Computational Geometry
Given an $x$-monotone polygonal chain $T$ with $n$ vertices, and an integer $k$, we consider the problem of finding the lowest horizontal line $L$ lying above $T$ with $k$ point guards lying on $L$, so that every point on the chain is…
Nonograms are a popular type of puzzle, where an arrangement of curves in the plane (in the classic version, a rectangular grid) is given together with a series of hints, indicating which cells of the subdivision are to be colored. The…
We introduce Topological Offsets, a novel approach to generate manifold and self-intersection-free offset surfaces that are topologically equivalent to an offset infinitesimally close to the surface. Our approach, by construction, creates a…
Efficient methods to determine the relative position of two conics are of great interest for applications in robotics, computer animation, CAGD, computational physics, and other areas. We present a method to obtain the relative position of…
Volume calculation of configurational spaces acts as a vital part in configurational entropy calculation, which contributes towards calculating free energy landscape for molecular systems. In this article, we present our sampling-based…
A subset $S$ of vertices in a planar graph $G$ is a free set if, for every set $P$ of $|S|$ points in the plane, there exists a straight-line crossing-free drawing of $G$ in which vertices of $S$ are mapped to distinct points in $P$. In…
Given two convex polygons $P$ and $Q$ with $n$ and $m$ edges, the maximum overlap problem is to find a translation of $P$ that maximizes the area of its intersection with $Q$. We give the first randomized algorithm for this problem with…
We study the smallest intersecting and enclosing ball problems in Euclidean spaces for input objects that are compact and convex. They link and unify many problems in computational geometry and machine learning. We show that both problems…
Geometric data sets arising in modern applications are often very large and change dynamically over time. A popular framework for dealing with such data sets is the evolving data framework, where a discrete structure continuously varies…
$\newcommand{\popt}{{\mathcal{p}}} \newcommand{\Re}{\mathbb{R}}\newcommand{\N}{{\mathcal{N}}} \newcommand{\BX}{\mathcal{B}} \newcommand{\bb}{\mathsf{b}} \newcommand{\eps}{\varepsilon} \newcommand{\polylog}{\mathrm{polylog}} $ Let…
Many application areas collect unstructured trajectory data. In subtrajectory clustering, one is interested to find patterns in this data using a hybrid combination of segmentation and clustering. We analyze two variants of this problem…
Tilt models offer intuitive and clean definitions of complex systems in which particles are influenced by global control commands. Despite a wide range of applications, there has been almost no theoretical investigation into the associated…
Geometric hitting set problems, in which we seek a smallest set of points that collectively hit a given set of ranges, are ubiquitous in computational geometry. Most often, the set is discrete and is given explicitly. We propose new…
In this paper, we study estimators for geometric optimization problems in the sublinear geometric model. In this model, we have oracle access to a point set with size $n$ in a discrete space $[\Delta]^d$, where queries can be made to an…
The medial axis of a smoothly embedded surface in $\mathbb{R}^3$ consists of all points for which the Euclidean distance function on the surface has at least two minima. We generalize this notion to the mid-sphere axis, which consists of…
We study three fundamental three-dimensional (3D) geometric packing problems: 3D (Geometric) Bin Packing (3D-BP), 3D Strip Packing (3D-SP), and Minimum Volume Bounding Box (3D-MVBB), where given a set of 3D (rectangular) cuboids, the goal…
Let $P$ be a polygon with $k$ vertices. Let $R$ and $B$ be two simple, interior disjoint curves on the boundary of $P$, with $n$ and $m$ vertices. We show how to compute the Fr\'echet distance between $R$ and $B$ using the geodesic…
We present polynomial-time approximation schemes based on local search} technique for both geometric (discrete) independent set (\mdis) and geometric (discrete) dominating set (\mdds) problems, where the objects are arbitrary radii disks…
The problem of finding a path between two points while avoiding obstacles is critical in robotic path planning. We focus on the feasibility problem: determining whether such a path exists. We model the robot as a query-specific rectangular…
The traveling salesman problem (TSP) famously asks for a shortest tour that a salesperson can take to visit a given set of cities in any order. In this paper, we ask how much faster $k \ge 2$ salespeople can visit the cities if they divide…