approximate_sqrt(chord_vec.squared_length()));
        for(const halfedge_descriptor& he : chord) {
          Vector_3 vec = vector_functor(pt_begin, m_vpoint_map[target(he, *m_ptm)]);
          vec = cross_product_functor(chord_vec, vec);
          const FT dist = CGAL::approximate_sqrt(vec.squared_length());
          if (dist > dist_max) {
            dist_max = dist;
            he_max = he;
          }
        }
      }
      else {
        FT dist_max(0.0);
        for(const halfedge_descriptor& he : chord) {
          const FT dist = CGAL::approximate_sqrt(CGAL::squared_distance(
            pt_begin, m_vpoint_map[target(he, *m_ptm)]));
          if (dist > dist_max) {
            dist_max = dist;
            he_max = he;
          }
        }
      }

      // add one anchors to this boundary cycle
      attach_anchor(he_max);
      const_cast<Boundary_cycle &>(bcycle).num_anchors++;
    }
  }

  /*!
   * @brief runs the pseudo Constrained Delaunay Triangulation at each proxy region,
   * and stores the extracted indexed triangles in `tris`.
   * @pre all anchors are found, i.e. all boundary cycles have been visited
   * and attached with at least 3 anchors.
   */
  void pseudo_cdt() {
    // subgraph attached with vertex anchor status and edge weight
    typedef boost::property<boost::vertex_index1_t, std::size_t,
      boost::property<boost::vertex_index2_t, std::size_t> > VertexProperty;
    typedef boost::property<boost::edge_weight_t, FT,
      boost::property<boost::edge_index_t, std::size_t> > EdgeProperty;
    typedef boost::subgraph<boost::adjacency_list<
      boost::listS, boost::vecS,
      boost::undirectedS,
      VertexProperty, EdgeProperty> > SubGraph;
    typedef typename boost::property_map<SubGraph, boost::vertex_index1_t>::type VertexIndex1Map;
    typedef typename boost::property_map<SubGraph, boost::vertex_index2_t>::type VertexIndex2Map;
    typedef typename boost::property_map<SubGraph, boost::edge_weight_t>::type EdgeWeightMap;
    typedef typename SubGraph::vertex_descriptor sg_vertex_descriptor;
    typedef std::vector<sg_vertex_descriptor> VertexVector;

    typedef std::unordered_map<vertex_descriptor, sg_vertex_descriptor> VertexMap;
    typedef boost::associative_property_map<VertexMap> ToSGVertexMap;
    VertexMap vmap;
    ToSGVertexMap to_sgv_map(vmap);

    // mapping the TriangleMesh mesh into a SubGraph
    SubGraph gmain;
    VertexIndex1Map global_vanchor_map = get(boost::vertex_index1, gmain);
    VertexIndex2Map global_vtag_map = get(boost::vertex_index2, gmain);
    EdgeWeightMap global_eweight_map = get(boost::edge_weight, gmain);
    for(vertex_descriptor v : vertices(*m_ptm)) {
      sg_vertex_descriptor sgv = add_vertex(gmain);
      global_vanchor_map[sgv] = get(m_vanchor_map, v);
      global_vtag_map[sgv] = get(m_vanchor_map, v);
      vmap.insert(std::pair<vertex_descriptor, sg_vertex_descriptor>(v, sgv));
    }
    for(edge_descriptor e : edges(*m_ptm)) {
      const vertex_descriptor vs = source(e, *m_ptm);
      const vertex_descriptor vt = target(e, *m_ptm);
      const FT len = CGAL::approximate_sqrt(CGAL::squared_distance(
        m_vpoint_map[vs], m_vpoint_map[vt]));
      add_edge(to_sgv_map[vs], to_sgv_map[vt], len, gmain);
    }

    std::vector<VertexVector> vertex_patches(m_proxies.size());
    for(vertex_descriptor v : vertices(*m_ptm)) {
      std::set<std::size_t> px_set;
      for(face_descriptor f : faces_around_target(halfedge(v, *m_ptm), *m_ptm)) {
        if (f != boost::graph_traits<TriangleMesh>::null_face())
          px_set.insert(get(m_fproxy_map, f));
      }
      for(std::size_t p : px_set)
        vertex_patches[p].push_back(to_sgv_map[v]);
    }
    for(VertexVector& vpatch : vertex_patches) {
      // add a super vertex connecting to its boundary anchors in each patch
      const sg_vertex_descriptor superv = add_vertex(gmain);
      global_vanchor_map[superv] = CGAL_VSA_INVALID_TAG;
      global_vtag_map[superv] = CGAL_VSA_INVALID_TAG;
      int anchor_count = 0;
      for(sg_vertex_descriptor v : vpatch) {
        if (is_anchor_attached(v, global_vanchor_map)) {
          add_edge(superv, v, FT(0.0), gmain);
          anchor_count++;
        }
      }
      CGAL_assertion(anchor_count >=3 || anchor_count == 0);
      // ball patch has no boundary or anchor, usually are small floating parts
      // push dummy source vertex in the patch
      if (anchor_count == 0)
        vpatch.push_back(boost::graph_traits<SubGraph>::null_vertex());
      else
        vpatch.push_back(superv);
    }

    // multi-source Dijkstra's shortest path algorithm applied to each proxy patch
    for(VertexVector& vpatch : vertex_patches) {
      // ignore ball patch
      if (vpatch.back() == boost::graph_traits<SubGraph>::null_vertex())
        continue;

      // construct subgraph
      SubGraph &glocal = gmain.create_subgraph();
      for(sg_vertex_descriptor v : vpatch)
        add_vertex(v, glocal);

      // most subgraph functions work with local descriptors
      VertexIndex1Map local_vanchor_map = get(boost::vertex_index1, glocal);
      VertexIndex2Map local_vtag_map = get(boost::vertex_index2, glocal);
      EdgeWeightMap local_eweight_map = get(boost::edge_weight, glocal);

      const sg_vertex_descriptor source = glocal.global_to_local(vpatch.back());
      VertexVector pred(num_vertices(glocal),
        boost::graph_traits<SubGraph>::null_vertex());
      boost::dijkstra_shortest_paths(glocal, source,
        boost::predecessor_map(&pred[0]).weight_map(local_eweight_map));

      // backtrack to the anchor and tag each vertex in the local patch graph
      for(sg_vertex_descriptor v : make_range(vertices(glocal))) {
        // skip the added super source vertex in the patch
        if (v == source)
          continue;
        sg_vertex_descriptor curr = v;
        while (!is_anchor_attached(curr, local_vanchor_map))
          curr = pred[curr];
        local_vtag_map[v] = local_vanchor_map[curr];
      }
    }

    // tag all boundary chords
    for(const Boundary_cycle& bcycle : m_bcycles) {
      halfedge_descriptor he = bcycle.he_head;
      do {
        Boundary_chord chord;
        walk_to_next_anchor(he, chord);

        std::vector<FT> vdist;
        vdist.push_back(FT(0.0));
        for(halfedge_descriptor h : chord) {
          FT elen = global_eweight_map[edge(
            to_sgv_map[source(h, *m_ptm)],
            to_sgv_map[target(h, *m_ptm)],
            gmain).first];
          vdist.push_back(vdist.back() + elen);
        }

        FT half_chord_len = vdist.back() / FT(2.0);
        const std::size_t anchorleft = get(m_vanchor_map, source(chord.front(), *m_ptm));
        const std::size_t anchorright = get(m_vanchor_map, target(chord.back(), *m_ptm));
        typename std::vector<FT>::iterator ditr = vdist.begin() + 1;
        for (Boundary_chord_iterator citr = chord.begin(); citr != chord.end() - 1; ++citr, ++ditr) {
          if (*ditr < half_chord_len)
            global_vtag_map[to_sgv_map[target(*citr, *m_ptm)]] = anchorleft;
          else
            global_vtag_map[to_sgv_map[target(*citr, *m_ptm)]] = anchorright;
        }
      } while (he != bcycle.he_head);
    }

    // collect triangles
    for(face_descriptor f : faces(*m_ptm)) {
      halfedge_descriptor he = halfedge(f, *m_ptm);
      std::size_t i = global_vtag_map[to_sgv_map[source(he, *m_ptm)]];
      std::size_t j = global_vtag_map[to_sgv_map[target(he, *m_ptm)]];
      std::size_t k = global_vtag_map[to_sgv_map[target(next(he, *m_ptm), *m_ptm)]];
      if (i != j && i != k && j != k) {
        std::array<std::size_t, 3> t;
        t[0] = i;
        t[1] = j;
        t[2] = k;
        m_tris.push_back(t);
      }
    }
  }

  /*!
   * @brief walks along the region boundary cycle to the first halfedge
   * pointing to a vertex associated with an anchor.
   * @param[in,out] he_start region boundary halfedge
   */
  void walk_to_first_anchor(halfedge_descriptor &he_start) {
    const halfedge_descriptor start_mark = he_start;
    while (!is_anchor_attached(he_start)) {
      // no anchor attached to the halfedge target
      walk_to_next_border_halfedge(he_start);
      if (he_start == start_mark) // back to where started, a boundary cycle without anchor
        return;
    }
  }

  /*!
   * @brief walks along the region boundary cycle to the next anchor
   * and records the path as a `Boundary_chord`.
   * @param[in,out] he_start starting region boundary halfedge
   * pointing to a vertex associated with an anchor
   * @param[out] chord recorded path chord
   */
  void walk_to_next_anchor(halfedge_descriptor &he_start, Boundary_chord &chord) const {
    do {
      walk_to_next_border_halfedge(he_start);
      chord.push_back(he_start);
    } while (!is_anchor_attached(he_start));
  }

  /*!
   * @brief walks to the next boundary cycle halfedge.
   * @param[in,out] he_start region boundary halfedge
   */
  void walk_to_next_border_halfedge(halfedge_descriptor &he_start) const {
    const std::size_t px_idx = get(m_fproxy_map, face(he_start, *m_ptm));
    for(halfedge_descriptor h : halfedges_around_target(he_start, *m_ptm)) {
      if (CGAL::is_border(h, *m_ptm) || get(m_fproxy_map, face(h, *m_ptm)) != px_idx) {
        he_start = opposite(h, *m_ptm);
        return;
      }
    }
  }

  /*!
   * @brief subdivides a chord recursively in range `[chord_begin, chord_end)`.
   * @param chord_begin begin iterator of the chord
   * @param chord_end end iterator of the chord
   * @param subdivision_ratio the chord recursive split error threshold
   * @param boundary_subdivision_ratio the chord subdivision ratio threshold to the chord length or average edge length for boundary edges
   * @param relative_to_chord set `true` if the `subdivision_ratio` is relative to the chord length (relative sense),
   * otherwise it is relative to the average edge length (absolute sense).
   * @param with_dihedral_angle if set to `true` add dihedral angle weight to the distance.
   * @return the number of anchors of the chord apart from the first one
   */
  std::size_t subdivide_chord(
    const Boundary_chord_iterator &chord_begin,
    const Boundary_chord_iterator &chord_end,
    const FT subdivision_ratio,
    const FT boundary_subdivision_ratio,
    const bool relative_to_chord,
    const bool with_dihedral_angle) {
    const std::size_t chord_size = std::distance(chord_begin, chord_end);
    const halfedge_descriptor he_first = *chord_begin;
    const halfedge_descriptor he_last = *(chord_end - 1);
    const std::size_t anchor_first = get(m_vanchor_map, source(he_first, *m_ptm));
    const std::size_t anchor_last = get(m_vanchor_map, target(he_last, *m_ptm));

    bool is_boundary = is_border_edge(he_first, *m_ptm);

    if(is_boundary && boundary_subdivision_ratio == 0){
      for (Boundary_chord_iterator citr = chord_begin; *citr != he_last; ++citr) {
        attach_anchor(*citr);
      }
    }

    // do not subdivide trivial non-circular chord
    if ((anchor_first != anchor_last) && (chord_size < 2))
      return 1;

    bool if_subdivide = false;
    Boundary_chord_iterator chord_max = chord_begin;
    const Point_3 &pt_begin = m_vpoint_map[source(he_first, *m_ptm)];
    const Point_3 &pt_end = m_vpoint_map[target(he_last, *m_ptm)];
    if (anchor_first == anchor_last) {
      // circular chord
      CGAL_assertion(chord_size > 2);

      FT dist_max(0.0);
      for (Boundary_chord_iterator citr = chord_begin; citr != chord_end; ++citr) {
        const FT dist = CGAL::approximate_sqrt(CGAL::squared_distance(
          pt_begin, m_vpoint_map[target(*citr, *m_ptm)]));
        if (dist > dist_max) {
          chord_max = citr;
          dist_max = dist;
        }
      }

      if_subdivide = true;
    }
    else {
      FT dist_max(0.0);
      Vector_3 chord_vec = vector_functor(pt_begin, pt_end);
      const FT chord_len = CGAL::approximate_sqrt(chord_vec.squared_length());
      bool degenerate_chord = false;
      if (chord_len > FT(0.0)) {
        Segment_3 seg(pt_begin, pt_end);
        for (Boundary_chord_iterator citr = chord_begin; citr != chord_end; ++citr) {
          const FT dist = CGAL::approximate_sqrt(CGAL::squared_distance(m_vpoint_map[target(*citr, *m_ptm)], seg));
          if (dist > dist_max) {
            chord_max = citr;
            dist_max = dist;
          }
        }
      }
      else {
        degenerate_chord = true;
        for (Boundary_chord_iterator citr = chord_begin; citr != chord_end; ++citr) {
          const FT dist = CGAL::approximate_sqrt(CGAL::squared_distance(
            pt_begin, m_vpoint_map[target(*citr, *m_ptm)]));
          if (dist > dist_max) {
            chord_max = citr;
            dist_max = dist;
          }
        }
      }

      FT criterion = dist_max;
      if (relative_to_chord && !degenerate_chord)
        criterion /= chord_len;
      else
        criterion /= m_average_edge_length;

      if (with_dihedral_angle) {
        // suppose the proxy normal angle is acute
        std::size_t px_left = get(m_fproxy_map, face(he_first, *m_ptm));
        std::size_t px_right = px_left;
        if (!CGAL::is_border(opposite(he_first, *m_ptm), *m_ptm))
          px_right = get(m_fproxy_map, face(opposite(he_first, *m_ptm), *m_ptm));
        FT norm_sin(1.0);
        if (!CGAL::is_border(opposite(he_first, *m_ptm), *m_ptm)) {
          Vector_3 vec = cross_product_functor(
            m_px_planes[px_left].normal, m_px_planes[px_right].normal);
          norm_sin = CGAL::approximate_sqrt(vec.squared_length());
        }
        criterion *= norm_sin;
      }
      if (is_boundary) {
        if (criterion > boundary_subdivision_ratio)
          if_subdivide = true;
      }
      else {
        if (criterion > subdivision_ratio)
          if_subdivide = true;
      }
    }
    if (if_subdivide) {
      // subdivide at the most remote vertex
      attach_anchor(*chord_max);

      const std::size_t num_left = subdivide_chord(chord_begin, chord_max + 1,
        subdivision_ratio, boundary_subdivision_ratio, relative_to_chord, with_dihedral_angle);
      const std::size_t num_right = subdivide_chord(chord_max + 1, chord_end,
        subdivision_ratio, boundary_subdivision_ratio, relative_to_chord, with_dihedral_angle);

      return num_left + num_right;
    }

    return 1;
  }

  /*!
   * @brief tests if the target vertex of a halfedge is attached with an anchor.
   * @param he a halfedge descriptor
   * @return `true` is attached with an anchor, and `false` otherwise.
   */
  bool is_anchor_attached(const halfedge_descriptor he) const {
    return is_anchor_attached(target(he, *m_ptm), m_vanchor_map);
  }

  /*!
   * @brief checks if a vertex is attached with an anchor.
   * @tparam VertexAnchorIndexMap `ReadablePropertyMap`
   * with `boost::graph_traits<TriangleMesh>::vertex_descriptor` as key and `std::size_t` as value type
   * @param vtx a vertex descriptor
   * @param vanchor_map vertex anchor index map
   */
  template<typename VertexAnchorIndexMap>
  bool is_anchor_attached(
    const typename boost::property_traits<VertexAnchorIndexMap>::key_type vtx,
    const VertexAnchorIndexMap &vanchor_map) const {
    return get(vanchor_map, vtx) != CGAL_VSA_INVALID_TAG;
  }

  /*!
   * @brief attaches an anchor to the vertex.
   * @param vtx vertex
   */
  void attach_anchor(const vertex_descriptor vtx) {
    put(m_vanchor_map, vtx, m_anchors.size());
    // default anchor location is the vertex point
    m_anchors.push_back(Anchor(vtx, m_vpoint_map[vtx]));
  }

  /*!
   * @brief attaches an anchor to the target vertex of the halfedge.
   * @param he halfedge
   */
  void attach_anchor(const halfedge_descriptor he) {
    attach_anchor(target(he, *m_ptm));
  }

  /*!
   * @brief optimizes the anchor location by averaging the projection points of
   * the anchor vertex to the incident proxy plane.
   */
  void optimize_anchor_location(bool optimize_boundary_anchor_location) {
    for(Anchor& a : m_anchors) {
      const vertex_descriptor v = a.vtx;

      if(! optimize_boundary_anchor_location && is_border(v,*m_ptm)){
        a.pos  = m_vpoint_map[v];
        continue;
      }

      // incident proxy set
      std::set<std::size_t> px_set;
      for(halfedge_descriptor h : halfedges_around_target(v, *m_ptm)) {
        if (!CGAL::is_border(h, *m_ptm))
          px_set.insert(get(m_fproxy_map, face(h, *m_ptm)));
      }

      // projection
      // todo: replace averaging by qem/svd ?  Mael?
      FT sum_area(0.0);
      Vector_3 vec = CGAL::NULL_VECTOR;
      const Point_3 vtx_pt = m_vpoint_map[v];
      for(const std::size_t px_idx : px_set) {
        const Vector_3 proj = vector_functor(
          CGAL::ORIGIN, m_px_planes[px_idx].plane.projection(vtx_pt));
        const FT area = m_px_planes[px_idx].area;
        vec = sum_functor(vec, scale_functor(proj, area));
        sum_area += area;
      }
      if (sum_area > FT(0.0))
        a.pos = translate_point_functor(
          CGAL::ORIGIN,
          scale_functor(vec, FT(1.0) / sum_area));
      else
        a.pos = vtx_pt;
    }
  }

  /*!
   * @brief calculates the averaged edge length of a triangle mesh.
   * @param tm the input triangle mesh
   * @param vpoint_map vertex point map
   * @return averaged edge length
   */
  FT compute_averaged_edge_length(const TriangleMesh &tm, const VertexPointMap &vpoint_map) const {
    // compute average edge length
    FT sum(0.0);
    for(edge_descriptor e : edges(tm)) {
      const vertex_descriptor vs = source(e, tm);
      const vertex_descriptor vt = target(e, tm);
      sum += CGAL::approximate_sqrt(CGAL::squared_distance(
        vpoint_map[vs], vpoint_map[vt]));
    }
    return sum / num_edges(tm);
  }

  /*!
   * @brief fits an area averaged plane from a range of faces.
   * @tparam FaceIterator face_descriptor container iterator
   * @param beg container begin
   * @param end container end
   * @return fitted plane
   */
  template <typename FaceIterator>
  Plane_3 fit_plane_area_averaged(const FaceIterator &beg, const FaceIterator &end) {
    CGAL_assertion(beg != end);
    // area average normal and centroid
    Vector_3 norm = CGAL::NULL_VECTOR;
    Vector_3 cent = CGAL::NULL_VECTOR;
    FT sum_area(0.0);
    for (FaceIterator fitr = beg; fitr != end; ++fitr) {
      const halfedge_descriptor he = halfedge(*fitr, *m_ptm);
      const Point_3 &p0 = m_vpoint_map[source(he, *m_ptm)];
      const Point_3 &p1 = m_vpoint_map[target(he, *m_ptm)];
      const Point_3 &p2 = m_vpoint_map[target(next(he, *m_ptm), *m_ptm)];

      Vector_3 vec = vector_functor(CGAL::ORIGIN, CGAL::centroid(p0, p1, p2));
      const FT farea = CGAL::approximate_sqrt(CGAL::squared_area(p0, p1, p2));
      Vector_3 fnorm =
        collinear_functor(p0, p1, p2) ? CGAL::NULL_VECTOR : CGAL::unit_normal(p0, p1, p2);

      norm = sum_functor(norm, scale_functor(fnorm, farea));
      cent = sum_functor(cent, scale_functor(vec, farea));
      sum_area += farea;
    }
    if (norm.squared_length() > FT(0.0))
      norm = scale_functor(norm, FT(1.0) / CGAL::approximate_sqrt(norm.squared_length()));
    else
      norm = Vector_3(FT(0.0), FT(0.0), FT(1.0));
    if (sum_area > FT(0.0))
      cent = scale_functor(cent, FT(1.0) / sum_area);

    return Plane_3(CGAL::ORIGIN + cent, norm);
  }

  /*!
   * @brief fits a plane from a range of faces with PCA algorithm.
   * @tparam FaceIterator face_descriptor container iterator
   * @param beg container begin
   * @param end container end
   * @return fitted plane
   */
  template <typename FaceIterator>
  Plane_3 fit_plane_pca(const FaceIterator &beg, const FaceIterator &end) {
    CGAL_assertion(beg != end);

    typedef typename Geom_traits::Triangle_3 Triangle_3;
    std::list<Triangle_3> tri_list;
    for (FaceIterator fitr = beg; fitr != end; ++fitr) {
      halfedge_descriptor he = halfedge(*fitr, *m_ptm);
      const Point_3 &p0 = m_vpoint_map[source(he, *m_ptm)];
      const Point_3 &p1 = m_vpoint_map[target(he, *m_ptm)];
      const Point_3 &p2 = m_vpoint_map[target(next(he, *m_ptm), *m_ptm)];
      tri_list.push_back(Triangle_3(p0, p1, p2));
    }

    // construct and fit proxy plane
    Plane_3 fit_plane;
    CGAL::linear_least_squares_fitting_3(
      tri_list.begin(),
      tri_list.end(),
      fit_plane,
      CGAL::Dimension_tag<2>());

    return fit_plane;
  }
};

} // end namespace CGAL

#undef CGAL_VSA_INVALID_TAG

#endif // CGAL_VARIATIONAL_SHAPE_APPROXIMATION_H
