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 -
TY - JOUR
AB - Representing an atom by a solid sphere in 3-dimensional Euclidean space, we get the space-filling diagram of a molecule by taking the union. Molecular dynamics simulates its motion subject to bonds and other forces, including the solvation free energy. The morphometric approach [12, 17] writes the latter 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 mean curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [3], and the weighted Gaussian curvature [1], this yields the derivative of the morphometric expression of the solvation free energy.
AU - Akopyan, Arseniy
AU - Edelsbrunner, Herbert
ID - 9157
IS - 1
JF - Computational and Mathematical Biophysics
SN - 2544-7297
TI - The weighted mean curvature derivative of a space-filling diagram
VL - 8
ER -
TY - JOUR
AB - Rhombic dodecahedron is a space filling polyhedron which represents the close packing of spheres in 3D space and the Voronoi structures of the face centered cubic (FCC) lattice. In this paper, we describe a new coordinate system where every 3-integer coordinates grid point corresponds to a rhombic dodecahedron centroid. In order to illustrate the interest of the new coordinate system, we propose the characterization of 3D digital plane with its topological features, such as the interrelation between the thickness of the digital plane and the separability constraint we aim to obtain. We also present the characterization of 3D digital lines and study it as the intersection of multiple digital planes. Characterization of 3D digital sphere with relevant topological features is proposed as well along with the 48-symmetry appearing in the new coordinate system.
AU - Biswas, Ranita
AU - Largeteau-Skapin, Gaëlle
AU - Zrour, Rita
AU - Andres, Eric
ID - 9249
IS - 1
JF - Mathematical Morphology - Theory and Applications
SN - 2353-3390
TI - Digital objects in rhombic dodecahedron grid
VL - 4
ER -
TY - CONF
AB - We call a multigraph non-homotopic if it can be drawn in the plane in such a way that no two edges connecting the same pair of vertices can be continuously transformed into each other without passing through a vertex, and no loop can be shrunk to its end-vertex in the same way. It is easy to see that a non-homotopic multigraph on n>1 vertices can have arbitrarily many edges. We prove that the number of crossings between the edges of a non-homotopic multigraph with n vertices and m>4n edges is larger than cm2n for some constant c>0 , and that this bound is tight up to a polylogarithmic factor. We also show that the lower bound is not asymptotically sharp as n is fixed and m⟶∞ .
AU - Pach, János
AU - Tardos, Gábor
AU - Tóth, Géza
ID - 9299
SN - 0302-9743
T2 - 28th International Symposium on Graph Drawing and Network Visualization
TI - Crossings between non-homotopic edges
VL - 12590
ER -
TY - CONF
AB - We present LiveTraVeL (Live Transit Vehicle Labeling), a real-time system to label a stream of noisy observations of transit vehicle trajectories with the transit routes they are serving (e.g., northbound bus #5). In order to scale efficiently to large transit networks, our system first retrieves a small set of candidate routes from a geometrically indexed data structure, then applies a fine-grained scoring step to choose the best match. Given that real-time data remains unavailable for the majority of the world’s transit agencies, these inferences can help feed a real-time map of a transit system’s trips, infer transit trip delays in real time, or measure and correct noisy transit tracking data. This system can run on vehicle observations from a variety of sources that don’t attach route information to vehicle observations, such as public imagery streams or user-contributed transit vehicle sightings.We abstract away the specifics of the sensing system and demonstrate the effectiveness of our system on a "semisynthetic" dataset of all New York City buses, where we simulate sensed trajectories by starting with fully labeled vehicle trajectories reported via the GTFS-Realtime protocol, removing the transit route IDs, and perturbing locations with synthetic noise. Using just the geometric shapes of the trajectories, we demonstrate that our system converges on the correct route ID within a few minutes, even after a vehicle switches from serving one trip to the next.
AU - Osang, Georg F
AU - Cook, James
AU - Fabrikant, Alex
AU - Gruteser, Marco
ID - 7216
SN - 9781538670248
T2 - 2019 IEEE Intelligent Transportation Systems Conference
TI - LiveTraVeL: Real-time matching of transit vehicle trajectories to transit routes at scale
ER -
TY - GEN
AB - The input to the token swapping problem is a graph with vertices v1, v2, . . . , vn, and n tokens with labels 1,2, . . . , n, one on each vertex. The goal is to get token i to vertex vi for all i= 1, . . . , n using a minimum number of swaps, where a swap exchanges the tokens on the endpoints of an edge.Token swapping on a tree, also known as “sorting with a transposition tree,” is not known to be in P nor NP-complete. We present some partial results:
1. An optimum swap sequence may need to perform a swap on a leaf vertex that has the correct token (a “happy leaf”), disproving a conjecture of Vaughan.
2. Any algorithm that fixes happy leaves—as all known approximation algorithms for the problem do—has approximation factor at least 4/3. Furthermore, the two best-known 2-approximation algorithms have approximation factor exactly 2.
3. A generalized problem—weighted coloured token swapping—is NP-complete on trees, but solvable in polynomial time on paths and stars. In this version, tokens and vertices have colours, and colours have weights. The goal is to get every token to a vertex of the same colour, and the cost of a swap is the sum of the weights of the two tokens involved.
AU - Biniaz, Ahmad
AU - Jain, Kshitij
AU - Lubiw, Anna
AU - Masárová, Zuzana
AU - Miltzow, Tillmann
AU - Mondal, Debajyoti
AU - Naredla, Anurag Murty
AU - Tkadlec, Josef
AU - Turcotte, Alexi
ID - 7950
T2 - arXiv
TI - Token swapping on trees
ER -
TY - JOUR
AB - The order-k Voronoi tessellation of a locally finite set 𝑋⊆ℝ𝑛 decomposes ℝ𝑛 into convex domains whose points have the same k nearest neighbors in X. Assuming X is a stationary Poisson point process, we give explicit formulas for the expected number and total area of faces of a given dimension per unit volume of space. We also develop a relaxed version of discrete Morse theory and generalize by counting only faces, for which the k nearest points in X are within a given distance threshold.
AU - Edelsbrunner, Herbert
AU - Nikitenko, Anton
ID - 5678
IS - 4
JF - Discrete and Computational Geometry
SN - 01795376
TI - Poisson–Delaunay Mosaics of Order k
VL - 62
ER -
TY - JOUR
AB - We answer a question of David Hilbert: given two circles it is not possible in general to construct their centers using only a straightedge. On the other hand, we give infinitely many families of pairs of circles for which such construction is possible.
AU - Akopyan, Arseniy
AU - Fedorov, Roman
ID - 6050
JF - Proceedings of the American Mathematical Society
TI - Two circles and only a straightedge
VL - 147
ER -
TY - JOUR
AB - We give non-degeneracy criteria for Riemannian simplices based on simplices in spaces of constant sectional curvature. It extends previous work on Riemannian simplices, where we developed Riemannian simplices with respect to Euclidean reference simplices. The criteria we give in this article are in terms of quality measures for spaces of constant curvature that we develop here. We see that simplices in spaces that have nearly constant curvature, are already non-degenerate under very weak quality demands. This is of importance because it allows for sampling of Riemannian manifolds based on anisotropy of the manifold and not (absolute) curvature.
AU - Dyer, Ramsay
AU - Vegter, Gert
AU - Wintraecken, Mathijs
ID - 6515
IS - 1
JF - Journal of Computational Geometry
SN - 1920-180X
TI - Simplices modelled on spaces of constant curvature
VL - 10
ER -