Chasing Puppies: Mobile Beacon Routing on Closed Curves

We solve an open problem posed by Michael Biro at CCCG 2013 that was inspired by his and others' work on beacon-based routing. Consider a human and a puppy on a simple closed curve in the plane. The human can walk along the curve at bounded speed and change direction as desired. The puppy runs with unbounded speed along the curve as long as the Euclidean straight-line distance to the human is decreasing, so that it is always at a point on the curve where the distance is locally minimal. Assuming that the curve is smooth (with some mild genericity constraints) or a simple polygon, we prove that the human can always catch the puppy in finite time.Comment: Full version of a SOCG 2021 paper, 28 pages, 27 figure

Computing the Fréchet distance between uncertain curves in one dimension.

We consider the problem of computing the Fréchet distance between two curves for which the exact locations of the vertices are unknown. Each vertex may be placed in a given uncertainty region for that vertex, and the objective is to place vertices so as to minimise the Fréchet distance. This problem was recently shown to be NP-hard in 2D, and it is unclear how to compute an optimal vertex placement at all. We present the first general algorithmic framework for this problem. We prove that it results in a polynomial-time algorithm for curves in 1D with intervals as uncertainty regions. In contrast, we show that the problem is NP-hard in 1D in the case that vertices are placed to maximise the Fréchet distance. We also study the weak Fréchet distance between uncertain curves. While finding the optimal placement of vertices seems more difficult than the regular Fréchet distance—and indeed we can easily prove that the problem is NP-hard in 2D—the optimal placement of vertices in 1D can be computed in polynomial time. Finally, we investigate the discrete weak Fréchet distance, for which, somewhat surprisingly, the problem is NP-hard already in 1D

Mapping Multiple Regions to the Grid with Bounded Hausdorff Distance

We study a problem motivated by digital geometry: given a set of disjoint geometric regions, assign each region Ri a set of grid cells Pi, so that Pi is connected, similar to Ri, and does not touch any grid cell assigned to another region. Similarity is measured using the Hausdorff distance. We analyze the achievable Hausdorff distance in terms of the number of input regions, and prove asymptotically tight bounds for several classes of input regions

On Geometric Measures and Their Computation

In this thesis, we explore geometric measures and how to compute them. First, we use similarity measures to determine the quality of the output of a transformation. Second, we explore new algorithms and data structures that allow us to calculate these distance measures. Third, we present a new distance measure and investigate its computation. Last, we deal with diversity measures by researching how to subdivide a set into diverse subsets. To be more precise, in Chapters 2 and 3, we transform a vector graphic into a raster image and measure the quality of the output using the Hausdorff distance. In Chapter 2, the vector graphic we transform is a set of regions. We investigate restrictions on the regions, like convexity or fatness, and prove worst case lower bounds of the Hausdorff distance between the regions and their corresponding grid polygons under these conditions. We finish by describing algorithms that match these bounds. In Chapter 3, the regions we transform into a raster image are points. Even for this simple shape we were able to prove that determining pixels with the smallest possible Hausdorff distance to the input points is NP-hard. Nonetheless, we are able to find two polynomial-time algorithms that determine a set of pixels corresponding to the input points: first, we show an algorithm where the Hausdorff distance between the points and the pixels is at most a small constant factor larger than the optimal solution. Second, we show an algorithm where the Hausdorff distance between the points and the pixels is at most a small additive constant larger than the optimal solution. In Chapter 4, we consider the computation of the Hausdorff distance. We present a data structure on a set of segments that allows queries of the following type: for a given segment, we can quickly determine the Hausdorff distance between the query segment and the set of segments the data structure is built on. We provided a mechanism to balance between space usage and preprocessing time for that data structure. In Chapter 5, we explore a new similarity measure, the k-Fréchet distance. We define the two variants: the cover distance and the cut distance. Both bridge between the Hausdorff and the Fréchet distance and measure the similarity between two curves. Even though both the Fréchet distance and the Hausdorff distance are easy to compute, the k-Fréchet distances are NP-hard to compute. Still, we present efficient algorithms that compute the cover or cut distance with specific restrictions. In Chapter 6, we focus on diversity measures instead of similarity measures like in the previous chapters. For sets of colored points we use the species richness measure or the Shannon index to determine its diversity. We investigate the problem of how to subdivide a set of colored points into subsets such that the subsets are diverse. We present results for multiple variants: we subdivide either into convex subsets or using a Voronoi diagram, and investigate the problem both in 1D and in 2D

β-Stars or On Extending a Drawing of a Connected Subgraph

We consider the problem of extending the drawing of a subgraph of a given plane graph to a drawing of the entire graph using straight-line and polyline edges. We define the notion of star complexity of a polygon and show that a drawing Γ_H of an induced connected subgraph H can be extended with at most min{h/2, β+log_2(h)+1} bends per edge, where β is the largest star complexity of a face of Γ_H and h is the size of the largest face of H. This result significantly improves the previously known upper bound of 72|V(H)| [5] for the case where H is connected. We also show that our bound is worst case optimal up to a small additive constant. Additionally, we provide an indication of complexity of the problem of testing whether a star-shaped inner face can be extended to a straight-line drawing of the graph; this is in contrast to the fact that the same problem is solvable in linear time for the case of star-shaped outer face [9] and convex inner face [12]

Spiroplots: a New Discrete-time Dynamical System to Generate Curve Patterns

We introduce a new procedural dynamic system that can generate a variety of shapes that often appear as curves, but technically, the figures are plots of many points. We name them spiroplots and show how this new system relates to other procedures or processes that generate figures. Spiroplots are an extremely simple process but with a surprising visual variety. We prove some fundamental properties and analyze some instances to see how the geometry or topology of the input determines the generated figures. We show that some spiroplots have a finite cycle and return to the initial situation, whereas others will produce new points infinitely often. This paper is accompanied by a JavaScript app that allows anyone to generate spiroplots

Obstructing Classification via Projection

Machine learning and data mining techniques are effective tools to classify large amounts of data. But they tend to preserve any inherent bias in the data, for example, with regards to gender or race. Removing such bias from data or the learned representations is quite challenging. In this paper we study a geometric problem which models a possible approach for bias removal. Our input is a set of points P in Euclidean space R^d and each point is labeled with k binary-valued properties. A priori we assume that it is `easy to classify the data according to each property. Our goal is to obstruct the classification according to one property by a suitable projection to a lower-dimensional Euclidean space R^m (m < d), while classification according to all other properties remains easy. What it means for classification to be easy depends on the classification model used. We first consider classification by linear separability as employed by support vector machines. We use Kirchberger's Theorem to show that, under certain conditions, a simple projection to R^(d-1) suffices to eliminate the linear separability of one of the properties whilst maintaining the linear separability of the other properties. We also study the problem of maximizing the linear ``inseparability of the chosen property. Second, we consider more complex forms of separability and prove a connection between the number of projections required to obstruct classification and the Helly-type properties of such separabilities