Computational Geometry
We study the Tukey layers and convex layers of a planar point set, which consists of $n$ points independently and uniformly sampled from a convex polygon with $k$ vertices. We show that the expected number of vertices on the first $t$ Tukey…
We study the problem of minimum enclosing rectangle with outliers, which asks to find, for a given set of $n$ planar points, a rectangle with minimum area that encloses at least $(n-t)$ points. The uncovered points are regarded as outliers.…
Assume you have a 2-dimensional pizza with $2n$ ingredients that you want to share with your friend. For this you are allowed to cut the pizza using several straight cuts, and then give every second piece to your friend. You want to do this…
We present simpler algorithms for two closely related morphing problems, both based on the barycentric interpolation paradigm introduced by Floater and Gotsman, which is in turn based on Floater's asymmetric extension of Tutte's classical…
This paper studies the problem of computing quasi-upward planar drawings of bimodal plane digraphs with minimum curve complexity, i.e., drawings such that the maximum number of bends per edge is minimized. We prove that every bimodal plane…
This is the arXiv index for the electronic proceedings of GD 2021, which contains the peer-reviewed and revised accepted papers with an optional appendix. Proceedings (without appendices) are also to be published by Springer in the Lecture…
The mapper construction is a powerful tool from topological data analysis that is designed for the analysis and visualization of multivariate data. In this paper, we investigate a method for stitching a pair of univariate mappers together…
In this paper, we prove that the Max-Morse Matching Problem is approximable, thus resolving an open problem posed by Joswig and Pfetsch. We describe two different approximation algorithms for the Max-Morse Matching Problem. For…
Point feature labeling is a classical problem in cartography and GIS that has been extensively studied for geospatial point data. At the same time, word clouds are a popular visualization tool to show the most important words in text data…
We consider the online search problem in which a server starting at the origin of a $d$-dimensional Euclidean space has to find an arbitrary hyperplane. The best-possible competitive ratio and the length of the shortest curve from which…
We construct a family of 17 disjoint axis-parallel line segments in the plane that do not admit a circumscribing polygon.
Deciding whether a family of disjoint axis-parallel line segments in the plane can be linked into a simple polygon (or a simple polygonal chain) by adding segments between their endpoints is NP-hard.
A graph is 1-planar if it can be drawn in the plane so that each edge is crossed at most once. However, there are 1-planar graphs which do not admit a straight-line 1-planar drawing. We show that every 1-planar graph has a straight-line…
The it Convex Hull Membership(CHM) problem is: Given a point $p$ and a subset $S$ of $n$ points in $\mathbb{R}^m$, is $p \in conv(S)$? CHM is not only a fundamental problem in Linear Programming, Computational Geometry, Machine Learning and…
We study several problems on geometric packing and covering with movement. Given a family $\mathcal{I}$ of $n$ intervals of $\kappa$ distinct lengths, and another interval $B$, can we pack the intervals in $\mathcal{I}$ inside $B$…
We introduce a novel gradient descent algorithm extending the well-known Gradient Sampling methodology to the class of stratifiably smooth objective functions, which are defined as locally Lipschitz functions that are smooth on some regular…
Deciding whether a family of disjoint line segments in the plane can be linked into a simple polygon (or a simple polygonal chain) by adding segments between their endpoints is NP-hard.
We prove that circle graphs (intersection graphs of circle chords) can be embedded as intersection graphs of rays in the plane with polynomial-size bit complexity. We use this embedding to show that the global curve simplification problem…
Let $V_n$ be a set of $n$ points in the plane and let $x \notin V_n$. An $x$-loop is a continuous closed curve not containing any point of $V_n$. We say that two $x$-loops are non-homotopic if they cannot be transformed continuously into…
A drawing of a graph is fan-planar if the edges intersecting a common edge $a$ share a vertex $A$ on the same side of $a$. More precisely, orienting $e$ arbitrarily and the other edges towards $A$ results in a consistent orientation of the…