Merge branch 'master' into edge_collapse_integration_vincent

34 files changed, 620 insertions, 167 deletions

diff --git a/src/Alpha_complex/utilities/alpha_complex_persistence.cpp b/src/Alpha_complex/utilities/alpha_complex_persistence.cpp
index 7c898dfd..e17831d9 100644
--- a/src/Alpha_complex/utilities/alpha_complex_persistence.cpp
+++ b/src/Alpha_complex/utilities/alpha_complex_persistence.cpp
@@ -11,6 +11,7 @@
 #include <boost/program_options.hpp>
 
 #include <CGAL/Epick_d.h>
+#include <CGAL/Epeck_d.h>
 
 #include <gudhi/Alpha_complex.h>
 #include <gudhi/Persistent_cohomology.h>
diff --git a/src/cmake/modules/GUDHI_third_party_libraries.cmake b/src/cmake/modules/GUDHI_third_party_libraries.cmake
index 0abe66b7..a56a2756 100644
--- a/src/cmake/modules/GUDHI_third_party_libraries.cmake
+++ b/src/cmake/modules/GUDHI_third_party_libraries.cmake
@@ -68,7 +68,10 @@ if(CGAL_FOUND)
 endif()
 
 # For those who dislike bundled dependencies, this indicates where to find a preinstalled Hera.
-set(HERA_WASSERSTEIN_INCLUDE_DIR ${CMAKE_SOURCE_DIR}/ext/hera/wasserstein/include CACHE PATH "Directory where one can find Hera's wasserstein.h")
+set(HERA_WASSERSTEIN_INTERNAL_INCLUDE_DIR ${CMAKE_SOURCE_DIR}/ext/hera/wasserstein/include)
+set(HERA_WASSERSTEIN_INCLUDE_DIR ${HERA_WASSERSTEIN_INTERNAL_INCLUDE_DIR} CACHE PATH "Directory where one can find Hera's wasserstein.h")
+set(HERA_BOTTLENECK_INTERNAL_INCLUDE_DIR ${CMAKE_SOURCE_DIR}/ext/hera/bottleneck/include)
+set(HERA_BOTTLENECK_INCLUDE_DIR ${HERA_BOTTLENECK_INTERNAL_INCLUDE_DIR} CACHE PATH "Directory where one can find Hera's bottleneck.h")
 
 option(WITH_GUDHI_USE_TBB "Build with Intel TBB parallelization" ON)
 
@@ -199,7 +202,7 @@ if(PYTHONINTERP_FOUND AND CYTHON_FOUND)
     if(NOT SPHINX_PATH)
       if(PYTHON_VERSION_MAJOR EQUAL 3)
         # In Python3, just hack sphinx-build if it does not exist
-        set(SPHINX_PATH "${PYTHON_EXECUTABLE}" "${CMAKE_CURRENT_SOURCE_DIR}/${GUDHI_PYTHON_PATH}/doc/python3-sphinx-build.py")
+        set(SPHINX_PATH "${PYTHON_EXECUTABLE}" "-m" "sphinx.cmd.build")
       endif(PYTHON_VERSION_MAJOR EQUAL 3)
     endif(NOT SPHINX_PATH)
   endif(SPHINX_FOUND)
diff --git a/src/cmake/modules/GUDHI_user_version_target.cmake b/src/cmake/modules/GUDHI_user_version_target.cmake
index 9cf648e3..e4f39aae 100644
--- a/src/cmake/modules/GUDHI_user_version_target.cmake
+++ b/src/cmake/modules/GUDHI_user_version_target.cmake
@@ -49,8 +49,17 @@ add_custom_command(TARGET user_version PRE_BUILD COMMAND ${CMAKE_COMMAND} -E
 add_custom_command(TARGET user_version PRE_BUILD COMMAND ${CMAKE_COMMAND} -E
                    copy ${CMAKE_SOURCE_DIR}/CMakeGUDHIVersion.txt ${GUDHI_USER_VERSION_DIR}/CMakeGUDHIVersion.txt)
 
-add_custom_command(TARGET user_version PRE_BUILD COMMAND ${CMAKE_COMMAND} -E
-                   copy_directory ${CMAKE_SOURCE_DIR}/${GUDHI_PYTHON_PATH} ${GUDHI_USER_VERSION_DIR}/python)
+# As cython generates .cpp files in source, we have to copy all except cpp files from python directory
+file(GLOB_RECURSE PYTHON_FILES ${CMAKE_SOURCE_DIR}/${GUDHI_PYTHON_PATH}/*)
+foreach(PYTHON_FILE ${PYTHON_FILES})
+  get_filename_component(PYTHON_FILE_EXT ${PYTHON_FILE} EXT)
+  if (NOT "${PYTHON_FILE_EXT}" STREQUAL ".cpp")
+    string(REPLACE "${CMAKE_SOURCE_DIR}/${GUDHI_PYTHON_PATH}/" "" RELATIVE_PYTHON_FILE ${PYTHON_FILE})
+    add_custom_command(TARGET user_version PRE_BUILD COMMAND ${CMAKE_COMMAND} -E
+                   copy ${PYTHON_FILE} ${GUDHI_USER_VERSION_DIR}/python/${RELATIVE_PYTHON_FILE})
+  endif()
+endforeach()
+
 add_custom_command(TARGET user_version PRE_BUILD COMMAND ${CMAKE_COMMAND} -E
                    copy_directory ${CMAKE_SOURCE_DIR}/data ${GUDHI_USER_VERSION_DIR}/data)
 add_custom_command(TARGET user_version PRE_BUILD COMMAND ${CMAKE_COMMAND} -E
@@ -58,8 +67,11 @@ add_custom_command(TARGET user_version PRE_BUILD COMMAND ${CMAKE_COMMAND} -E
 add_custom_command(TARGET user_version PRE_BUILD COMMAND ${CMAKE_COMMAND} -E
                    copy_directory ${CMAKE_SOURCE_DIR}/src/GudhUI ${GUDHI_USER_VERSION_DIR}/GudhUI)
 
-add_custom_command(TARGET user_version PRE_BUILD COMMAND ${CMAKE_COMMAND} -E
-                   copy_directory ${CMAKE_SOURCE_DIR}/ext/hera/wasserstein/include ${GUDHI_USER_VERSION_DIR}/ext/hera/wasserstein/include)
+if(HERA_WASSERSTEIN_INCLUDE_DIR STREQUAL HERA_WASSERSTEIN_INTERNAL_INCLUDE_DIR OR
+    HERA_BOTTLENECK_INCLUDE_DIR STREQUAL HERA_BOTTLENECK_INTERNAL_INCLUDE_DIR)
+  add_custom_command(TARGET user_version PRE_BUILD COMMAND ${CMAKE_COMMAND} -E
+                     copy_directory ${CMAKE_SOURCE_DIR}/ext/hera ${GUDHI_USER_VERSION_DIR}/ext/hera)
+endif()
 
 set(GUDHI_DIRECTORIES "doc;example;concept;utilities")
 
diff --git a/src/python/CMakeLists.txt b/src/python/CMakeLists.txt
index ab08cd6d..81be1b76 100644
--- a/src/python/CMakeLists.txt
+++ b/src/python/CMakeLists.txt
@@ -58,6 +58,7 @@ if(PYTHONINTERP_FOUND)
     set(GUDHI_PYTHON_MODULES_EXTRA "${GUDHI_PYTHON_MODULES_EXTRA}'wasserstein', ")
     set(GUDHI_PYTHON_MODULES_EXTRA "${GUDHI_PYTHON_MODULES_EXTRA}'point_cloud', ")
     set(GUDHI_PYTHON_MODULES_EXTRA "${GUDHI_PYTHON_MODULES_EXTRA}'weighted_rips_complex', ")
+    set(GUDHI_PYTHON_MODULES_EXTRA "${GUDHI_PYTHON_MODULES_EXTRA}'dtm_rips_complex', ")
 
     add_gudhi_debug_info("Python version ${PYTHON_VERSION_STRING}")
     add_gudhi_debug_info("Cython version ${CYTHON_VERSION}")
@@ -129,7 +130,8 @@ if(PYTHONINTERP_FOUND)
     set(GUDHI_CYTHON_MODULES "${GUDHI_CYTHON_MODULES}'reader_utils', ")
     set(GUDHI_CYTHON_MODULES "${GUDHI_CYTHON_MODULES}'witness_complex', ")
     set(GUDHI_CYTHON_MODULES "${GUDHI_CYTHON_MODULES}'strong_witness_complex', ")
-    set(GUDHI_PYBIND11_MODULES "${GUDHI_PYBIND11_MODULES}'hera', ")
+    set(GUDHI_PYBIND11_MODULES "${GUDHI_PYBIND11_MODULES}'hera/wasserstein', ")
+    set(GUDHI_PYBIND11_MODULES "${GUDHI_PYBIND11_MODULES}'hera/bottleneck', ")
     if (NOT CGAL_VERSION VERSION_LESS 4.11.0)
       set(GUDHI_PYBIND11_MODULES "${GUDHI_PYBIND11_MODULES}'bottleneck', ")
       set(GUDHI_CYTHON_MODULES "${GUDHI_CYTHON_MODULES}'nerve_gic', ")
@@ -234,6 +236,11 @@ if(PYTHONINTERP_FOUND)
     file(COPY "gudhi/wasserstein" DESTINATION "${CMAKE_CURRENT_BINARY_DIR}/gudhi")
     file(COPY "gudhi/point_cloud" DESTINATION "${CMAKE_CURRENT_BINARY_DIR}/gudhi")
     file(COPY "gudhi/weighted_rips_complex.py" DESTINATION "${CMAKE_CURRENT_BINARY_DIR}/gudhi")
+    file(COPY "gudhi/dtm_rips_complex.py" DESTINATION "${CMAKE_CURRENT_BINARY_DIR}/gudhi")
+    file(COPY "gudhi/hera/__init__.py" DESTINATION "${CMAKE_CURRENT_BINARY_DIR}/gudhi/hera")
+
+    # Some files for pip package
+    file(COPY "introduction.rst" DESTINATION "${CMAKE_CURRENT_BINARY_DIR}/")
 
     add_custom_command(
         OUTPUT gudhi.so
@@ -353,7 +360,9 @@ if(PYTHONINTERP_FOUND)
                COMMAND ${CMAKE_COMMAND} -E env "PYTHONPATH=${CMAKE_CURRENT_BINARY_DIR}"
                ${PYTHON_EXECUTABLE} "${CMAKE_CURRENT_SOURCE_DIR}/example/bottleneck_basic_example.py")
 
-      add_gudhi_py_test(test_bottleneck_distance)
+      if (PYBIND11_FOUND)
+        add_gudhi_py_test(test_bottleneck_distance)
+      endif()
 
       # Cover complex
       file(COPY ${CMAKE_SOURCE_DIR}/data/points/human.off DESTINATION ${CMAKE_CURRENT_BINARY_DIR}/)
@@ -492,9 +501,15 @@ if(PYTHONINTERP_FOUND)
 
     # Weighted Rips
     if(SCIPY_FOUND)
-      add_gudhi_py_test(test_weighted_rips)
+      add_gudhi_py_test(test_weighted_rips_complex)
+    endif()
+
+    # DTM Rips
+    if(SCIPY_FOUND)
+      add_gudhi_py_test(test_dtm_rips_complex)
     endif()
 
+
     # Set missing or not modules
     set(GUDHI_MODULES ${GUDHI_MODULES} "python" CACHE INTERNAL "GUDHI_MODULES")
   else(CYTHON_FOUND)
diff --git a/src/python/doc/alpha_complex_sum.inc b/src/python/doc/alpha_complex_sum.inc
index 3aba0d71..aeab493f 100644
--- a/src/python/doc/alpha_complex_sum.inc
+++ b/src/python/doc/alpha_complex_sum.inc
@@ -3,15 +3,13 @@
 
    +----------------------------------------------------------------+-------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+
    | .. figure::                                                    | Alpha complex is a simplicial complex constructed from the finite       | :Author: Vincent Rouvreau                                                                                                 |
-   |      ../../doc/Alpha_complex/alpha_complex_representation.png  | cells of a Delaunay Triangulation.                                      |                                                                                                                           |
-   |      :alt: Alpha complex representation                        |                                                                         | :Since: GUDHI 2.0.0                                                                                                       |
-   |      :figclass: align-center                                   | The filtration value of each simplex is computed as the **square** of   |                                                                                                                           |
-   |                                                                | the circumradius of the simplex if the circumsphere is empty (the       | :License: MIT (`GPL v3 </licensing/>`_)                                                                                   |
-   |                                                                | simplex is then said to be Gabriel), and as the minimum of the          |                                                                                                                           |
-   |                                                                | filtration values of the codimension 1 cofaces that make it not         | :Requires: `Eigen <installation.html#eigen>`_ :math:`\geq` 3.1.0 and `CGAL <installation.html#cgal>`_ :math:`\geq` 4.11.0 |
-   |                                                                | Gabriel otherwise.                                                      |                                                                                                                           |
+   |      ../../doc/Alpha_complex/alpha_complex_representation.png  | cells of a Delaunay Triangulation. It has the same persistent homology  |                                                                                                                           |
+   |      :alt: Alpha complex representation                        | as the Čech complex and is significantly smaller.                       | :Since: GUDHI 2.0.0                                                                                                       |
+   |      :figclass: align-center                                   |                                                                         |                                                                                                                           |
+   |                                                                |                                                                         | :License: MIT (`GPL v3 </licensing/>`_)                                                                                   |
+   |                                                                |                                                                         |                                                                                                                           |
+   |                                                                |                                                                         | :Requires: `Eigen <installation.html#eigen>`_ :math:`\geq` 3.1.0 and `CGAL <installation.html#cgal>`_ :math:`\geq` 4.11.0 |
    |                                                                |                                                                         |                                                                                                                           |
-   |                                                                | For performances reasons, it is advised to use CGAL :math:`\geq` 5.0.0. |                                                                                                                           |
    +----------------------------------------------------------------+-------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+
    | * :doc:`alpha_complex_user`                                    | * :doc:`alpha_complex_ref`                                                                                                                                                                          |
    +----------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
diff --git a/src/python/doc/alpha_complex_user.rst b/src/python/doc/alpha_complex_user.rst
index d49f45b4..097470c1 100644
--- a/src/python/doc/alpha_complex_user.rst
+++ b/src/python/doc/alpha_complex_user.rst
@@ -9,15 +9,33 @@ Definition
 
 .. include:: alpha_complex_sum.inc
 
-`AlphaComplex` is constructing a :doc:`SimplexTree <simplex_tree_ref>` using
+:doc:`AlphaComplex <alpha_complex_ref>` is constructing a :doc:`SimplexTree <simplex_tree_ref>` using
 `Delaunay Triangulation  <http://doc.cgal.org/latest/Triangulation/index.html#Chapter_Triangulations>`_
 :cite:`cgal:hdj-t-19b` from the `Computational Geometry Algorithms Library <http://www.cgal.org/>`_
 :cite:`cgal:eb-19b`.
 
 Remarks
 ^^^^^^^
-When an :math:`\alpha`-complex is constructed with an infinite value of :math:`\alpha^2`,
-the complex is a Delaunay complex (with special filtration values).
+* When an :math:`\alpha`-complex is constructed with an infinite value of :math:`\alpha^2`, the complex is a Delaunay
+  complex (with special filtration values). The Delaunay complex without filtration values is also available by
+  passing :code:`default_filtration_value = True` to :func:`~gudhi.AlphaComplex.create_simplex_tree`.
+* For people only interested in the topology of the Alpha complex (for instance persistence), Alpha complex is
+  equivalent to the `Čech complex <https://gudhi.inria.fr/doc/latest/group__cech__complex.html>`_ and much smaller if
+  you do not bound the radii. `Čech complex <https://gudhi.inria.fr/doc/latest/group__cech__complex.html>`_ can still
+  make sense in higher dimension precisely because you can bound the radii.
+* Using the default :code:`precision = 'safe'` makes the construction safe.
+  If you pass :code:`precision = 'exact'` to :func:`~gudhi.AlphaComplex.__init__`, the filtration values are the exact
+  ones converted to float. This can be very slow.
+  If you pass :code:`precision = 'safe'` (the default), the filtration values are only
+  guaranteed to have a small multiplicative error compared to the exact value.
+  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 :code:`precision = 'fast'` 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 `CGAL <installation.html#cgal>`_ :math:`\geq` 5.0.0.
+
+For performances reasons, it is advised to use CGAL :math:`\geq` 5.0.0.
 
 Example from points
 -------------------
diff --git a/src/python/doc/bottleneck_distance_user.rst b/src/python/doc/bottleneck_distance_user.rst
index 89da89d3..6c6e08d9 100644
--- a/src/python/doc/bottleneck_distance_user.rst
+++ b/src/python/doc/bottleneck_distance_user.rst
@@ -9,14 +9,23 @@ Definition
 
 .. include:: bottleneck_distance_sum.inc
 
-This implementation is based on ideas from "Geometry Helps in Bottleneck Matching and Related Problems"
-:cite:`DBLP:journals/algorithmica/EfratIK01`. Another relevant publication, although it was not used is
-"Geometry Helps to Compare Persistence Diagrams" :cite:`Kerber:2017:GHC:3047249.3064175`.
+This implementation by François Godi is based on ideas from "Geometry Helps in Bottleneck Matching and Related Problems"
+:cite:`DBLP:journals/algorithmica/EfratIK01` and requires `CGAL <installation.html#cgal>`_ (`GPL v3 </licensing/>`_).
 
-Function
---------
 .. autofunction:: gudhi.bottleneck_distance
 
+This other implementation comes from `Hera
+<https://bitbucket.org/grey_narn/hera/src/master/>`_ (BSD-3-Clause) which is
+based on "Geometry Helps to Compare Persistence Diagrams"
+:cite:`Kerber:2017:GHC:3047249.3064175` by Michael Kerber, Dmitriy
+Morozov, and Arnur Nigmetov.
+
+.. warning::
+   Beware that its approximation allows for a multiplicative error, while the function above uses an additive error.
+
+.. autofunction:: gudhi.hera.bottleneck_distance
+
+
 Distance computation
 --------------------
 
diff --git a/src/python/doc/cubical_complex_sum.inc b/src/python/doc/cubical_complex_sum.inc
index 28bf8e94..87db184d 100644
--- a/src/python/doc/cubical_complex_sum.inc
+++ b/src/python/doc/cubical_complex_sum.inc
@@ -2,9 +2,9 @@
    :widths: 30 40 30
 
    +--------------------------------------------------------------------------+----------------------------------------------------------------------+-----------------------------+
-   | .. figure::                                                              | The cubical complex is an example of a structured complex useful in  | :Author: Pawel Dlotko       |
-   |      ../../doc/Bitmap_cubical_complex/Cubical_complex_representation.png | computational mathematics (specially rigorous numerics) and image    |                             |
-   |      :alt: Cubical complex representation                                | analysis.                                                            | :Since: GUDHI 2.0.0         |
+   | .. figure::                                                              | The cubical complex represents a grid as a cell complex with         | :Author: Pawel Dlotko       |
+   |      ../../doc/Bitmap_cubical_complex/Cubical_complex_representation.png | cells of all dimensions.                                             |                             |
+   |      :alt: Cubical complex representation                                |                                                                      | :Since: GUDHI 2.0.0         |
    |      :figclass: align-center                                             |                                                                      |                             |
    |                                                                          |                                                                      | :License: MIT               |
    |                                                                          |                                                                      |                             |
diff --git a/src/python/doc/index.rst b/src/python/doc/index.rst
index 13e51047..05bc18b4 100644
--- a/src/python/doc/index.rst
+++ b/src/python/doc/index.rst
@@ -53,8 +53,8 @@ Tangential complex
 Topological descriptors computation
 ***********************************
 
-Persistence cohomology
-======================
+Persistent cohomology
+=====================
 
 .. include:: persistent_cohomology_sum.inc
 
diff --git a/src/python/doc/installation.rst b/src/python/doc/installation.rst
index de09c5b3..525ca84e 100644
--- a/src/python/doc/installation.rst
+++ b/src/python/doc/installation.rst
@@ -5,20 +5,47 @@
 Installation
 ############
 
-Conda
-*****
-The easiest way to install the Python version of GUDHI is using
-`conda <https://gudhi.inria.fr/conda/>`_.
+Packages
+********
+The easiest way to install the Python version of GUDHI is using pre-built packages.
+We recommend `conda <https://gudhi.inria.fr/conda/>`_
+
+.. code-block:: bash
+
+    conda install -c conda-forge gudhi
+
+Gudhi is also available on `PyPI <https://pypi.org/project/gudhi/>`_
+
+.. code-block:: bash
+
+    pip install gudhi
+
+Third party packages are also available, for instance on Debian or Ubuntu
+
+.. code-block:: bash
+
+    apt install python3-gudhi
+
+In all cases, you may still want to install some of the optional `run time dependencies`_.
 
 Compiling
 *********
+These instructions are for people who want to compile gudhi from source, they are
+unnecessary if you installed a binary package of Gudhi as above. They assume that
+you have downloaded a `release <https://github.com/GUDHI/gudhi-devel/releases>`_,
+with a name like `gudhi.3.2.0.tar.gz`, then run `tar xf gudhi.3.2.0.tar.gz`, which
+created a directory `gudhi.3.2.0`, hereinafter referred to as `/path-to-gudhi/`.
+If you are instead using a git checkout, beware that the paths are a bit
+different, and in particular the `python/` subdirectory is actually `src/python/`
+there.
+
 The library uses c++14 and requires `Boost <https://www.boost.org/>`_ :math:`\geq` 1.56.0,
 `CMake <https://www.cmake.org/>`_ :math:`\geq` 3.1  to generate makefiles,
 `NumPy <http://numpy.org>`_, `Cython <https://www.cython.org/>`_ and
 `pybind11 <https://github.com/pybind/pybind11>`_ to compile
 the GUDHI Python module.
 It is a multi-platform library and compiles on Linux, Mac OSX and Visual
-Studio 2017.
+Studio 2017 or later.
 
 On `Windows <https://wiki.python.org/moin/WindowsCompilers>`_ , only Python
 :math:`\geq` 3.5 are available because of the required Visual Studio version.
@@ -260,6 +287,13 @@ a flag `enable_autodiff=True` is used). In order to reduce code duplication, we
 use `EagerPy <https://eagerpy.jonasrauber.de/>`_ which wraps arrays from
 PyTorch, TensorFlow and JAX in a common interface.
 
+Joblib
+------
+
+`Joblib <https://joblib.readthedocs.io/>`_ is used both as a dependency of `Scikit-learn`_,
+and directly for parallelism in some modules (:class:`~gudhi.point_cloud.knn.KNearestNeighbors`,
+:func:`~gudhi.representations.metrics.pairwise_persistence_diagram_distances`).
+
 Hnswlib
 -------
 
diff --git a/src/python/doc/nerve_gic_complex_user.rst b/src/python/doc/nerve_gic_complex_user.rst
index 0e67fc78..0b820abf 100644
--- a/src/python/doc/nerve_gic_complex_user.rst
+++ b/src/python/doc/nerve_gic_complex_user.rst
@@ -50,7 +50,7 @@ The cover C comes from the preimages of intervals (10 intervals with gain 0.3)
 covering the height function (coordinate 2),
 which are then refined into their connected components using the triangulation of the .OFF file.
 
-.. testcode::
+.. code-block:: python
 
     import gudhi
     nerve_complex = gudhi.CoverComplex()
@@ -99,9 +99,6 @@ the program output is:
       [-0.171433, 0.367393]
       [-0.909111, 0.745853]
     0 interval(s) in dimension 1:
-
-.. testoutput::
-
     Nerve is of dimension 1 - 41 simplices - 21 vertices.
     [0]
     [1]
diff --git a/src/python/doc/persistent_cohomology_sum.inc b/src/python/doc/persistent_cohomology_sum.inc
index a1ff2eee..58e44b8a 100644
--- a/src/python/doc/persistent_cohomology_sum.inc
+++ b/src/python/doc/persistent_cohomology_sum.inc
@@ -6,9 +6,7 @@
    |      ../../doc/Persistent_cohomology/3DTorus_poch.png           | a sequence of (homology) groups, capturing global topological         |                                               |
    |      :figclass: align-center                                    | features like connected components, holes, cavities, etc. Persistent  | :Since: GUDHI 2.0.0                           |
    |                                                                 | homology studies the evolution -- birth, life and death -- of these   |                                               |
-   |      Rips Persistent Cohomology on a 3D                         | features when the topological space is changing. Consequently, the    | :License: MIT                                 |
-   |      Torus                                                      | theory is essentially composed of three elements: topological spaces, |                                               |
-   |                                                                 | their homology groups and an evolution scheme.                        |                                               |
+   |      Rips Persistent Cohomology on a 3D Torus                   | features when the topological space is changing.                      | :License: MIT                                 |
    |                                                                 |                                                                       |                                               |
    |                                                                 | Computation of persistent cohomology using the algorithm of           |                                               |
    |                                                                 | :cite:`DBLP:journals/dcg/SilvaMV11` and                               |                                               |
diff --git a/src/python/doc/python3-sphinx-build.py b/src/python/doc/python3-sphinx-build.py
deleted file mode 100755
index 84d158cf..00000000
--- a/src/python/doc/python3-sphinx-build.py
+++ /dev/null
@@ -1,11 +0,0 @@
-#!/usr/bin/env python3
-
-"""
-Emulate sphinx-build for python3
-"""
-
-from sys import exit, argv
-from sphinx import main
-
-if __name__ == '__main__':
-    exit(main(argv))
diff --git a/src/python/doc/rips_complex_ref.rst b/src/python/doc/rips_complex_ref.rst
index 5f3e46c1..f0582d5c 100644
--- a/src/python/doc/rips_complex_ref.rst
+++ b/src/python/doc/rips_complex_ref.rst
@@ -13,8 +13,6 @@ Rips complex reference manual
 
    .. automethod:: gudhi.RipsComplex.__init__
 
-.. _weighted-rips-complex-reference-manual:
-
 ======================================
 Weighted Rips complex reference manual
 ======================================
@@ -25,3 +23,14 @@ Weighted Rips complex reference manual
    :show-inheritance:
 
    .. automethod:: gudhi.weighted_rips_complex.WeightedRipsComplex.__init__
+
+=================================
+DTM Rips complex reference manual
+=================================
+
+.. autoclass:: gudhi.dtm_rips_complex.DTMRipsComplex
+   :members:
+   :undoc-members:
+   :show-inheritance:
+
+   .. automethod:: gudhi.dtm_rips_complex.DTMRipsComplex.__init__
\ No newline at end of file
diff --git a/src/python/doc/rips_complex_sum.inc b/src/python/doc/rips_complex_sum.inc
index f7580714..c123ea2a 100644
--- a/src/python/doc/rips_complex_sum.inc
+++ b/src/python/doc/rips_complex_sum.inc
@@ -2,18 +2,13 @@
    :widths: 30 40 30
 
    +----------------------------------------------------------------+------------------------------------------------------------------------+----------------------------------------------------------------------+
-   | .. figure::                                                    | Rips complex is a simplicial complex constructed from a one skeleton   | :Authors: Clément Maria, Pawel Dlotko, Vincent Rouvreau, Marc Glisse |
-   |      ../../doc/Rips_complex/rips_complex_representation.png    | graph.                                                                 |                                                                      |
+   | .. figure::                                                    | The Vietoris-Rips complex is a simplicial complex built as the         | :Authors: Clément Maria, Pawel Dlotko, Vincent Rouvreau, Marc Glisse |
+   |      ../../doc/Rips_complex/rips_complex_representation.png    | clique-complex of a proximity graph.                                   |                                                                      |
    |      :figclass: align-center                                   |                                                                        | :Since: GUDHI 2.0.0                                                  |
-   |                                                                | The filtration value of each edge is computed from a user-given        |                                                                      |
-   |                                                                | distance function and is inserted until a user-given threshold         | :License: MIT                                                        |
-   |                                                                | value.                                                                 |                                                                      |
+   |                                                                | We also provide sparse approximations, to speed-up the computation     |                                                                      |
+   |                                                                | of persistent homology, and weighted versions, which are more robust   | :License: MIT                                                        |
+   |                                                                | to outliers.                                                           |                                                                      |
    |                                                                |                                                                        |                                                                      |
-   |                                                                | This complex can be built from a point cloud and a distance function,  |                                                                      |
-   |                                                                | or from a distance matrix.                                             |                                                                      |
-   |                                                                |                                                                        |                                                                      |
-   |                                                                | Weighted Rips complex constructs a simplicial complex from a distance  |                                                                      |
-   |                                                                | matrix and weights on vertices.                                        |                                                                      |
    +----------------------------------------------------------------+------------------------------------------------------------------------+----------------------------------------------------------------------+
    | * :doc:`rips_complex_user`                                     | * :doc:`rips_complex_ref`                                                                                                                     |
    +----------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------+
diff --git a/src/python/doc/rips_complex_user.rst b/src/python/doc/rips_complex_user.rst
index 819568be..6048cc4e 100644
--- a/src/python/doc/rips_complex_user.rst
+++ b/src/python/doc/rips_complex_user.rst
@@ -378,6 +378,7 @@ Example from a point cloud combined with DistanceToMeasure
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
 
 Combining with DistanceToMeasure, one can compute the DTM-filtration of a point set, as in `this notebook <https://github.com/GUDHI/TDA-tutorial/blob/master/Tuto-GUDHI-DTM-filtrations.ipynb>`_. 
+Remark that `DTMRipsComplex <rips_complex_user.html#dtm-rips-complex>`_ class provides exactly this function.
 
 .. testcode::
 
@@ -398,3 +399,24 @@ The output is:
 .. testoutput::
 
     [(0, (3.1622776601683795, inf)), (0, (3.1622776601683795, 5.39834563766817)), (0, (3.1622776601683795, 5.39834563766817))]
+
+DTM Rips Complex
+----------------
+
+:class:`~gudhi.dtm_rips_complex.DTMRipsComplex` builds a simplicial complex from a point set or a full distance matrix (in the form of ndarray), as described in the above example.
+This class constructs a weighted Rips complex giving larger weights to outliers, which reduces their impact on the persistence diagram. See `this notebook <https://github.com/GUDHI/TDA-tutorial/blob/master/Tuto-GUDHI-DTM-filtrations.ipynb>`_ for some experiments.
+
+.. testcode::
+
+    import numpy as np
+    from gudhi.dtm_rips_complex import DTMRipsComplex
+    pts = np.array([[2.0, 2.0], [0.0, 1.0], [3.0, 4.0]])
+    dtm_rips = DTMRipsComplex(points=pts, k=2)
+    st = dtm_rips.create_simplex_tree(max_dimension=2)
+    print(st.persistence())
+
+The output is:
+
+.. testoutput::
+
+    [(0, (3.1622776601683795, inf)), (0, (3.1622776601683795, 5.39834563766817)), (0, (3.1622776601683795, 5.39834563766817))]
diff --git a/src/python/doc/tangential_complex_sum.inc b/src/python/doc/tangential_complex_sum.inc
index 22314a2d..2f330a07 100644
--- a/src/python/doc/tangential_complex_sum.inc
+++ b/src/python/doc/tangential_complex_sum.inc
@@ -3,10 +3,10 @@
 
    +----------------------------------------------------------------+------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+
    | .. figure::                                                    | A Tangential Delaunay complex is a simplicial complex designed to      | :Author:    Clément Jamin                                                                                                 |
-   |      ../../doc/Tangential_complex/tc_examples.png              | reconstruct a :math:`k`-dimensional manifold embedded in :math:`d`-    |                                                                                                                           |
-   |      :figclass: align-center                                   | dimensional Euclidean space. The input is a point sample coming from   | :Since:    GUDHI 2.0.0                                                                                                    |
-   |                                                                | an unknown manifold. The running time depends only linearly on the     |                                                                                                                           |
-   |                                                                | extrinsic dimension :math:`d` and exponentially on the intrinsic       | :License: MIT (`GPL v3 </licensing/>`_)                                                                                   |
+   |      ../../doc/Tangential_complex/tc_examples.png              | reconstruct a :math:`k`-dimensional manifold embedded in               |                                                                                                                           |
+   |      :figclass: align-center                                   | :math:`d`-dimensional Euclidean space. The input is a point sample     | :Since:    GUDHI 2.0.0                                                                                                    |
+   |                                                                | coming from an unknown manifold. The running time depends only linearly|                                                                                                                           |
+   |                                                                | on the extrinsic dimension :math:`d` and exponentially on the intrinsic| :License: MIT (`GPL v3 </licensing/>`_)                                                                                   |
    |                                                                | dimension :math:`k`.                                                   |                                                                                                                           |
    |                                                                |                                                                        | :Requires: `Eigen <installation.html#eigen>`_ :math:`\geq` 3.1.0 and `CGAL <installation.html#cgal>`_ :math:`\geq` 4.11.0 |
    +----------------------------------------------------------------+------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------+
diff --git a/src/python/gudhi/alpha_complex.pyx b/src/python/gudhi/alpha_complex.pyx
index d75e374a..a356384d 100644
--- a/src/python/gudhi/alpha_complex.pyx
+++ b/src/python/gudhi/alpha_complex.pyx
@@ -27,11 +27,11 @@ __license__ = "GPL v3"
 
 cdef extern from "Alpha_complex_interface.h" namespace "Gudhi":
     cdef cppclass Alpha_complex_interface "Gudhi::alpha_complex::Alpha_complex_interface":
-        Alpha_complex_interface(vector[vector[double]] points) nogil except +
+        Alpha_complex_interface(vector[vector[double]] points, bool fast_version) nogil except +
         # bool from_file is a workaround for cython to find the correct signature
-        Alpha_complex_interface(string off_file, bool from_file) nogil except +
+        Alpha_complex_interface(string off_file, bool fast_version, bool from_file) nogil except +
         vector[double] get_point(int vertex) nogil except +
-        void create_simplex_tree(Simplex_tree_interface_full_featured* simplex_tree, double max_alpha_square) nogil except +
+        void create_simplex_tree(Simplex_tree_interface_full_featured* simplex_tree, double max_alpha_square, bool exact_version, bool default_filtration_value) nogil except +
 
 # AlphaComplex python interface
 cdef class AlphaComplex:
@@ -53,10 +53,11 @@ cdef class AlphaComplex:
 
     """
 
-    cdef Alpha_complex_interface * thisptr
+    cdef Alpha_complex_interface * this_ptr
+    cdef bool exact
 
     # Fake constructor that does nothing but documenting the constructor
-    def __init__(self, points=None, off_file=''):
+    def __init__(self, points=None, off_file='', precision='safe'):
         """AlphaComplex constructor.
 
         :param points: A list of points in d-Dimension.
@@ -66,15 +67,21 @@ cdef class AlphaComplex:
 
         :param off_file: An OFF file style name.
         :type off_file: string
+
+        :param precision: Alpha complex precision can be 'fast', 'safe' or 'exact'. Default is 'safe'.
+        :type precision: string
         """
 
     # The real cython constructor
-    def __cinit__(self, points = None, off_file = ''):
+    def __cinit__(self, points = None, off_file = '', precision = 'safe'):
+        assert precision in ['fast', 'safe', 'exact'], "Alpha complex precision can only be 'fast', 'safe' or 'exact'"
+        cdef bool fast = precision == 'fast'
+        self.exact = precision == 'exact'
+
         cdef vector[vector[double]] pts
         if off_file:
             if os.path.isfile(off_file):
-                self.thisptr = new Alpha_complex_interface(
-                    off_file.encode('utf-8'), True)
+                self.this_ptr = new Alpha_complex_interface(off_file.encode('utf-8'), fast, True)
             else:
                 print("file " + off_file + " not found.")
         else:
@@ -83,17 +90,16 @@ cdef class AlphaComplex:
                 points=[]
             pts = points
             with nogil:
-                self.thisptr = new Alpha_complex_interface(pts)
-                
+                self.this_ptr = new Alpha_complex_interface(pts, fast)
 
     def __dealloc__(self):
-        if self.thisptr != NULL:
-            del self.thisptr
+        if self.this_ptr != NULL:
+            del self.this_ptr
 
     def __is_defined(self):
         """Returns true if AlphaComplex pointer is not NULL.
          """
-        return self.thisptr != NULL
+        return self.this_ptr != NULL
 
     def get_point(self, vertex):
         """This function returns the point corresponding to a given vertex.
@@ -103,21 +109,24 @@ cdef class AlphaComplex:
         :rtype: list of float
         :returns: the point.
         """
-        return self.thisptr.get_point(vertex)
+        return self.this_ptr.get_point(vertex)
 
-    def create_simplex_tree(self, max_alpha_square = float('inf')):
+    def create_simplex_tree(self, max_alpha_square = float('inf'), default_filtration_value = False):
         """
-        :param max_alpha_square: The maximum alpha square threshold the
-            simplices shall not exceed. Default is set to infinity, and
-            there is very little point using anything else since it does
-            not save time.
+        :param max_alpha_square: The maximum alpha square threshold the simplices shall not exceed. Default is set to
+            infinity, and there is very little point using anything else since it does not save time.
         :type max_alpha_square: float
+        :param default_filtration_value: Set this value to `True` if filtration values are not needed to be computed
+            (will be set to `NaN`). Default value is `False` (which means compute the filtration values).
+        :type default_filtration_value: bool
         :returns: A simplex tree created from the Delaunay Triangulation.
         :rtype: SimplexTree
         """
         stree = SimplexTree()
         cdef double mas = max_alpha_square
         cdef intptr_t stree_int_ptr=stree.thisptr
+        cdef bool compute_filtration = default_filtration_value == True
         with nogil:
-            self.thisptr.create_simplex_tree(<Simplex_tree_interface_full_featured*>stree_int_ptr, mas)
+            self.this_ptr.create_simplex_tree(<Simplex_tree_interface_full_featured*>stree_int_ptr,
+                                              mas, self.exact, compute_filtration)
         return stree
diff --git a/src/python/gudhi/bottleneck.cc b/src/python/gudhi/bottleneck.cc
index 732cb9a8..8a3d669a 100644
--- a/src/python/gudhi/bottleneck.cc
+++ b/src/python/gudhi/bottleneck.cc
@@ -12,24 +12,29 @@
 
 #include <pybind11_diagram_utils.h>
 
-double bottleneck(Dgm d1, Dgm d2, double epsilon)
+// For compatibility with older versions, we want to support e=None.
+// In C++17, the recommended way is std::optional<double>.
+double bottleneck(Dgm d1, Dgm d2, py::object epsilon)
 {
+  double e = (std::numeric_limits<double>::min)();
+  if (!epsilon.is_none()) e = epsilon.cast<double>();
   // I *think* the call to request() has to be before releasing the GIL.
   auto diag1 = numpy_to_range_of_pairs(d1);
   auto diag2 = numpy_to_range_of_pairs(d2);
 
   py::gil_scoped_release release;
 
-  return Gudhi::persistence_diagram::bottleneck_distance(diag1, diag2, epsilon);
+  return Gudhi::persistence_diagram::bottleneck_distance(diag1, diag2, e);
 }
 
 PYBIND11_MODULE(bottleneck, m) {
       m.attr("__license__") = "GPL v3";
       m.def("bottleneck_distance", &bottleneck,
           py::arg("diagram_1"), py::arg("diagram_2"),
-          py::arg("e") = (std::numeric_limits<double>::min)(),
+          py::arg("e") = py::none(),
           R"pbdoc(
-    This function returns the point corresponding to a given vertex.
+    Compute the Bottleneck distance between two diagrams.
+    Points at infinity and on the diagonal are supported.
 
     :param diagram_1: The first diagram.
     :type diagram_1: numpy array of shape (m,2)
@@ -42,7 +47,7 @@ PYBIND11_MODULE(bottleneck, m) {
         bits of the mantissa may be wrong). This version of the algorithm takes
         advantage of the limited precision of `double` and is usually a lot
         faster to compute, whatever the value of `e`.
-        Thus, by default, `e` is the smallest positive double.
+        Thus, by default (`e=None`), `e` is the smallest positive double.
     :type e: float
     :rtype: float
     :returns: the bottleneck distance.
diff --git a/src/python/gudhi/dtm_rips_complex.py b/src/python/gudhi/dtm_rips_complex.py
new file mode 100644
index 00000000..63c9b138
--- /dev/null
+++ b/src/python/gudhi/dtm_rips_complex.py
@@ -0,0 +1,51 @@
+# 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):       Yuichi Ike, Raphaël Tinarrage
+#
+# Copyright (C) 2020 Inria, Copyright (C) 2020 FUjitsu Laboratories Ltd.
+#
+# Modification(s):
+#   - YYYY/MM Author: Description of the modification
+
+
+from gudhi.weighted_rips_complex import WeightedRipsComplex
+from gudhi.point_cloud.dtm import DistanceToMeasure
+from scipy.spatial.distance import cdist
+
+class DTMRipsComplex(WeightedRipsComplex):
+    """
+    Class to generate a DTM Rips complex from a distance matrix or a point set, 
+    in the way described in :cite:`dtmfiltrations`.
+    Remark that all the filtration values are doubled compared to the definition in the paper 
+    for the consistency with RipsComplex.
+    :Requires: `SciPy <installation.html#scipy>`_
+    """
+    def __init__(self, 
+                 points=None, 
+                 distance_matrix=None, 
+                 k=1, 
+                 q=2,
+                 max_filtration=float('inf')):
+        """
+        Args:
+            points (numpy.ndarray): array of points.
+            distance_matrix (numpy.ndarray): full distance matrix.
+            k (int): number of neighbors for the computation of DTM. Defaults to 1, which is equivalent to the usual Rips complex.
+            q (float): order used to compute the distance to measure. Defaults to 2.
+            max_filtration (float): specifies the maximal filtration value to be considered.      
+        """
+        if distance_matrix is None:
+            if points is None:
+                # Empty Rips construction
+                points=[]
+            distance_matrix = cdist(points,points)
+        self.distance_matrix = distance_matrix
+
+        # TODO: address the error when k is too large 
+        if k <= 1:
+            self.weights = [0] * len(distance_matrix)
+        else:
+            dtm = DistanceToMeasure(k, q=q, metric="precomputed")        
+            self.weights = dtm.fit_transform(distance_matrix)
+        self.max_filtration = max_filtration
+        
diff --git a/src/python/gudhi/hera/__init__.py b/src/python/gudhi/hera/__init__.py
new file mode 100644
index 00000000..f70b92b9
--- /dev/null
+++ b/src/python/gudhi/hera/__init__.py
@@ -0,0 +1,7 @@
+from .wasserstein import wasserstein_distance
+from .bottleneck import bottleneck_distance
+
+
+__author__ = "Marc Glisse"
+__copyright__ = "Copyright (C) 2020 Inria"
+__license__ = "MIT"
diff --git a/src/python/gudhi/hera/bottleneck.cc b/src/python/gudhi/hera/bottleneck.cc
new file mode 100644
index 00000000..0cb562ce
--- /dev/null
+++ b/src/python/gudhi/hera/bottleneck.cc
@@ -0,0 +1,54 @@
+/*    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):       Marc Glisse
+ *
+ *    Copyright (C) 2020 Inria
+ *
+ *    Modification(s):
+ *      - YYYY/MM Author: Description of the modification
+ */
+
+#include <pybind11_diagram_utils.h>
+
+#ifdef _MSC_VER
+// https://github.com/grey-narn/hera/issues/3
+// ssize_t is a non-standard type (well, posix)
+using py::ssize_t;
+#endif
+
+#include <bottleneck.h> // Hera
+
+double bottleneck_distance(Dgm d1, Dgm d2, double delta)
+{
+  // I *think* the call to request() has to be before releasing the GIL.
+  auto diag1 = numpy_to_range_of_pairs(d1);
+  auto diag2 = numpy_to_range_of_pairs(d2);
+
+  py::gil_scoped_release release;
+
+  if (delta == 0)
+    return hera::bottleneckDistExact(diag1, diag2);
+  else
+    return hera::bottleneckDistApprox(diag1, diag2, delta);
+}
+
+PYBIND11_MODULE(bottleneck, m) {
+      m.def("bottleneck_distance", &bottleneck_distance,
+          py::arg("X"), py::arg("Y"),
+          py::arg("delta") = .01,
+          R"pbdoc(
+        Compute the Bottleneck distance between two diagrams.
+        Points at infinity are supported.
+
+        .. note::
+           Points on the diagonal are not supported and must be filtered out before calling this function.
+
+        Parameters:
+            X (n x 2 numpy array): First diagram
+            Y (n x 2 numpy array): Second diagram
+            delta (float): Relative error 1+delta
+
+        Returns:
+            float: (approximate) bottleneck distance d_B(X,Y)
+    )pbdoc");
+}
diff --git a/src/python/gudhi/hera.cc b/src/python/gudhi/hera/wasserstein.cc
index ea80a9a8..1a21f02f 100644
--- a/src/python/gudhi/hera.cc
+++ b/src/python/gudhi/hera/wasserstein.cc
@@ -33,7 +33,7 @@ double wasserstein_distance(
   return hera::wasserstein_dist(diag1, diag2, params);
 }
 
-PYBIND11_MODULE(hera, m) {
+PYBIND11_MODULE(wasserstein, m) {
       m.def("wasserstein_distance", &wasserstein_distance,
           py::arg("X"), py::arg("Y"),
           py::arg("order") = 1,
diff --git a/src/python/gudhi/representations/kernel_methods.py b/src/python/gudhi/representations/kernel_methods.py
index 596f4f07..23fd23c7 100644
--- a/src/python/gudhi/representations/kernel_methods.py
+++ b/src/python/gudhi/representations/kernel_methods.py
@@ -10,7 +10,7 @@
 import numpy as np
 from sklearn.base import BaseEstimator, TransformerMixin
 from sklearn.metrics import pairwise_distances, pairwise_kernels
-from .metrics import SlicedWassersteinDistance, PersistenceFisherDistance, _sklearn_wrapper, pairwise_persistence_diagram_distances, _sliced_wasserstein_distance, _persistence_fisher_distance
+from .metrics import SlicedWassersteinDistance, PersistenceFisherDistance, _sklearn_wrapper, _pairwise, pairwise_persistence_diagram_distances, _sliced_wasserstein_distance, _persistence_fisher_distance
 from .preprocessing import Padding
 
 #############################################
@@ -60,7 +60,7 @@ def _persistence_scale_space_kernel(D1, D2, kernel_approx=None, bandwidth=1.):
     weight_pss = lambda x: 1 if x[1] >= x[0] else -1
     return 0.5 * _persistence_weighted_gaussian_kernel(DD1, DD2, weight=weight_pss, kernel_approx=kernel_approx, bandwidth=bandwidth)
 
-def pairwise_persistence_diagram_kernels(X, Y=None, kernel="sliced_wasserstein", **kwargs):
+def pairwise_persistence_diagram_kernels(X, Y=None, kernel="sliced_wasserstein", n_jobs=None, **kwargs):
     """
     This function computes the kernel matrix between two lists of persistence diagrams given as numpy arrays of shape (nx2).
 
@@ -68,38 +68,41 @@ def pairwise_persistence_diagram_kernels(X, Y=None, kernel="sliced_wasserstein",
         X (list of n numpy arrays of shape (numx2)): first list of persistence diagrams. 
         Y (list of m numpy arrays of shape (numx2)): second list of persistence diagrams (optional). If None, pairwise kernel values are computed from the first list only.
         kernel: kernel to use. It can be either a string ("sliced_wasserstein", "persistence_scale_space", "persistence_weighted_gaussian", "persistence_fisher") or a function taking two numpy arrays of shape (nx2) and (mx2) as inputs. If it is a function, make sure that it is symmetric.
+        n_jobs (int): number of jobs to use for the computation. This uses joblib.Parallel(prefer="threads"), so kernels that do not release the GIL may not scale unless run inside a `joblib.parallel_backend <https://joblib.readthedocs.io/en/latest/parallel.html#joblib.parallel_backend>`_ block.
         **kwargs: optional keyword parameters. Any further parameters are passed directly to the kernel function. See the docs of the various kernel classes in this module.
 
     Returns: 
         numpy array of shape (nxm): kernel matrix.
     """    
     XX = np.reshape(np.arange(len(X)), [-1,1])
-    YY = None if Y is None else np.reshape(np.arange(len(Y)), [-1,1])
+    YY = None if Y is None or Y is X else np.reshape(np.arange(len(Y)), [-1,1])
     if kernel == "sliced_wasserstein":
-        return np.exp(-pairwise_persistence_diagram_distances(X, Y, metric="sliced_wasserstein", num_directions=kwargs["num_directions"]) / kwargs["bandwidth"])
+        return np.exp(-pairwise_persistence_diagram_distances(X, Y, metric="sliced_wasserstein", num_directions=kwargs["num_directions"], n_jobs=n_jobs) / kwargs["bandwidth"])
     elif kernel == "persistence_fisher":
-        return np.exp(-pairwise_persistence_diagram_distances(X, Y, metric="persistence_fisher", kernel_approx=kwargs["kernel_approx"], bandwidth=kwargs["bandwidth"]) / kwargs["bandwidth_fisher"])
+        return np.exp(-pairwise_persistence_diagram_distances(X, Y, metric="persistence_fisher", kernel_approx=kwargs["kernel_approx"], bandwidth=kwargs["bandwidth"], n_jobs=n_jobs) / kwargs["bandwidth_fisher"])
     elif kernel == "persistence_scale_space":
-        return pairwise_kernels(XX, YY, metric=_sklearn_wrapper(_persistence_scale_space_kernel, X, Y, **kwargs))
+        return _pairwise(pairwise_kernels, False, XX, YY, metric=_sklearn_wrapper(_persistence_scale_space_kernel, X, Y, **kwargs), n_jobs=n_jobs)
     elif kernel == "persistence_weighted_gaussian":
-        return pairwise_kernels(XX, YY, metric=_sklearn_wrapper(_persistence_weighted_gaussian_kernel, X, Y, **kwargs))
+        return _pairwise(pairwise_kernels, False, XX, YY, metric=_sklearn_wrapper(_persistence_weighted_gaussian_kernel, X, Y, **kwargs), n_jobs=n_jobs)
     else:
-        return pairwise_kernels(XX, YY, metric=_sklearn_wrapper(metric, **kwargs))
+        return _pairwise(pairwise_kernels, False, XX, YY, metric=_sklearn_wrapper(metric, **kwargs), n_jobs=n_jobs)
 
 class SlicedWassersteinKernel(BaseEstimator, TransformerMixin):
     """
     This is a class for computing the sliced Wasserstein kernel matrix from a list of persistence diagrams. The sliced Wasserstein kernel is computed by exponentiating the corresponding sliced Wasserstein distance with a Gaussian kernel. See http://proceedings.mlr.press/v70/carriere17a.html for more details. 
     """
-    def __init__(self, num_directions=10, bandwidth=1.0):
+    def __init__(self, num_directions=10, bandwidth=1.0, n_jobs=None):
         """
         Constructor for the SlicedWassersteinKernel class.
 
         Parameters:
             bandwidth (double): bandwidth of the Gaussian kernel applied to the sliced Wasserstein distance (default 1.).
             num_directions (int): number of lines evenly sampled from [-pi/2,pi/2] in order to approximate and speed up the kernel computation (default 10).
+            n_jobs (int): number of jobs to use for the computation. See :func:`pairwise_persistence_diagram_kernels` for details.
         """
         self.bandwidth = bandwidth
         self.num_directions = num_directions
+        self.n_jobs = n_jobs
 
     def fit(self, X, y=None):
         """
@@ -122,7 +125,7 @@ class SlicedWassersteinKernel(BaseEstimator, TransformerMixin):
         Returns:
             numpy array of shape (number of diagrams in **diagrams**) x (number of diagrams in X): matrix of pairwise sliced Wasserstein kernel values.
         """
-        return pairwise_persistence_diagram_kernels(X, self.diagrams_, kernel="sliced_wasserstein", bandwidth=self.bandwidth, num_directions=self.num_directions) 
+        return pairwise_persistence_diagram_kernels(X, self.diagrams_, kernel="sliced_wasserstein", bandwidth=self.bandwidth, num_directions=self.num_directions, n_jobs=self.n_jobs)
 
     def __call__(self, diag1, diag2):
         """
@@ -141,7 +144,7 @@ class PersistenceWeightedGaussianKernel(BaseEstimator, TransformerMixin):
     """
     This is a class for computing the persistence weighted Gaussian kernel matrix from a list of persistence diagrams. The persistence weighted Gaussian kernel is computed by convolving the persistence diagram points with weighted Gaussian kernels. See http://proceedings.mlr.press/v48/kusano16.html for more details. 
     """
-    def __init__(self, bandwidth=1., weight=lambda x: 1, kernel_approx=None):
+    def __init__(self, bandwidth=1., weight=lambda x: 1, kernel_approx=None, n_jobs=None):
         """
         Constructor for the PersistenceWeightedGaussianKernel class.
   
@@ -149,9 +152,11 @@ class PersistenceWeightedGaussianKernel(BaseEstimator, TransformerMixin):
             bandwidth (double): bandwidth of the Gaussian kernel with which persistence diagrams will be convolved (default 1.)
             weight (function): weight function for the persistence diagram points (default constant function, ie lambda x: 1). This function must be defined on 2D points, ie lists or numpy arrays of the form [p_x,p_y].
             kernel_approx (class): kernel approximation class used to speed up computation (default None). Common kernel approximations classes can be found in the scikit-learn library (such as RBFSampler for instance).
+            n_jobs (int): number of jobs to use for the computation. See :func:`pairwise_persistence_diagram_kernels` for details.
         """
         self.bandwidth, self.weight = bandwidth, weight
         self.kernel_approx = kernel_approx
+        self.n_jobs = n_jobs
 
     def fit(self, X, y=None):
         """
@@ -174,7 +179,7 @@ class PersistenceWeightedGaussianKernel(BaseEstimator, TransformerMixin):
         Returns:
             numpy array of shape (number of diagrams in **diagrams**) x (number of diagrams in X): matrix of pairwise persistence weighted Gaussian kernel values.
         """
-        return pairwise_persistence_diagram_kernels(X, self.diagrams_, kernel="persistence_weighted_gaussian", bandwidth=self.bandwidth, weight=self.weight, kernel_approx=self.kernel_approx) 
+        return pairwise_persistence_diagram_kernels(X, self.diagrams_, kernel="persistence_weighted_gaussian", bandwidth=self.bandwidth, weight=self.weight, kernel_approx=self.kernel_approx, n_jobs=self.n_jobs)
 
     def __call__(self, diag1, diag2):
         """
@@ -193,15 +198,17 @@ class PersistenceScaleSpaceKernel(BaseEstimator, TransformerMixin):
     """
     This is a class for computing the persistence scale space kernel matrix from a list of persistence diagrams. The persistence scale space kernel is computed by adding the symmetric to the diagonal of each point in each persistence diagram, with negative weight, and then convolving the points with a Gaussian kernel. See https://www.cv-foundation.org/openaccess/content_cvpr_2015/papers/Reininghaus_A_Stable_Multi-Scale_2015_CVPR_paper.pdf for more details. 
     """
-    def __init__(self, bandwidth=1., kernel_approx=None):
+    def __init__(self, bandwidth=1., kernel_approx=None, n_jobs=None):
         """
         Constructor for the PersistenceScaleSpaceKernel class.
   
         Parameters:
             bandwidth (double): bandwidth of the Gaussian kernel with which persistence diagrams will be convolved (default 1.)
             kernel_approx (class): kernel approximation class used to speed up computation (default None). Common kernel approximations classes can be found in the scikit-learn library (such as RBFSampler for instance).
+            n_jobs (int): number of jobs to use for the computation. See :func:`pairwise_persistence_diagram_kernels` for details.
         """
         self.bandwidth, self.kernel_approx = bandwidth, kernel_approx
+        self.n_jobs = n_jobs
 
     def fit(self, X, y=None):
         """
@@ -224,7 +231,7 @@ class PersistenceScaleSpaceKernel(BaseEstimator, TransformerMixin):
         Returns:
             numpy array of shape (number of diagrams in **diagrams**) x (number of diagrams in X): matrix of pairwise persistence scale space kernel values.
         """
-        return pairwise_persistence_diagram_kernels(X, self.diagrams_, kernel="persistence_scale_space", bandwidth=self.bandwidth, kernel_approx=self.kernel_approx)
+        return pairwise_persistence_diagram_kernels(X, self.diagrams_, kernel="persistence_scale_space", bandwidth=self.bandwidth, kernel_approx=self.kernel_approx, n_jobs=self.n_jobs)
 
     def __call__(self, diag1, diag2):
         """
@@ -243,7 +250,7 @@ class PersistenceFisherKernel(BaseEstimator, TransformerMixin):
     """
     This is a class for computing the persistence Fisher kernel matrix from a list of persistence diagrams. The persistence Fisher kernel is computed by exponentiating the corresponding persistence Fisher distance with a Gaussian kernel. See papers.nips.cc/paper/8205-persistence-fisher-kernel-a-riemannian-manifold-kernel-for-persistence-diagrams for more details. 
     """
-    def __init__(self, bandwidth_fisher=1., bandwidth=1., kernel_approx=None):
+    def __init__(self, bandwidth_fisher=1., bandwidth=1., kernel_approx=None, n_jobs=None):
         """
         Constructor for the PersistenceFisherKernel class.
 
@@ -251,9 +258,11 @@ class PersistenceFisherKernel(BaseEstimator, TransformerMixin):
             bandwidth (double): bandwidth of the Gaussian kernel applied to the persistence Fisher distance (default 1.).
             bandwidth_fisher (double): bandwidth of the Gaussian kernel used to turn persistence diagrams into probability distributions by PersistenceFisherDistance class (default 1.).
             kernel_approx (class): kernel approximation class used to speed up computation (default None). Common kernel approximations classes can be found in the scikit-learn library (such as RBFSampler for instance).
+            n_jobs (int): number of jobs to use for the computation. See :func:`pairwise_persistence_diagram_kernels` for details.
         """
         self.bandwidth = bandwidth
         self.bandwidth_fisher, self.kernel_approx = bandwidth_fisher, kernel_approx
+        self.n_jobs = n_jobs
 
     def fit(self, X, y=None):
         """
@@ -276,7 +285,7 @@ class PersistenceFisherKernel(BaseEstimator, TransformerMixin):
         Returns:
             numpy array of shape (number of diagrams in **diagrams**) x (number of diagrams in X): matrix of pairwise persistence Fisher kernel values.
         """
-        return pairwise_persistence_diagram_kernels(X, self.diagrams_, kernel="persistence_fisher", bandwidth=self.bandwidth, bandwidth_fisher=self.bandwidth_fisher, kernel_approx=self.kernel_approx)
+        return pairwise_persistence_diagram_kernels(X, self.diagrams_, kernel="persistence_fisher", bandwidth=self.bandwidth, bandwidth_fisher=self.bandwidth_fisher, kernel_approx=self.kernel_approx, n_jobs=self.n_jobs)
 
     def __call__(self, diag1, diag2):
         """
diff --git a/src/python/gudhi/representations/metrics.py b/src/python/gudhi/representations/metrics.py
index 8a32f7e9..cf2e0879 100644
--- a/src/python/gudhi/representations/metrics.py
+++ b/src/python/gudhi/representations/metrics.py
@@ -12,6 +12,7 @@ from sklearn.base import BaseEstimator, TransformerMixin
 from sklearn.metrics import pairwise_distances
 from gudhi.hera import wasserstein_distance as hera_wasserstein_distance
 from .preprocessing import Padding
+from joblib import Parallel, delayed
 
 #############################################
 # Metrics ###################################
@@ -116,6 +117,20 @@ def _persistence_fisher_distance(D1, D2, kernel_approx=None, bandwidth=1.):
             vectorj = vectorj/vectorj_sum
         return np.arccos(  min(np.dot(np.sqrt(vectori), np.sqrt(vectorj)), 1.)  )
 
+def _pairwise(fallback, skipdiag, X, Y, metric, n_jobs):
+    if Y is not None:
+        return fallback(X, Y, metric=metric, n_jobs=n_jobs)
+    triu = np.triu_indices(len(X), k=skipdiag)
+    tril = (triu[1], triu[0])
+    par = Parallel(n_jobs=n_jobs, prefer="threads")
+    d = par(delayed(metric)([triu[0][i]], [triu[1][i]]) for i in range(len(triu[0])))
+    m = np.empty((len(X), len(X)))
+    m[triu] = d
+    m[tril] = d
+    if skipdiag:
+        np.fill_diagonal(m, 0)
+    return m
+
 def _sklearn_wrapper(metric, X, Y, **kwargs):
     """
     This function is a wrapper for any metric between two persistence diagrams that takes two numpy arrays of shapes (nx2) and (mx2) as arguments.
@@ -134,7 +149,7 @@ PAIRWISE_DISTANCE_FUNCTIONS = {
     "persistence_fisher": _persistence_fisher_distance,
 }
 
-def pairwise_persistence_diagram_distances(X, Y=None, metric="bottleneck", **kwargs):
+def pairwise_persistence_diagram_distances(X, Y=None, metric="bottleneck", n_jobs=None, **kwargs):
     """
     This function computes the distance matrix between two lists of persistence diagrams given as numpy arrays of shape (nx2).
 
@@ -142,48 +157,51 @@ def pairwise_persistence_diagram_distances(X, Y=None, metric="bottleneck", **kwa
         X (list of n numpy arrays of shape (numx2)): first list of persistence diagrams. 
         Y (list of m numpy arrays of shape (numx2)): second list of persistence diagrams (optional). If None, pairwise distances are computed from the first list only.
         metric: distance to use. It can be either a string ("sliced_wasserstein", "wasserstein", "hera_wasserstein" (Wasserstein distance computed with Hera---note that Hera is also used for the default option "wasserstein"), "pot_wasserstein" (Wasserstein distance computed with POT), "bottleneck", "persistence_fisher") or a function taking two numpy arrays of shape (nx2) and (mx2) as inputs. If it is a function, make sure that it is symmetric and that it outputs 0 if called on the same two arrays. 
+        n_jobs (int): number of jobs to use for the computation. This uses joblib.Parallel(prefer="threads"), so metrics that do not release the GIL may not scale unless run inside a `joblib.parallel_backend <https://joblib.readthedocs.io/en/latest/parallel.html#joblib.parallel_backend>`_ block.
         **kwargs: optional keyword parameters. Any further parameters are passed directly to the distance function. See the docs of the various distance classes in this module.
 
     Returns: 
         numpy array of shape (nxm): distance matrix
     """
     XX = np.reshape(np.arange(len(X)), [-1,1])
-    YY = None if Y is None else np.reshape(np.arange(len(Y)), [-1,1]) 
+    YY = None if Y is None or Y is X else np.reshape(np.arange(len(Y)), [-1,1])
     if metric == "bottleneck":
         try: 
             from .. import bottleneck_distance
-            return pairwise_distances(XX, YY, metric=_sklearn_wrapper(bottleneck_distance, X, Y, **kwargs))
+            return _pairwise(pairwise_distances, True, XX, YY, metric=_sklearn_wrapper(bottleneck_distance, X, Y, **kwargs), n_jobs=n_jobs)
         except ImportError:
             print("Gudhi built without CGAL")
             raise
     elif metric == "pot_wasserstein":
         try:
             from gudhi.wasserstein import wasserstein_distance as pot_wasserstein_distance
-            return pairwise_distances(XX, YY, metric=_sklearn_wrapper(pot_wasserstein_distance,  X, Y, **kwargs))
+            return _pairwise(pairwise_distances, True, XX, YY, metric=_sklearn_wrapper(pot_wasserstein_distance,  X, Y, **kwargs), n_jobs=n_jobs)
         except ImportError:
             print("POT (Python Optimal Transport) is not installed. Please install POT or use metric='wasserstein' or metric='hera_wasserstein'")
             raise
     elif metric == "sliced_wasserstein":
         Xproj = _compute_persistence_diagram_projections(X, **kwargs)
         Yproj = None if Y is None else _compute_persistence_diagram_projections(Y, **kwargs)
-        return pairwise_distances(XX, YY, metric=_sklearn_wrapper(_sliced_wasserstein_distance_on_projections, Xproj, Yproj))
+        return _pairwise(pairwise_distances, True, XX, YY, metric=_sklearn_wrapper(_sliced_wasserstein_distance_on_projections, Xproj, Yproj), n_jobs=n_jobs)
     elif type(metric) == str:
-        return pairwise_distances(XX, YY, metric=_sklearn_wrapper(PAIRWISE_DISTANCE_FUNCTIONS[metric], X, Y, **kwargs))
+        return _pairwise(pairwise_distances, True, XX, YY, metric=_sklearn_wrapper(PAIRWISE_DISTANCE_FUNCTIONS[metric], X, Y, **kwargs), n_jobs=n_jobs)
     else:
-        return pairwise_distances(XX, YY, metric=_sklearn_wrapper(metric, X, Y, **kwargs))
+        return _pairwise(pairwise_distances, True, XX, YY, metric=_sklearn_wrapper(metric, X, Y, **kwargs), n_jobs=n_jobs)
 
 class SlicedWassersteinDistance(BaseEstimator, TransformerMixin):
     """
     This is a class for computing the sliced Wasserstein distance matrix from a list of persistence diagrams. The Sliced Wasserstein distance is computed by projecting the persistence diagrams onto lines, comparing the projections with the 1-norm, and finally integrating over all possible lines. See http://proceedings.mlr.press/v70/carriere17a.html for more details. 
     """
-    def __init__(self, num_directions=10):
+    def __init__(self, num_directions=10, n_jobs=None):
         """
         Constructor for the SlicedWassersteinDistance class.
 
         Parameters:
             num_directions (int): number of lines evenly sampled from [-pi/2,pi/2] in order to approximate and speed up the distance computation (default 10). 
+            n_jobs (int): number of jobs to use for the computation. See :func:`pairwise_persistence_diagram_distances` for details.
         """
         self.num_directions = num_directions
+        self.n_jobs = n_jobs
 
     def fit(self, X, y=None):
         """
@@ -206,7 +224,7 @@ class SlicedWassersteinDistance(BaseEstimator, TransformerMixin):
         Returns:
             numpy array of shape (number of diagrams in **diagrams**) x (number of diagrams in X): matrix of pairwise sliced Wasserstein distances.
         """
-        return pairwise_persistence_diagram_distances(X, self.diagrams_, metric="sliced_wasserstein", num_directions=self.num_directions)
+        return pairwise_persistence_diagram_distances(X, self.diagrams_, metric="sliced_wasserstein", num_directions=self.num_directions, n_jobs=self.n_jobs)
 
     def __call__(self, diag1, diag2):
         """
@@ -227,14 +245,16 @@ class BottleneckDistance(BaseEstimator, TransformerMixin):
 
     :Requires: `CGAL <installation.html#cgal>`_ :math:`\geq` 4.11.0
     """
-    def __init__(self, epsilon=None):
+    def __init__(self, epsilon=None, n_jobs=None):
         """
         Constructor for the BottleneckDistance class.
 
         Parameters:
             epsilon (double): absolute (additive) error tolerated on the distance (default is the smallest positive float), see :func:`gudhi.bottleneck_distance`.
+            n_jobs (int): number of jobs to use for the computation. See :func:`pairwise_persistence_diagram_distances` for details.
         """
         self.epsilon = epsilon
+        self.n_jobs = n_jobs
 
     def fit(self, X, y=None):
         """
@@ -257,7 +277,7 @@ class BottleneckDistance(BaseEstimator, TransformerMixin):
         Returns:
             numpy array of shape (number of diagrams in **diagrams**) x (number of diagrams in X): matrix of pairwise bottleneck distances.
         """
-        Xfit = pairwise_persistence_diagram_distances(X, self.diagrams_, metric="bottleneck", e=self.epsilon)
+        Xfit = pairwise_persistence_diagram_distances(X, self.diagrams_, metric="bottleneck", e=self.epsilon, n_jobs=self.n_jobs)
         return Xfit
 
     def __call__(self, diag1, diag2):
@@ -282,15 +302,17 @@ class PersistenceFisherDistance(BaseEstimator, TransformerMixin):
     """
     This is a class for computing the persistence Fisher distance matrix from a list of persistence diagrams. The persistence Fisher distance is obtained by computing the original Fisher distance between the probability distributions associated to the persistence diagrams given by convolving them with a Gaussian kernel. See http://papers.nips.cc/paper/8205-persistence-fisher-kernel-a-riemannian-manifold-kernel-for-persistence-diagrams for more details. 
     """
-    def __init__(self, bandwidth=1., kernel_approx=None):
+    def __init__(self, bandwidth=1., kernel_approx=None, n_jobs=None):
         """
         Constructor for the PersistenceFisherDistance class.
 
         Parameters:
             bandwidth (double): bandwidth of the Gaussian kernel used to turn persistence diagrams into probability distributions (default 1.).
             kernel_approx (class): kernel approximation class used to speed up computation (default None). Common kernel approximations classes can be found in the scikit-learn library (such as RBFSampler for instance).   
+            n_jobs (int): number of jobs to use for the computation. See :func:`pairwise_persistence_diagram_distances` for details.
         """
         self.bandwidth, self.kernel_approx = bandwidth, kernel_approx
+        self.n_jobs = n_jobs
 
     def fit(self, X, y=None):
         """
@@ -313,7 +335,7 @@ class PersistenceFisherDistance(BaseEstimator, TransformerMixin):
         Returns:
             numpy array of shape (number of diagrams in **diagrams**) x (number of diagrams in X): matrix of pairwise persistence Fisher distances.
         """
-        return pairwise_persistence_diagram_distances(X, self.diagrams_, metric="persistence_fisher", bandwidth=self.bandwidth, kernel_approx=self.kernel_approx)
+        return pairwise_persistence_diagram_distances(X, self.diagrams_, metric="persistence_fisher", bandwidth=self.bandwidth, kernel_approx=self.kernel_approx, n_jobs=self.n_jobs)
 
     def __call__(self, diag1, diag2):
         """
@@ -332,7 +354,7 @@ class WassersteinDistance(BaseEstimator, TransformerMixin):
     """
     This is a class for computing the Wasserstein distance matrix from a list of persistence diagrams. 
     """
-    def __init__(self, order=2, internal_p=2, mode="pot", delta=0.01):
+    def __init__(self, order=2, internal_p=2, mode="pot", delta=0.01, n_jobs=None):
         """
         Constructor for the WassersteinDistance class.
 
@@ -341,10 +363,12 @@ class WassersteinDistance(BaseEstimator, TransformerMixin):
             internal_p (int): ground metric on the (upper-half) plane (i.e. norm l_p in R^2), default value is 2 (euclidean norm), see :func:`gudhi.wasserstein.wasserstein_distance`.
             mode (str): method for computing Wasserstein distance. Either "pot" or "hera".
             delta (float): relative error 1+delta. Used only if mode == "hera".
+            n_jobs (int): number of jobs to use for the computation. See :func:`pairwise_persistence_diagram_distances` for details.
         """
         self.order, self.internal_p, self.mode = order, internal_p, mode
         self.metric = "pot_wasserstein" if mode == "pot" else "hera_wasserstein"
         self.delta = delta
+        self.n_jobs = n_jobs
 
     def fit(self, X, y=None):
         """
@@ -368,9 +392,9 @@ class WassersteinDistance(BaseEstimator, TransformerMixin):
             numpy array of shape (number of diagrams in **diagrams**) x (number of diagrams in X): matrix of pairwise Wasserstein distances.
         """
         if self.metric == "hera_wasserstein":
-            Xfit = pairwise_persistence_diagram_distances(X, self.diagrams_, metric=self.metric, order=self.order, internal_p=self.internal_p, delta=self.delta)
+            Xfit = pairwise_persistence_diagram_distances(X, self.diagrams_, metric=self.metric, order=self.order, internal_p=self.internal_p, delta=self.delta, n_jobs=self.n_jobs)
         else:
-            Xfit = pairwise_persistence_diagram_distances(X, self.diagrams_, metric=self.metric, order=self.order, internal_p=self.internal_p, matching=False)
+            Xfit = pairwise_persistence_diagram_distances(X, self.diagrams_, metric=self.metric, order=self.order, internal_p=self.internal_p, matching=False, n_jobs=self.n_jobs)
         return Xfit
 
     def __call__(self, diag1, diag2):
diff --git a/src/python/include/Alpha_complex_interface.h b/src/python/include/Alpha_complex_interface.h
index 40de88f3..3ac5db1f 100644
--- a/src/python/include/Alpha_complex_interface.h
+++ b/src/python/include/Alpha_complex_interface.h
@@ -23,45 +23,74 @@
 #include <iostream>
 #include <vector>
 #include <string>
+#include <memory>  // for std::unique_ptr
 
 namespace Gudhi {
 
 namespace alpha_complex {
 
 class Alpha_complex_interface {
-  using Dynamic_kernel = CGAL::Epeck_d< CGAL::Dynamic_dimension_tag >;
-  using Point_d = Dynamic_kernel::Point_d;
+ private:
+  using Exact_kernel = CGAL::Epeck_d<CGAL::Dynamic_dimension_tag>;
+  using Inexact_kernel = CGAL::Epick_d<CGAL::Dynamic_dimension_tag>;
+  using Point_exact_kernel = typename Exact_kernel::Point_d;
+  using Point_inexact_kernel = typename Inexact_kernel::Point_d;
 
- public:
-  Alpha_complex_interface(const std::vector<std::vector<double>>& points) {
-    auto mkpt = [](std::vector<double> const& vec){
-      return Point_d(vec.size(), vec.begin(), vec.end());
-    };
-    alpha_complex_ = new Alpha_complex<Dynamic_kernel>(boost::adaptors::transform(points, mkpt));
+  template <typename CgalPointType>
+  std::vector<double> pt_cgal_to_cython(CgalPointType& point) {
+    std::vector<double> vd;
+    for (auto coord = point.cartesian_begin(); coord != point.cartesian_end(); coord++)
+      vd.push_back(CGAL::to_double(*coord));
+    return vd;
   }
 
-  Alpha_complex_interface(const std::string& off_file_name, bool from_file = true) {
-    alpha_complex_ = new Alpha_complex<Dynamic_kernel>(off_file_name);
+  template <typename CgalPointType>
+  static CgalPointType pt_cython_to_cgal(std::vector<double> const& vec) {
+    return CgalPointType(vec.size(), vec.begin(), vec.end());
   }
 
-  ~Alpha_complex_interface() {
-    delete alpha_complex_;
+ public:
+  Alpha_complex_interface(const std::vector<std::vector<double>>& points, bool fast_version)
+      : fast_version_(fast_version) {
+    if (fast_version_) {
+      ac_inexact_ptr_ = std::make_unique<Alpha_complex<Inexact_kernel>>(
+          boost::adaptors::transform(points, pt_cython_to_cgal<Point_inexact_kernel>));
+    } else {
+      ac_exact_ptr_ = std::make_unique<Alpha_complex<Exact_kernel>>(
+          boost::adaptors::transform(points, pt_cython_to_cgal<Point_exact_kernel>));
+    }
+  }
+
+  Alpha_complex_interface(const std::string& off_file_name, bool fast_version, bool from_file = true)
+      : fast_version_(fast_version) {
+    if (fast_version_)
+      ac_inexact_ptr_ = std::make_unique<Alpha_complex<Inexact_kernel>>(off_file_name);
+    else
+      ac_exact_ptr_ = std::make_unique<Alpha_complex<Exact_kernel>>(off_file_name);
   }
 
   std::vector<double> get_point(int vh) {
-    std::vector<double> vd;
-    Point_d const& ph = alpha_complex_->get_point(vh);
-    for (auto coord = ph.cartesian_begin(); coord != ph.cartesian_end(); coord++)
-      vd.push_back(CGAL::to_double(*coord));
-    return vd;
+    if (fast_version_) {
+      Point_inexact_kernel const& point = ac_inexact_ptr_->get_point(vh);
+      return pt_cgal_to_cython(point);
+    } else {
+      Point_exact_kernel const& point = ac_exact_ptr_->get_point(vh);
+      return pt_cgal_to_cython(point);
+    }
   }
 
-  void create_simplex_tree(Simplex_tree_interface<>* simplex_tree, double max_alpha_square) {
-    alpha_complex_->create_complex(*simplex_tree, max_alpha_square);
+  void create_simplex_tree(Simplex_tree_interface<>* simplex_tree, double max_alpha_square, bool exact_version,
+                           bool default_filtration_value) {
+    if (fast_version_)
+      ac_inexact_ptr_->create_complex(*simplex_tree, max_alpha_square, exact_version, default_filtration_value);
+    else
+      ac_exact_ptr_->create_complex(*simplex_tree, max_alpha_square, exact_version, default_filtration_value);
   }
 
  private:
-  Alpha_complex<Dynamic_kernel>* alpha_complex_;
+  bool fast_version_;
+  std::unique_ptr<Alpha_complex<Exact_kernel>> ac_exact_ptr_;
+  std::unique_ptr<Alpha_complex<Inexact_kernel>> ac_inexact_ptr_;
 };
 
 }  // namespace alpha_complex
diff --git a/src/python/include/pybind11_diagram_utils.h b/src/python/include/pybind11_diagram_utils.h
index d9627258..2d5194f4 100644
--- a/src/python/include/pybind11_diagram_utils.h
+++ b/src/python/include/pybind11_diagram_utils.h
@@ -18,8 +18,8 @@ namespace py = pybind11;
 typedef py::array_t<double> Dgm;
 
 // Get m[i,0] and m[i,1] as a pair
-static auto pairify(void* p, ssize_t h, ssize_t w) {
-  return [=](ssize_t i){
+static auto pairify(void* p, py::ssize_t h, py::ssize_t w) {
+  return [=](py::ssize_t i){
     char* birth = (char*)p + i * h;
     char* death = birth + w;
     return std::make_pair(*(double*)birth, *(double*)death);
@@ -32,8 +32,8 @@ inline auto numpy_to_range_of_pairs(py::array_t<double> dgm) {
   if((buf.ndim!=2 || buf.shape[1]!=2) && (buf.ndim!=1 || buf.shape[0]!=0))
     throw std::runtime_error("Diagram must be an array of size n x 2");
   // In the case of shape (0), avoid reading non-existing strides[1] even if we won't use it.
-  ssize_t stride1 = buf.ndim == 2 ? buf.strides[1] : 0;
-  auto cnt = boost::counting_range<ssize_t>(0, buf.shape[0]);
+  py::ssize_t stride1 = buf.ndim == 2 ? buf.strides[1] : 0;
+  auto cnt = boost::counting_range<py::ssize_t>(0, buf.shape[0]);
   return boost::adaptors::transform(cnt, pairify(buf.ptr, buf.strides[0], stride1));
   // Be careful that the returned range cannot contain references to dead temporaries.
 }
diff --git a/src/python/introduction.rst b/src/python/introduction.rst
new file mode 100644
index 00000000..11c06ac5
--- /dev/null
+++ b/src/python/introduction.rst
@@ -0,0 +1,24 @@
+The Gudhi library is an open source library for Computational Topology and
+Topological Data Analysis (TDA). It offers state-of-the-art algorithms
+to construct various types of simplicial complexes, data structures to
+represent them, and algorithms to compute geometric approximations of shapes
+and persistent homology.
+
+The GUDHI library offers the following interoperable modules:
+
+* Complexes:
+   * Cubical
+   * Simplicial: Rips, Witness, Alpha and Čech complexes
+   * Cover: Nerve and Graph induced complexes
+* Data structures and basic operations:
+   * Simplex tree, Skeleton blockers and Toplex map
+   * Construction, update, filtration and simplification
+* Topological descriptors computation
+* Manifold reconstruction
+* Topological descriptors tools:
+   * Bottleneck distance
+   * Statistical tools
+   * Persistence diagram and barcode
+
+For more information about Topological Data Analysis and its workflow, please
+refer to the `Wikipedia TDA dedicated page <https://en.wikipedia.org/wiki/Topological_data_analysis>`_.
diff --git a/src/python/setup.py.in b/src/python/setup.py.in
index b9f4e3f0..98d058fc 100644
--- a/src/python/setup.py.in
+++ b/src/python/setup.py.in
@@ -48,10 +48,12 @@ ext_modules = cythonize(ext_modules)
 
 for module in pybind11_modules:
     my_include_dirs = include_dirs + [pybind11.get_include(False), pybind11.get_include(True)]
-    if module == 'hera':
+    if module == 'hera/wasserstein':
         my_include_dirs = ['@HERA_WASSERSTEIN_INCLUDE_DIR@'] + my_include_dirs
+    elif module == 'hera/bottleneck':
+        my_include_dirs = ['@HERA_BOTTLENECK_INCLUDE_DIR@'] + my_include_dirs
     ext_modules.append(Extension(
-        'gudhi.' + module,
+        'gudhi.' + module.replace('/', '.'),
         sources = [source_dir + module + '.cc'],
         language = 'c++',
         include_dirs = my_include_dirs,
@@ -62,14 +64,29 @@ for module in pybind11_modules:
         runtime_library_dirs=runtime_library_dirs,
         ))
 
+# read the contents of introduction.rst
+with open("introduction.rst", "r") as fh:
+    long_description = fh.read()
+
 setup(
     name = 'gudhi',
     packages=find_packages(), # find_namespace_packages(include=["gudhi*"])
     author='GUDHI Editorial Board',
     author_email='gudhi-contact@lists.gforge.inria.fr',
     version='@GUDHI_VERSION@',
-    url='http://gudhi.gforge.inria.fr/',
+    url='https://gudhi.inria.fr/',
+    project_urls={
+        'Bug Tracker': 'https://github.com/GUDHI/gudhi-devel/issues',
+        'Documentation': 'https://gudhi.inria.fr/python/latest/',
+        'Source Code': 'https://github.com/GUDHI/gudhi-devel',
+        'License': 'https://gudhi.inria.fr/licensing/'
+        },
+    description='The Gudhi library is an open source library for ' \
+        'Computational Topology and Topological Data Analysis (TDA).',
+    long_description_content_type='text/x-rst',
+    long_description=long_description,
     ext_modules = ext_modules,
-    install_requires = ['cython','numpy >= 1.9',],
-    setup_requires = ['numpy >= 1.9','pybind11',],
+    install_requires = ['numpy >= 1.9',],
+    setup_requires = ['cython','numpy >= 1.9','pybind11',],
+    package_data={"": ["*.dll"], },
 )
diff --git a/src/python/test/test_alpha_complex.py b/src/python/test/test_alpha_complex.py
index 77121302..943ad2c4 100755
--- a/src/python/test/test_alpha_complex.py
+++ b/src/python/test/test_alpha_complex.py
@@ -24,14 +24,18 @@ __copyright__ = "Copyright (C) 2016 Inria"
 __license__ = "MIT"
 
 
-def test_empty_alpha():
-    alpha_complex = AlphaComplex(points=[[0, 0]])
+def _empty_alpha(precision):
+    alpha_complex = AlphaComplex(points=[[0, 0]], precision = precision)
     assert alpha_complex.__is_defined() == True
 
+def test_empty_alpha():
+    _empty_alpha('fast')
+    _empty_alpha('safe')
+    _empty_alpha('exact')
 
-def test_infinite_alpha():
+def _infinite_alpha(precision):
     point_list = [[0, 0], [1, 0], [0, 1], [1, 1]]
-    alpha_complex = AlphaComplex(points=point_list)
+    alpha_complex = AlphaComplex(points=point_list, precision = precision)
     assert alpha_complex.__is_defined() == True
 
     simplex_tree = alpha_complex.create_simplex_tree()
@@ -79,10 +83,14 @@ def test_infinite_alpha():
     else:
         assert False
 
+def test_infinite_alpha():
+    _infinite_alpha('fast')
+    _infinite_alpha('safe')
+    _infinite_alpha('exact')
 
-def test_filtered_alpha():
+def _filtered_alpha(precision):
     point_list = [[0, 0], [1, 0], [0, 1], [1, 1]]
-    filtered_alpha = AlphaComplex(points=point_list)
+    filtered_alpha = AlphaComplex(points=point_list, precision = precision)
 
     simplex_tree = filtered_alpha.create_simplex_tree(max_alpha_square=0.25)
 
@@ -119,7 +127,12 @@ def test_filtered_alpha():
     assert simplex_tree.get_star([0]) == [([0], 0.0), ([0, 1], 0.25), ([0, 2], 0.25)]
     assert simplex_tree.get_cofaces([0], 1) == [([0, 1], 0.25), ([0, 2], 0.25)]
 
-def test_safe_alpha_persistence_comparison():
+def test_filtered_alpha():
+    _filtered_alpha('fast')
+    _filtered_alpha('safe')
+    _filtered_alpha('exact')
+
+def _safe_alpha_persistence_comparison(precision):
     #generate periodic signal
     time = np.arange(0, 10, 1)
     signal = [math.sin(x) for x in time]
@@ -131,10 +144,10 @@ def test_safe_alpha_persistence_comparison():
     embedding2 = [[signal[i], delayed[i]] for i in range(len(time))]
     
     #build alpha complex and simplex tree
-    alpha_complex1 = AlphaComplex(points=embedding1)
+    alpha_complex1 = AlphaComplex(points=embedding1, precision = precision)
     simplex_tree1 = alpha_complex1.create_simplex_tree()
     
-    alpha_complex2 = AlphaComplex(points=embedding2)
+    alpha_complex2 = AlphaComplex(points=embedding2, precision = precision)
     simplex_tree2 = alpha_complex2.create_simplex_tree()
     
     diag1 = simplex_tree1.persistence()
@@ -143,3 +156,47 @@ def test_safe_alpha_persistence_comparison():
     for (first_p, second_p) in zip_longest(diag1, diag2):
         assert first_p[0] == pytest.approx(second_p[0])
         assert first_p[1] == pytest.approx(second_p[1])
+
+
+def test_safe_alpha_persistence_comparison():
+    # Won't work for 'fast' version
+    _safe_alpha_persistence_comparison('safe')
+    _safe_alpha_persistence_comparison('exact')
+
+def _delaunay_complex(precision):
+    point_list = [[0, 0], [1, 0], [0, 1], [1, 1]]
+    filtered_alpha = AlphaComplex(points=point_list, precision = precision)
+
+    simplex_tree = filtered_alpha.create_simplex_tree(default_filtration_value = True)
+
+    assert simplex_tree.num_simplices() == 11
+    assert simplex_tree.num_vertices() == 4
+
+    assert point_list[0] == filtered_alpha.get_point(0)
+    assert point_list[1] == filtered_alpha.get_point(1)
+    assert point_list[2] == filtered_alpha.get_point(2)
+    assert point_list[3] == filtered_alpha.get_point(3)
+    try:
+        filtered_alpha.get_point(4) == []
+    except IndexError:
+        pass
+    else:
+        assert False
+    try:
+        filtered_alpha.get_point(125) == []
+    except IndexError:
+        pass
+    else:
+        assert False
+
+    for filtered_value in simplex_tree.get_filtration():
+        assert math.isnan(filtered_value[1])
+    for filtered_value in simplex_tree.get_star([0]):
+        assert math.isnan(filtered_value[1])
+    for filtered_value in simplex_tree.get_cofaces([0], 1):
+        assert math.isnan(filtered_value[1])
+
+def test_delaunay_complex():
+    _delaunay_complex('fast')
+    _delaunay_complex('safe')
+    _delaunay_complex('exact')
diff --git a/src/python/test/test_bottleneck_distance.py b/src/python/test/test_bottleneck_distance.py
index 70b2abad..6915bea8 100755
--- a/src/python/test/test_bottleneck_distance.py
+++ b/src/python/test/test_bottleneck_distance.py
@@ -9,6 +9,8 @@
 """
 
 import gudhi
+import gudhi.hera
+import pytest
 
 __author__ = "Vincent Rouvreau"
 __copyright__ = "Copyright (C) 2016 Inria"
@@ -19,5 +21,7 @@ def test_basic_bottleneck():
     diag1 = [[2.7, 3.7], [9.6, 14.0], [34.2, 34.974], [3.0, float("Inf")]]
     diag2 = [[2.8, 4.45], [9.5, 14.1], [3.2, float("Inf")]]
 
-    assert gudhi.bottleneck_distance(diag1, diag2, 0.1) == 0.8081763781405569
     assert gudhi.bottleneck_distance(diag1, diag2) == 0.75
+    assert gudhi.bottleneck_distance(diag1, diag2, 0.1) == pytest.approx(0.75, abs=0.1)
+    assert gudhi.hera.bottleneck_distance(diag1, diag2, 0) == 0.75
+    assert gudhi.hera.bottleneck_distance(diag1, diag2, 0.1) == pytest.approx(0.75, rel=0.1)
diff --git a/src/python/test/test_dtm_rips_complex.py b/src/python/test/test_dtm_rips_complex.py
new file mode 100644
index 00000000..e1c0ee44
--- /dev/null
+++ b/src/python/test/test_dtm_rips_complex.py
@@ -0,0 +1,32 @@
+""" 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):       Yuichi Ike
+
+    Copyright (C) 2020 Inria, Copyright (C) 2020 FUjitsu Laboratories Ltd.
+
+    Modification(s):
+      - YYYY/MM Author: Description of the modification
+"""
+
+from gudhi.dtm_rips_complex import DTMRipsComplex
+from gudhi import RipsComplex
+import numpy as np
+from math import sqrt
+import pytest
+
+def test_dtm_rips_complex():
+    pts = np.array([[2.0, 2.0], [0.0, 1.0], [3.0, 4.0]])
+    dtm_rips = DTMRipsComplex(points=pts, k=2)
+    st = dtm_rips.create_simplex_tree(max_dimension=2)
+    st.persistence()
+    persistence_intervals0 = st.persistence_intervals_in_dimension(0)
+    assert persistence_intervals0 == pytest.approx(np.array([[3.16227766, 5.39834564],[3.16227766, 5.39834564], [3.16227766, float("inf")]]))
+    
+def test_compatibility_with_rips():
+    distance_matrix = np.array([[0, 1, 1, sqrt(2)], [1, 0, sqrt(2), 1], [1, sqrt(2), 0, 1], [sqrt(2), 1, 1, 0]])
+    dtm_rips = DTMRipsComplex(distance_matrix=distance_matrix, max_filtration=42)
+    st = dtm_rips.create_simplex_tree(max_dimension=1)
+    rips_complex = RipsComplex(distance_matrix=distance_matrix, max_edge_length=42)
+    st_from_rips = rips_complex.create_simplex_tree(max_dimension=1)
+    assert list(st.get_filtration()) == list(st_from_rips.get_filtration())
+
diff --git a/src/python/test/test_representations.py b/src/python/test/test_representations.py
index dba7f952..589cee00 100755
--- a/src/python/test/test_representations.py
+++ b/src/python/test/test_representations.py
@@ -1,12 +1,43 @@
 import os
 import sys
 import matplotlib.pyplot as plt
+import numpy as np
+import pytest
+
 
 def test_representations_examples():
     # Disable graphics for testing purposes
-    plt.show = lambda:None
+    plt.show = lambda: None
     here = os.path.dirname(os.path.realpath(__file__))
     sys.path.append(here + "/../example")
     import diagram_vectorizations_distances_kernels
 
     return None
+
+
+from gudhi.representations.metrics import *
+from gudhi.representations.kernel_methods import *
+
+
+def _n_diags(n):
+    l = []
+    for _ in range(n):
+        a = np.random.rand(50, 2)
+        a[:, 1] += a[:, 0]  # So that y >= x
+        l.append(a)
+    return l
+
+
+def test_multiple():
+    l1 = _n_diags(9)
+    l2 = _n_diags(11)
+    l1b = l1.copy()
+    d1 = pairwise_persistence_diagram_distances(l1, e=0.00001, n_jobs=4)
+    d2 = BottleneckDistance(epsilon=0.00001).fit_transform(l1)
+    d3 = pairwise_persistence_diagram_distances(l1, l1b, e=0.00001, n_jobs=4)
+    assert d1 == pytest.approx(d2)
+    assert d3 == pytest.approx(d2, abs=1e-5) # Because of 0 entries (on the diagonal)
+    d1 = pairwise_persistence_diagram_distances(l1, l2, metric="wasserstein", order=2, internal_p=2)
+    d2 = WassersteinDistance(order=2, internal_p=2, n_jobs=4).fit(l2).transform(l1)
+    print(d1.shape, d2.shape)
+    assert d1 == pytest.approx(d2, rel=.02)
diff --git a/src/python/test/test_weighted_rips.py b/src/python/test/test_weighted_rips_complex.py
index 7ef48333..7ef48333 100644
--- a/src/python/test/test_weighted_rips.py
+++ b/src/python/test/test_weighted_rips_complex.py