arXiv subjects

Carola Wenk

Publications and source records attributed to Carola Wenk.

At least 19 recordsLinked to original sources

arXiv2026

Perfect Rectangular Tilings with Two Colors

We study a finite tiling problem, where tiles are unit squares whose four edges are colored with one of two colors. We ask whether a given rectangle admits a perfect rectangular tiling: every cell of the rectangle is occupied by one tile, neighboring edge colors match, and exactly $n_i$ tiles of type $i$ are used, where rotations of the tiles are allowed. Our problem is related to classical Wang tilings, more general finite tile-placement problems, and edge placement puzzles. But in our problem, the multiplicities of the tile types are part of the input and the tile alphabet is fixed and extremely small; thus the complexity of the problem arises from the interaction between the rectangle dimensions and the prescribed tile multiplicities. We provide a comprehensive study of the perfect rectangular tiling problem. For this we consider all classes of subsets of the six possible tile types for two-colored edges, and we characterize for each class whether multiplicities either always allow a perfect rectangular tiling or whether their existence can be decided efficiently.

cs.CG↗

arXiv2026

HD-GEN: A High-Performance Software System for Human Mobility Data Generation Based on Patterns of Life

Understanding individual-level human mobility is critical for a wide range of applications. Real-world trajectory datasets provide valuable insights into movement behaviors and patterns of life but are often constrained by data sparsity and participation bias. Synthetic data, by contrast, offers scalability and flexibility but frequently lacks realism. % To address this gap, we introduce a comprehensive software pipeline for generating, calibrating, processing, and visualizing large-scale individual-level human mobility datasets that combine the realism of empirical data with the control and extensibility simulations. % Our system consists of four integrated components: (1) a data generation engine that constructs geographically grounded simulations using OpenStreetMap data to produce diverse mobility logs; (2) a genetic algorithm--based calibration module that fine-tunes simulation parameters to align with real-world mobility characteristics; (3) a data processing suite that transforms raw simulation logs into structured formats suitable for downstream applications; and (4) a visualization module that extracts and presents key mobility patterns and insights from the processed datasets for improved interpretability. Evaluation of generated trajectory datasets for the Atlanta, Georgia, USA region show realistic behavior that, despite emerging from a simulation without any reference to real human individuals, exhibits realistic human behavior that closely matches aggregate metrics of real-world datasets. We also provide a sensitivity analysis to study what simulation parameters affect simulation runtime. Code and simulated datasets are shared to provide the broad research community with large-scale dataset that, albeit not real, exhibit realistic human behavior while being orders of magnitudes larger than any open real-world mobility dataset.

cs.SE↗

arXiv2026

SF-LIFE: A Large-Scale Simulated Movement Dataset for the San Francisco Bay Area

We introduce SF-LIFE, a large-scale simulated movement dataset designed to accelerate research in transportation, mobility, and machine learning. The dataset contains 3,024,000,000,000 location records capturing complete, noise-free, multi-modality trajectories of 500,000 simulated agents observed at a 1Hz frequency navigating the San Francisco Bay Area network over a 70-day period. The data captures (1) needs-driven daily agendas of individual agents generated by an agent-based simulation of human patterns of life and (2) detailed kinematic trajectories moving agents across the OpenStreetMap representation of San Francisco using data from 40+ transit agencies across 9 counties. SF-LIFE provides unprecedented scale and detail as trajectories are based on real transit infrastructure using San Francisco General Transit Feed Specification (GTFS) data, having agent movements across multiple modalities, including bus, rail, bike, automobile, and walking. For this high-fidelity simulated representation of San Francisco, we provide (1) the full trajectory data annotated with transportation mode labels, (2) reduced-size versions of the trajectory data with reduced temporal frequency, (3) agent activity information describing the causal activity why an agent visits a place, (4) agent demographic data, and (5) the underlying OSM road network and building data. As the first dataset of its scale and level of detail, SF-LIFE overcomes the privacy, noise, and completeness limitations inherent in real-world tracking data, providing a robust and ethically sourced resource for research in transit optimization, human mobility analysis, and urban computing.

physics.soc-ph↗

arXiv2026

The Kinetic Hourglass Data Structure for Computing the Bottleneck Distance of Dynamic Data

The kinetic data structure (KDS) framework is a powerful tool for maintaining various geometric configurations of continuously moving objects. In this work, we introduce the kinetic hourglass, a novel KDS implementation designed to compute the bottleneck distance for geometric matching problems. We detail the events and updates required for handling general graphs, accompanied by a complexity analysis. Furthermore, we demonstrate the utility of the kinetic hourglass by applying it to compute the bottleneck distance between two persistent homology transforms (PHTs) derived from shapes in $\mathbb{R}^2$, which are topological summaries obtained by computing persistent homology from every direction in $\mathbb{S}^1$.

cs.DS↗

arXiv2024

The Patterns of Life Human Mobility Simulation

We demonstrate the Patterns of Life Simulation to create realistic simulations of human mobility in a city. This simulation has recently been used to generate massive amounts of trajectory and check-in data. Our demonstration focuses on using the simulation twofold: (1) using the graphical user interface (GUI), and (2) running the simulation headless by disabling the GUI for faster data generation. We further demonstrate how the Patterns of Life simulation can be used to simulate any region on Earth by using publicly available data from OpenStreetMap. Finally, we also demonstrate recent improvements to the scalability of the simulation allows simulating up to 100,000 individual agents for years of simulation time. During our demonstration, as well as offline using our guides on GitHub, participants will learn: (1) The theories of human behavior driving the Patters of Life simulation, (2) how to simulate to generate massive amounts of synthetic yet realistic trajectory data, (3) running the simulation for a region of interest chosen by participants using OSM data, (4) learn the scalability of the simulation and understand the properties of generated data, and (5) manage thousands of parallel simulation instances running concurrently.

cs.MA↗

arXiv2024

Rapid and Precise Topological Comparison with Merge Tree Neural Networks

Merge trees are a valuable tool in the scientific visualization of scalar fields; however, current methods for merge tree comparisons are computationally expensive, primarily due to the exhaustive matching between tree nodes. To address this challenge, we introduce the Merge Tree Neural Network (MTNN), a learned neural network model designed for merge tree comparison. The MTNN enables rapid and high-quality similarity computation. We first demonstrate how to train graph neural networks, which emerged as effective encoders for graphs, in order to produce embeddings of merge trees in vector spaces for efficient similarity comparison. Next, we formulate the novel MTNN model that further improves the similarity comparisons by integrating the tree and node embeddings with a new topological attention mechanism. We demonstrate the effectiveness of our model on real-world data in different domains and examine our model's generalizability across various datasets. Our experimental analysis demonstrates our approach's superiority in accuracy and efficiency. In particular, we speed up the prior state-of-the-art by more than $100\times$ on the benchmark datasets while maintaining an error rate below $0.1\%$.

cs.LG↗

arXiv2024

Towards Mobility Data Science (Vision Paper)

Mobility data captures the locations of moving objects such as humans, animals, and cars. With the availability of GPS-equipped mobile devices and other inexpensive location-tracking technologies, mobility data is collected ubiquitously. In recent years, the use of mobility data has demonstrated significant impact in various domains including traffic management, urban planning, and health sciences. In this paper, we present the emerging domain of mobility data science. Towards a unified approach to mobility data science, we envision a pipeline having the following components: mobility data collection, cleaning, analysis, management, and privacy. For each of these components, we explain how mobility data science differs from general data science, we survey the current state of the art and describe open challenges for the research community in the coming years.

cs.OH↗

arXiv2023

Realizability of Free Spaces of Curves

The free space diagram is a popular tool to compute the well-known Fréchet distance. As the Fréchet distance is used in many different fields, many variants have been established to cover the specific needs of these applications. Often, the question arises whether a certain pattern in the free space diagram is "realizable", i.e., whether there exists a pair of polygonal chains whose free space diagram corresponds to it. The answer to this question may help in deciding the computational complexity of these distance measures, as well as allowing to design more efficient algorithms for restricted input classes that avoid certain free space patterns. Therefore, we study the inverse problem: Given a potential free space diagram, do there exist curves that generate this diagram? Our problem of interest is closely tied to the classic Distance Geometry problem. We settle the complexity of Distance Geometry in $\mathbb{R}^{> 2}$, showing $\exists\mathbb{R}$-hardness. We use this to show that for curves in $\mathbb{R}^{\ge 2}$, the realizability problem is $\exists\mathbb{R}$-complete, both for continuous and for discrete Fréchet distance. We prove that the continuous case in $\mathbb{R}^1$ is only weakly NP-hard, and we provide a pseudo-polynomial time algorithm and show that it is fixed-parameter tractable. Interestingly, for the discrete case in $\mathbb{R}^1$, we show that the problem becomes solvable in polynomial time.

cs.CG↗

arXiv2023

Visualizing Topological Importance: A Class-Driven Approach

This paper presents the first approach to visualize the importance of topological features that define classes of data. Topological features, with their ability to abstract the fundamental structure of complex data, are an integral component of visualization and analysis pipelines. Although not all topological features present in data are of equal importance. To date, the default definition of feature importance is often assumed and fixed. This work shows how proven explainable deep learning approaches can be adapted for use in topological classification. In doing so, it provides the first technique that illuminates what topological structures are important in each dataset in regards to their class label. In particular, the approach uses a learned metric classifier with a density estimator of the points of a persistence diagram as input. This metric learns how to reweigh this density such that classification accuracy is high. By extracting this weight, an importance field on persistent point density can be created. This provides an intuitive representation of persistence point importance that can be used to drive new visualizations. This work provides two examples: Visualization on each diagram directly and, in the case of sublevel set filtrations on images, directly on the images themselves. This work highlights real-world examples of this approach visualizing the important topological features in graph, 3D shape, and medical image data.

cs.LG↗

arXiv2023

From Curves to Words and Back Again: Geometric Computation of Minimum-Area Homotopy

Let $γ$ be a generic closed curve in the plane. Samuel Blank, in his 1967 Ph.D. thesis, determined if $γ$ is self-overlapping by geometrically constructing a combinatorial word from $γ$. More recently, Zipei Nie, in an unpublished manuscript, computed the minimum homotopy area of $γ$ by constructing a combinatorial word algebraically. We provide a unified framework for working with both words and determine the settings under which Blank's word and Nie's word are equivalent. Using this equivalence, we give a new geometric proof for the correctness of Nie's algorithm. Unlike previous work, our proof is constructive which allows us to naturally compute the actual homotopy that realizes the minimum area. Furthermore, we contribute to the theory of self-overlapping curves by providing the first polynomial-time algorithm to compute a self-overlapping decomposition of any closed curve $γ$ with minimum area.

cs.CG↗

arXiv2023

Metric and Path-Connectedness Properties of the Frechet Distance for Paths and Graphs

The Frechet distance is often used to measure distances between paths, with applications in areas ranging from map matching to GPS trajectory analysis to handwriting recognition. More recently, the Frechet distance has been generalized to a distance between two copies of the same graph embedded or immersed in a metric space; this more general setting opens up a wide range of more complex applications in graph analysis. In this paper, we initiate a study of some of the fundamental topological properties of spaces of paths and of graphs mapped to R^n under the Frechet distance, in an effort to lay the theoretical groundwork for understanding how these distances can be used in practice. In particular, we prove whether or not these spaces, and the metric balls therein, are path-connected.

cs.CG↗

arXiv2022

Distance Measures for Geometric Graphs

A geometric graph is a combinatorial graph, endowed with a geometry that is inherited from its embedding in a Euclidean space. Formulation of a meaningful measure of (dis-)similarity in both the combinatorial and geometric structures of two such geometric graphs is a challenging problem in pattern recognition. We study two notions of distance measures for geometric graphs, called the geometric edit distance (GED) and geometric graph distance (GGD). While the former is based on the idea of editing one graph to transform it into the other graph, the latter is inspired by inexact matching of the graphs. For decades, both notions have been lending themselves well as measures of similarity between attributed graphs. If used without any modification, however, they fail to provide a meaningful distance measure for geometric graphs -- even cease to be a metric. We have curated their associated cost functions for the context of geometric graphs. Alongside studying the metric properties of GED and GGD, we investigate how the two notions compare. We further our understanding of the computational aspects of GGD by showing that the distance is $\mathcal{NP}$-hard to compute, even if the graphs are planar and arbitrary cost coefficients are allowed.

cs.CG↗

arXiv2022

On the Reconstruction of Geodesic Subspaces of $\mathbb{R}^N$

We consider the topological and geometric reconstruction of a geodesic subspace of $\mathbb{R}^N$ both from the Čech and Vietoris-Rips filtrations on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique leverages the intrinsic length metric induced by the geodesics on the subspace. We consider the distortion and convexity radius as our sampling parameters for a successful reconstruction. For a geodesic subspace with finite distortion and positive convexity radius, we guarantee a correct computation of its homotopy and homology groups from the sample. For geodesic subspaces of $\mathbb{R}^2$, we also devise an algorithm to output a homotopy equivalent geometric complex that has a very small Hausdorff distance to the unknown shape of interest.

math.AT↗

arXiv2022

Approximating Gromov-Hausdorff Distance in Euclidean Space

The Gromov-Hausdorff distance $(d_{GH})$ proves to be a useful distance measure between shapes. In order to approximate $d_{GH}$ for compact subsets $X,Y\subset\mathbb{R}^d$, we look into its relationship with $d_{H,iso}$, the infimum Hausdorff distance under Euclidean isometries. As already known for dimension $d\geq 2$, the $d_{H,iso}$ cannot be bounded above by a constant factor times $d_{GH}$. For $d=1$, however, we prove that $d_{H,iso}\leq\frac{5}{4}d_{GH}$. We also show that the bound is tight. In effect, this gives rise to an $O(n\log{n})$-time algorithm to approximate $d_{GH}$ with an approximation factor of $\left(1+\frac{1}{4}\right)$.

math.MG↗

arXiv2021

Minimum-Complexity Graph Simplification under Fréchet-Like Distances

Simplifying graphs is a very applicable problem in numerous domains, especially in computational geometry. Given a geometric graph and a threshold, the minimum-complexity graph simplification asks for computing an alternative graph of minimum complexity so that the distance between the two graphs remains at most the threshold. In this paper, we propose several NP-hardness and algorithmic results depending on the type of input and simplified graphs, the vertex placement of the simplified graph, and the distance measures between them (graph and traversal distances [1,2]). In general, we show that for arbitrary input and output graphs, the problem is NP-hard under some specific vertex-placement of the simplified graph. When the input and output are trees, and the graph distance is applied from the simplified tree to the input tree, we give an $O(kn^5)$ time algorithm, where $k$ is the number of the leaves of the two trees that are identical and $n$ is the number of vertices of the input.

cs.CG↗

arXiv2021

A Domain-Oblivious Approach for Learning Concise Representations of Filtered Topological Spaces for Clustering

Persistence diagrams have been widely used to quantify the underlying features of filtered topological spaces in data visualization. In many applications, computing distances between diagrams is essential; however, computing these distances has been challenging due to the computational cost. In this paper, we propose a persistence diagram hashing framework that learns a binary code representation of persistence diagrams, which allows for fast computation of distances. This framework is built upon a generative adversarial network (GAN) with a diagram distance loss function to steer the learning process. Instead of using standard representations, we hash diagrams into binary codes, which have natural advantages in large-scale tasks. The training of this model is domain-oblivious in that it can be computed purely from synthetic, randomly created diagrams. As a consequence, our proposed method is directly applicable to various datasets without the need for retraining the model. These binary codes, when compared using fast Hamming distance, better maintain topological similarity properties between datasets than other vectorized representations. To evaluate this method, we apply our framework to the problem of diagram clustering and we compare the quality and performance of our approach to the state-of-the-art. In addition, we show the scalability of our approach on a dataset with 10k persistence diagrams, which is not possible with current techniques. Moreover, our experimental results demonstrate that our method is significantly faster with the potential of less memory usage, while retaining comparable or better quality comparisons.

cs.CG↗

arXiv2020

Combinatorial Properties of Self-Overlapping Curves and Interior Boundaries

We study the interplay between the recently defined concept of minimum homotopy area and the classical topic of self-overlapping curves. The latter are plane curves which are the image of the boundary of an immersed disk. Our first contribution is to prove new sufficient combinatorial conditions for a curve to be self-overlapping. We show that a curve $γ$ with Whitney index 1 and without any self-overlapping subcurves is self-overlapping. As a corollary, we obtain sufficient conditions for self-overlappingness solely in terms of the Whitney index of the curve and its subcurves. These results follow from our second contribution, which shows that any plane curve $γ$, modulo a basepoint condition, is transformed into an interior boundary by wrapping around $γ$ with Jordan curves. Equivalently, the minimum homotopy area of $γ$ is reduced to the minimal possible threshold, namely the winding area, through wrapping. In fact, we show that $n+1$ wraps suffice, where $γ$ has $n$ vertices. Our third contribution is to prove the equivalence of various definitions of self-overlapping curves and interior boundaries, often implicit in the literature. We also introduce and characterize zero-obstinance curves, further generalizations of interior boundaries defined by optimality in minimum homotopy area.

cs.CG↗

arXiv2020

Global Curve Simplification

Due to its many applications, \emph{curve simplification} is a long-studied problem in computational geometry and adjacent disciplines, such as graphics, geographical information science, etc. Given a polygonal curve $P$ with $n$ vertices, the goal is to find another polygonal curve $P'$ with a smaller number of vertices such that $P'$ is sufficiently similar to $P$. Quality guarantees of a simplification are usually given in a \emph{local} sense, bounding the distance between a shortcut and its corresponding section of the curve. In this work, we aim to provide a systematic overview of curve simplification problems under \emph{global} distance measures that bound the distance between $P$ and $P'$. We consider six different curve distance measures: three variants of the \emph{Hausdorff} distance and three variants of the \emph{Fréchet} distance. And we study different restrictions on the choice of vertices for $P'$. We provide polynomial-time algorithms for some variants of the global curve simplification problem and show NP-hardness for other variants. Through this systematic study we observe, for the first time, some surprising patterns, and suggest directions for future research in this important area.

cs.CG↗