Simplex_tree.h

/*    This file is part of the Gudhi Library. The Gudhi library
 *    (Geometric Understanding in Higher Dimensions) is a generic C++
 *    library for computational topology.
 *
 *    Author(s):       Clément Maria
 *
 *    Copyright (C) 2014 Inria
 *
 *    This program is free software: you can redistribute it and/or modify
 *    it under the terms of the GNU General Public License as published by
 *    the Free Software Foundation, either version 3 of the License, or
 *    (at your option) any later version.
 *
 *    This program is distributed in the hope that it will be useful,
 *    but WITHOUT ANY WARRANTY; without even the implied warranty of
 *    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *    GNU General Public License for more details.
 *
 *    You should have received a copy of the GNU General Public License
 *    along with this program.  If not, see <http://www.gnu.org/licenses/>.
 */

#ifndef SIMPLEX_TREE_H_
#define SIMPLEX_TREE_H_

#include <gudhi/Simplex_tree/Simplex_tree_node_explicit_storage.h>
#include <gudhi/Simplex_tree/Simplex_tree_siblings.h>
#include <gudhi/Simplex_tree/Simplex_tree_iterators.h>
#include <gudhi/Simplex_tree/indexing_tag.h>

#include <gudhi/reader_utils.h>
#include <gudhi/graph_simplicial_complex.h>
#include <gudhi/Debug_utils.h>

#include <boost/container/flat_map.hpp>
#include <boost/iterator/transform_iterator.hpp>
#include <boost/graph/adjacency_list.hpp>
#include <boost/range/adaptor/reversed.hpp>

#ifdef GUDHI_USE_TBB
#include <tbb/parallel_sort.h>
#endif

#include <utility>
#include <vector>
#include <functional>  // for greater<>
#include <stdexcept>
#include <limits>  // Inf
#include <initializer_list>
#include <algorithm>  // for std::max
#include <cstdint>  // for std::uint32_t
#include <iterator>  // for std::distance

namespace Gudhi {

struct Simplex_tree_options_full_featured;

template<typename SimplexTreeOptions = Simplex_tree_options_full_featured>
class Simplex_tree {
 public:
  typedef SimplexTreeOptions Options;
  typedef typename Options::Indexing_tag Indexing_tag;
  typedef typename Options::Filtration_value Filtration_value;
  typedef typename Options::Simplex_key Simplex_key;
  typedef typename Options::Vertex_handle Vertex_handle;

  /* Type of node in the simplex tree. */
  typedef Simplex_tree_node_explicit_storage<Simplex_tree> Node;
  /* Type of dictionary Vertex_handle -> Node for traversing the simplex tree. */
  // Note: this wastes space when Vertex_handle is 32 bits and Node is aligned on 64 bits. It would be better to use a
  // flat_set (with our own comparator) where we can control the layout of the struct (put Vertex_handle and
  // Simplex_key next to each other).
  typedef typename boost::container::flat_map<Vertex_handle, Node> Dictionary;

  /* \brief Set of nodes sharing a same parent in the simplex tree. */
  /* \brief Set of nodes sharing a same parent in the simplex tree. */
  typedef Simplex_tree_siblings<Simplex_tree, Dictionary> Siblings;

  struct Key_simplex_base_real {
    Key_simplex_base_real() : key_(-1) {}
    void assign_key(Simplex_key k) { key_ = k; }
    Simplex_key key() const { return key_; }
   private:
    Simplex_key key_;
  };
  struct Key_simplex_base_dummy {
    Key_simplex_base_dummy() {}
    // Undefined so it will not link
    void assign_key(Simplex_key);
    Simplex_key key() const;
  };
  typedef typename std::conditional<Options::store_key, Key_simplex_base_real, Key_simplex_base_dummy>::type
      Key_simplex_base;

  struct Filtration_simplex_base_real {
    Filtration_simplex_base_real() : filt_(0) {}
    void assign_filtration(Filtration_value f) { filt_ = f; }
    Filtration_value filtration() const { return filt_; }
   private:
    Filtration_value filt_;
  };
  struct Filtration_simplex_base_dummy {
    Filtration_simplex_base_dummy() {}
    void assign_filtration(Filtration_value GUDHI_CHECK_code(f)) { GUDHI_CHECK(f == 0, "filtration value specified for a complex that does not store them"); }
    Filtration_value filtration() const { return 0; }
  };
  typedef typename std::conditional<Options::store_filtration, Filtration_simplex_base_real,
    Filtration_simplex_base_dummy>::type Filtration_simplex_base;

 public:
  typedef typename Dictionary::iterator Simplex_handle;

 private:
  typedef typename Dictionary::iterator Dictionary_it;
  typedef typename Dictionary_it::value_type Dit_value_t;

  struct return_first {
    Vertex_handle operator()(const Dit_value_t& p_sh) const {
      return p_sh.first;
    }
  };

 public:
  typedef boost::transform_iterator<return_first, Dictionary_it> Complex_vertex_iterator;
  typedef boost::iterator_range<Complex_vertex_iterator> Complex_vertex_range;
  typedef Simplex_tree_simplex_vertex_iterator<Simplex_tree> Simplex_vertex_iterator;
  typedef boost::iterator_range<Simplex_vertex_iterator> Simplex_vertex_range;
  typedef std::vector<Simplex_handle> Cofaces_simplex_range;
  typedef Simplex_tree_boundary_simplex_iterator<Simplex_tree> Boundary_simplex_iterator;
  typedef boost::iterator_range<Boundary_simplex_iterator> Boundary_simplex_range;
  typedef Simplex_tree_complex_simplex_iterator<Simplex_tree> Complex_simplex_iterator;
  typedef boost::iterator_range<Complex_simplex_iterator> Complex_simplex_range;
  typedef Simplex_tree_skeleton_simplex_iterator<Simplex_tree> Skeleton_simplex_iterator;
  typedef boost::iterator_range<Skeleton_simplex_iterator> Skeleton_simplex_range;
  typedef std::vector<Simplex_handle> Filtration_simplex_range;
  typedef typename Filtration_simplex_range::const_iterator Filtration_simplex_iterator;

  /* @} */  // end name range and iterator types
  Complex_vertex_range complex_vertex_range() {
    return Complex_vertex_range(
                                boost::make_transform_iterator(root_.members_.begin(), return_first()),
                                boost::make_transform_iterator(root_.members_.end(), return_first()));
  }

  Complex_simplex_range complex_simplex_range() {
    return Complex_simplex_range(Complex_simplex_iterator(this),
                                 Complex_simplex_iterator());
  }

  Skeleton_simplex_range skeleton_simplex_range(int dim) {
    return Skeleton_simplex_range(Skeleton_simplex_iterator(this, dim),
                                  Skeleton_simplex_iterator());
  }

  Filtration_simplex_range const& filtration_simplex_range(Indexing_tag = Indexing_tag()) {
    if (filtration_vect_.empty()) {
      initialize_filtration();
    }
    return filtration_vect_;
  }

  Simplex_vertex_range simplex_vertex_range(Simplex_handle sh) {
    assert(sh != null_simplex());  // Empty simplex
    return Simplex_vertex_range(Simplex_vertex_iterator(this, sh),
                                Simplex_vertex_iterator(this));
  }

  template<class SimplexHandle>
  Boundary_simplex_range boundary_simplex_range(SimplexHandle sh) {
    return Boundary_simplex_range(Boundary_simplex_iterator(this, sh),
                                  Boundary_simplex_iterator(this));
  }
  // end range and iterator methods
  Simplex_tree()
      : null_vertex_(-1),
      root_(nullptr, null_vertex_),
      filtration_vect_(),
      dimension_(-1) { }

  Simplex_tree(const Simplex_tree& simplex_source)
      : null_vertex_(simplex_source.null_vertex_),
      root_(nullptr, null_vertex_ , simplex_source.root_.members_),
      filtration_vect_(),
      dimension_(simplex_source.dimension_) {
    auto root_source = simplex_source.root_;
    rec_copy(&root_, &root_source);
  }

  void rec_copy(Siblings *sib, Siblings *sib_source) {
    for (auto sh = sib->members().begin(), sh_source = sib_source->members().begin();
         sh != sib->members().end(); ++sh, ++sh_source) {
      if (has_children(sh_source)) {
        Siblings * newsib = new Siblings(sib, sh_source->first);
        newsib->members_.reserve(sh_source->second.children()->members().size());
        for (auto & child : sh_source->second.children()->members())
          newsib->members_.emplace_hint(newsib->members_.end(), child.first, Node(newsib, child.second.filtration()));
        rec_copy(newsib, sh_source->second.children());
        sh->second.assign_children(newsib);
      }
    }
  }

  Simplex_tree(Simplex_tree && old)
      : null_vertex_(std::move(old.null_vertex_)),
      root_(std::move(old.root_)),
      filtration_vect_(std::move(old.filtration_vect_)),
      dimension_(std::move(old.dimension_)) {
    old.dimension_ = -1;
    old.root_ = Siblings(nullptr, null_vertex_);
  }

  ~Simplex_tree() {
    for (auto sh = root_.members().begin(); sh != root_.members().end(); ++sh) {
      if (has_children(sh)) {
        rec_delete(sh->second.children());
      }
    }
  }  // end constructor/destructor
 private:
  // Recursive deletion
  void rec_delete(Siblings * sib) {
    for (auto sh = sib->members().begin(); sh != sib->members().end(); ++sh) {
      if (has_children(sh)) {
        rec_delete(sh->second.children());
      }
    }
    delete sib;
  }

 public:
  bool operator==(Simplex_tree& st2) {
    if ((null_vertex_ != st2.null_vertex_) ||
        (dimension_ != st2.dimension_))
      return false;
    return rec_equal(&root_, &st2.root_);
  }

  bool operator!=(Simplex_tree& st2) {
    return (!(*this == st2));
  }

 private:
  bool rec_equal(Siblings* s1, Siblings* s2) {
    if (s1->members().size() != s2->members().size())
      return false;
    for (auto sh1 = s1->members().begin(), sh2 = s2->members().begin();
         (sh1 != s1->members().end() && sh2 != s2->members().end()); ++sh1, ++sh2) {
      if (sh1->first != sh2->first || sh1->second.filtration() != sh2->second.filtration())
        return false;
      if (has_children(sh1) != has_children(sh2))
        return false;
      // Recursivity on children only if both have children
      else if (has_children(sh1))
        if (!rec_equal(sh1->second.children(), sh2->second.children()))
          return false;
    }
    return true;
  }

 public:
  static Simplex_key key(Simplex_handle sh) {
    return sh->second.key();
  }

  Simplex_handle simplex(Simplex_key idx) const {
    return filtration_vect_[idx];
  }

  static Filtration_value filtration(Simplex_handle sh) {
    if (sh != null_simplex()) {
      return sh->second.filtration();
    } else {
      return std::numeric_limits<Filtration_value>::infinity();
    }
  }

  void assign_filtration(Simplex_handle sh, Filtration_value fv) {
    GUDHI_CHECK(sh != null_simplex(),
                std::invalid_argument("Simplex_tree::assign_filtration - cannot assign filtration on null_simplex"));
    sh->second.assign_filtration(fv);
  }

  static Simplex_handle null_simplex() {
    return Dictionary_it(nullptr);
  }

  static Simplex_key null_key() {
    return -1;
  }

  Vertex_handle null_vertex() const {
    return null_vertex_;
  }

  size_t num_vertices() const {
    return root_.members_.size();
  }

 public:
  size_t num_simplices() {
    return num_simplices(&root_);
  }

 private:
  size_t num_simplices(Siblings * sib) {
    auto sib_begin = sib->members().begin();
    auto sib_end = sib->members().end();
    size_t simplices_number = sib_end - sib_begin;
    for (auto sh = sib_begin; sh != sib_end; ++sh) {
      if (has_children(sh)) {
        simplices_number += num_simplices(sh->second.children());
      }
    }
    return simplices_number;
  }

 public:
  int dimension(Simplex_handle sh) {
    Siblings * curr_sib = self_siblings(sh);
    int dim = 0;
    while (curr_sib != nullptr) {
      ++dim;
      curr_sib = curr_sib->oncles();
    }
    return dim - 1;
  }

  int upper_bound_dimension() const {
    return dimension_;
  }

  int dimension() {
    if (dimension_to_be_lowered_)
      lower_upper_bound_dimension();
    return dimension_;
  }

  template<class SimplexHandle>
  bool has_children(SimplexHandle sh) const {
    // Here we rely on the root using null_vertex(), which cannot match any real vertex.
    return (sh->second.children()->parent() == sh->first);
  }

  template<class InputVertexRange = std::initializer_list<Vertex_handle>>
  Simplex_handle find(const InputVertexRange & s) {
    auto first = std::begin(s);
    auto last = std::end(s);

    if (first == last)
      return null_simplex();  // ----->>

    // Copy before sorting
    std::vector<Vertex_handle> copy(first, last);
    std::sort(std::begin(copy), std::end(copy));
    return find_simplex(copy);
  }

 private:
  Simplex_handle find_simplex(const std::vector<Vertex_handle> & simplex) {
    Siblings * tmp_sib = &root_;
    Dictionary_it tmp_dit;
    auto vi = simplex.begin();
    if (Options::contiguous_vertices) {
      // Equivalent to the first iteration of the normal loop
      GUDHI_CHECK(contiguous_vertices(), "non-contiguous vertices");
      Vertex_handle v = *vi++;
      if(v < 0 || v >= static_cast<Vertex_handle>(root_.members_.size()))
        return null_simplex();
      tmp_dit = root_.members_.begin() + v;
      if (vi == simplex.end())
        return tmp_dit;
      if (!has_children(tmp_dit))
        return null_simplex();
      tmp_sib = tmp_dit->second.children();
    }
    for (;;) {
      tmp_dit = tmp_sib->members_.find(*vi++);
      if (tmp_dit == tmp_sib->members_.end())
        return null_simplex();
      if (vi == simplex.end())
        return tmp_dit;
      if (!has_children(tmp_dit))
        return null_simplex();
      tmp_sib = tmp_dit->second.children();
    }
  }

  Simplex_handle find_vertex(Vertex_handle v) {
    if (Options::contiguous_vertices) {
      assert(contiguous_vertices());
      return root_.members_.begin() + v;
    } else {
      return root_.members_.find(v);
    }
  }

 public:
  bool contiguous_vertices() const {
    if (root_.members_.empty()) return true;
    if (root_.members_.begin()->first != 0) return false;
    if (std::prev(root_.members_.end())->first != static_cast<Vertex_handle>(root_.members_.size() - 1)) return false;
    return true;
  }

 private:
  std::pair<Simplex_handle, bool> insert_vertex_vector(const std::vector<Vertex_handle>& simplex,
                                                     Filtration_value filtration) {
    Siblings * curr_sib = &root_;
    std::pair<Simplex_handle, bool> res_insert;
    auto vi = simplex.begin();
    for (; vi != simplex.end() - 1; ++vi) {
      GUDHI_CHECK(*vi != null_vertex(), "cannot use the dummy null_vertex() as a real vertex");
      res_insert = curr_sib->members_.emplace(*vi, Node(curr_sib, filtration));
      if (!(has_children(res_insert.first))) {
        res_insert.first->second.assign_children(new Siblings(curr_sib, *vi));
      }
      curr_sib = res_insert.first->second.children();
    }
    GUDHI_CHECK(*vi != null_vertex(), "cannot use the dummy null_vertex() as a real vertex");
    res_insert = curr_sib->members_.emplace(*vi, Node(curr_sib, filtration));
    if (!res_insert.second) {
      // if already in the complex
      if (res_insert.first->second.filtration() > filtration) {
        // if filtration value modified
        res_insert.first->second.assign_filtration(filtration);
        return res_insert;
      }
      // if filtration value unchanged
      return std::pair<Simplex_handle, bool>(null_simplex(), false);
    }
    // otherwise the insertion has succeeded - size is a size_type
    if (static_cast<int>(simplex.size()) - 1 > dimension_) {
      // Update dimension if needed
      dimension_ = static_cast<int>(simplex.size()) - 1;
    }
    return res_insert;
  }

 public:
  template<class InputVertexRange = std::initializer_list<Vertex_handle>>
  std::pair<Simplex_handle, bool> insert_simplex(const InputVertexRange & simplex,
                                                 Filtration_value filtration = 0) {
    auto first = std::begin(simplex);
    auto last = std::end(simplex);

    if (first == last)
      return std::pair<Simplex_handle, bool>(null_simplex(), true);  // ----->>

    // Copy before sorting
    std::vector<Vertex_handle> copy(first, last);
    std::sort(std::begin(copy), std::end(copy));
    return insert_vertex_vector(copy, filtration);
  }

  template<class InputVertexRange = std::initializer_list<Vertex_handle>>
  std::pair<Simplex_handle, bool> insert_simplex_and_subfaces(const InputVertexRange& Nsimplex,
                    Filtration_value filtration = 0) {
    auto first = std::begin(Nsimplex);
    auto last = std::end(Nsimplex);

    if (first == last)
      return { null_simplex(), true };  // ----->>

    // Copy before sorting
    // Thread local is not available on XCode version < V.8 - It will slow down computation
#ifdef GUDHI_CAN_USE_CXX11_THREAD_LOCAL
    thread_local
#endif  // GUDHI_CAN_USE_CXX11_THREAD_LOCAL
    std::vector<Vertex_handle> copy;
    copy.clear();
    copy.insert(copy.end(), first, last);
    std::sort(std::begin(copy), std::end(copy));
    GUDHI_CHECK_code(
      for (Vertex_handle v : copy)
        GUDHI_CHECK(v != null_vertex(), "cannot use the dummy null_vertex() as a real vertex");
    )

    return insert_simplex_and_subfaces_sorted(copy, filtration);
  }

 private:
  template<class ForwardVertexRange = std::initializer_list<Vertex_handle>>
  std::pair<Simplex_handle, bool> insert_simplex_and_subfaces_sorted(const ForwardVertexRange& Nsimplex, Filtration_value filt = 0) {
    auto first = std::begin(Nsimplex);
    auto last = std::end(Nsimplex);
    if (first == last)
      return { null_simplex(), true }; // FIXME: false would make more sense to me.
    GUDHI_CHECK(std::is_sorted(first, last), "simplex vertices listed in unsorted order");
    // Update dimension if needed. We could wait to see if the insertion succeeds, but I doubt there is much to gain.
    dimension_ = (std::max)(dimension_, static_cast<int>(std::distance(first, last)) - 1);
    return rec_insert_simplex_and_subfaces_sorted(root(), first, last, filt);
  }
  // To insert {1,2,3,4}, we insert {2,3,4} twice, once at the root, and once below 1.
  template<class ForwardVertexIterator>
  std::pair<Simplex_handle, bool> rec_insert_simplex_and_subfaces_sorted(Siblings* sib, ForwardVertexIterator first, ForwardVertexIterator last, Filtration_value filt) {
    // An alternative strategy would be:
    // - try to find the complete simplex, if found (and low filtration) exit
    // - insert all the vertices at once in sib
    // - loop over those (new or not) simplices, with a recursive call(++first, last)
    Vertex_handle vertex_one = *first;
    auto&& dict = sib->members();
    auto insertion_result = dict.emplace(vertex_one, Node(sib, filt));
    Simplex_handle simplex_one = insertion_result.first;
    bool one_is_new = insertion_result.second;
    if (!one_is_new) {
      if (filtration(simplex_one) > filt) {
    assign_filtration(simplex_one, filt);
      } else {
    // FIXME: this interface makes no sense, and it doesn't seem to be tested.
    insertion_result.first = null_simplex();
      }
    }
    if (++first == last) return insertion_result;
    if (!has_children(simplex_one))
      // TODO: have special code here, we know we are building the whole subtree from scratch.
      simplex_one->second.assign_children(new Siblings(sib, vertex_one));
    auto res = rec_insert_simplex_and_subfaces_sorted(simplex_one->second.children(), first, last, filt);
    // No need to continue if the full simplex was already there with a low enough filtration value.
    if (res.first != null_simplex()) rec_insert_simplex_and_subfaces_sorted(sib, first, last, filt);
    return res;
  }

 public:
  void assign_key(Simplex_handle sh, Simplex_key key) {
    sh->second.assign_key(key);
  }

  std::pair<Simplex_handle, Simplex_handle> endpoints(Simplex_handle sh) {
    assert(dimension(sh) == 1);
    return { find_vertex(sh->first), find_vertex(self_siblings(sh)->parent()) };
  }

  template<class SimplexHandle>
  Siblings* self_siblings(SimplexHandle sh) {
    if (sh->second.children()->parent() == sh->first)
      return sh->second.children()->oncles();
    else
      return sh->second.children();
  }

 public:
  Siblings * root() {
    return &root_;
  }

  void set_dimension(int dimension) {
    dimension_to_be_lowered_ = false;
    dimension_ = dimension;
  }

 public:
  void initialize_filtration() {
    filtration_vect_.clear();
    filtration_vect_.reserve(num_simplices());
    for (Simplex_handle sh : complex_simplex_range())
      filtration_vect_.push_back(sh);

    /* We use stable_sort here because with libstdc++ it is faster than sort.
     * is_before_in_filtration is now a total order, but we used to call
     * stable_sort for the following heuristic:
     * The use of a depth-first traversal of the simplex tree, provided by
     * complex_simplex_range(), combined with a stable sort is meant to
     * optimize the order of simplices with same filtration value. The
     * heuristic consists in inserting the cofaces of a simplex as soon as
     * possible.
     */
#ifdef GUDHI_USE_TBB
    tbb::parallel_sort(filtration_vect_.begin(), filtration_vect_.end(), is_before_in_filtration(this));
#else
    std::stable_sort(filtration_vect_.begin(), filtration_vect_.end(), is_before_in_filtration(this));
#endif
  }

 private:
  void rec_coface(std::vector<Vertex_handle> &vertices, Siblings *curr_sib, int curr_nbVertices,
                  std::vector<Simplex_handle>& cofaces, bool star, int nbVertices) {
    if (!(star || curr_nbVertices <= nbVertices))  // dimension of actual simplex <= nbVertices
      return;
    for (Simplex_handle simplex = curr_sib->members().begin(); simplex != curr_sib->members().end(); ++simplex) {
      if (vertices.empty()) {
        // If we reached the end of the vertices, and the simplex has more vertices than the given simplex
        // => we found a coface

        // Add a coface if we wan't the star or if the number of vertices of the current simplex matches with nbVertices
        bool addCoface = (star || curr_nbVertices == nbVertices);
        if (addCoface)
          cofaces.push_back(simplex);
        if ((!addCoface || star) && has_children(simplex))  // Rec call
          rec_coface(vertices, simplex->second.children(), curr_nbVertices + 1, cofaces, star, nbVertices);
      } else {
        if (simplex->first == vertices.back()) {
          // If curr_sib matches with the top vertex
          bool equalDim = (star || curr_nbVertices == nbVertices);  // dimension of actual simplex == nbVertices
          bool addCoface = vertices.size() == 1 && equalDim;
          if (addCoface)
            cofaces.push_back(simplex);
          if ((!addCoface || star) && has_children(simplex)) {
            // Rec call
            Vertex_handle tmp = vertices.back();
            vertices.pop_back();
            rec_coface(vertices, simplex->second.children(), curr_nbVertices + 1, cofaces, star, nbVertices);
            vertices.push_back(tmp);
          }
        } else if (simplex->first > vertices.back()) {
          return;
        } else {
          // (simplex->first < vertices.back()
          if (has_children(simplex))
            rec_coface(vertices, simplex->second.children(), curr_nbVertices + 1, cofaces, star, nbVertices);
        }
      }
    }
  }

 public:
  Cofaces_simplex_range star_simplex_range(const Simplex_handle simplex) {
    return cofaces_simplex_range(simplex, 0);
  }

  Cofaces_simplex_range cofaces_simplex_range(const Simplex_handle simplex, int codimension) {
    Cofaces_simplex_range cofaces;
    // codimension must be positive or null integer
    assert(codimension >= 0);
    Simplex_vertex_range rg = simplex_vertex_range(simplex);
    std::vector<Vertex_handle> copy(rg.begin(), rg.end());
    if (codimension + static_cast<int>(copy.size()) > dimension_ + 1 ||
        (codimension == 0 && static_cast<int>(copy.size()) > dimension_))  // n+codimension greater than dimension_
      return cofaces;
    // must be sorted in decreasing order
    assert(std::is_sorted(copy.begin(), copy.end(), std::greater<Vertex_handle>()));
    bool star = codimension == 0;
    rec_coface(copy, &root_, 1, cofaces, star, codimension + static_cast<int>(copy.size()));
    return cofaces;
  }

 private:
  bool reverse_lexicographic_order(Simplex_handle sh1, Simplex_handle sh2) {
    Simplex_vertex_range rg1 = simplex_vertex_range(sh1);
    Simplex_vertex_range rg2 = simplex_vertex_range(sh2);
    Simplex_vertex_iterator it1 = rg1.begin();
    Simplex_vertex_iterator it2 = rg2.begin();
    while (it1 != rg1.end() && it2 != rg2.end()) {
      if (*it1 == *it2) {
        ++it1;
        ++it2;
      } else {
        return *it1 < *it2;
      }
    }
    return ((it1 == rg1.end()) && (it2 != rg2.end()));
  }

  struct is_before_in_filtration {
    explicit is_before_in_filtration(Simplex_tree * st)
        : st_(st) { }

    bool operator()(const Simplex_handle sh1, const Simplex_handle sh2) const {
      // Not using st_->filtration(sh1) because it uselessly tests for null_simplex.
      if (sh1->second.filtration() != sh2->second.filtration()) {
        return sh1->second.filtration() < sh2->second.filtration();
      }
      // is sh1 a proper subface of sh2
      return st_->reverse_lexicographic_order(sh1, sh2);
    }

    Simplex_tree * st_;
  };

 public:
  template<class OneSkeletonGraph>
  void insert_graph(const OneSkeletonGraph& skel_graph) {
    // the simplex tree must be empty
    assert(num_simplices() == 0);

    if (boost::num_vertices(skel_graph) == 0) {
      return;
    }
    if (num_edges(skel_graph) == 0) {
      dimension_ = 0;
    } else {
      dimension_ = 1;
    }

    root_.members_.reserve(boost::num_vertices(skel_graph));

    typename boost::graph_traits<OneSkeletonGraph>::vertex_iterator v_it,
        v_it_end;
    for (std::tie(v_it, v_it_end) = boost::vertices(skel_graph); v_it != v_it_end;
         ++v_it) {
      root_.members_.emplace_hint(
                                  root_.members_.end(), *v_it,
                                  Node(&root_, boost::get(vertex_filtration_t(), skel_graph, *v_it)));
    }
    typename boost::graph_traits<OneSkeletonGraph>::edge_iterator e_it,
        e_it_end;
    for (std::tie(e_it, e_it_end) = boost::edges(skel_graph); e_it != e_it_end;
         ++e_it) {
      auto u = source(*e_it, skel_graph);
      auto v = target(*e_it, skel_graph);
      if (u == v) throw "Self-loops are not simplicial";
      // We cannot skip edges with the wrong orientation and expect them to
      // come a second time with the right orientation, that does not always
      // happen in practice. emplace() should be a NOP when an element with the
      // same key is already there, so seeing the same edge multiple times is
      // ok.
      // Should we actually forbid multiple edges? That would be consistent
      // with rejecting self-loops.
      if (v < u) std::swap(u, v);
      auto sh = find_vertex(u);
      if (!has_children(sh)) {
        sh->second.assign_children(new Siblings(&root_, sh->first));
      }

      sh->second.children()->members().emplace(v,
          Node(sh->second.children(), boost::get(edge_filtration_t(), skel_graph, *e_it)));
    }
  }

  void expansion(int max_dim) {
    dimension_ = max_dim;
    for (Dictionary_it root_it = root_.members_.begin();
         root_it != root_.members_.end(); ++root_it) {
      if (has_children(root_it)) {
        siblings_expansion(root_it->second.children(), max_dim - 1);
      }
    }
    dimension_ = max_dim - dimension_;
  }

 private:
  void siblings_expansion(Siblings * siblings,  // must contain elements
                          int k) {
    if (dimension_ > k) {
      dimension_ = k;
    }
    if (k == 0)
      return;
    Dictionary_it next = siblings->members().begin();
    ++next;

#ifdef GUDHI_CAN_USE_CXX11_THREAD_LOCAL
    thread_local
#endif  // GUDHI_CAN_USE_CXX11_THREAD_LOCAL
    std::vector<std::pair<Vertex_handle, Node> > inter;
    for (Dictionary_it s_h = siblings->members().begin();
         s_h != siblings->members().end(); ++s_h, ++next) {
      Simplex_handle root_sh = find_vertex(s_h->first);
      if (has_children(root_sh)) {
        intersection(
                     inter,  // output intersection
                     next,  // begin
                     siblings->members().end(),  // end
                     root_sh->second.children()->members().begin(),
                     root_sh->second.children()->members().end(),
                     s_h->second.filtration());
        if (inter.size() != 0) {
          Siblings * new_sib = new Siblings(siblings,  // oncles
                                            s_h->first,  // parent
                                            inter);  // boost::container::ordered_unique_range_t
          inter.clear();
          s_h->second.assign_children(new_sib);
          siblings_expansion(new_sib, k - 1);
        } else {
          // ensure the children property
          s_h->second.assign_children(siblings);
          inter.clear();
        }
      }
    }
  }

  static void intersection(std::vector<std::pair<Vertex_handle, Node> >& intersection,
                           Dictionary_it begin1, Dictionary_it end1,
                           Dictionary_it begin2, Dictionary_it end2,
                           Filtration_value filtration_) {
    if (begin1 == end1 || begin2 == end2)
      return;  // ----->>
    while (true) {
      if (begin1->first == begin2->first) {
        Filtration_value filt = (std::max)({begin1->second.filtration(), begin2->second.filtration(), filtration_});
        intersection.emplace_back(begin1->first, Node(nullptr, filt));
        if (++begin1 == end1 || ++begin2 == end2)
          return;  // ----->>
      } else if (begin1->first < begin2->first) {
        if (++begin1 == end1)
          return;
      } else /* begin1->first > begin2->first */ {
        if (++begin2 == end2)
          return;  // ----->>
      }
    }
  }

 public:
  template< typename Blocker >
  void expansion_with_blockers(int max_dim, Blocker block_simplex) {
    // Loop must be from the end to the beginning, as higher dimension simplex are always on the left part of the tree
    for (auto& simplex : boost::adaptors::reverse(root_.members())) {
      if (has_children(&simplex)) {
        siblings_expansion_with_blockers(simplex.second.children(), max_dim, max_dim - 1, block_simplex);
      }
    }
  }

 private:
  template< typename Blocker >
  void siblings_expansion_with_blockers(Siblings* siblings, int max_dim, int k, Blocker block_simplex) {
    if (dimension_ < max_dim - k) {
      dimension_ = max_dim - k;
    }
    if (k == 0)
      return;
    // No need to go deeper
    if (siblings->members().size() < 2)
      return;
    // Reverse loop starting before the last one for 'next' to be the last one
    for (auto simplex = siblings->members().rbegin() + 1; simplex != siblings->members().rend(); simplex++) {
      std::vector<std::pair<Vertex_handle, Node> > intersection;
      for(auto next = siblings->members().rbegin(); next != simplex; next++) {
        bool to_be_inserted = true;
        Filtration_value filt = simplex->second.filtration();
        // If all the boundaries are present, 'next' needs to be inserted
        for (Simplex_handle border : boundary_simplex_range(simplex)) {
          Simplex_handle border_child = find_child(border, next->first);
          if (border_child == null_simplex()) {
            to_be_inserted=false;
            break;
          }
          filt = (std::max)(filt, filtration(border_child));
        }
        if (to_be_inserted) {
          intersection.emplace_back(next->first, Node(nullptr, filt));
        }
      }
      if (intersection.size() != 0) {
        // Reverse the order to insert
        Siblings * new_sib = new Siblings(siblings,  // oncles
                                          simplex->first,  // parent
                                          boost::adaptors::reverse(intersection));  // boost::container::ordered_unique_range_t
        std::vector<Vertex_handle> blocked_new_sib_vertex_list;
        // As all intersections are inserted, we can call the blocker function on all new_sib members
        for (auto new_sib_member = new_sib->members().begin();
             new_sib_member != new_sib->members().end();
             new_sib_member++) {
           bool blocker_result = block_simplex(new_sib_member);
           // new_sib member has been blocked by the blocker function
           // add it to the list to be removed - do not perform it while looping on it
           if (blocker_result) {
             blocked_new_sib_vertex_list.push_back(new_sib_member->first);
           }
        }
        if (blocked_new_sib_vertex_list.size() == new_sib->members().size()) {
          // Specific case where all have to be deleted
          delete new_sib;
          // ensure the children property
          simplex->second.assign_children(siblings);
        } else {
          for (auto& blocked_new_sib_member : blocked_new_sib_vertex_list) {
            new_sib->members().erase(blocked_new_sib_member);
          }
          // ensure recursive call
          simplex->second.assign_children(new_sib);
          siblings_expansion_with_blockers(new_sib, max_dim, k - 1, block_simplex);
        }
      } else {
        // ensure the children property
        simplex->second.assign_children(siblings);
      }
    }
  }

  /* \private Returns the Simplex_handle composed of the vertex list (from the Simplex_handle), plus the given
   * Vertex_handle if the Vertex_handle is found in the Simplex_handle children list.
   * Returns null_simplex() if it does not exist
  */
  Simplex_handle find_child(Simplex_handle sh, Vertex_handle vh) const {
    if (!has_children(sh))
      return null_simplex();

    Simplex_handle child = sh->second.children()->find(vh);
    // Specific case of boost::flat_map does not find, returns boost::flat_map::end()
    // in simplex tree we want a null_simplex()
    if (child == sh->second.children()->members().end())
      return null_simplex();

    return child;
  }

 public:
  void print_hasse(std::ostream& os) {
    os << num_simplices() << " " << std::endl;
    for (auto sh : filtration_simplex_range()) {
      os << dimension(sh) << " ";
      for (auto b_sh : boundary_simplex_range(sh)) {
        os << key(b_sh) << " ";
      }
      os << filtration(sh) << " \n";
    }
  }

 public:
  bool make_filtration_non_decreasing() {
    bool modified = false;
    // Loop must be from the end to the beginning, as higher dimension simplex are always on the left part of the tree
    for (auto& simplex : boost::adaptors::reverse(root_.members())) {
      if (has_children(&simplex)) {
        modified |= rec_make_filtration_non_decreasing(simplex.second.children());
      }
    }
    return modified;
  }

 private:
  bool rec_make_filtration_non_decreasing(Siblings * sib) {
    bool modified = false;

    // Loop must be from the end to the beginning, as higher dimension simplex are always on the left part of the tree
    for (auto& simplex : boost::adaptors::reverse(sib->members())) {
      // Find the maximum filtration value in the border
      Boundary_simplex_range boundary = boundary_simplex_range(&simplex);
      Boundary_simplex_iterator max_border = std::max_element(std::begin(boundary), std::end(boundary),
                                                              [](Simplex_handle sh1, Simplex_handle sh2) {
                                                                return filtration(sh1) < filtration(sh2);
                                                              });

      Filtration_value max_filt_border_value = filtration(*max_border);
      if (simplex.second.filtration() < max_filt_border_value) {
        // Store the filtration modification information
        modified = true;
        simplex.second.assign_filtration(max_filt_border_value);
      }
      if (has_children(&simplex)) {
        modified |= rec_make_filtration_non_decreasing(simplex.second.children());
      }
    }
    // Make the modified information to be traced by upper call
    return modified;
  }

 public:
  bool prune_above_filtration(Filtration_value filtration) {
    return rec_prune_above_filtration(root(), filtration);
  }

 private:
  bool rec_prune_above_filtration(Siblings* sib, Filtration_value filt) {
    auto&& list = sib->members();
    auto last = std::remove_if(list.begin(), list.end(), [=](Dit_value_t& simplex) {
        if (simplex.second.filtration() <= filt) return false;
        if (has_children(&simplex)) rec_delete(simplex.second.children());
        // dimension may need to be lowered
        dimension_to_be_lowered_ = true;
        return true;
      });

    bool modified = (last != list.end());
    if (last == list.begin() && sib != root()) {
      // Removing the whole siblings, parent becomes a leaf.
      sib->oncles()->members()[sib->parent()].assign_children(sib->oncles());
      delete sib;
      // dimension may need to be lowered
      dimension_to_be_lowered_ = true;
      return true;
    } else {
      // Keeping some elements of siblings. Remove the others, and recurse in the remaining ones.
      list.erase(last, list.end());
      for (auto&& simplex : list)
        if (has_children(&simplex))
          modified |= rec_prune_above_filtration(simplex.second.children(), filt);
    }
    return modified;
  }

 private:
  bool lower_upper_bound_dimension() {
    // reset automatic detection to recompute
    dimension_to_be_lowered_ = false;
    int new_dimension = -1;
    // Browse the tree from the left to the right as higher dimension cells are more likely on the left part of the tree
    for (Simplex_handle sh : complex_simplex_range()) {
#ifdef DEBUG_TRACES
      for (auto vertex : simplex_vertex_range(sh)) {
        std::cout << " " << vertex;
      }
      std::cout << std::endl;
#endif  // DEBUG_TRACES

      int sh_dimension = dimension(sh);
      if (sh_dimension >= dimension_)
        // Stop browsing as soon as the dimension is reached, no need to go furter
        return false;
      new_dimension = (std::max)(new_dimension, sh_dimension);
    }
    dimension_ = new_dimension;
    return true;
  }


 public:
  void remove_maximal_simplex(Simplex_handle sh) {
    // Guarantee the simplex has no children
    GUDHI_CHECK(!has_children(sh),
                std::invalid_argument("Simplex_tree::remove_maximal_simplex - argument has children"));

    // Simplex is a leaf, it means the child is the Siblings owning the leaf
    Siblings* child = sh->second.children();

    if ((child->size() > 1) || (child == root())) {
      // Not alone, just remove it from members
      // Special case when child is the root of the simplex tree, just remove it from members
      child->erase(sh);
    } else {
      // Sibling is emptied : must be deleted, and its parent must point on his own Sibling
      child->oncles()->members().at(child->parent()).assign_children(child->oncles());
      delete child;
      // dimension may need to be lowered
      dimension_to_be_lowered_ = true;
    }
  }

 private:
  Vertex_handle null_vertex_;
  Siblings root_;
  std::vector<Simplex_handle> filtration_vect_;
  int dimension_;
  bool dimension_to_be_lowered_ = false;
};

// Print a Simplex_tree in os.
template<typename...T>
std::ostream& operator<<(std::ostream & os, Simplex_tree<T...> & st) {
  for (auto sh : st.filtration_simplex_range()) {
    os << st.dimension(sh) << " ";
    for (auto v : st.simplex_vertex_range(sh)) {
      os << v << " ";
    }
    os << st.filtration(sh) << "\n";  // TODO(VR): why adding the key ?? not read ?? << "     " << st.key(sh) << " \n";
  }
  return os;
}

template<typename...T>
std::istream& operator>>(std::istream & is, Simplex_tree<T...> & st) {
  typedef Simplex_tree<T...> ST;
  std::vector<typename ST::Vertex_handle> simplex;
  typename ST::Filtration_value fil;
  int max_dim = -1;
  while (read_simplex(is, simplex, fil)) {
    // read all simplices in the file as a list of vertices
    // Warning : simplex_size needs to be casted in int - Can be 0
    int dim = static_cast<int> (simplex.size() - 1);
    if (max_dim < dim) {
      max_dim = dim;
    }
    // insert every simplex in the simplex tree
    st.insert_simplex(simplex, fil);
    simplex.clear();
  }
  st.set_dimension(max_dim);

  return is;
}

struct Simplex_tree_options_full_featured {
  typedef linear_indexing_tag Indexing_tag;
  typedef int Vertex_handle;
  typedef double Filtration_value;
  typedef std::uint32_t Simplex_key;
  static const bool store_key = true;
  static const bool store_filtration = true;
  static const bool contiguous_vertices = false;
};

struct Simplex_tree_options_fast_persistence {
  typedef linear_indexing_tag Indexing_tag;
  typedef int Vertex_handle;
  typedef float Filtration_value;
  typedef std::uint32_t Simplex_key;
  static const bool store_key = true;
  static const bool store_filtration = true;
  static const bool contiguous_vertices = true;
};
  // end defgroup simplex_tree

}  // namespace Gudhi

#endif  // SIMPLEX_TREE_H_