TY - THES
AB - This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph.
For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton.
In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars.
AU - Masárová, Zuzana
ID - 7944
KW - reconfiguration
KW - reconfiguration graph
KW - triangulations
KW - flip
KW - constrained triangulations
KW - shellability
KW - piecewise-linear balls
KW - token swapping
KW - trees
KW - coloured weighted token swapping
SN - 978-3-99078-005-3
TI - Reconfiguration problems
ER -
TY - CONF
AB - Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f: ℝ^d → ℝ^(d-n). A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation 𝒯 of the ambient space ℝ^d. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine triangulation 𝒯. This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary.
AU - Boissonnat, Jean-Daniel
AU - Wintraecken, Mathijs
ID - 7952
SN - 1868-8969
T2 - 36th International Symposium on Computational Geometry
TI - The topological correctness of PL-approximations of isomanifolds
VL - 164
ER -
TY - JOUR
AB - We quantise Whitney’s construction to prove the existence of a triangulation for any C^2 manifold, so that we get an algorithm with explicit bounds. We also give a new elementary proof, which is completely geometric.
AU - Boissonnat, Jean-Daniel
AU - Kachanovich, Siargey
AU - Wintraecken, Mathijs
ID - 8940
JF - Discrete & Computational Geometry
KW - Theoretical Computer Science
KW - Computational Theory and Mathematics
KW - Geometry and Topology
KW - Discrete Mathematics and Combinatorics
SN - 0179-5376
TI - Triangulating submanifolds: An elementary and quantified version of Whitney’s method
ER -
TY - JOUR
AB - We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space X equipped with a continuous function f:X→R. We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line R. We then introduce a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium on point-based graphics, 2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of (X,f) when it is applied to points randomly sampled from a probability density function concentrated on (X,f). Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (In: 32nd international symposium on computational geometry, volume 51 of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany, pp 53:1–53:16, 2016), we first show that the mapper graph of (X,f), a constructible R-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of (X,f) to the mapper of a super-level set of a probability density function concentrated on (X,f). Finally, building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data.
AU - Brown, Adam
AU - Bobrowski, Omer
AU - Munch, Elizabeth
AU - Wang, Bei
ID - 9111
JF - Journal of Applied and Computational Topology
SN - 2367-1726
TI - Probabilistic convergence and stability of random mapper graphs
ER -
TY - JOUR
AB - The morphometric approach [11, 14] writes the solvation free energy as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted Gaussian curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [4], and the weighted mean curvature in [1], this yields the derivative of the morphometric expression of solvation free energy.
AU - Akopyan, Arseniy
AU - Edelsbrunner, Herbert
ID - 9156
IS - 1
JF - Computational and Mathematical Biophysics
SN - 2544-7297
TI - The weighted Gaussian curvature derivative of a space-filling diagram
VL - 8
ER -