计算几何
Given a positive integer $k$, we study the problem of finding a convex polygon of minimum perimeter that encloses exactly $k$ points of $\mathbf{Z}^2$. We show that an optimal polygon is contained in a circular annulus of width…
Rectilinear matching to the integer grid asks to assign each of $n$ points in $\mathbb R^2$ to a distinct point of $\mathbb Z^2$, minimizing total $\ell_1$ movement. The main difficulty is that the target set is infinite: one must first…
For an edge-weighted graph $G=(V,E)$ and a stretch parameter $t\geq 1$, a $t$-spanner is a subgraph $H\subseteq G$ such that the shortest path distances in $G$ and $H$ satisfy $\delta_H(u,v)\leq t\, \delta_G(u,v)$ for all $u,v\in V$. In…
We consider covering and partitioning a simple polygon into pieces which either have unit geodesic radius or unit geodesic diameter, using the $\ell_2$-metric for distances. There is no known method for finding an exact solution to these…
We give linear-time, and thus optimal, $(1+\varepsilon)$-approximation algorithms for numerous variants of the Frechet distance between $c$-packed curves (where $c \in O(1)$), removing an additional log factor that was present in previous…
Research on D\"urer's problem focuses on edge unfoldings of convex polyhedra that avoid overlap. We invert the goal and find unfoldings that overlap at some point to any given thickness t. We have two main results. The first is that, if we…
The Fr\'echet distance is a well-studied distance measure for paths in a metric space. It is mostly studied for paths in $d$-dimensional Euclidean space. Here, computing the Fr\'echet distance between two polylines takes time roughly…
Geometric covering problems ask for a small family of geometric objects whose union covers a given point set. We study the more restrictive \emph{boundary covering} variant, where every point must lie on the boundary of a chosen object.…
Spherical range queries are a fundamental primitive for working with spatial data. Many spatial data structures have been developed to answer these queries, but choosing the optimal one for a specific application is a difficult task. This…
We give randomized $(5+\epsilon)$-approximation algorithms for both the continuous and discrete Fr\'echet distances on arbitrary two polygonal curves $\tau$ and $\sigma$ in $\mathbb R^d$ for fixed $d$, with $n$ and $m\le n$ vertices…
In 1942, Freudenthal showed that a simplex in Euclidean space can be subdivided such that the quality (well-shapedness of the simplex, quantified in terms of e.g. fatness) of the simplices in the subdivision is lower bounded. This answered…
A circle graph is the intersection graph of a set of chords in a circle. A dominating set of a graph $G=(V,E)$ is a subset $D\subseteq V$ such that every vertex in $V\setminus D$ is adjacent to at least one vertex of $D$. Computing a…
A drawing of a graph is {\em $x$-monotone} if every vertical line intersects each edge of the graph at most once. We present an $O(n^5)$ time algorithm for deciding whether a simple drawing of the complete graph $K_n$ is weakly isomorphic…
We introduce generalized altitudes of a simplex, extending the usual vertex-to-opposite-face altitude to arbitrary pairs of opposite faces. These quantities encode the relative position of the affine spans of such faces and yield a uniform…
The shifting technique of Hochbaum and Maass [J.ACM'85] produces PTASes with the fastest known running times $n^{O(1/\varepsilon^{d-1})}$ for several $d$-dimensional geometric problems. However, it is only known, due to Marx [FOCS'07], that…
Abstract Voronoi diagrams are defined in terms of a given system of planar bisecting curves satisfying some simple combinatorial properties. They offer a unifying framework for a wide range of concrete Voronoi instances on generalized sites…
We study the problem of reconfiguring a crossing-free embedding of a graph on a surface, with edges represented as curves, into another crossing-free embedding of the same graph on the same surface with the same fixed vertex positions. In…
Oriented interval graphs, a recent generalization of interval graphs introduced by Gutowski et al. [GD 2022], are intersection graphs of intervals, each of which is oriented either left or right. Such a representation defines a mixed…
We present a method for constructing intrinsic triangulations of closed discrete surfaces, in which edges correspond to shortest geodesic paths and faces decompose into geometric primitives inherited from the underlying mesh. Starting from…
This paper introduces an exact $k$-cell decomposition for visibility planning in polygonal environments for agents equipped with $k$-modems, devices that can see through up to $k$ walls. Unlike prior decompositions that may include…