Related papers: On Separating Points by Lines

Given a set of N points, we have discovered an algorithm that can separate these points from one another by n-dimensional planes. Each point is chosen at random and put into a set S and planes which separate them are determined and put into…

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…

Let $P$ be a set of $n$ points in the plane that determines at most $n/5$ distinct distances. We show that no line can contain more than $O(n^{43/52}{\rm polylog}(n))$ points of $P$. We also show a similar result for rectangular distances,…

Kelly's theorem states that a set of $n$ points affinely spanning $\mathbb{C}^3$ must determine at least one ordinary complex line (a line passing through exactly two of the points). Our main theorem shows that such sets determine at least…

We consider the problem of identifying n points in the plane using disks, i.e., minimizing the number of disks so that each point is contained in a disk and no two points are in exactly the same set of disks. This problem can be seen as an…

Two elements, $x$ and $y$, are separated by a set $S$ if it contains exactly one of $x$ and $y$. We prove that any set of $n$ points in general position in the plane can be separated by $O(n\log\log n/\log n)$ convex sets, and for some…

We show that for $m$ points and $n$ lines in the real plane, the number of distinct distances between the points and the lines is $\Omega(m^{1/5}n^{3/5})$, as long as $m^{1/2}\le n\le m^2$. We also prove that for any $m$ points in the…

Let $P$ be a set of $n$ points in the real plane contained in an algebraic curve $C$ of degree $d$. We prove that the number of distinct distances determined by $P$ is at least $c_d n^{4/3}$, unless $C$ contains a line or a circle. We also…

Let $P$ be a set of $n$ points in the plane. In this paper we study a new variant of the circular separability problem in which a point set $P$ is preprocessed so that one can quickly answer queries of the following form: Given a geometric…

We seek for lines of minimal distance to finitely many points in the plane. The distance between a line and a set of points is defined by the L^p-norm, 1\leq p\leq \infty, of the vector of vertical or orthogonal distances from the single…

Given two points in the plane, and a set of "obstacles" given as curves through the plane with assigned weights, we consider the point-separation problem, which asks for the minimum-weight subset of the obstacles separating the two points.…

We study the following geometric separation problem: Given a set $R$ of red points and a set $B$ of blue points in the plane, find a minimum-size set of lines that separate $R$ from $B$. We show that, in its full generality, parameterized…

We consider the minimal number of points on a regular grid on the plane that generates $n$ line segments of points of exactly length $k$. We illustrate how this is related to the $n$-queens problem on the toroidal chessboard and show that…

We show that the number of unit-area triangles determined by a set $S$ of $n$ points in the plane is $O(n^{20/9})$, improving the earlier bound $O(n^{9/4})$ of Apfelbaum and Sharir [Discrete Comput. Geom., 2010]. We also consider two…

According to a result of Arkin~\etal~(2016), given $n$ point pairs in the plane, there exists a simple polygonal cycle that separates the two points in each pair to different sides; moreover, a $O(\sqrt{n})$-factor approximation with…

A classic theorem of Euclidean geometry asserts that any noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chv\'atal conjectured that this holds for an arbitrary finite metric space, with a certain…

Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of $n$ points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$ outside of the segment delimited by $x$…

Let $\{p_1, \ldots , p_n \} \subset {\Bbb{R}}^2$ be a separated point set, i.e., any two points have a distance at least $1$. Let $k \ge 1$ be an integer, and $1 \le t_1 < \ldots < t_k$ be real numbers. Let $\delta > 0$. Suppose for all $1…

Two planar sets are circularly separable if there exists a circle enclosing one of the sets and whose open interior disk does not intersect the other set. This paper studies two problems related to circular separability. A linear-time…

We show that for every $n \in \mathbb N$ there is a collection of points $p_1, \ldots, p_n$ and lines $\ell_1, \ldots, \ell_n$ in the unit square such that for any $i$ we have $p_i \in \ell_i$ and the distance from $p_i$ to any other line…