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PHAT (Persistent Homology Algorithm Toolbox) - Version 1.0
Copyright 2013 IST Austria
Authors:
Ulrich Bauer (ulrich.bauer@ist.ac.at)
Michael Kerber (mkerber@mpi-inf.mpg.de)
Jan Reininghaus (jan.reininghaus@ist.ac.at)
Description:
This software library contains methods for computing the persistence pairs of a
filtered cell complex represented by an ordered boundary matrix with Z_2 coefficients.
For an introduction to persistent homology, see the textbook [1]. This software package
contains code for several algorithmic variants:
- The standard algorithm (see [1], p.153)
- The "row algorithm" from [2] (called pHrow in there)
- The "twist algorithm", as described in [3] (default algorithm)
- The "chunk algorithm" presented in [4]
The last two algorithms exploit the special structure of the boundary matrix
to take shortcuts in the computation. The chunk algorithm makes use of multiple
CPU cores if it is compiled with OpenMP support.
All algorithms are implemented as classes with a common interface:
they provide a member function "reduce" which takes a Boundary_matrix object
as input (to be defined below) and manipulate it to reduced form.
From this reduced form one can then easily extract the persistence pairs.
Alternatively, the algorithms can compute the associated "Persistence_pairs"
directly using the "get_pairs" method.
The algorithm classes, as well as the Boundary_matrix class, are templates
that take a "Representation" class as parameter. This representation defines
how columns of the matrix are represented and how low-level operations
(e.g., column additions) are performed. The right choice of the representation
class can be as important for the performance of the program as choosing the
algorithm. We provide the following choices of representation classes:
- "vector_vector": Each column is represented as a sorted std::vector of integers,
containing the indices of the non-zero entries of the column. The matrix
itself is a std::vector of such columns.
- "vector_set": Each column is a std::set of integers, with the same meaning
as before. The matrix is stored as a std::vector of such columns.
- "sparse_pivot_column" (default representation): The matrix is stored as in the
vector_vector representation. However, when a column is manipulated, it is first
converted into a std::set, using an extra data field called the "pivot column".
When another column is manipulated later, the pivot column is converted back to
the std::vector representation. This can lead to speed improvements when many columns
are added to a given pivot column consecutively. In a multicore setup, there is one
pivot column per core.
- "full_pivot_column": The same idea as in the sparse version. However,
instead of a std::set, the pivot column is expanded into a bit vector of size n
(the dimension of the matrix). To avoid costly initializations,
the class remembers which entries have been manipulated for a pivot column
and updates only those entries when another column becomes the pivot.
There are two ways to interface with the library:
A) using files:
1) write the boundary matrix / filtration into a file "input" (see below for the file format).
2) compile src/phat.cpp and run it:
phat input output (if you use the binary encoding, see below)
phat --ascii input output (if you use ascii encoding, see below)
3) read the resulting persistence pairs into your program
B) using the C++ library interface:
0) include all headers found in src/phat.cpp
1) define a boundary matrix object, e.g.
phat::boundary_matrix< full_pivot_column > boundary_matrix;
2) set the number of columns:
boundary_matrix.set_num_cols(...);
3) initialize each column using
boundary_matrix.set_col(...) and boundary_matrix.set_dim(...)
4) define an object to hold the result:
phat::persistence_pairs pairs;
5) run an algorithm like this:
phat::compute_persistence_pairs< phat::chunk_reduction >( pairs, boundary_matrix );
6) use pairs.get_num_pairs() and pairs.get_pair(...) to examine the result
A simple example that demonstrates this functionality can be found in src/simple_example.cpp
File Formats:
The library supports input and output in ascii and binary format
through the methods "[load|save]_[ascii|binary]" in the classes boundary_matrix
and persistence_pairs. The file formats are defined as follows:
boundary_matrix - ascii:
The file represents the filtration of the cell complex, containing one cell
per line (empty lines and lines starting with "#" are ignored). A cell is given by
a sequence of integers, separated by spaces, where the first integer denotes the
dimension of the cell, and all following integers give the indices
of the cells that form its boundary (the index of a cell is its position
in the filtration, starting with 0).
A sample file "single_triangle.dat" can be found in the examples folder.
boundary_matrix - binary:
I binary format, the file is simply interpreted as a sequence of 64 bit signed integer
numbers. The first number is interpreted as the number of cells of the complex. The
descriptions of the cells is expected to follow, with the first number representing the
dimension of the cell, the next number, say N, representing the size of the boundary,
followed by N numbers denoting the indices of the boundary cells.
A sample file "single_triangle.bin" can be found in the examples folder.
persistence_pairs - ascii:
The file contains the persistence pairs, sorted by birth index. The first integer in the
file is equal to the number of pairs. It is followed by pairs of integers encode the
respective birth and death indices.
A sample file "single_triangle_persistence_pairs.dat" can be found in the examples folder.
persistence_pairs - binary:
Same as ascii format, see above. Only now the integers are encoded as 64bit signed integers.
A sample file "single_triangle_persistence_pairs.bin" can be found in the examples folder.
References:
[1] H.Edelsbrunner, J.Harer: Computational Topology, An Introduction. American Mathematical Society, 2010, ISBN 0-8218-4925-5
[2] V.de Silva, D.Morozov, M.Vejdemo-Johansson: Dualities in persistent (co)homology. Inverse Problems 27, 2011
[3] C.Chen, M.Kerber: Persistent Homology Computation With a Twist. 27th European Workshop on Computational Geometry, 2011.
[4] U.Bauer, M.Kerber, J.Reininghaus: Clear and Compress: Computing Persistent Homology in Chunks. arxiv: ????.????
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