Related papers: Minimum-Width Double-Strip and Parallelogram Annul…
Let $P$ be a set of $n$ points in the plane. We show how to find, for a given integer $k>0$, the smallest-area axis-parallel rectangle that covers $k$ points of $P$ in $O(nk^2 \log n+ n\log^2 n)$ time. We also consider the problem of, given…
We present a fully dynamic data structure for the maintenance of lower envelopes of pseudo-lines. The structure has $O(\log^2 n)$ update time and $O(\log n)$ vertical ray shooting query time. To achieve this performance, we devise a new…
We consider the Vertex Cover problem in intersection graphs of axis-parallel rectangles on the plane. We present two algorithms: The first is an EPTAS for non-crossing rectangle families, rectangle families $\calR$ where $R_1 \setminus R_2$…
In this work, we carry out structural and algorithmic studies of a problem of barrier forming: selecting theminimum number of straight line segments (barriers) that separate several sets of mutually disjoint objects in the plane. The…
Consider a M\"obius strip with $n$ chosen points on its edge. A triangulation is a maximal collection of arcs among these points and cuts the strip into triangles. In this paper, we proved the number of all triangulations that one can…
Topological drawings are representations of graphs in the plane, where vertices are represented by points, and edges by simple curves connecting the points. A drawing is simple if two edges intersect at most in a single point, either at a…
We study a fundamental problem in Computational Geometry, the planar two-center problem. In this problem, the input is a set $S$ of $n$ points in the plane and the goal is to find two smallest congruent disks whose union contains all points…
We show how to compute for $n$-vertex planar graphs in $O(n^{11/6}{\rm polylog}(n))$ expected time the diameter and the sum of the pairwise distances. The algorithms work for directed graphs with real weights and no negative cycles. In…
We consider the planar two-center problem for a convex polygon: given a convex polygon in the plane, find two congruent disks of minimum radius whose union contains the polygon. We present an $O(n\log n)$-time algorithm for the two-center…
Let P be a set of n points in R^3. The 2-center problem for P is to find two congruent balls of minimum radius whose union covers P. We present two randomized algorithms for computing a 2-center of P. The first algorithm runs in O(n^3 log^5…
Given two disjoint sets $W_1$ and $W_2$ of points in the plane, the Optimal Discretization problem asks for the minimum size of a family of horizontal and vertical lines that separate $W_1$ from $W_2$, that is, in every region into which…
A set of n segments in the plane may form a Euclidean TSP tour, a tree, or a matching, among others. Optimal TSP tours as well as minimum spanning trees and perfect matchings have no crossing segments, but several heuristics and…
The present paper is devoted to studying of minimal parametric fillings of finite metric spaces (a version of optimal connection problem) by methods of Linear Programming. The estimate on the multiplicity of multi-tours appearing in the…
Venn diagrams are a graphical way to represent a set system. Each of the n sets is represented by a simple closed curve. The n curves subdivide the plane into 2^n open connected regions, each of which represents the intersection of its…
Parallelograms are one of the basic building blocks in two-dimensional tiling. They have important applications in a wide variety of science and engineering fields, such as wireless communication networks, urban transportation, operations…
In this paper, we consider the problems for covering multiple intervals on a line. Given a set $B$ of $m$ line segments (called "barriers") on a horizontal line $L$ and another set $S$ of $n$ horizontal line segments of the same length in…
The problem of parallel thread mapping is studied for the case of discrete orthogonal $m$-simplices. The possibility of a $O(1)$ time recursive block-space map $\lambda: \mathbb{Z}^m \mapsto \mathbb{Z}^m$ is analyzed from the point of view…
In this paper we show a deterministic parallel all-pairs shortest paths algorithm for real-weighted directed graphs. The algorithm has $\tilde{O}(nm+(n/d)^3)$ work and $\tilde{O}(d)$ depth for any depth parameter $d\in [1,n]$. To the best…
We study the problem of computing the diameter and the mean distance of a continuous graph, i.e., a connected graph where all points along the edges, instead of only the vertices, must be taken into account. It is known that for continuous…
Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets $\mathcal{P}$, which are sets of $n$ points in the plane with pairwise integral distances where not all the…