Related papers: Computing the Geometric Intersection Number of Cur…
In this article, we provide new structural results and algorithms for the Homotopy Height problem. In broad terms, this problem quantifies how much a curve on a surface needs to be stretched to sweep continuously between two positions. More…
We give a lower bound on the number of non-simple closed curves on a hyperbolic surface, given upper bounds on both length and self-intersection number. In particular, we carefully show how to construct closed geodesics on pairs of pants,…
We consider the problem of computing the topology and describing the geometry of a parametric curve in $\mathbb{R}^n$. We present an algorithm, PTOPO, that constructs an abstract graph that is isotopic to the curve in the embedding space.…
Let $\cC$ be a set of curves in the plane such that no three curves in $\cC$ intersect at a single point and every pair of curves in $\cC$ intersect at exactly one point which is either a crossing or a touching point. J\'anos Pach…
A collection $ \Delta $ of simple closed curves on an orientable surface is an algebraic $ k $-system if the algebraic intersection number $\langle \alpha,\beta \rangle$ is equal to $k $ in absolute value for every $ \alpha , \beta \in…
We consider the problem of answering connectivity queries on a real algebraic curve. The curve is given as the real trace of an algebraic curve, assumed to be in generic position, and being defined by some rational parametrizations. The…
In this work we study quantum algorithms for Hopcroft's problem which is a fundamental problem in computational geometry. Given $n$ points and $n$ lines in the plane, the task is to determine whether there is a point-line incidence. The…
In this paper we present the first deterministic polynomial time algorithm for determining the existence of a Hamiltonian cycle and finding a Hamiltonian cycle in general graphs. Our algorithm can also solve the Hamiltonian path problem in…
Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of non-trivial cycles in G, using different techniques depending on whether or not G is an undirected graph. If G is…
Measuring the similarity of curves is a fundamental problem arising in many application fields. There has been considerable interest in several such measures, both in Euclidean space and in more general setting such as curves on Riemannian…
We present an approach of computing the intersection curve $\mathcal{C}$ of two rational parametric surface $\S_1(u,s)$ and $\S_2(v,t)$, one being projectable and hence can easily be implicitized. Plugging the parametric surface to the…
Given in the plane a set $S$ of $n$ points and a set of disks centered at these points, the disk graph $G(S)$ induced by these disks has vertex set $S$ and an edge between two vertices if their disks intersect. Note that the disks may have…
We describe a $O(\log n )$-approximation algorithm for computing the homotopic \Frechet distance between two polygonal curves that lie on the boundary of a triangulated topological disk. Prior to this work, algorithms were known only for…
The graph homomorphism problem (HOM) asks whether the vertices of a given $n$-vertex graph $G$ can be mapped to the vertices of a given $h$-vertex graph $H$ such that each edge of $G$ is mapped to an edge of $H$. The problem generalizes the…
This paper presents the first purely numerical (i.e., non-algebraic) subdivision algorithm for the isotopic approximation of a simple arrangement of curves. The arrangement is "simple" in the sense that any three curves have no common…
Clustering trajectories is a central challenge when faced with large amounts of movement data such as GPS data. We study a clustering problem that can be stated as a geometric set cover problem: Given a polygonal curve of complexity $n$,…
We design the first subexponential-time (parameterized) algorithms for several cut and cycle-hitting problems on $H$-minor free graphs. In particular, we obtain the following results (where $k$ is the solution-size parameter). 1.…
We describe a polynomial-time algorithm to compute a (tight) geodesic between two curves in the curve graph. As well as enabling us to compute the distance between a pair of curves, this has several applications to mapping classes. For…
Given a graph $G$, an {\em obstacle representation} of $G$ is a set of points in the plane representing the vertices of $G$, together with a set of connected obstacles such that two vertices of $G$ are joined by an edge if and only if the…
We consider the following question: Given $n$ lines and $n$ circles in $\mathbb{R}^3$, what is the maximum number of intersection points lying on at least one line and on at least one circle of these families. We prove that if there are no…