Related papers: Efficient computations of discrete cubical homolog…
We prove that if G is a graph without 3-cycles and 4-cycles, then the discrete cubical homology of G is trivial in dimension d, for all d\ge 2. We also construct a sequence { G_d } of graphs such that this homology is non-trivial in…
We first prove a one-to-one correspondence between finding Hamiltonian cycles in a cubic planar graphs and finding trees with specific properties in dual graphs. Using this information, we construct an exact algorithm for finding…
This work arose from efforts to generalise the usual cubical boundary by using different 'weights' for opposite faces, but still to obtain a chain complex, and this method was found to generalise. We describe a variant of the classical…
We develop the partitioning technique for quantum discrete systems. The graph consists of several subgraphs: a central graph and several branch graphs, with each branch graph being rooted by an individual node on the central one. We show…
Cohomology operations (including the cohomology ring) of a geometric object are finer algebraic invariants than the homology of it. In the literature, there exist various algorithms for computing the homology groups of simplicial complexes…
Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically -- as long as the algorithm does not…
Let $V$ be a finite set. Let $\mathcal{K}$ be a simplicial complex with its vertices in $V$. In this paper, we discuss some differential calculus on $V$. We construct some constrained homology groups of $\mathcal{K}$ by using the…
To compute the persistent homology of a grayscale digital image one needs to build a simplicial or cubical complex from it. For cubical complexes, the two commonly used constructions (corresponding to direct and indirect digital…
The singular cubical homology theory for the category of quivers or digraphs can be constructed similarly to the classical singular homology theory for topological spaces. The case of digraphs and quivers differs from the topological case…
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…
Fast exact algorithms are known for Hamiltonian paths in undirected and directed bipartite graphs through elegant though involved algorithms that are quite different from each other. We devise algorithms that are simple and similar to each…
We outline a novel clustering scheme for simplicial complexes that produces clusters of simplices in a way that is sensitive to the homology of the complex. The method is inspired by, and can be seen as a higher-dimensional version of,…
The diameter of a graph is among its most basic parameters. Since a few years, it moreover became a key issue to compute it for massive graphs in the context of complex network analysis. However, known algorithms, including the ones…
We present an efficient method for the computation of homogenized coefficients of divergence-form operators with random coefficients. The approach is based on a multiscale representation of the homogenized coefficients. We then implement…
We present a parallelizable algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then…
Numerous models for supervised and reinforcement learning benefit from combinations of discrete and continuous model components. End-to-end learnable discrete-continuous models are compositional, tend to generalize better, and are more…
Many computational problems admit fast algorithms on special inputs, however, the required properties might be quite restrictive. E.g., many graph problems can be solved much faster on interval or cographs, or on graphs of small…
We formulate a simple algorithm for computing global exact symmetries of closed discrete curves in plane. The method is based on a suitable trigonometric interpolation of vertices of the given polyline and consequent computation of the…
In this paper, we introduce a new method to compute magnitude homology of general graphs. To each direct sum component of magnitude chain complexes, we assign a pair of simplicial complexes whose simplicial chain complex is isomorphic to…
We describe algorithms for finding harmonic cochains, an essential ingredient for solving elliptic partial differential equations in exterior calculus. Harmonic cochains are also useful in computational topology and computer graphics. We…