About

Oineus is an implementation of shared-memory parallel computation of persistent homology published in D. Morozov and A. Nigmetov. "Towards lockfree persistent homology." Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures. 2020.

Currently it supports computation of lower-star persistence of scalar functions on regular grids or of user-defined filtrations (where each simplex needs to be created manually). It is written in C++ with python bindings (pybind11).

Compilation

Oineus requires C++17 standard and python3. Pybind11 is included as a submodule. Compilation is standard:

$ git clone --recurse-submodules [email protected]:anigmetov/oineus.git
$ cd oineus
$ mkdir build
$ cd build
$ cmake ..
$ make -j4

Usage

Compiled Oineys python package is located in [build_directory]/python/bindings. Functions are given as NumPy arrays of either np.float32 or np.float64 dtype.

compute_diagrams_ls function arguments:

• data: 1D, 2D or 3D array with function values on a grid.
• negate: if True, compute upper-star persistence, default: False.
• wrap: if True, domain is treated as torus (periodic boundary conditions), default: False.
• params: settings used to run reduction, params.n_threads specifies how many threads to use.
• max_dim: maximum dimension to compute diagrams (filtration will be one dimension higher: to get persistence diagrams in dimension 1, we need 2-cells).
• n_threads: number of threads to use, default: 1.
• return: Diagrams in all dimensions. Diagrams in each dimension will be returned by in_dimension function as 2D numpy arrays [(b_1, d_1), (b_2, d_2), ... ].

>>> import numpy as np
>>> import oineus as oin
>>> f = np.random.randn(48, 48, 48)
>>> params = oin.ReductionParams()
>>> params.n_threads = 16
>>> dgms = oin.compute_diagrams_ls(data=f, negate=False, wrap=False, params=params, include_inf_points=True, max_dim=2)
>>> dgm = dgms.in_dimension(0)

Kernel, image and cokernel persistence

Oineus can compute the kernel, image and cokernel persistence diagrams as in "Persistent Homology for Kernels, Images, and Cokernels" by D. Cohen-Steiner, H. Edelsbrunner, D. Morozov. We first perform the required reductions using compute_kernel_image_cokernel_diagrams, which has arguments:

• K the simplicial complex with function values, as a list with an element per simplex in the format [simplex_id, vertices, value], where vertices is a list containing the ids of the vertices, and value is the value under the function f.
• L the simplicial sub-complex with function values, as a list with an element per simplex in the format [simplex_id, vertices, value], where vertices is a list containing the ids of the vertices, and value is the value under the function g.
• L_to_K a list which maps the cells in L to their corresponding cells in K,
• n_threads the number of threads you want to use,
• return an object which contains the kernel, image and cokernel diagrams, as well as the reduced matrices.

To obtain the different diagrams, we use kernel(), image(), cokernel(), and then we can use in_dimension to get the sepcific diagram in a specific dimension.

Note: aside from the number of threads, all other parameters are set already.

Example

Suppose we have a simplicial complex $K$ with a function $f$ on it, and a subcomplex $L \subset K$ with a function $g$ on it. In this example, $g = f|_L$. We then perform the 5 necessary reductions and compute the persistence diagrams using compute_kernel_image_cokernel_diagrams, and then access the 3 sets of diagrams using kernel(), image(), cokernel() respectively. After which we can obtain a diagram in a specific dimension $i$ using in_dimension(i).

>>> import oineus as oin
>>> n_threads = 4
>>> K = [[0, [0], 10], [1,[1],50], [2,[2], 10], [3, [3], 10], [4,[0,1], 50], [5, [1,2], 50], [6,[0,3], 10], [7, [2,3], 10]]
>>> L = [[0, [0], 10], [1,[1],50], [2,[2], 10], [3, [0,1], 50], [4,[1,2],50]]
>>> L_to_K = [0,1,2,4,5]
>>> ker_im_cok_dgms = oin.compute_kernel_image_cokernel_diagrams(K, L, L_to_K, n_threads)
>>> ker_dgms = ker_im_cok_dgms.kernel()
>>> im_dgms = ker_im_cok_dgms.image()
>>> cok_dgms = ker_im_cok_dgms.cokernel()
>>> ker_dgms.in_dimension(0)

License

Oineus is a free program distributed under modified BSD license. See legal.txt for details.