arXiv subjects

Hannah Schreiber

Publications and source records attributed to Hannah Schreiber.

5 recordsLinked to original sources

arXiv2022

On Complexity of Computing Bottleneck and Lexicographic Optimal Cycles in a Homology Class

Homology features of spaces which appear in applications, for instance 3D meshes, are among the most important topological properties of these objects. Given a non-trivial cycle in a homology class, we consider the problem of computing a representative in that homology class which is optimal. We study two measures of optimality, namely, the lexicographic order of cycles (the lex-optimal cycle) and the bottleneck norm (a bottleneck-optimal cycle). We give a simple algorithm for computing the lex-optimal cycle for a 1-homology lass in a closed orientable surface. In contrast to this, our main result is that, in the case of 3-Manifolds of size $n^2$ in the Euclidean 3-space, the problem of finding a bottleneck optimal cycle cannot be solved more efficiently than solving a system of linear equations with an $n \times n$ sparse matrix. From this reduction, we deduce several hardness results. Most notably, we show that for 3-manifolds given as a subset of the 3-space of size $n^2$, persistent homology computations are at least as hard as rank computation (for sparse matrices) while ordinary homology computations can be done in $O(n^2 \log n)$ time. This is the first such distinction between these two computations. Moreover, it follows that the same disparity exists between the height persistent homology computation and general sub-level set persistent homology computation for simplicial complexes in the 3-space.

cs.CG↗

arXiv2019

Discrete Morse Theory for Computing Zigzag Persistence

We introduce a theoretical and computational framework to use discrete Morse theory as an efficient preprocessing in order to compute zigzag persistent homology. From a zigzag filtration of complexes $(K_i)$, we introduce a zigzag Morse filtration whose complexes $(A_i)$ are Morse reductions of the original complexes $(K_i)$, and we prove that they both have same persistent homology. This zigzag Morse filtration generalizes the filtered Morse complex of Mischaikow and Nanda, defined for standard persistence. The maps in the zigzag Morse filtration are forward and backward inclusions, as is standard in zigzag persistence, as well as a new type of map inducing non trivial changes in the boundary operator of the Morse complex. We study in details this last map, and design algorithms to compute the update both at the complex level and at the homology matrix level when computing zigzag persistence. We deduce an algorithm to compute the zigzag persistence of a filtration that depends mostly on the number of critical cells of the complexes, and show experimentally that it performs better in practice.

cs.CG↗

arXiv2018

Distinguishing locally finite trees

The distinguishing number $D(G)$ of a graph $G$ is the smallest number of colors that is needed to color the vertices of $G$ such that the only color preserving automorphism is the identity. For infinite graphs $D(G)$ is bounded by the supremum of the valences, and for finite graphs by $Δ(G)+1$, where $Δ(G)$ is the maximum valence. Given a finite or infinite tree $T$ of bounded finite valence $k$ and an integer $c$, where $2 \leq c \leq k$, we are interested in coloring the vertices of $T$ by $c$ colors, such that every color preserving automorphism fixes as many vertices as possible. In this sense we show that there always exists a $c$-coloring for which all vertices whose distance from the next leaf is at least $\lceil\log_ck\rceil$ are fixed by any color preserving automorphism, and that one can do much better in many cases.

math.CO↗

arXiv2017

Barcodes of Towers and a Streaming Algorithm for Persistent Homology

A tower is a sequence of simplicial complexes connected by simplicial maps. We show how to compute a filtration, a sequence of nested simplicial complexes, with the same persistent barcode as the tower. Our approach is based on the coning strategy by Dey et al. (SoCG 2014). We show that a variant of this approach yields a filtration that is asymptotically only marginally larger than the tower and can be efficiently computed by a streaming algorithm, both in theory and in practice. Furthermore, we show that our approach can be combined with a streaming algorithm to compute the barcode of the tower via matrix reduction. The space complexity of the algorithm does not depend on the length of the tower, but the maximal size of any subcomplex within the tower.

math.AT↗

arXiv2017

Distinguishing graphs of maximum valence 3

The distinguishing number $D(G)$ of a graph $G$ is the smallest number of colors that is needed to color $G$ such that the only color preserving automorphism is the identity. We give a complete classification for all connected graphs $G$ of maximum valence $\triangle(G)=3$ and distinguishing number $D(G) = 3$. As one of the consequences we get that all infinite connected graphs with $\triangle(G)=3$ are 2-distinguishable.

math.CO↗