A library for exploring persistent homology
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Aleph — A Library for Exploring Persistent Homology

Aleph is a C++ library for exploring and extending usages of persistent homology. Its main goal is to provide users with a versatile, simple-to-use implementation that quickly permits building prototype applications.

Aleph is inspired by DIPHA and PHAT. In particular, Aleph borrowed the idea of keeping the representation of a boundary matrix separate from the implementation.

For more information, please read the original paper describing PHAT.

Since its inception in late 2016, Aleph has been used to support the following papers:

Please refer to the list of publications in the documentation of Aleph for more details. The documentation covers how to reproduce a subset of the results reported in the papers above.

If you want to contribute, please see the contribution guidelines for more details.


Aleph contains numerous algorithms and helper classes that simplify working with persistent homology. Here is a brief selection of the most important ones:

  • Easy-to-use and expressive simplex and simplicial complex class
  • Support for different input formats to read simplicial complexes from a variety of input files
    • 1D functions
    • Edge lists
    • GraphML
    • GML
    • HDF5
    • Lexicographic triangulations
    • Matrices
    • NET (Pajek graphs)
    • PLY
    • VTK
  • Standard algorithm and twisted reduction algorithm for calculating persistent homology
  • Support for dualized variants of both algorithms
  • Support for different boundary matrix representations
  • Persistence diagram class
  • Distance and kernel measures
    • Bottleneck distance
    • Multi-scale smoothing kernel
    • Wasserstein distance
  • Algorithms for computing intersection homology and persistent intersection homology
  • Basic support for Čech complexes
  • Support for Dowker complexes


Documentation of the main features, including some tutorials, is available on GitHub. If you want to delve into the code, the examples subdirectory is a good starting point.


Aleph uses the MIT license. Please see the file LICENSE.md in the main directory of the repository for more details.


  • A recent C++ compiler with support for C++11
  • CMake, preferably a recent version >= 3.2
  • Several Boost dependencies for some of the data structures:
    • Boost.Functional
    • Boost.Iterator
    • Boost.MultiIndex

Optional dependencies

  • Eigen3 for some auxiliary mathematical functions
  • FLANN for fast nearest-neighbour queries
  • HDF5 for parsing HDF5 input files
  • pybind11 for building the Python bindings
  • RapidJSON for parsing JSON input files
  • TinyXML2 for parsing GraphML input files

Building Aleph

Aleph is meant to be used as a header-only library on top of which you can develop your own projects based on persistent homology. However, Aleph ships with numerous unit tests, some example programs, and tools required for my current research. For building them, please clone the repository to some local directory on your computer. Running the following commands within this directory should be sufficient in most cases:

$ mkdir build
$ cd build
$ cmake ../
$ make

It is advisable to test that Aleph works correctly on your system, so you can run the unit tests with:

$ make test

Please submit any issues you may encounter.

For more information, including how to run tests, please refer to the detailed build instructions in the documentation.

Installing Aleph

If you want to install Aleph from source, simply issue

$ make install

from the compilation directory. In general, this will require root privileges, unless you change the CMAKE_INSTALL_PREFIX variable to a local directory.

It is easier to install Aleph as a package. Currently, only packages for Arch Linux are available. Use your favourite AUR helper tool for installing Aleph:

$ pacaur -S aleph-git # pacaur (deprecated)
$ trizen -S aleph-git # trizen
$ yaourt -S aleph-git # yaourt

If you want to volunteer and submit a package for your favourite Linux distribution, please take a look at issue #27 and add your comments.

Installing the Python bindings

If your build instructions are configured to build the Python bindings, i.e. BUILD_PYTHON_BINDINGS follow these instructions to install them:

$ cd build/bindings/python/aleph
$ pthon3 setup.py install

Note that this uses the old setuptools approach for installing the package. A simpler installation based on pip is forthcoming.

Contact & contributors

For general discussion, questions, and comments, please contact the principal developer and maintainer Bastian Rieck (bastian.rieck@iwr.uni-heidelberg.de).

The following people have contributed code to Aleph: