41 files changed, 625 insertions, 352 deletions
diff --git a/src/Alpha_complex/benchmark/Alpha_complex_3d_benchmark.cpp b/src/Alpha_complex/benchmark/Alpha_complex_3d_benchmark.cpp
index 005a712a..99ad94b9 100644
--- a/src/Alpha_complex/benchmark/Alpha_complex_3d_benchmark.cpp
+++ b/src/Alpha_complex/benchmark/Alpha_complex_3d_benchmark.cpp
@@ -115,7 +115,7 @@ void benchmark_weighted_points_on_torus_3D(const std::string& msg) {
     std::cout << "  Alpha complex 3d on torus with " << nb_points << " points." << std::endl;
     std::vector<K::Point_d> points_on_torus = Gudhi::generate_points_on_torus_3D<K>(nb_points, 1.0, 0.5);
 
-    using Point = typename Weighted_alpha_complex_3d::Point_3;
+    using Point = typename Weighted_alpha_complex_3d::Bare_point_3;
     using Weighted_point = typename Weighted_alpha_complex_3d::Weighted_point_3;
 
     std::vector<Weighted_point> points;
@@ -206,7 +206,7 @@ void benchmark_weighted_periodic_points(const std::string& msg) {
     std::cout << "  Weighted periodic alpha complex 3d with " << nb_points * nb_points * nb_points << " points."
               << std::endl;
 
-    using Point = typename Weighted_periodic_alpha_complex_3d::Point_3;
+    using Point = typename Weighted_periodic_alpha_complex_3d::Bare_point_3;
     using Weighted_point = typename Weighted_periodic_alpha_complex_3d::Weighted_point_3;
     std::vector<Weighted_point> points;
 
diff --git a/src/Alpha_complex/doc/Intro_alpha_complex.h b/src/Alpha_complex/doc/Intro_alpha_complex.h
index b075d1fc..adc1378f 100644
--- a/src/Alpha_complex/doc/Intro_alpha_complex.h
+++ b/src/Alpha_complex/doc/Intro_alpha_complex.h
@@ -51,6 +51,17 @@ namespace alpha_complex {
  * - For people only interested in the topology of the \ref alpha_complex (for instance persistence),
  * \ref alpha_complex is equivalent to the \ref cech_complex and much smaller if you do not bound the radii.
  * \ref cech_complex can still make sense in higher dimension precisely because you can bound the radii.
+ * - Using the default `CGAL::Epeck_d` makes the construction safe. If you pass exact=true to create_complex, the
+ * filtration values are the exact ones converted to the filtration value type of the simplicial complex. This can be
+ * very slow. If you pass exact=false (the default), the filtration values are only guaranteed to have a small
+ * multiplicative error compared to the exact value, see <code><a class="el" target="_blank"
+ * href="https://doc.cgal.org/latest/Number_types/classCGAL_1_1Lazy__exact__nt.html">
+ * CGAL::Lazy_exact_nt<NT>::set_relative_precision_of_to_double</a></code> for details. A drawback, when computing
+ * persistence, is that an empty exact interval [10^12,10^12] may become a non-empty approximate interval
+ * [10^12,10^12+10^6]. Using `CGAL::Epick_d` makes the computations slightly faster, and the combinatorics are still
+ * exact, but the computation of filtration values can exceptionally be arbitrarily bad. In all cases, we still
+ * guarantee that the output is a valid filtration (faces have a filtration value no larger than their cofaces).
+ * - For performances reasons, it is advised to use `Alpha_complex` with \ref cgal &ge; 5.0.0.
  *
  * \section pointsexample Example from points
  *
diff --git a/src/Alpha_complex/example/Alpha_complex_from_off.cpp b/src/Alpha_complex/example/Alpha_complex_from_off.cpp
index d411e90a..220a66de 100644
--- a/src/Alpha_complex/example/Alpha_complex_from_off.cpp
+++ b/src/Alpha_complex/example/Alpha_complex_from_off.cpp
@@ -2,8 +2,6 @@
 // to construct a simplex_tree from alpha complex
 #include <gudhi/Simplex_tree.h>
 
-#include <CGAL/Epick_d.h>
-
 #include <iostream>
 #include <string>
 
@@ -23,22 +21,21 @@ int main(int argc, char **argv) {
   // ----------------------------------------------------------------------------
   // Init of an alpha complex from an OFF file
   // ----------------------------------------------------------------------------
-  using Kernel = CGAL::Epick_d< CGAL::Dynamic_dimension_tag >;
-  Gudhi::alpha_complex::Alpha_complex<Kernel> alpha_complex_from_file(off_file_name);
+  Gudhi::alpha_complex::Alpha_complex<> alpha_complex_from_file(off_file_name);
 
-  std::streambuf* streambufffer;
+  std::streambuf* streambuffer;
   std::ofstream ouput_file_stream;
 
   if (argc == 4) {
     ouput_file_stream.open(std::string(argv[3]));
-    streambufffer = ouput_file_stream.rdbuf();
+    streambuffer = ouput_file_stream.rdbuf();
   } else {
-    streambufffer = std::cout.rdbuf();
+    streambuffer = std::cout.rdbuf();
   }
 
   Gudhi::Simplex_tree<> simplex;
   if (alpha_complex_from_file.create_complex(simplex, alpha_square_max_value)) {
-    std::ostream output_stream(streambufffer);
+    std::ostream output_stream(streambuffer);
 
     // ----------------------------------------------------------------------------
     // Display information about the alpha complex
diff --git a/src/Alpha_complex/example/Alpha_complex_from_points.cpp b/src/Alpha_complex/example/Alpha_complex_from_points.cpp
index 981aa470..6526ca3a 100644
--- a/src/Alpha_complex/example/Alpha_complex_from_points.cpp
+++ b/src/Alpha_complex/example/Alpha_complex_from_points.cpp
@@ -2,12 +2,13 @@
 // to construct a simplex_tree from alpha complex
 #include <gudhi/Simplex_tree.h>
 
-#include <CGAL/Epick_d.h>
+#include <CGAL/Epeck_d.h>
 
 #include <iostream>
 #include <vector>
 
-using Kernel = CGAL::Epick_d< CGAL::Dimension_tag<2> >;
+// Explicit dimension 2 Epeck_d kernel
+using Kernel = CGAL::Epeck_d< CGAL::Dimension_tag<2> >;
 using Point = Kernel::Point_d;
 using Vector_of_points = std::vector<Point>;
 
diff --git a/src/Alpha_complex/example/CMakeLists.txt b/src/Alpha_complex/example/CMakeLists.txt
index b069b443..b0337934 100644
--- a/src/Alpha_complex/example/CMakeLists.txt
+++ b/src/Alpha_complex/example/CMakeLists.txt
@@ -5,9 +5,12 @@ if (NOT CGAL_WITH_EIGEN3_VERSION VERSION_LESS 4.11.0)
   target_link_libraries(Alpha_complex_example_from_points  ${CGAL_LIBRARY})
   add_executable ( Alpha_complex_example_from_off Alpha_complex_from_off.cpp )
   target_link_libraries(Alpha_complex_example_from_off ${CGAL_LIBRARY})
+  add_executable ( Alpha_complex_example_fast_from_off Fast_alpha_complex_from_off.cpp )
+  target_link_libraries(Alpha_complex_example_fast_from_off ${CGAL_LIBRARY})
   if (TBB_FOUND)
     target_link_libraries(Alpha_complex_example_from_points ${TBB_LIBRARIES})
     target_link_libraries(Alpha_complex_example_from_off ${TBB_LIBRARIES})
+    target_link_libraries(Alpha_complex_example_fast_from_off ${TBB_LIBRARIES})
   endif()
 
   add_test(NAME Alpha_complex_example_from_points COMMAND $<TARGET_FILE:Alpha_complex_example_from_points>)
@@ -16,7 +19,13 @@ if (NOT CGAL_WITH_EIGEN3_VERSION VERSION_LESS 4.11.0)
       "${CMAKE_SOURCE_DIR}/data/points/alphacomplexdoc.off" "60.0" "${CMAKE_CURRENT_BINARY_DIR}/alphaoffreader_result_60.txt")
   add_test(NAME Alpha_complex_example_from_off_32 COMMAND $<TARGET_FILE:Alpha_complex_example_from_off>
       "${CMAKE_SOURCE_DIR}/data/points/alphacomplexdoc.off" "32.0" "${CMAKE_CURRENT_BINARY_DIR}/alphaoffreader_result_32.txt")
-  if (DIFF_PATH)
+
+  add_test(NAME Alpha_complex_example_fast_from_off_60 COMMAND $<TARGET_FILE:Alpha_complex_example_fast_from_off>
+      "${CMAKE_SOURCE_DIR}/data/points/alphacomplexdoc.off" "60.0" "${CMAKE_CURRENT_BINARY_DIR}/fastalphaoffreader_result_60.txt")
+  add_test(NAME Alpha_complex_example_fast_from_off_32 COMMAND $<TARGET_FILE:Alpha_complex_example_fast_from_off>
+      "${CMAKE_SOURCE_DIR}/data/points/alphacomplexdoc.off" "32.0" "${CMAKE_CURRENT_BINARY_DIR}/fastalphaoffreader_result_32.txt")
+
+if (DIFF_PATH)
     # Do not forget to copy test results files in current binary dir
     file(COPY "alphaoffreader_for_doc_32.txt" DESTINATION ${CMAKE_CURRENT_BINARY_DIR}/)
     file(COPY "alphaoffreader_for_doc_60.txt" DESTINATION ${CMAKE_CURRENT_BINARY_DIR}/)
@@ -25,6 +34,11 @@ if (NOT CGAL_WITH_EIGEN3_VERSION VERSION_LESS 4.11.0)
         ${CMAKE_CURRENT_BINARY_DIR}/alphaoffreader_result_60.txt ${CMAKE_CURRENT_BINARY_DIR}/alphaoffreader_for_doc_60.txt)
     add_test(Alpha_complex_example_from_off_32_diff_files ${DIFF_PATH}
         ${CMAKE_CURRENT_BINARY_DIR}/alphaoffreader_result_32.txt ${CMAKE_CURRENT_BINARY_DIR}/alphaoffreader_for_doc_32.txt)
+
+    add_test(Alpha_complex_example_fast_from_off_60_diff_files ${DIFF_PATH}
+        ${CMAKE_CURRENT_BINARY_DIR}/fastalphaoffreader_result_60.txt ${CMAKE_CURRENT_BINARY_DIR}/alphaoffreader_for_doc_60.txt)
+    add_test(Alpha_complex_example_fast_from_off_32_diff_files ${DIFF_PATH}
+        ${CMAKE_CURRENT_BINARY_DIR}/fastalphaoffreader_result_32.txt ${CMAKE_CURRENT_BINARY_DIR}/alphaoffreader_for_doc_32.txt)
   endif()
 
   add_executable ( Alpha_complex_example_weighted_3d_from_points Weighted_alpha_complex_3d_from_points.cpp )
diff --git a/src/Alpha_complex/example/Fast_alpha_complex_from_off.cpp b/src/Alpha_complex/example/Fast_alpha_complex_from_off.cpp
new file mode 100644
index 00000000..f181005a
--- /dev/null
+++ b/src/Alpha_complex/example/Fast_alpha_complex_from_off.cpp
@@ -0,0 +1,65 @@
+#include <gudhi/Alpha_complex.h>
+// to construct a simplex_tree from alpha complex
+#include <gudhi/Simplex_tree.h>
+
+#include <CGAL/Epick_d.h>
+
+#include <iostream>
+#include <string>
+
+void usage(int nbArgs, char * const progName) {
+  std::cerr << "Error: Number of arguments (" << nbArgs << ") is not correct\n";
+  std::cerr << "Usage: " << progName << " filename.off alpha_square_max_value [ouput_file.txt]\n";
+  std::cerr << "       i.e.: " << progName << " ../../data/points/alphacomplexdoc.off 60.0\n";
+  exit(-1);  // ----- >>
+}
+
+int main(int argc, char **argv) {
+  if ((argc != 3) && (argc != 4)) usage(argc, (argv[0] - 1));
+
+  std::string off_file_name {argv[1]};
+  double alpha_square_max_value {atof(argv[2])};
+
+  // WARNING : CGAL::Epick_d is fast but not safe (unlike CGAL::Epeck_d)
+  // (i.e. when the points are on a grid)
+  using Fast_kernel = CGAL::Epick_d<CGAL::Dynamic_dimension_tag>;
+  // ----------------------------------------------------------------------------
+  // Init of an alpha complex from an OFF file
+  // ----------------------------------------------------------------------------
+  Gudhi::alpha_complex::Alpha_complex<Fast_kernel> alpha_complex_from_file(off_file_name);
+
+  std::streambuf* streambuffer;
+  std::ofstream ouput_file_stream;
+
+  if (argc == 4) {
+    ouput_file_stream.open(std::string(argv[3]));
+    streambuffer = ouput_file_stream.rdbuf();
+  } else {
+    streambuffer = std::cout.rdbuf();
+  }
+
+  Gudhi::Simplex_tree<> simplex;
+  if (alpha_complex_from_file.create_complex(simplex, alpha_square_max_value)) {
+    std::ostream output_stream(streambuffer);
+
+    // ----------------------------------------------------------------------------
+    // Display information about the alpha complex
+    // ----------------------------------------------------------------------------
+    output_stream << "Alpha complex is of dimension " << simplex.dimension() <<
+        " - " << simplex.num_simplices() << " simplices - " <<
+        simplex.num_vertices() << " vertices." << std::endl;
+
+    output_stream << "Iterator on alpha complex simplices in the filtration order, with [filtration value]:" <<
+        std::endl;
+    for (auto f_simplex : simplex.filtration_simplex_range()) {
+      output_stream << "   ( ";
+      for (auto vertex : simplex.simplex_vertex_range(f_simplex)) {
+        output_stream << vertex << " ";
+      }
+      output_stream << ") -> " << "[" << simplex.filtration(f_simplex) << "] ";
+      output_stream << std::endl;
+    }
+  }
+  ouput_file_stream.close();
+  return 0;
+}
diff --git a/src/Alpha_complex/example/Weighted_alpha_complex_3d_from_points.cpp b/src/Alpha_complex/example/Weighted_alpha_complex_3d_from_points.cpp
index ac11b68c..fcf80802 100644
--- a/src/Alpha_complex/example/Weighted_alpha_complex_3d_from_points.cpp
+++ b/src/Alpha_complex/example/Weighted_alpha_complex_3d_from_points.cpp
@@ -10,7 +10,7 @@
 // Complexity = FAST, weighted = true, periodic = false
 using Weighted_alpha_complex_3d =
     Gudhi::alpha_complex::Alpha_complex_3d<Gudhi::alpha_complex::complexity::SAFE, true, false>;
-using Point = Weighted_alpha_complex_3d::Point_3;
+using Bare_point = Weighted_alpha_complex_3d::Bare_point_3;
 using Weighted_point = Weighted_alpha_complex_3d::Weighted_point_3;
 
 int main(int argc, char **argv) {
@@ -18,11 +18,11 @@ int main(int argc, char **argv) {
   // Init of a list of points and weights from a small molecule
   // ----------------------------------------------------------------------------
   std::vector<Weighted_point> weighted_points;
-  weighted_points.push_back(Weighted_point(Point(1, -1, -1), 4.));
-  weighted_points.push_back(Weighted_point(Point(-1, 1, -1), 4.));
-  weighted_points.push_back(Weighted_point(Point(-1, -1, 1), 4.));
-  weighted_points.push_back(Weighted_point(Point(1, 1, 1), 4.));
-  weighted_points.push_back(Weighted_point(Point(2, 2, 2), 1.));
+  weighted_points.push_back(Weighted_point(Bare_point(1, -1, -1), 4.));
+  weighted_points.push_back(Weighted_point(Bare_point(-1, 1, -1), 4.));
+  weighted_points.push_back(Weighted_point(Bare_point(-1, -1, 1), 4.));
+  weighted_points.push_back(Weighted_point(Bare_point(1, 1, 1), 4.));
+  weighted_points.push_back(Weighted_point(Bare_point(2, 2, 2), 1.));
 
   // ----------------------------------------------------------------------------
   // Init of an alpha complex from the list of points
diff --git a/src/Alpha_complex/include/gudhi/Alpha_complex.h b/src/Alpha_complex/include/gudhi/Alpha_complex.h
index 8919cdb9..0c2569c8 100644
--- a/src/Alpha_complex/include/gudhi/Alpha_complex.h
+++ b/src/Alpha_complex/include/gudhi/Alpha_complex.h
@@ -20,11 +20,12 @@
 #include <math.h>  // isnan, fmax
 
 #include <CGAL/Delaunay_triangulation.h>
-#include <CGAL/Epick_d.h>
+#include <CGAL/Epeck_d.h>  // For EXACT or SAFE version
+#include <CGAL/Epick_d.h>  // For FAST version
 #include <CGAL/Spatial_sort_traits_adapter_d.h>
 #include <CGAL/property_map.h>  // for CGAL::Identity_property_map
-#include <CGAL/NT_converter.h>
 #include <CGAL/version.h>  // for CGAL_VERSION_NR
+#include <CGAL/NT_converter.h>
 
 #include <Eigen/src/Core/util/Macros.h>  // for EIGEN_VERSION_AT_LEAST
 
@@ -39,17 +40,20 @@
 
 // Make compilation fail - required for external projects - https://github.com/GUDHI/gudhi-devel/issues/10
 #if CGAL_VERSION_NR < 1041101000
-# error Alpha_complex_3d is only available for CGAL >= 4.11
+# error Alpha_complex is only available for CGAL >= 4.11
 #endif
 
 #if !EIGEN_VERSION_AT_LEAST(3,1,0)
-# error Alpha_complex_3d is only available for Eigen3 >= 3.1.0 installed with CGAL
+# error Alpha_complex is only available for Eigen3 >= 3.1.0 installed with CGAL
 #endif
 
 namespace Gudhi {
 
 namespace alpha_complex {
 
+template<typename D> struct Is_Epeck_D { static const bool value = false; };
+template<typename D> struct Is_Epeck_D<CGAL::Epeck_d<D>> { static const bool value = true; };
+
 /**
  * \class Alpha_complex Alpha_complex.h gudhi/Alpha_complex.h
  * \brief Alpha complex data structure.
@@ -63,17 +67,31 @@ namespace alpha_complex {
  * 
  * Please refer to \ref alpha_complex for examples.
  *
- * The complex is a template class requiring an Epick_d <a target="_blank"
+ * The complex is a template class requiring an <a target="_blank"
+ * href="https://doc.cgal.org/latest/Kernel_d/structCGAL_1_1Epeck__d.html">CGAL::Epeck_d</a>,
+ * or an <a target="_blank"
+ * href="https://doc.cgal.org/latest/Kernel_d/structCGAL_1_1Epick__d.html">CGAL::Epick_d</a> <a target="_blank"
  * href="http://doc.cgal.org/latest/Kernel_d/index.html#Chapter_dD_Geometry_Kernel">dD Geometry Kernel</a>
  * \cite cgal:s-gkd-15b from CGAL as template, default value is <a target="_blank"
- * href="http://doc.cgal.org/latest/Kernel_d/classCGAL_1_1Epick__d.html">CGAL::Epick_d</a>
+ * href="https://doc.cgal.org/latest/Kernel_d/structCGAL_1_1Epeck__d.html">CGAL::Epeck_d</a>
  * < <a target="_blank" href="http://doc.cgal.org/latest/Kernel_23/classCGAL_1_1Dynamic__dimension__tag.html">
  * CGAL::Dynamic_dimension_tag </a> >
  * 
- * \remark When Alpha_complex is constructed with an infinite value of alpha, the complex is a Delaunay complex.
- * 
+ * \remark
+ * - When Alpha_complex is constructed with an infinite value of alpha, the complex is a Delaunay complex.
+ * - Using the default `CGAL::Epeck_d` makes the construction safe. If you pass exact=true to create_complex, the
+ * filtration values are the exact ones converted to the filtration value type of the simplicial complex. This can be
+ * very slow. If you pass exact=false (the default), the filtration values are only guaranteed to have a small
+ * multiplicative error compared to the exact value, see <code><a class="el" target="_blank"
+ * href="https://doc.cgal.org/latest/Number_types/classCGAL_1_1Lazy__exact__nt.html">
+ * CGAL::Lazy_exact_nt<NT>::set_relative_precision_of_to_double</a></code> for details. A drawback, when computing
+ * persistence, is that an empty exact interval [10^12,10^12] may become a non-empty approximate interval
+ * [10^12,10^12+10^6]. Using `CGAL::Epick_d` makes the computations slightly faster, and the combinatorics are still
+ * exact, but the computation of filtration values can exceptionally be arbitrarily bad. In all cases, we still
+ * guarantee that the output is a valid filtration (faces have a filtration value no larger than their cofaces).
+ * - For performances reasons, it is advised to use `Alpha_complex` with \ref cgal &ge; 5.0.0.
  */
-template<class Kernel = CGAL::Epick_d<CGAL::Dynamic_dimension_tag>>
+template<class Kernel = CGAL::Epeck_d<CGAL::Dynamic_dimension_tag>>
 class Alpha_complex {
  public:
   // Add an int in TDS to save point index in the structure
@@ -184,6 +202,12 @@ class Alpha_complex {
  private:
   template<typename InputPointRange >
   void init_from_range(const InputPointRange& points) {
+    #if CGAL_VERSION_NR < 1050000000
+    if (Is_Epeck_D<Kernel>::value)
+      std::cerr << "It is strongly advised to use a CGAL version >= 5.0 with Epeck_d Kernel for performance reasons."
+                << std::endl;
+    #endif
+
     auto first = std::begin(points);
     auto last = std::end(points);
 
@@ -237,6 +261,8 @@ class Alpha_complex {
    * @param[in] complex SimplicialComplexForAlpha to be created.
    * @param[in] max_alpha_square maximum for alpha square value. Default value is +\f$\infty\f$, and there is very
    * little point using anything else since it does not save time.
+   * @param[in] exact Exact filtration values computation. Not exact if `Kernel` is not <a target="_blank"
+   * href="https://doc.cgal.org/latest/Kernel_d/structCGAL_1_1Epeck__d.html">CGAL::Epeck_d</a>.
    * 
    * @return true if creation succeeds, false otherwise.
    * 
@@ -248,7 +274,8 @@ class Alpha_complex {
   template <typename SimplicialComplexForAlpha,
             typename Filtration_value = typename SimplicialComplexForAlpha::Filtration_value>
   bool create_complex(SimplicialComplexForAlpha& complex,
-                      Filtration_value max_alpha_square = std::numeric_limits<Filtration_value>::infinity()) {
+                      Filtration_value max_alpha_square = std::numeric_limits<Filtration_value>::infinity(),
+                      bool exact = false) {
     // From SimplicialComplexForAlpha type required to insert into a simplicial complex (with or without subfaces).
     typedef typename SimplicialComplexForAlpha::Vertex_handle Vertex_handle;
     typedef typename SimplicialComplexForAlpha::Simplex_handle Simplex_handle;
@@ -324,9 +351,13 @@ class Alpha_complex {
             if (f_simplex_dim > 0) {
               // squared_radius function initialization
               Squared_Radius squared_radius = kernel_.compute_squared_radius_d_object();
-              CGAL::NT_converter<typename Geom_traits::FT, Filtration_value> cv;
 
-              alpha_complex_filtration = cv(squared_radius(pointVector.begin(), pointVector.end()));
+              CGAL::NT_converter<typename Geom_traits::FT, Filtration_value> cv;
+              auto sqrad = squared_radius(pointVector.begin(), pointVector.end());
+#if CGAL_VERSION_NR >= 1050000000
+              if(exact) CGAL::exact(sqrad);
+#endif
+              alpha_complex_filtration = cv(sqrad);
             }
             complex.assign_filtration(f_simplex, alpha_complex_filtration);
 #ifdef DEBUG_TRACES
diff --git a/src/Alpha_complex/include/gudhi/Alpha_complex_3d.h b/src/Alpha_complex/include/gudhi/Alpha_complex_3d.h
index 13ebb9c1..7f96c94c 100644
--- a/src/Alpha_complex/include/gudhi/Alpha_complex_3d.h
+++ b/src/Alpha_complex/include/gudhi/Alpha_complex_3d.h
@@ -43,7 +43,7 @@
 #include <vector>
 #include <unordered_map>
 #include <stdexcept>
-#include <cstddef>
+#include <cstddef>  // for std::size_t
 #include <memory>       // for std::unique_ptr
 #include <type_traits>  // for std::conditional and std::enable_if
 #include <limits>  // for numeric_limits<>
@@ -97,7 +97,7 @@ struct Value_from_iterator<complexity::EXACT> {
  * \details
  * The data structure is constructing a <a href="https://doc.cgal.org/latest/Alpha_shapes_3/index.html">CGAL 3D Alpha
  * Shapes</a> from a range of points (can be read from an OFF file, cf. Points_off_reader).
- * Duplicate points are inserted once in the Alpha_complex. This is the reason why the vertices may be not contiguous.
+ * Duplicate points are inserted once in the Alpha_complex.
  *
  * \tparam Complexity shall be `Gudhi::alpha_complex::complexity` type. Default value is
  * `Gudhi::alpha_complex::complexity::SAFE`.
@@ -225,23 +225,23 @@ class Alpha_complex_3d {
    * Must be compatible with double. */
   using FT = typename Alpha_shape_3::FT;
 
-  /** \brief Gives public access to the Point_3 type. Here is a Point_3 constructor example:
+  /** \brief Gives public access to the Bare_point_3 (bare aka. unweighed) type.
+   * Here is a Bare_point_3 constructor example:
 \code{.cpp}
 using Alpha_complex_3d = Gudhi::alpha_complex::Alpha_complex_3d<Gudhi::alpha_complex::complexity::SAFE, false, false>;
 
 // x0 = 1., y0 = -1.1, z0 = -1..
-Alpha_complex_3d::Point_3 p0(1., -1.1, -1.);
+Alpha_complex_3d::Bare_point_3 p0(1., -1.1, -1.);
 \endcode
    * */
-  using Point_3 = typename Kernel::Point_3;
+  using Bare_point_3 = typename Kernel::Point_3;
 
   /** \brief Gives public access to the Weighted_point_3 type. A Weighted point can be constructed as follows:
 \code{.cpp}
-using Weighted_alpha_complex_3d =
-  Gudhi::alpha_complex::Alpha_complex_3d<Gudhi::alpha_complex::complexity::SAFE, true, false>;
+using Weighted_alpha_complex_3d = Gudhi::alpha_complex::Alpha_complex_3d<Gudhi::alpha_complex::complexity::SAFE, true, false>;
 
 // x0 = 1., y0 = -1.1, z0 = -1., weight = 4.
-Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Point_3(1., -1.1, -1.), 4.);
+Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Bare_point_3(1., -1.1, -1.), 4.);
 \endcode
  *
  * Note: This type is defined to void if Alpha complex is not weighted.
@@ -249,6 +249,11 @@ Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Point
  * */
   using Weighted_point_3 = typename Triangulation_3<Kernel, Tds, Weighted, Periodic>::Weighted_point_3;
 
+  /** \brief `Alpha_complex_3d::Point_3` type is either a `Alpha_complex_3d::Bare_point_3` (Weighted = false) or a
+   * `Alpha_complex_3d::Weighted_point_3` (Weighted = true).
+   */
+  using Point_3 = typename Alpha_shape_3::Point;
+
  private:
   using Dispatch =
       CGAL::Dispatch_output_iterator<CGAL::cpp11::tuple<CGAL::Object, FT>,
@@ -264,13 +269,12 @@ Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Point
  public:
   /** \brief Alpha_complex constructor from a list of points.
    *
-   * @param[in] points Range of points to triangulate. Points must be in `Alpha_complex_3d::Point_3` or
-   * `Alpha_complex_3d::Weighted_point_3`.
+   * @param[in] points Range of points to triangulate. Points must be in `Alpha_complex_3d::Point_3`.
    *
    * @pre Available if Alpha_complex_3d is not Periodic.
    *
    * The type InputPointRange must be a range for which std::begin and std::end return input iterators on a
-   * `Alpha_complex_3d::Point_3` or a `Alpha_complex_3d::Weighted_point_3`.
+   * `Alpha_complex_3d::Point_3`.
    */
   template <typename InputPointRange>
   Alpha_complex_3d(const InputPointRange& points) {
@@ -284,13 +288,13 @@ Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Point
    *
    * @exception std::invalid_argument In debug mode, if points and weights do not have the same size.
    *
-   * @param[in] points Range of points to triangulate. Points must be in `Alpha_complex_3d::Point_3`.
+   * @param[in] points Range of points to triangulate. Points must be in `Alpha_complex_3d::Bare_point_3`.
    * @param[in] weights Range of weights on points. Weights shall be in double.
    *
    * @pre Available if Alpha_complex_3d is Weighted and not Periodic.
    *
    * The type InputPointRange must be a range for which std::begin and
-   * std::end return input iterators on a `Alpha_complex_3d::Point_3`.
+   * std::end return input iterators on a `Alpha_complex_3d::Bare_point_3`.
    * The type WeightRange must be a range for which std::begin and
    * std::end return an input iterator on a double.
    */
@@ -318,8 +322,7 @@ Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Point
    *
    * @exception std::invalid_argument In debug mode, if the size of the cuboid in every directions is not the same.
    *
-   * @param[in] points Range of points to triangulate. Points must be in `Alpha_complex_3d::Point_3` or
-   * `Alpha_complex_3d::Weighted_point_3`.
+   * @param[in] points Range of points to triangulate. Points must be in `Alpha_complex_3d::Point_3`.
    * @param[in] x_min Iso-oriented cuboid x_min.
    * @param[in] y_min Iso-oriented cuboid y_min.
    * @param[in] z_min Iso-oriented cuboid z_min.
@@ -330,7 +333,7 @@ Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Point
    * @pre Available if Alpha_complex_3d is Periodic.
    *
    * The type InputPointRange must be a range for which std::begin and std::end return input iterators on a
-   * `Alpha_complex_3d::Point_3` or a `Alpha_complex_3d::Weighted_point_3`.
+   * `Alpha_complex_3d::Point_3`.
    *
    * @note In weighted version, please check weights are greater than zero, and lower than 1/64*cuboid length
    * squared.
@@ -366,7 +369,7 @@ Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Point
    * @exception std::invalid_argument In debug mode, if a weight is negative, zero, or greater than 1/64*cuboid length
    * squared.
    *
-   * @param[in] points Range of points to triangulate. Points must be in `Alpha_complex_3d::Point_3`.
+   * @param[in] points Range of points to triangulate. Points must be in `Alpha_complex_3d::Bare_point_3`.
    * @param[in] weights Range of weights on points. Weights shall be in double.
    * @param[in] x_min Iso-oriented cuboid x_min.
    * @param[in] y_min Iso-oriented cuboid y_min.
@@ -378,7 +381,7 @@ Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Point
    * @pre Available if Alpha_complex_3d is Weighted and Periodic.
    *
    * The type InputPointRange must be a range for which std::begin and
-   * std::end return input iterators on a `Alpha_complex_3d::Point_3`.
+   * std::end return input iterators on a `Alpha_complex_3d::Bare_point_3`.
    * The type WeightRange must be a range for which std::begin and
    * std::end return an input iterator on a double.
    * The type of x_min, y_min, z_min, x_max, y_max and z_max must be a double.
@@ -452,9 +455,7 @@ Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Point
       return false;  // ----- >>
     }
 
-    // using Filtration_value = typename SimplicialComplexForAlpha3d::Filtration_value;
     using Complex_vertex_handle = typename SimplicialComplexForAlpha3d::Vertex_handle;
-    using Alpha_shape_simplex_tree_map = std::unordered_map<Alpha_vertex_handle, Complex_vertex_handle>;
     using Simplex_tree_vector_vertex = std::vector<Complex_vertex_handle>;
 
 #ifdef DEBUG_TRACES
@@ -474,7 +475,6 @@ Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Point
     std::cout << "filtration_with_alpha_values returns : " << objects.size() << " objects" << std::endl;
 #endif  // DEBUG_TRACES
 
-    Alpha_shape_simplex_tree_map map_cgal_simplex_tree;
     using Alpha_value_iterator = typename std::vector<FT>::const_iterator;
     Alpha_value_iterator alpha_value_iterator = alpha_values.begin();
     for (auto object_iterator : objects) {
@@ -484,7 +484,8 @@ Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Point
       if (const Cell_handle* cell = CGAL::object_cast<Cell_handle>(&object_iterator)) {
         for (auto i = 0; i < 4; i++) {
 #ifdef DEBUG_TRACES
-          std::cout << "from cell[" << i << "]=" << (*cell)->vertex(i)->point() << std::endl;
+          std::cout << "from cell[" << i << "] - Point coordinates (" << (*cell)->vertex(i)->point() << ")"
+                    << std::endl;
 #endif  // DEBUG_TRACES
           vertex_list.push_back((*cell)->vertex(i));
         }
@@ -495,7 +496,8 @@ Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Point
         for (auto i = 0; i < 4; i++) {
           if ((*facet).second != i) {
 #ifdef DEBUG_TRACES
-            std::cout << "from facet=[" << i << "]" << (*facet).first->vertex(i)->point() << std::endl;
+            std::cout << "from facet=[" << i << "] - Point coordinates (" << (*facet).first->vertex(i)->point() << ")"
+                      << std::endl;
 #endif  // DEBUG_TRACES
             vertex_list.push_back((*facet).first->vertex(i));
           }
@@ -506,7 +508,8 @@ Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Point
       } else if (const Edge* edge = CGAL::object_cast<Edge>(&object_iterator)) {
         for (auto i : {(*edge).second, (*edge).third}) {
 #ifdef DEBUG_TRACES
-          std::cout << "from edge[" << i << "]=" << (*edge).first->vertex(i)->point() << std::endl;
+          std::cout << "from edge[" << i << "] - Point coordinates (" << (*edge).first->vertex(i)->point() << ")"
+                    << std::endl;
 #endif  // DEBUG_TRACES
           vertex_list.push_back((*edge).first->vertex(i));
         }
@@ -516,7 +519,7 @@ Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Point
       } else if (const Alpha_vertex_handle* vertex = CGAL::object_cast<Alpha_vertex_handle>(&object_iterator)) {
 #ifdef DEBUG_TRACES
         count_vertices++;
-        std::cout << "from vertex=" << (*vertex)->point() << std::endl;
+        std::cout << "from vertex - Point coordinates (" << (*vertex)->point() << ")" << std::endl;
 #endif  // DEBUG_TRACES
         vertex_list.push_back((*vertex));
       }
@@ -528,7 +531,8 @@ Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Point
           // alpha shape not found
           Complex_vertex_handle vertex = map_cgal_simplex_tree.size();
 #ifdef DEBUG_TRACES
-          std::cout << "vertex [" << the_alpha_shape_vertex->point() << "] not found - insert " << vertex << std::endl;
+          std::cout << "Point (" << the_alpha_shape_vertex->point() << ") not found - insert new vertex id " << vertex
+                    << std::endl;
 #endif  // DEBUG_TRACES
           the_simplex.push_back(vertex);
           map_cgal_simplex_tree.emplace(the_alpha_shape_vertex, vertex);
@@ -536,7 +540,7 @@ Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Point
           // alpha shape found
           Complex_vertex_handle vertex = the_map_iterator->second;
 #ifdef DEBUG_TRACES
-          std::cout << "vertex [" << the_alpha_shape_vertex->point() << "] found in " << vertex << std::endl;
+          std::cout << "Point (" << the_alpha_shape_vertex->point() << ") found as vertex id " << vertex << std::endl;
 #endif  // DEBUG_TRACES
           the_simplex.push_back(vertex);
         }
@@ -567,9 +571,32 @@ Weighted_alpha_complex_3d::Weighted_point_3 wp0(Weighted_alpha_complex_3d::Point
     return true;
   }
 
+  /** \brief get_point returns the point corresponding to the vertex given as parameter.
+   *
+   * @param[in] vertex Vertex handle of the point to retrieve.
+   * @return The point found.
+   * @exception std::out_of_range In case vertex is not found (cf. std::vector::at).
+   */
+  const Point_3& get_point(std::size_t vertex) {
+    if (map_cgal_simplex_tree.size() != cgal_vertex_iterator_vector.size()) {
+      cgal_vertex_iterator_vector.resize(map_cgal_simplex_tree.size());
+      for (auto map_iterator : map_cgal_simplex_tree) {
+        cgal_vertex_iterator_vector[map_iterator.second] = map_iterator.first;
+      }
+
+    }
+    auto cgal_vertex_iterator = cgal_vertex_iterator_vector.at(vertex);
+    return cgal_vertex_iterator->point();
+  }
+
  private:
   // use of a unique_ptr on cgal Alpha_shape_3, as copy and default constructor is not available - no need to be freed
   std::unique_ptr<Alpha_shape_3> alpha_shape_3_ptr_;
+
+  // Map type to switch from CGAL vertex iterator to simplex tree vertex handle.
+  std::unordered_map<Alpha_vertex_handle, std::size_t> map_cgal_simplex_tree;
+  // Vector type to switch from simplex tree vertex handle to CGAL vertex iterator.
+  std::vector<Alpha_vertex_handle> cgal_vertex_iterator_vector;
 };
 
 }  // namespace alpha_complex
diff --git a/src/Alpha_complex/test/Alpha_complex_3d_unit_test.cpp b/src/Alpha_complex/test/Alpha_complex_3d_unit_test.cpp
index 1102838a..cd698a27 100644
--- a/src/Alpha_complex/test/Alpha_complex_3d_unit_test.cpp
+++ b/src/Alpha_complex/test/Alpha_complex_3d_unit_test.cpp
@@ -56,21 +56,52 @@ BOOST_AUTO_TEST_CASE(Alpha_complex_3d_from_points) {
   // -----------------
   std::cout << "Fast alpha complex 3d" << std::endl;
 
-  Fast_alpha_complex_3d alpha_complex(get_points<Fast_alpha_complex_3d::Point_3>());
+  std::vector<Fast_alpha_complex_3d::Bare_point_3> points = get_points<Fast_alpha_complex_3d::Bare_point_3>();
+  Fast_alpha_complex_3d alpha_complex(points);
 
   Gudhi::Simplex_tree<> stree;
   alpha_complex.create_complex(stree);
 
+  for (std::size_t index = 0; index < points.size(); index++) {
+    bool found = false;
+    for (auto point : points) {
+      if (point == alpha_complex.get_point(index)) {
+        found = true;
+        break;
+      }
+    }
+    // Check all points from alpha complex are found in the input point cloud
+    BOOST_CHECK(found);
+  }
+  // Exception if we go out of range
+  BOOST_CHECK_THROW(alpha_complex.get_point(points.size()), std::out_of_range);
+
   // -----------------
   // Exact version
   // -----------------
   std::cout << "Exact alpha complex 3d" << std::endl;
 
-  Exact_alpha_complex_3d exact_alpha_complex(get_points<Exact_alpha_complex_3d::Point_3>());
+  std::vector<Exact_alpha_complex_3d::Bare_point_3> exact_points = get_points<Exact_alpha_complex_3d::Bare_point_3>();
+  Exact_alpha_complex_3d exact_alpha_complex(exact_points);
 
   Gudhi::Simplex_tree<> exact_stree;
   exact_alpha_complex.create_complex(exact_stree);
 
+  for (std::size_t index = 0; index < exact_points.size(); index++) {
+    bool found = false;
+    Exact_alpha_complex_3d::Bare_point_3 ap = exact_alpha_complex.get_point(index);
+    for (auto point : points) {
+      if ((point.x() == ap.x()) && (point.y() == ap.y()) && (point.z() == ap.z())) {
+        found = true;
+        break;
+      }
+    }
+    // Check all points from alpha complex are found in the input point cloud
+    BOOST_CHECK(found);
+  }
+  // Exception if we go out of range
+  BOOST_CHECK_THROW(exact_alpha_complex.get_point(exact_points.size()), std::out_of_range);
+
   // ---------------------
   // Compare both versions
   // ---------------------
@@ -110,11 +141,27 @@ BOOST_AUTO_TEST_CASE(Alpha_complex_3d_from_points) {
   // -----------------
   std::cout << "Safe alpha complex 3d" << std::endl;
 
-  Safe_alpha_complex_3d safe_alpha_complex(get_points<Safe_alpha_complex_3d::Point_3>());
+  std::vector<Safe_alpha_complex_3d::Bare_point_3> safe_points = get_points<Safe_alpha_complex_3d::Bare_point_3>();
+  Safe_alpha_complex_3d safe_alpha_complex(safe_points);
 
   Gudhi::Simplex_tree<> safe_stree;
   safe_alpha_complex.create_complex(safe_stree);
 
+  for (std::size_t index = 0; index < safe_points.size(); index++) {
+    bool found = false;
+    Safe_alpha_complex_3d::Bare_point_3 ap = safe_alpha_complex.get_point(index);
+    for (auto point : points) {
+      if ((point.x() == ap.x()) && (point.y() == ap.y()) && (point.z() == ap.z())) {
+        found = true;
+        break;
+      }
+    }
+    // Check all points from alpha complex are found in the input point cloud
+    BOOST_CHECK(found);
+  }
+  // Exception if we go out of range
+  BOOST_CHECK_THROW(safe_alpha_complex.get_point(safe_points.size()), std::out_of_range);
+
   // ---------------------
   // Compare both versions
   // ---------------------
diff --git a/src/Alpha_complex/test/Alpha_complex_unit_test.cpp b/src/Alpha_complex/test/Alpha_complex_unit_test.cpp
index 01e4cee3..40b3fe09 100644
--- a/src/Alpha_complex/test/Alpha_complex_unit_test.cpp
+++ b/src/Alpha_complex/test/Alpha_complex_unit_test.cpp
@@ -15,6 +15,7 @@
 
 #include <CGAL/Delaunay_triangulation.h>
 #include <CGAL/Epick_d.h>
+#include <CGAL/Epeck_d.h>
 
 #include <cmath>  // float comparison
 #include <limits>
@@ -28,12 +29,16 @@
 #include <gudhi/Unitary_tests_utils.h>
 
 // Use dynamic_dimension_tag for the user to be able to set dimension
-typedef CGAL::Epick_d< CGAL::Dynamic_dimension_tag > Kernel_d;
+typedef CGAL::Epeck_d< CGAL::Dynamic_dimension_tag > Exact_kernel_d;
 // Use static dimension_tag for the user not to be able to set dimension
-typedef CGAL::Epick_d< CGAL::Dimension_tag<3> > Kernel_s;
+typedef CGAL::Epeck_d< CGAL::Dimension_tag<3> > Exact_kernel_s;
+// Use dynamic_dimension_tag for the user to be able to set dimension
+typedef CGAL::Epick_d< CGAL::Dynamic_dimension_tag > Inexact_kernel_d;
+// Use static dimension_tag for the user not to be able to set dimension
+typedef CGAL::Epick_d< CGAL::Dimension_tag<3> > Inexact_kernel_s;
 // The triangulation uses the default instantiation of the TriangulationDataStructure template parameter
 
-typedef boost::mpl::list<Kernel_d, Kernel_s> list_of_kernel_variants;
+typedef boost::mpl::list<Exact_kernel_d, Exact_kernel_s, Inexact_kernel_d, Inexact_kernel_s> list_of_kernel_variants;
 
 BOOST_AUTO_TEST_CASE_TEMPLATE(Alpha_complex_from_OFF_file, TestedKernel, list_of_kernel_variants) {
   // ----------------------------------------------------------------------------
@@ -86,7 +91,7 @@ BOOST_AUTO_TEST_CASE_TEMPLATE(Alpha_complex_from_OFF_file, TestedKernel, list_of
 }
 
 // Use static dimension_tag for the user not to be able to set dimension
-typedef CGAL::Epick_d< CGAL::Dimension_tag<4> > Kernel_4;
+typedef CGAL::Epeck_d< CGAL::Dimension_tag<4> > Kernel_4;
 typedef Kernel_4::Point_d Point_4;
 typedef std::vector<Point_4> Vector_4_Points;
 
diff --git a/src/Alpha_complex/test/Periodic_alpha_complex_3d_unit_test.cpp b/src/Alpha_complex/test/Periodic_alpha_complex_3d_unit_test.cpp
index ac3791a4..731763fa 100644
--- a/src/Alpha_complex/test/Periodic_alpha_complex_3d_unit_test.cpp
+++ b/src/Alpha_complex/test/Periodic_alpha_complex_3d_unit_test.cpp
@@ -44,11 +44,11 @@ typedef boost::mpl::list<Fast_periodic_alpha_complex_3d, Safe_periodic_alpha_com
 
 BOOST_AUTO_TEST_CASE_TEMPLATE(Alpha_complex_periodic_throw, Periodic_alpha_complex_3d, periodic_variants_type_list) {
   std::cout << "Periodic alpha complex 3d exception throw" << std::endl;
-  using Point_3 = typename Periodic_alpha_complex_3d::Point_3;
-  std::vector<Point_3> p_points;
+  using Bare_point_3 = typename Periodic_alpha_complex_3d::Bare_point_3;
+  std::vector<Bare_point_3> p_points;
 
   // Not important, this is not what we want to check
-  p_points.push_back(Point_3(0.0, 0.0, 0.0));
+  p_points.push_back(Bare_point_3(0.0, 0.0, 0.0));
 
   std::cout << "Check exception throw in debug mode" << std::endl;
   // Check it throws an exception when the cuboid is not iso
@@ -73,13 +73,13 @@ BOOST_AUTO_TEST_CASE(Alpha_complex_periodic) {
   // ---------------------
   std::cout << "Fast periodic alpha complex 3d" << std::endl;
 
-  using Creator = CGAL::Creator_uniform_3<double, Fast_periodic_alpha_complex_3d::Point_3>;
+  using Creator = CGAL::Creator_uniform_3<double, Fast_periodic_alpha_complex_3d::Bare_point_3>;
   CGAL::Random random(7);
-  CGAL::Random_points_in_cube_3<Fast_periodic_alpha_complex_3d::Point_3, Creator> in_cube(1, random);
-  std::vector<Fast_periodic_alpha_complex_3d::Point_3> p_points;
+  CGAL::Random_points_in_cube_3<Fast_periodic_alpha_complex_3d::Bare_point_3, Creator> in_cube(1, random);
+  std::vector<Fast_periodic_alpha_complex_3d::Bare_point_3> p_points;
 
   for (int i = 0; i < 50; i++) {
-    Fast_periodic_alpha_complex_3d::Point_3 p = *in_cube++;
+    Fast_periodic_alpha_complex_3d::Bare_point_3 p = *in_cube++;
     p_points.push_back(p);
   }
 
@@ -88,15 +88,30 @@ BOOST_AUTO_TEST_CASE(Alpha_complex_periodic) {
   Gudhi::Simplex_tree<> stree;
   periodic_alpha_complex.create_complex(stree);
 
+  for (std::size_t index = 0; index < p_points.size(); index++) {
+    bool found = false;
+    Fast_periodic_alpha_complex_3d::Bare_point_3 ap = periodic_alpha_complex.get_point(index);
+    for (auto point : p_points) {
+      if ((point.x() == ap.x()) && (point.y() == ap.y()) && (point.z() == ap.z())) {
+        found = true;
+        break;
+      }
+    }
+    // Check all points from alpha complex are found in the input point cloud
+    BOOST_CHECK(found);
+  }
+  // Exception if we go out of range
+  BOOST_CHECK_THROW(periodic_alpha_complex.get_point(p_points.size()), std::out_of_range);
+
   // ----------------------
   // Exact periodic version
   // ----------------------
   std::cout << "Exact periodic alpha complex 3d" << std::endl;
 
-  std::vector<Exact_periodic_alpha_complex_3d::Point_3> e_p_points;
+  std::vector<Exact_periodic_alpha_complex_3d::Bare_point_3> e_p_points;
 
   for (auto p : p_points) {
-    e_p_points.push_back(Exact_periodic_alpha_complex_3d::Point_3(p[0], p[1], p[2]));
+    e_p_points.push_back(Exact_periodic_alpha_complex_3d::Bare_point_3(p[0], p[1], p[2]));
   }
 
   Exact_periodic_alpha_complex_3d exact_alpha_complex(e_p_points, -1., -1., -1., 1., 1., 1.);
@@ -142,10 +157,10 @@ BOOST_AUTO_TEST_CASE(Alpha_complex_periodic) {
   // ----------------------
   std::cout << "Safe periodic alpha complex 3d" << std::endl;
 
-  std::vector<Safe_periodic_alpha_complex_3d::Point_3> s_p_points;
+  std::vector<Safe_periodic_alpha_complex_3d::Bare_point_3> s_p_points;
 
   for (auto p : p_points) {
-    s_p_points.push_back(Safe_periodic_alpha_complex_3d::Point_3(p[0], p[1], p[2]));
+    s_p_points.push_back(Safe_periodic_alpha_complex_3d::Bare_point_3(p[0], p[1], p[2]));
   }
 
   Safe_periodic_alpha_complex_3d safe_alpha_complex(s_p_points, -1., -1., -1., 1., 1., 1.);
diff --git a/src/Alpha_complex/test/Weighted_alpha_complex_3d_unit_test.cpp b/src/Alpha_complex/test/Weighted_alpha_complex_3d_unit_test.cpp
index 44deb930..8035f6e8 100644
--- a/src/Alpha_complex/test/Weighted_alpha_complex_3d_unit_test.cpp
+++ b/src/Alpha_complex/test/Weighted_alpha_complex_3d_unit_test.cpp
@@ -43,14 +43,14 @@ typedef boost::mpl::list<Fast_weighted_alpha_complex_3d, Safe_weighted_alpha_com
 
 #ifdef GUDHI_DEBUG
 BOOST_AUTO_TEST_CASE_TEMPLATE(Alpha_complex_weighted_throw, Weighted_alpha_complex_3d, weighted_variants_type_list) {
-  using Point_3 = typename Weighted_alpha_complex_3d::Point_3;
-  std::vector<Point_3> w_points;
-  w_points.push_back(Point_3(0.0, 0.0, 0.0));
-  w_points.push_back(Point_3(0.0, 0.0, 0.2));
-  w_points.push_back(Point_3(0.2, 0.0, 0.2));
-  // w_points.push_back(Point_3(0.6, 0.6, 0.0));
-  // w_points.push_back(Point_3(0.8, 0.8, 0.2));
-  // w_points.push_back(Point_3(0.2, 0.8, 0.6));
+  using Bare_point_3 = typename Weighted_alpha_complex_3d::Bare_point_3;
+  std::vector<Bare_point_3> w_points;
+  w_points.push_back(Bare_point_3(0.0, 0.0, 0.0));
+  w_points.push_back(Bare_point_3(0.0, 0.0, 0.2));
+  w_points.push_back(Bare_point_3(0.2, 0.0, 0.2));
+  // w_points.push_back(Bare_point_3(0.6, 0.6, 0.0));
+  // w_points.push_back(Bare_point_3(0.8, 0.8, 0.2));
+  // w_points.push_back(Bare_point_3(0.2, 0.8, 0.6));
 
   // weights size is different from w_points size to make weighted Alpha_complex_3d throw in debug mode
   std::vector<double> weights = {0.01, 0.005, 0.006, 0.01, 0.009, 0.001};
@@ -62,14 +62,14 @@ BOOST_AUTO_TEST_CASE_TEMPLATE(Alpha_complex_weighted_throw, Weighted_alpha_compl
 
 BOOST_AUTO_TEST_CASE_TEMPLATE(Alpha_complex_weighted, Weighted_alpha_complex_3d, weighted_variants_type_list) {
   std::cout << "Weighted alpha complex 3d from points and weights" << std::endl;
-  using Point_3 = typename Weighted_alpha_complex_3d::Point_3;
-  std::vector<Point_3> w_points;
-  w_points.push_back(Point_3(0.0, 0.0, 0.0));
-  w_points.push_back(Point_3(0.0, 0.0, 0.2));
-  w_points.push_back(Point_3(0.2, 0.0, 0.2));
-  w_points.push_back(Point_3(0.6, 0.6, 0.0));
-  w_points.push_back(Point_3(0.8, 0.8, 0.2));
-  w_points.push_back(Point_3(0.2, 0.8, 0.6));
+  using Bare_point_3 = typename Weighted_alpha_complex_3d::Bare_point_3;
+  std::vector<Bare_point_3> w_points;
+  w_points.push_back(Bare_point_3(0.0, 0.0, 0.0));
+  w_points.push_back(Bare_point_3(0.0, 0.0, 0.2));
+  w_points.push_back(Bare_point_3(0.2, 0.0, 0.2));
+  w_points.push_back(Bare_point_3(0.6, 0.6, 0.0));
+  w_points.push_back(Bare_point_3(0.8, 0.8, 0.2));
+  w_points.push_back(Bare_point_3(0.2, 0.8, 0.6));
 
   // weights size is different from w_points size to make weighted Alpha_complex_3d throw in debug mode
   std::vector<double> weights = {0.01, 0.005, 0.006, 0.01, 0.009, 0.001};
@@ -91,6 +91,24 @@ BOOST_AUTO_TEST_CASE_TEMPLATE(Alpha_complex_weighted, Weighted_alpha_complex_3d,
   Gudhi::Simplex_tree<> stree_bis;
   alpha_complex_w_p.create_complex(stree_bis);
 
+  for (std::size_t index = 0; index < weighted_points.size(); index++) {
+    bool found = false;
+    Weighted_point_3 awp = alpha_complex_w_p.get_point(index);
+    for (auto weighted_point : weighted_points) {
+      if ((weighted_point.weight() == awp.weight()) &&
+          (weighted_point.x() == awp.x()) &&
+          (weighted_point.y() == awp.y()) &&
+          (weighted_point.z() == awp.z())) {
+        found = true;
+        break;
+      }
+    }
+    // Check all points from alpha complex are found in the input point cloud
+    BOOST_CHECK(found);
+  }
+  // Exception if we go out of range
+  BOOST_CHECK_THROW(alpha_complex_w_p.get_point(weighted_points.size()), std::out_of_range);
+
   // ---------------------
   // Compare both versions
   // ---------------------
diff --git a/src/Alpha_complex/test/Weighted_periodic_alpha_complex_3d_unit_test.cpp b/src/Alpha_complex/test/Weighted_periodic_alpha_complex_3d_unit_test.cpp
index 670c7799..b09e92d5 100644
--- a/src/Alpha_complex/test/Weighted_periodic_alpha_complex_3d_unit_test.cpp
+++ b/src/Alpha_complex/test/Weighted_periodic_alpha_complex_3d_unit_test.cpp
@@ -47,13 +47,13 @@ BOOST_AUTO_TEST_CASE_TEMPLATE(Alpha_complex_weighted_periodic_throw, Weighted_pe
                               wp_variants_type_list) {
   std::cout << "Weighted periodic alpha complex 3d exception throw" << std::endl;
 
-  using Creator = CGAL::Creator_uniform_3<double, typename Weighted_periodic_alpha_complex_3d::Point_3>;
+  using Creator = CGAL::Creator_uniform_3<double, typename Weighted_periodic_alpha_complex_3d::Bare_point_3>;
   CGAL::Random random(7);
-  CGAL::Random_points_in_cube_3<typename Weighted_periodic_alpha_complex_3d::Point_3, Creator> in_cube(1, random);
-  std::vector<typename Weighted_periodic_alpha_complex_3d::Point_3> wp_points;
+  CGAL::Random_points_in_cube_3<typename Weighted_periodic_alpha_complex_3d::Bare_point_3, Creator> in_cube(1, random);
+  std::vector<typename Weighted_periodic_alpha_complex_3d::Bare_point_3> wp_points;
 
   for (int i = 0; i < 50; i++) {
-    typename Weighted_periodic_alpha_complex_3d::Point_3 p = *in_cube++;
+    typename Weighted_periodic_alpha_complex_3d::Bare_point_3 p = *in_cube++;
     wp_points.push_back(p);
   }
   std::vector<double> p_weights;
@@ -117,13 +117,13 @@ BOOST_AUTO_TEST_CASE(Alpha_complex_weighted_periodic) {
   // ---------------------
   std::cout << "Fast weighted periodic alpha complex 3d" << std::endl;
 
-  using Creator = CGAL::Creator_uniform_3<double, Fast_weighted_periodic_alpha_complex_3d::Point_3>;
+  using Creator = CGAL::Creator_uniform_3<double, Fast_weighted_periodic_alpha_complex_3d::Bare_point_3>;
   CGAL::Random random(7);
-  CGAL::Random_points_in_cube_3<Fast_weighted_periodic_alpha_complex_3d::Point_3, Creator> in_cube(1, random);
-  std::vector<Fast_weighted_periodic_alpha_complex_3d::Point_3> p_points;
+  CGAL::Random_points_in_cube_3<Fast_weighted_periodic_alpha_complex_3d::Bare_point_3, Creator> in_cube(1, random);
+  std::vector<Fast_weighted_periodic_alpha_complex_3d::Bare_point_3> p_points;
 
   for (int i = 0; i < 50; i++) {
-    Fast_weighted_periodic_alpha_complex_3d::Point_3 p = *in_cube++;
+    Fast_weighted_periodic_alpha_complex_3d::Bare_point_3 p = *in_cube++;
     p_points.push_back(p);
   }
   std::vector<double> p_weights;
@@ -142,10 +142,10 @@ BOOST_AUTO_TEST_CASE(Alpha_complex_weighted_periodic) {
   // ----------------------
   std::cout << "Exact weighted periodic alpha complex 3d" << std::endl;
 
-  std::vector<Exact_weighted_periodic_alpha_complex_3d::Point_3> e_p_points;
+  std::vector<Exact_weighted_periodic_alpha_complex_3d::Bare_point_3> e_p_points;
 
   for (auto p : p_points) {
-    e_p_points.push_back(Exact_weighted_periodic_alpha_complex_3d::Point_3(p[0], p[1], p[2]));
+    e_p_points.push_back(Exact_weighted_periodic_alpha_complex_3d::Bare_point_3(p[0], p[1], p[2]));
   }
 
   Exact_weighted_periodic_alpha_complex_3d exact_alpha_complex(e_p_points, p_weights, -1., -1., -1., 1., 1., 1.);
@@ -191,10 +191,10 @@ BOOST_AUTO_TEST_CASE(Alpha_complex_weighted_periodic) {
   // ----------------------
   std::cout << "Safe weighted periodic alpha complex 3d" << std::endl;
 
-  std::vector<Safe_weighted_periodic_alpha_complex_3d::Point_3> s_p_points;
+  std::vector<Safe_weighted_periodic_alpha_complex_3d::Bare_point_3> s_p_points;
 
   for (auto p : p_points) {
-    s_p_points.push_back(Safe_weighted_periodic_alpha_complex_3d::Point_3(p[0], p[1], p[2]));
+    s_p_points.push_back(Safe_weighted_periodic_alpha_complex_3d::Bare_point_3(p[0], p[1], p[2]));
   }
 
   Safe_weighted_periodic_alpha_complex_3d safe_alpha_complex(s_p_points, p_weights, -1., -1., -1., 1., 1., 1.);
diff --git a/src/Alpha_complex/utilities/CMakeLists.txt b/src/Alpha_complex/utilities/CMakeLists.txt
index 5295f3cd..57b92942 100644
--- a/src/Alpha_complex/utilities/CMakeLists.txt
+++ b/src/Alpha_complex/utilities/CMakeLists.txt
@@ -7,9 +7,19 @@ if (NOT CGAL_WITH_EIGEN3_VERSION VERSION_LESS 4.11.0)
   if (TBB_FOUND)
     target_link_libraries(alpha_complex_persistence ${TBB_LIBRARIES})
   endif(TBB_FOUND)
-  add_test(NAME Alpha_complex_utilities_alpha_complex_persistence COMMAND $<TARGET_FILE:alpha_complex_persistence>
-      "${CMAKE_SOURCE_DIR}/data/points/tore3D_300.off" "-p" "2" "-m" "0.45")
-
+  add_test(NAME Alpha_complex_utilities_safe_alpha_complex_persistence COMMAND $<TARGET_FILE:alpha_complex_persistence>
+      "${CMAKE_SOURCE_DIR}/data/points/tore3D_300.off" "-p" "2" "-m" "0.45" "-o" "safe.pers")
+  add_test(NAME Alpha_complex_utilities_fast_alpha_complex_persistence COMMAND $<TARGET_FILE:alpha_complex_persistence>
+      "${CMAKE_SOURCE_DIR}/data/points/tore3D_300.off" "-p" "2" "-m" "0.45" "-o" "fast.pers" "-f")
+  add_test(NAME Alpha_complex_utilities_exact_alpha_complex_persistence COMMAND $<TARGET_FILE:alpha_complex_persistence>
+      "${CMAKE_SOURCE_DIR}/data/points/tore3D_300.off" "-p" "2" "-m" "0.45" "-o" "exact.pers" "-e")
+  if (DIFF_PATH)
+    add_test(Alpha_complex_utilities_diff_exact_alpha_complex ${DIFF_PATH}
+             "exact.pers" "safe.pers")
+    add_test(Alpha_complex_utilities_diff_fast_alpha_complex ${DIFF_PATH}
+             "fast.pers" "safe.pers")
+  endif()
+  
   install(TARGETS alpha_complex_persistence DESTINATION bin)
 
   add_executable(alpha_complex_3d_persistence alpha_complex_3d_persistence.cpp)
@@ -20,21 +30,21 @@ if (NOT CGAL_WITH_EIGEN3_VERSION VERSION_LESS 4.11.0)
 
   add_test(NAME Alpha_complex_utilities_alpha_complex_3d COMMAND $<TARGET_FILE:alpha_complex_3d_persistence>
           "${CMAKE_SOURCE_DIR}/data/points/tore3D_300.off"
-          "-p" "2" "-m" "0.45" "-o" "safe.pers")
+          "-p" "2" "-m" "0.45" "-o" "safe_3d.pers")
 
   add_test(NAME Alpha_complex_utilities_exact_alpha_complex_3d COMMAND $<TARGET_FILE:alpha_complex_3d_persistence>
           "${CMAKE_SOURCE_DIR}/data/points/tore3D_300.off"
-          "-p" "2" "-m" "0.45" "-o" "exact.pers" "-e")
+          "-p" "2" "-m" "0.45" "-o" "exact_3d.pers" "-e")
 
   add_test(NAME Alpha_complex_utilities_safe_alpha_complex_3d COMMAND $<TARGET_FILE:alpha_complex_3d_persistence>
           "${CMAKE_SOURCE_DIR}/data/points/tore3D_300.off"
-          "-p" "2" "-m" "0.45" "-o" "fast.pers" "-f")
+          "-p" "2" "-m" "0.45" "-o" "fast_3d.pers" "-f")
 
   if (DIFF_PATH)
-    add_test(Alpha_complex_utilities_diff_alpha_complex_3d ${DIFF_PATH}
-            "exact.pers" "safe.pers")
-    add_test(Alpha_complex_utilities_diff_alpha_complex_3d ${DIFF_PATH}
-            "fast.pers" "safe.pers")
+    add_test(Alpha_complex_utilities_diff_exact_alpha_complex_3d ${DIFF_PATH}
+            "exact_3d.pers" "safe_3d.pers")
+    add_test(Alpha_complex_utilities_diff_fast_alpha_complex_3d ${DIFF_PATH}
+            "fast_3d.pers" "safe_3d.pers")
   endif()
 
   add_test(NAME Alpha_complex_utilities_periodic_alpha_complex_3d_persistence COMMAND $<TARGET_FILE:alpha_complex_3d_persistence>
diff --git a/src/Alpha_complex/utilities/alpha_complex_3d_persistence.cpp b/src/Alpha_complex/utilities/alpha_complex_3d_persistence.cpp
index 2272576e..929fc2e8 100644
--- a/src/Alpha_complex/utilities/alpha_complex_3d_persistence.cpp
+++ b/src/Alpha_complex/utilities/alpha_complex_3d_persistence.cpp
@@ -62,9 +62,9 @@ bool read_cuboid_file(const std::string &cuboid_file, double &x_min, double &y_m
 }
 
 template <typename AlphaComplex3d>
-std::vector<typename AlphaComplex3d::Point_3> read_off(const std::string &off_file_points) {
+std::vector<typename AlphaComplex3d::Bare_point_3> read_off(const std::string &off_file_points) {
   // Read the OFF file (input file name given as parameter) and triangulate points
-  Gudhi::Points_3D_off_reader<typename AlphaComplex3d::Point_3> off_reader(off_file_points);
+  Gudhi::Points_3D_off_reader<typename AlphaComplex3d::Bare_point_3> off_reader(off_file_points);
   // Check the read operation was correct
   if (!off_reader.is_valid()) {
     std::cerr << "Unable to read OFF file " << off_file_points << std::endl;
diff --git a/src/Alpha_complex/utilities/alpha_complex_persistence.cpp b/src/Alpha_complex/utilities/alpha_complex_persistence.cpp
index fab7bd30..486347cc 100644
--- a/src/Alpha_complex/utilities/alpha_complex_persistence.cpp
+++ b/src/Alpha_complex/utilities/alpha_complex_persistence.cpp
@@ -24,63 +24,84 @@
 using Simplex_tree = Gudhi::Simplex_tree<>;
 using Filtration_value = Simplex_tree::Filtration_value;
 
-void program_options(int argc, char *argv[], std::string &off_file_points, std::string &output_file_diag,
-                     Filtration_value &alpha_square_max_value, int &coeff_field_characteristic,
-                     Filtration_value &min_persistence);
+void program_options(int argc, char *argv[], std::string &off_file_points, bool &exact, bool &fast,
+                     std::string &output_file_diag, Filtration_value &alpha_square_max_value,
+                     int &coeff_field_characteristic, Filtration_value &min_persistence);
 
 int main(int argc, char **argv) {
   std::string off_file_points;
   std::string output_file_diag;
+  bool exact_version = false;
+  bool fast_version = false;
   Filtration_value alpha_square_max_value;
   int coeff_field_characteristic;
   Filtration_value min_persistence;
 
-  program_options(argc, argv, off_file_points, output_file_diag, alpha_square_max_value, coeff_field_characteristic,
-                  min_persistence);
+  program_options(argc, argv, off_file_points, exact_version, fast_version, output_file_diag, alpha_square_max_value,
+                  coeff_field_characteristic, min_persistence);
 
-  // ----------------------------------------------------------------------------
-  // Init of an alpha complex from an OFF file
-  // ----------------------------------------------------------------------------
-  using Kernel = CGAL::Epick_d<CGAL::Dynamic_dimension_tag>;
-  Gudhi::alpha_complex::Alpha_complex<Kernel> alpha_complex_from_file(off_file_points);
+  if ((exact_version) && (fast_version)) {
+    std::cerr << "You cannot set the exact and the fast version." << std::endl;
+    exit(-1);
+  }
 
   Simplex_tree simplex;
-  if (alpha_complex_from_file.create_complex(simplex, alpha_square_max_value)) {
-    // ----------------------------------------------------------------------------
-    // Display information about the alpha complex
-    // ----------------------------------------------------------------------------
-    std::cout << "Simplicial complex is of dimension " << simplex.dimension() << " - " << simplex.num_simplices()
-              << " simplices - " << simplex.num_vertices() << " vertices." << std::endl;
-
-    // Sort the simplices in the order of the filtration
-    simplex.initialize_filtration();
-
-    std::cout << "Simplex_tree dim: " << simplex.dimension() << std::endl;
-    // Compute the persistence diagram of the complex
-    Gudhi::persistent_cohomology::Persistent_cohomology<Simplex_tree, Gudhi::persistent_cohomology::Field_Zp> pcoh(
-        simplex);
-    // initializes the coefficient field for homology
-    pcoh.init_coefficients(coeff_field_characteristic);
-
-    pcoh.compute_persistent_cohomology(min_persistence);
-
-    // Output the diagram in filediag
-    if (output_file_diag.empty()) {
-      pcoh.output_diagram();
-    } else {
-      std::cout << "Result in file: " << output_file_diag << std::endl;
-      std::ofstream out(output_file_diag);
-      pcoh.output_diagram(out);
-      out.close();
+  if (fast_version) {
+    // WARNING : CGAL::Epick_d is fast but not safe (unlike CGAL::Epeck_d)
+    // (i.e. when the points are on a grid)
+    using Fast_kernel = CGAL::Epick_d<CGAL::Dynamic_dimension_tag>;
+
+    // Init of an alpha complex from an OFF file
+    Gudhi::alpha_complex::Alpha_complex<Fast_kernel> alpha_complex_from_file(off_file_points);
+
+    if (!alpha_complex_from_file.create_complex(simplex, alpha_square_max_value)) {
+      std::cerr << "Fast Alpha complex simplicial complex creation failed." << std::endl;
+      exit(-1);
     }
-  }
+  } else {
+    using Kernel = CGAL::Epeck_d<CGAL::Dynamic_dimension_tag>;
 
+    // Init of an alpha complex from an OFF file
+    Gudhi::alpha_complex::Alpha_complex<Kernel> alpha_complex_from_file(off_file_points);
+
+    if (!alpha_complex_from_file.create_complex(simplex, alpha_square_max_value, exact_version)) {
+      std::cerr << "Alpha complex simplicial complex creation failed." << std::endl;
+      exit(-1);
+    }
+  }
+  // ----------------------------------------------------------------------------
+  // Display information about the alpha complex
+  // ----------------------------------------------------------------------------
+  std::cout << "Simplicial complex is of dimension " << simplex.dimension() << " - " << simplex.num_simplices()
+            << " simplices - " << simplex.num_vertices() << " vertices." << std::endl;
+
+  // Sort the simplices in the order of the filtration
+  simplex.initialize_filtration();
+
+  std::cout << "Simplex_tree dim: " << simplex.dimension() << std::endl;
+  // Compute the persistence diagram of the complex
+  Gudhi::persistent_cohomology::Persistent_cohomology<Simplex_tree, Gudhi::persistent_cohomology::Field_Zp> pcoh(
+      simplex);
+  // initializes the coefficient field for homology
+  pcoh.init_coefficients(coeff_field_characteristic);
+
+  pcoh.compute_persistent_cohomology(min_persistence);
+
+  // Output the diagram in filediag
+  if (output_file_diag.empty()) {
+    pcoh.output_diagram();
+  } else {
+    std::cout << "Result in file: " << output_file_diag << std::endl;
+    std::ofstream out(output_file_diag);
+    pcoh.output_diagram(out);
+    out.close();
+  }
   return 0;
 }
 
-void program_options(int argc, char *argv[], std::string &off_file_points, std::string &output_file_diag,
-                     Filtration_value &alpha_square_max_value, int &coeff_field_characteristic,
-                     Filtration_value &min_persistence) {
+void program_options(int argc, char *argv[], std::string &off_file_points, bool &exact, bool &fast,
+                     std::string &output_file_diag, Filtration_value &alpha_square_max_value,
+                     int &coeff_field_characteristic, Filtration_value &min_persistence) {
   namespace po = boost::program_options;
   po::options_description hidden("Hidden options");
   hidden.add_options()("input-file", po::value<std::string>(&off_file_points),
@@ -88,6 +109,10 @@ void program_options(int argc, char *argv[], std::string &off_file_points, std::
 
   po::options_description visible("Allowed options", 100);
   visible.add_options()("help,h", "produce help message")(
+      "exact,e", po::bool_switch(&exact),
+      "To activate exact version of Alpha complex (default is false, not available if fast is set)")(
+      "fast,f", po::bool_switch(&fast),
+      "To activate fast version of Alpha complex (default is false, not available if exact is set)")(
       "output-file,o", po::value<std::string>(&output_file_diag)->default_value(std::string()),
       "Name of file in which the persistence diagram is written. Default print in std::cout")(
       "max-alpha-square-value,r", po::value<Filtration_value>(&alpha_square_max_value)
diff --git a/src/Alpha_complex/utilities/alphacomplex.md b/src/Alpha_complex/utilities/alphacomplex.md
index fcd16a3b..527598a9 100644
--- a/src/Alpha_complex/utilities/alphacomplex.md
+++ b/src/Alpha_complex/utilities/alphacomplex.md
@@ -46,6 +46,8 @@ for the Alpha complex construction.
 coefficient field Z/pZ for computing homology.
 * `-m [ --min-persistence ]` (default = 0) Minimal lifetime of homology feature
 to be recorded. Enter a negative value to see zero length intervals.
+* `-e [ --exact ]` for the exact computation version.
+* `-f [ --fast ]` for the fast computation version.
 
 **Example**
 
diff --git a/src/Persistent_cohomology/include/gudhi/Persistent_cohomology.h b/src/Persistent_cohomology/include/gudhi/Persistent_cohomology.h
index 944b6d35..0f1876d0 100644
--- a/src/Persistent_cohomology/include/gudhi/Persistent_cohomology.h
+++ b/src/Persistent_cohomology/include/gudhi/Persistent_cohomology.h
@@ -600,8 +600,10 @@ class Persistent_cohomology {
    * @return A vector of Betti numbers.
    */
   std::vector<int> betti_numbers() const {
+    // Don't allocate a vector of negative size for an empty complex
+    int siz = std::max(dim_max_, 0);
     // Init Betti numbers vector with zeros until Simplicial complex dimension
-    std::vector<int> betti_numbers(dim_max_, 0);
+    std::vector<int> betti_numbers(siz);
 
     for (auto pair : persistent_pairs_) {
       // Count never ended persistence intervals
@@ -639,8 +641,10 @@ class Persistent_cohomology {
    * @return A vector of persistent Betti numbers.
    */
   std::vector<int> persistent_betti_numbers(Filtration_value from, Filtration_value to) const {
+    // Don't allocate a vector of negative size for an empty complex
+    int siz = std::max(dim_max_, 0);
     // Init Betti numbers vector with zeros until Simplicial complex dimension
-    std::vector<int> betti_numbers(dim_max_, 0);
+    std::vector<int> betti_numbers(siz);
     for (auto pair : persistent_pairs_) {
       // Count persistence intervals that covers the given interval
       // null_simplex test : if the function is called with to=+infinity, we still get something useful. And it will
diff --git a/src/Persistent_cohomology/test/betti_numbers_unit_test.cpp b/src/Persistent_cohomology/test/betti_numbers_unit_test.cpp
index 0a08d200..b9f11607 100644
--- a/src/Persistent_cohomology/test/betti_numbers_unit_test.cpp
+++ b/src/Persistent_cohomology/test/betti_numbers_unit_test.cpp
@@ -284,4 +284,13 @@ BOOST_AUTO_TEST_CASE( betti_numbers )
   auto intervals_in_dimension_2 = pcoh.intervals_in_dimension(2);
   std::cout << "intervals_in_dimension_2.size() = " << intervals_in_dimension_2.size() << std::endl;
   BOOST_CHECK(intervals_in_dimension_2.size() == 0);
+
+  std::cout << "EMPTY COMPLEX" << std::endl;
+  Simplex_tree empty;
+  empty.initialize_filtration();
+  St_persistence pcoh_empty(empty, false);
+  pcoh_empty.init_coefficients(2);
+  pcoh_empty.compute_persistent_cohomology();
+  BOOST_CHECK(pcoh_empty.betti_numbers().size() == 0);
+  BOOST_CHECK(pcoh_empty.persistent_betti_numbers(0,1).size() == 0);
 }
diff --git a/src/Rips_complex/example/example_rips_complex_from_csv_distance_matrix_file.cpp b/src/Rips_complex/example/example_rips_complex_from_csv_distance_matrix_file.cpp
index 9e182f1e..b7040453 100644
--- a/src/Rips_complex/example/example_rips_complex_from_csv_distance_matrix_file.cpp
+++ b/src/Rips_complex/example/example_rips_complex_from_csv_distance_matrix_file.cpp
@@ -35,19 +35,19 @@ int main(int argc, char **argv) {
   Distance_matrix distances = Gudhi::read_lower_triangular_matrix_from_csv_file<Filtration_value>(csv_file_name);
   Rips_complex rips_complex_from_file(distances, threshold);
 
-  std::streambuf* streambufffer;
+  std::streambuf* streambuffer;
   std::ofstream ouput_file_stream;
 
   if (argc == 5) {
     ouput_file_stream.open(std::string(argv[4]));
-    streambufffer = ouput_file_stream.rdbuf();
+    streambuffer = ouput_file_stream.rdbuf();
   } else {
-    streambufffer = std::cout.rdbuf();
+    streambuffer = std::cout.rdbuf();
   }
 
   Simplex_tree stree;
   rips_complex_from_file.create_complex(stree, dim_max);
-  std::ostream output_stream(streambufffer);
+  std::ostream output_stream(streambuffer);
 
   // ----------------------------------------------------------------------------
   // Display information about the Rips complex
diff --git a/src/Rips_complex/example/example_rips_complex_from_off_file.cpp b/src/Rips_complex/example/example_rips_complex_from_off_file.cpp
index de2e4ea4..36b468a7 100644
--- a/src/Rips_complex/example/example_rips_complex_from_off_file.cpp
+++ b/src/Rips_complex/example/example_rips_complex_from_off_file.cpp
@@ -34,19 +34,19 @@ int main(int argc, char **argv) {
   Gudhi::Points_off_reader<Point> off_reader(off_file_name);
   Rips_complex rips_complex_from_file(off_reader.get_point_cloud(), threshold, Gudhi::Euclidean_distance());
 
-  std::streambuf* streambufffer;
+  std::streambuf* streambuffer;
   std::ofstream ouput_file_stream;
 
   if (argc == 5) {
     ouput_file_stream.open(std::string(argv[4]));
-    streambufffer = ouput_file_stream.rdbuf();
+    streambuffer = ouput_file_stream.rdbuf();
   } else {
-    streambufffer = std::cout.rdbuf();
+    streambuffer = std::cout.rdbuf();
   }
 
   Simplex_tree stree;
   rips_complex_from_file.create_complex(stree, dim_max);
-  std::ostream output_stream(streambufffer);
+  std::ostream output_stream(streambuffer);
 
   // ----------------------------------------------------------------------------
   // Display information about the Rips complex
diff --git a/src/common/doc/main_page.md b/src/common/doc/main_page.md
index d8cbf97f..e8d11fdf 100644
--- a/src/common/doc/main_page.md
+++ b/src/common/doc/main_page.md
@@ -45,6 +45,7 @@
     values of the codimension 1 cofaces that make it not Gabriel otherwise.
     All simplices that have a filtration value strictly greater than a given alpha squared value are not inserted into
     the complex.<br>
+    For performances reasons, it is advised to use \ref cgal &ge; 5.0.0.
     </td>
     <td width="15%">
       <b>Author:</b> Vincent Rouvreau<br>
diff --git a/src/python/gudhi/__init__.py.in b/src/python/gudhi/__init__.py.in
index 02888fff..0c462b02 100644
--- a/src/python/gudhi/__init__.py.in
+++ b/src/python/gudhi/__init__.py.in
@@ -1,14 +1,13 @@
 from importlib import import_module
 
-"""This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Vincent Rouvreau
-
-   Copyright (C) 2016  Inria
-
-   Modification(s):
-     - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+#  See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+#  Author(s):       Vincent Rouvreau
+#
+# Copyright (C) 2016  Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 __author__ = "GUDHI Editorial Board"
 __copyright__ = "Copyright (C) 2016 Inria"
diff --git a/src/python/gudhi/alpha_complex.pyx b/src/python/gudhi/alpha_complex.pyx
index 6d6309db..8f2c98d5 100644
--- a/src/python/gudhi/alpha_complex.pyx
+++ b/src/python/gudhi/alpha_complex.pyx
@@ -9,15 +9,14 @@ import os
 from gudhi.simplex_tree cimport *
 from gudhi.simplex_tree import SimplexTree
 
-""" This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Vincent Rouvreau
-
-    Copyright (C) 2016 Inria
-
-    Modification(s):
-      - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+# Author(s):       Vincent Rouvreau
+#
+# Copyright (C) 2016 Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 __author__ = "Vincent Rouvreau"
 __copyright__ = "Copyright (C) 2016 Inria"
@@ -68,7 +67,7 @@ cdef class AlphaComplex:
 
     # The real cython constructor
     def __cinit__(self, points=None, off_file=''):
-        if off_file is not '':
+        if off_file:
             if os.path.isfile(off_file):
                 self.thisptr = new Alpha_complex_interface(str.encode(off_file), True)
             else:
diff --git a/src/python/gudhi/bottleneck.pyx b/src/python/gudhi/bottleneck.pyx
index 4b378cbc..c2361024 100644
--- a/src/python/gudhi/bottleneck.pyx
+++ b/src/python/gudhi/bottleneck.pyx
@@ -3,15 +3,14 @@ from libcpp.vector cimport vector
 from libcpp.utility cimport pair
 import os
 
-""" This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Vincent Rouvreau
-
-    Copyright (C) 2016 Inria
-
-    Modification(s):
-      - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+# Author(s):       Vincent Rouvreau
+#
+# Copyright (C) 2016 Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 __author__ = "Vincent Rouvreau"
 __copyright__ = "Copyright (C) 2016 Inria"
diff --git a/src/python/gudhi/cubical_complex.pyx b/src/python/gudhi/cubical_complex.pyx
index 0dc133d1..011c407c 100644
--- a/src/python/gudhi/cubical_complex.pyx
+++ b/src/python/gudhi/cubical_complex.pyx
@@ -7,15 +7,14 @@ import os
 
 from numpy import array as np_array
 
-""" This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Vincent Rouvreau
-
-    Copyright (C) 2016 Inria
-
-    Modification(s):
-      - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+# Author(s):       Vincent Rouvreau
+#
+# Copyright (C) 2016 Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 __author__ = "Vincent Rouvreau"
 __copyright__ = "Copyright (C) 2016 Inria"
@@ -66,9 +65,9 @@ cdef class CubicalComplex:
     # The real cython constructor
     def __cinit__(self, dimensions=None, top_dimensional_cells=None,
                   perseus_file=''):
-        if (dimensions is not None) and (top_dimensional_cells is not None) and (perseus_file is ''):
+        if (dimensions is not None) and (top_dimensional_cells is not None) and (perseus_file == ''):
             self.thisptr = new Bitmap_cubical_complex_base_interface(dimensions, top_dimensional_cells)
-        elif (dimensions is None) and (top_dimensional_cells is None) and (perseus_file is not ''):
+        elif (dimensions is None) and (top_dimensional_cells is None) and (perseus_file != ''):
             if os.path.isfile(perseus_file):
                 self.thisptr = new Bitmap_cubical_complex_base_interface(str.encode(perseus_file))
             else:
diff --git a/src/python/gudhi/euclidean_strong_witness_complex.pyx b/src/python/gudhi/euclidean_strong_witness_complex.pyx
index 5d6e4fb9..e3f451f0 100644
--- a/src/python/gudhi/euclidean_strong_witness_complex.pyx
+++ b/src/python/gudhi/euclidean_strong_witness_complex.pyx
@@ -6,15 +6,14 @@ from libc.stdint cimport intptr_t
 from gudhi.simplex_tree cimport *
 from gudhi.simplex_tree import SimplexTree
 
-""" This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Vincent Rouvreau
-
-    Copyright (C) 2016 Inria
-
-    Modification(s):
-      - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+# Author(s):       Vincent Rouvreau
+#
+# Copyright (C) 2016 Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 __author__ = "Vincent Rouvreau"
 __copyright__ = "Copyright (C) 2016 Inria"
@@ -71,7 +70,7 @@ cdef class EuclideanStrongWitnessComplex:
         """
         stree = SimplexTree()
         cdef intptr_t stree_int_ptr=stree.thisptr
-        if limit_dimension is not -1:
+        if limit_dimension != -1:
             self.thisptr.create_simplex_tree(<Simplex_tree_interface_full_featured*>stree_int_ptr,
                 max_alpha_square, limit_dimension)
         else:
diff --git a/src/python/gudhi/euclidean_witness_complex.pyx b/src/python/gudhi/euclidean_witness_complex.pyx
index 2531919b..84a8ea1a 100644
--- a/src/python/gudhi/euclidean_witness_complex.pyx
+++ b/src/python/gudhi/euclidean_witness_complex.pyx
@@ -6,15 +6,14 @@ from libc.stdint cimport intptr_t
 from gudhi.simplex_tree cimport *
 from gudhi.simplex_tree import SimplexTree
 
-""" This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Vincent Rouvreau
-
-    Copyright (C) 2016 Inria
-
-    Modification(s):
-      - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+# Author(s):       Vincent Rouvreau
+#
+# Copyright (C) 2016 Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 __author__ = "Vincent Rouvreau"
 __copyright__ = "Copyright (C) 2016 Inria"
@@ -71,7 +70,7 @@ cdef class EuclideanWitnessComplex:
         """
         stree = SimplexTree()
         cdef intptr_t stree_int_ptr=stree.thisptr
-        if limit_dimension is not -1:
+        if limit_dimension != -1:
             self.thisptr.create_simplex_tree(<Simplex_tree_interface_full_featured*>stree_int_ptr,
                 max_alpha_square, limit_dimension)
         else:
diff --git a/src/python/gudhi/nerve_gic.pyx b/src/python/gudhi/nerve_gic.pyx
index 2b230b8c..acb78564 100644
--- a/src/python/gudhi/nerve_gic.pyx
+++ b/src/python/gudhi/nerve_gic.pyx
@@ -9,15 +9,14 @@ from libc.stdint cimport intptr_t
 from gudhi.simplex_tree cimport *
 from gudhi.simplex_tree import SimplexTree
 
-""" This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Vincent Rouvreau
-
-    Copyright (C) 2018 Inria
-
-    Modification(s):
-      - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+# Author(s):       Vincent Rouvreau
+#
+# Copyright (C) 2018 Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 __author__ = "Vincent Rouvreau"
 __copyright__ = "Copyright (C) 2018 Inria"
diff --git a/src/python/gudhi/off_reader.pyx b/src/python/gudhi/off_reader.pyx
index 9efd97ff..225e981c 100644
--- a/src/python/gudhi/off_reader.pyx
+++ b/src/python/gudhi/off_reader.pyx
@@ -3,15 +3,14 @@ from libcpp.vector cimport vector
 from libcpp.string cimport string
 import os
 
-""" This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Vincent Rouvreau
-
-    Copyright (C) 2016 Inria
-
-    Modification(s):
-      - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+# Author(s):       Vincent Rouvreau
+#
+# Copyright (C) 2016 Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 __author__ = "Vincent Rouvreau"
 __copyright__ = "Copyright (C) 2016 Inria"
@@ -29,7 +28,7 @@ def read_off(off_file=''):
     :returns:  The point set.
     :rtype: vector[vector[double]]
     """
-    if off_file is not '':
+    if off_file:
         if os.path.isfile(off_file):
             return read_points_from_OFF_file(str.encode(off_file))
         else:
diff --git a/src/python/gudhi/periodic_cubical_complex.pyx b/src/python/gudhi/periodic_cubical_complex.pyx
index 724fadd4..c89055db 100644
--- a/src/python/gudhi/periodic_cubical_complex.pyx
+++ b/src/python/gudhi/periodic_cubical_complex.pyx
@@ -7,15 +7,14 @@ import os
 
 from numpy import array as np_array
 
-""" This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Vincent Rouvreau
-
-    Copyright (C) 2016 Inria
-
-    Modification(s):
-      - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+# Author(s):       Vincent Rouvreau
+#
+# Copyright (C) 2016 Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 __author__ = "Vincent Rouvreau"
 __copyright__ = "Copyright (C) 2016 Inria"
@@ -68,9 +67,9 @@ cdef class PeriodicCubicalComplex:
     # The real cython constructor
     def __cinit__(self, dimensions=None, top_dimensional_cells=None,
                   periodic_dimensions=None, perseus_file=''):
-        if (dimensions is not None) and (top_dimensional_cells is not None) and (periodic_dimensions is not None) and (perseus_file is ''):
+        if (dimensions is not None) and (top_dimensional_cells is not None) and (periodic_dimensions is not None) and (perseus_file == ''):
             self.thisptr = new Periodic_cubical_complex_base_interface(dimensions, top_dimensional_cells, periodic_dimensions)
-        elif (dimensions is None) and (top_dimensional_cells is None) and (periodic_dimensions is None) and (perseus_file is not ''):
+        elif (dimensions is None) and (top_dimensional_cells is None) and (periodic_dimensions is None) and (perseus_file != ''):
             if os.path.isfile(perseus_file):
                 self.thisptr = new Periodic_cubical_complex_base_interface(str.encode(perseus_file))
             else:
diff --git a/src/python/gudhi/persistence_graphical_tools.py b/src/python/gudhi/persistence_graphical_tools.py
index 181bc8ea..43a00459 100644
--- a/src/python/gudhi/persistence_graphical_tools.py
+++ b/src/python/gudhi/persistence_graphical_tools.py
@@ -5,15 +5,14 @@ import numpy as np
 from gudhi.reader_utils import read_persistence_intervals_in_dimension
 from gudhi.reader_utils import read_persistence_intervals_grouped_by_dimension
 
-""" This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Vincent Rouvreau, Bertrand Michel
-
-    Copyright (C) 2016 Inria
-
-    Modification(s):
-      - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+# Author(s):       Vincent Rouvreau, Bertrand Michel
+#
+# Copyright (C) 2016 Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 __author__ = "Vincent Rouvreau, Bertrand Michel"
 __copyright__ = "Copyright (C) 2016 Inria"
diff --git a/src/python/gudhi/reader_utils.pyx b/src/python/gudhi/reader_utils.pyx
index 147fae71..6994c4f9 100644
--- a/src/python/gudhi/reader_utils.pyx
+++ b/src/python/gudhi/reader_utils.pyx
@@ -7,15 +7,14 @@ from libcpp.pair cimport pair
 from os import path
 from numpy import array as np_array
 
-""" This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Vincent Rouvreau
-
-    Copyright (C) 2017 Inria
-
-    Modification(s):
-      - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+# Author(s):       Vincent Rouvreau
+#
+# Copyright (C) 2017 Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 __author__ = "Vincent Rouvreau"
 __copyright__ = "Copyright (C) 2017 Inria"
@@ -37,7 +36,7 @@ def read_lower_triangular_matrix_from_csv_file(csv_file='', separator=';'):
     :returns:  The lower triangular matrix.
     :rtype: vector[vector[double]]
     """
-    if csv_file is not '':
+    if csv_file:
         if path.isfile(csv_file):
             return read_matrix_from_csv_file(str.encode(csv_file), ord(separator[0]))
     print("file " + csv_file + " not set or not found.")
@@ -56,7 +55,7 @@ def read_persistence_intervals_grouped_by_dimension(persistence_file=''):
     :returns:  The persistence pairs grouped by dimension.
     :rtype: map[int, vector[pair[double, double]]]
     """
-    if persistence_file is not '':
+    if persistence_file:
         if path.isfile(persistence_file):
             return read_pers_intervals_grouped_by_dimension(str.encode(persistence_file))
     print("file " + persistence_file + " not set or not found.")
@@ -79,7 +78,7 @@ def read_persistence_intervals_in_dimension(persistence_file='', only_this_dim=-
     :returns:  The persistence intervals.
     :rtype: numpy array of dimension 2
     """
-    if persistence_file is not '':
+    if persistence_file:
         if path.isfile(persistence_file):
             return np_array(read_pers_intervals_in_dimension(str.encode(
                 persistence_file), only_this_dim))
diff --git a/src/python/gudhi/rips_complex.pyx b/src/python/gudhi/rips_complex.pyx
index f2cd6a8d..cbbbab0d 100644
--- a/src/python/gudhi/rips_complex.pyx
+++ b/src/python/gudhi/rips_complex.pyx
@@ -8,15 +8,14 @@ from libc.stdint cimport intptr_t
 from gudhi.simplex_tree cimport *
 from gudhi.simplex_tree import SimplexTree
 
-""" This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Vincent Rouvreau
-
-    Copyright (C) 2016 Inria
-
-    Modification(s):
-      - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+# Author(s):       Vincent Rouvreau
+#
+# Copyright (C) 2016 Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 __author__ = "Vincent Rouvreau"
 __copyright__ = "Copyright (C) 2016 Inria"
diff --git a/src/python/gudhi/simplex_tree.pyx b/src/python/gudhi/simplex_tree.pyx
index 90ddf6dd..4a3cd9bc 100644
--- a/src/python/gudhi/simplex_tree.pyx
+++ b/src/python/gudhi/simplex_tree.pyx
@@ -2,15 +2,14 @@ from libc.stdint cimport intptr_t
 from numpy import array as np_array
 cimport simplex_tree
 
-""" This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Vincent Rouvreau
-
-    Copyright (C) 2016 Inria
-
-    Modification(s):
-      - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+# Author(s):       Vincent Rouvreau
+#
+# Copyright (C) 2016 Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 __author__ = "Vincent Rouvreau"
 __copyright__ = "Copyright (C) 2016 Inria"
@@ -75,13 +74,22 @@ cdef class SimplexTree:
         return self.get_ptr().simplex_filtration(simplex)
 
     def assign_filtration(self, simplex, filtration):
-        """This function assigns the simplicial complex filtration value for a
+        """This function assigns a new filtration value to a
         given N-simplex.
 
         :param simplex: The N-simplex, represented by a list of vertex.
         :type simplex: list of int.
-        :param filtration:  The simplicial complex filtration value.
+        :param filtration:  The new filtration value.
         :type filtration:  float
+
+        .. note::
+            Beware that after this operation, the structure may not be a valid
+            filtration anymore, a simplex could have a lower filtration value
+            than one of its faces. Callers are responsible for fixing this
+            (with more :meth:`assign_filtration` or
+            :meth:`make_filtration_non_decreasing` for instance) before calling
+            any function that relies on the filtration property, like
+            :meth:`initialize_filtration`.
         """
         self.get_ptr().assign_simplex_filtration(simplex, filtration)
 
diff --git a/src/python/gudhi/strong_witness_complex.pyx b/src/python/gudhi/strong_witness_complex.pyx
index e757abea..66d49b49 100644
--- a/src/python/gudhi/strong_witness_complex.pyx
+++ b/src/python/gudhi/strong_witness_complex.pyx
@@ -6,15 +6,14 @@ from libc.stdint cimport intptr_t
 from gudhi.simplex_tree cimport *
 from gudhi.simplex_tree import SimplexTree
 
-""" This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Vincent Rouvreau
-
-    Copyright (C) 2016 Inria
-
-    Modification(s):
-      - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+# Author(s):       Vincent Rouvreau
+#
+# Copyright (C) 2016 Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 __author__ = "Vincent Rouvreau"
 __copyright__ = "Copyright (C) 2016 Inria"
@@ -69,7 +68,7 @@ cdef class StrongWitnessComplex:
         """
         stree = SimplexTree()
         cdef intptr_t stree_int_ptr=stree.thisptr
-        if limit_dimension is not -1:
+        if limit_dimension != -1:
             self.thisptr.create_simplex_tree(<Simplex_tree_interface_full_featured*>stree_int_ptr,
                 max_alpha_square, limit_dimension)
         else:
diff --git a/src/python/gudhi/subsampling.pyx b/src/python/gudhi/subsampling.pyx
index 1135c1fb..e0cd1348 100644
--- a/src/python/gudhi/subsampling.pyx
+++ b/src/python/gudhi/subsampling.pyx
@@ -4,15 +4,14 @@ from libcpp.string cimport string
 from libcpp cimport bool
 import os
 
-""" This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Vincent Rouvreau
-
-    Copyright (C) 2016 Inria
-
-    Modification(s):
-      - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+# Author(s):       Vincent Rouvreau
+#
+# Copyright (C) 2016 Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 __author__ = "Vincent Rouvreau"
 __copyright__ = "Copyright (C) 2016 Inria"
@@ -44,15 +43,15 @@ def choose_n_farthest_points(points=None, off_file='', nb_points=0, starting_poi
     :param nb_points: Number of points of the subsample.
     :type nb_points: unsigned.
     :param starting_point: The iteration starts with the landmark `starting \
-    point`,which is the index of the poit to start with. If not set, this \
-    index is choosen randomly.
+    point`,which is the index of the point to start with. If not set, this \
+    index is chosen randomly.
     :type starting_point: unsigned.
     :returns:  The subsample point set.
     :rtype: vector[vector[double]]
     """
-    if off_file is not '':
+    if off_file:
         if os.path.isfile(off_file):
-            if starting_point is '':
+            if starting_point == '':
                 return subsampling_n_farthest_points_from_file(str.encode(off_file),
                                                                nb_points)
             else:
@@ -65,7 +64,7 @@ def choose_n_farthest_points(points=None, off_file='', nb_points=0, starting_poi
         if points is None:
             # Empty points
             points=[]
-        if starting_point is '':
+        if starting_point == '':
             return subsampling_n_farthest_points(points, nb_points)
         else:
             return subsampling_n_farthest_points(points, nb_points,
@@ -87,7 +86,7 @@ def pick_n_random_points(points=None, off_file='', nb_points=0):
     :returns:  The subsample point set.
     :rtype: vector[vector[double]]
     """
-    if off_file is not '':
+    if off_file:
         if os.path.isfile(off_file):
             return subsampling_n_random_points_from_file(str.encode(off_file),
                 nb_points)
@@ -117,7 +116,7 @@ def sparsify_point_set(points=None, off_file='', min_squared_dist=0.0):
     :returns:  The subsample point set.
     :rtype: vector[vector[double]]
     """
-    if off_file is not '':
+    if off_file:
         if os.path.isfile(off_file):
             return subsampling_sparsify_points_from_file(str.encode(off_file),
                                                          min_squared_dist)
diff --git a/src/python/gudhi/tangential_complex.pyx b/src/python/gudhi/tangential_complex.pyx
index 3a945fe2..b7678f4d 100644
--- a/src/python/gudhi/tangential_complex.pyx
+++ b/src/python/gudhi/tangential_complex.pyx
@@ -9,15 +9,14 @@ import os
 from gudhi.simplex_tree cimport *
 from gudhi.simplex_tree import SimplexTree
 
-""" This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Vincent Rouvreau
-
-    Copyright (C) 2016 Inria
-
-    Modification(s):
-      - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+# Author(s):       Vincent Rouvreau
+#
+# Copyright (C) 2016 Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 __author__ = "Vincent Rouvreau"
 __copyright__ = "Copyright (C) 2016 Inria"
@@ -65,7 +64,7 @@ cdef class TangentialComplex:
 
     # The real cython constructor
     def __cinit__(self, intrisic_dim, points=None, off_file=''):
-        if off_file is not '':
+        if off_file:
             if os.path.isfile(off_file):
                 self.thisptr = new Tangential_complex_interface(intrisic_dim, str.encode(off_file), True)
             else:
diff --git a/src/python/gudhi/wasserstein.py b/src/python/gudhi/wasserstein.py
index eba7c6d5..d8a3104c 100644
--- a/src/python/gudhi/wasserstein.py
+++ b/src/python/gudhi/wasserstein.py
@@ -5,15 +5,14 @@ try:
 except ImportError:
     print("POT (Python Optimal Transport) package is not installed. Try to run $ conda install -c conda-forge pot ; or $ pip install POT")
 
-""" This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Theo Lacombe
-
-    Copyright (C) 2019 Inria
-
-    Modification(s):
-      - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+# Author(s):       Theo Lacombe
+#
+# Copyright (C) 2019 Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 def _proj_on_diag(X):
     '''
diff --git a/src/python/gudhi/witness_complex.pyx b/src/python/gudhi/witness_complex.pyx
index baa70b7a..153fc615 100644
--- a/src/python/gudhi/witness_complex.pyx
+++ b/src/python/gudhi/witness_complex.pyx
@@ -6,15 +6,14 @@ from libc.stdint cimport intptr_t
 from gudhi.simplex_tree cimport *
 from gudhi.simplex_tree import SimplexTree
 
-""" This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
-    See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
-    Author(s):       Vincent Rouvreau
-
-    Copyright (C) 2016 Inria
-
-    Modification(s):
-      - YYYY/MM Author: Description of the modification
-"""
+# This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+# See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+# Author(s):       Vincent Rouvreau
+#
+# Copyright (C) 2016 Inria
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
 
 __author__ = "Vincent Rouvreau"
 __copyright__ = "Copyright (C) 2016 Inria"
@@ -69,7 +68,7 @@ cdef class WitnessComplex:
         """
         stree = SimplexTree()
         cdef intptr_t stree_int_ptr=stree.thisptr
-        if limit_dimension is not -1:
+        if limit_dimension != -1:
             self.thisptr.create_simplex_tree(<Simplex_tree_interface_full_featured*>stree_int_ptr,
                 max_alpha_square, limit_dimension)
         else: