arXiv subjects

Jorge Urrutia

Publications and source records attributed to Jorge Urrutia.

At least 19 recordsLinked to original sources

arXiv2025

Efficient computation of minimum-area rectilinear convex hull under rotation and generalizations

Let $P$ be a set of $n$ points in the plane. We compute the value of $θ\in [0,2π)$ for which the rectilinear convex hull of $P$, denoted by $\mathcal{RH}_θ(P)$, has minimum (or maximum) area in optimal $O(n\log n)$ time and $O(n)$ space, improving the previous $O(n^2)$ bound. Let $\mathcal{O}$ be a set of $k$ lines through the origin sorted by slope and let $α_i$ be the sizes of the $2k$ angles defined by pairs of two consecutive lines, $i=1, \ldots , 2k$. Let $Θ_{i}=π-α_i$ and $Θ=\min\{Θ_i \colon i=1,\ldots,2k\}$. We obtain: (1) Given a set $\mathcal{O}$ such that $Θ\ge\fracπ{2}$, we provide an algorithm to compute the $\mathcal{O}$-convex hull of $P$ in optimal $O(n\log n)$ time and $O(n)$ space; If $Θ< \fracπ{2}$, the time and space complexities are $O(\frac{n}Θ\log n)$ and $O(\frac{n}Θ)$ respectively. (2) Given a set $\mathcal{O}$ such that $Θ\ge\fracπ{2}$, we compute and maintain the boundary of the ${\mathcal{O}}_θ$-convex hull of $P$ for $θ\in [0,2π)$ in $O(kn\log n)$ time and $O(kn)$ space, or if $Θ< \fracπ{2}$, in $O(k\frac{n}Θ\log n)$ time and $O(k\frac{n}Θ)$ space. (3) Finally, given a set $\mathcal{O}$ such that $Θ\ge\fracπ{2}$, we compute, in $O(kn\log n)$ time and $O(kn)$ space, the angle $θ\in [0,2π)$ such that the $\mathcal{O}_θ$-convex hull of $P$ has minimum (or maximum) area over all $θ\in [0,2π)$.

cs.CG↗

arXiv2024

Maximum rectilinear convex subsets

Let $P$ be a set of $n$ points in the plane. We consider a variation of the classical Erdős-Szekeres problem, presenting efficient algorithms with $O(n^3)$ running time and $O(n^2)$ space complexity that compute: (1) A subset $S$ of $P$ such that the boundary of the rectilinear convex hull of $S$ has the maximum number of points from $P$, (2) a subset $S$ of $P$ such that the boundary of the rectilinear convex hull of $S$ has the maximum number of points from $P$ and its interior contains no element of $P$, (3) a subset $S$ of $P$ such that the rectilinear convex hull of $S$ has maximum area and its interior contains no element of $P$, and (4) when each point of $P$ is assigned a weight, positive or negative, a subset $S$ of $P$ that maximizes the total weight of the points in the rectilinear convex hull of $S$. We also revisit the problems of computing a maximum-area orthoconvex polygon and computing a maximum-area staircase polygon, amidst a point set in a rectangular domain. We obtain new and simpler algorithms to solve both problems with the same complexity as in the state of the art.

cs.CG↗

arXiv2023

Minimizing Visible Edges in Polyhedra

We prove that, given a polyhedron $\mathcal P$ in $\mathbb{R}^3$, every point in $\mathbb R^3$ that does not see any vertex of $\mathcal P$ must see eight or more edges of $\mathcal P$, and this bound is tight. More generally, this remains true if $\mathcal P$ is any finite arrangement of internally disjoint polygons in $\mathbb{R}^3$. We also prove that every point in $\mathbb{R}^3$ can see six or more edges of $\mathcal{P}$ (possibly only the endpoints of some these edges) and every point in the interior of $\mathcal{P}$ can see a positive portion of at least six edges of $\mathcal{P}$. These bounds are also tight.

cs.CG↗

arXiv2022

Crossing and intersecting families of geometric graphs on point sets

Let $S$ be a set of $n$ points in the plane in general position. Two line segments connecting pairs of points of $S$ cross if they have an interior point in common. Two vertex disjoint geometric graphs with vertices in $S$ cross if there are two edges, one from each graph, which cross. A set of vertex disjoint geometric graphs with vertices in $S$ is called mutually crossing if any two of them cross. We show that there exists a constant $c$ such that from any family of $n$ mutually crossing triangles, one can always obtain a family of at least $n^c$ mutually crossing $2$-paths (each of which is the result of deleting an edge from one of the triangles) and then provide an example that implies that $c$ cannot be taken to be larger than $2/3$. For every $n$ we determine the maximum number of crossings that a Hamiltonian cycle on a set of $n$ points might have. Next, we construct a point set whose longest perfect matching contains no crossings. We also consider edges consisting of a horizontal and a vertical line segment joining pairs of points of $S$, which we call elbows, and prove that in any point set $S$ there exists a family of $\lfloor n/4 \rfloor$ vertex disjoint mutually crossing elbows. Additionally, we show a point set that admits no more than $n/3$ mutually crossing elbows. Finally we study intersecting families of graphs, which are not necessarily vertex disjoint. A set of edge disjoint graphs with vertices in $S$ is called an intersecting family if for any two graphs in the set we can choose an edge in each of them such that they cross. We prove a conjecture by Lara and Rubio-Montiel, namely, that any set $S$ of $n$ points in general position admits a family of intersecting triangles with a quadratic number of elements. Some other results are obtained throughout this work.

math.CO↗

arXiv2022

Convex polygons and separation of convex sets

We prove that for any collection F of $n \ge 2$ pairwise disjoint compact convex sets in the plane there is a pair of sets A and B in F such that any line that separates A from B separates either A or B from a subcollection of F with at least n/18 sets.

math.CO↗

arXiv2022

Rectilinear Convex Hull of Points in 3D

Let $P$ be a set of $n$ points in $\mathbb{R}^3$ in general position, and let $RCH(P)$ be the rectilinear convex hull of $P$. In this paper we obtain an optimal $O(n\log n)$-time and $O(n)$-space algorithm to compute $RCH(P)$. We also obtain an efficient $O(n\log^2 n)$-time and $O(n\log n)$-space algorithm to compute and maintain the set of vertices of the rectilinear convex hull of $P$ as we rotate $\mathbb R^3$ around the $z$-axis. Finally we study some properties of the rectilinear convex hulls of point sets in $\mathbb{R}^3$.

cs.CG↗

arXiv2022

Separating bichromatic point sets in the plane by restricted orientation convex hulls

We explore the separability of point sets in the plane by a restricted-orientation convex hull, which is an orientation-dependent, possibly disconnected, and non-convex enclosing shape that generalizes the convex hull. Let $R$ and $B$ be two disjoint sets of red and blue points in the plane, and $\mathcal{O}$ be a set of $k \geq 2$ lines passing through the origin. We study the problem of computing the set of orientations of the lines of $\mathcal{O}$ for which the $\mathcal{O}$-convex hull of $R$ contains no points of $B$. For $k=2$ orthogonal lines we have the rectilinear convex hull. In optimal $O(n \log n)$ time and $O(n)$ space, $n = \vert R \vert + \vert B \vert$, we compute the set of rotation angles such that, after simultaneously rotating the lines of $\mathcal{O}$ around the origin in the same direction, the rectilinear convex hull of $R$ contains no points of $B$. We generalize this result to the case where $\mathcal{O}$ is formed by $k \geq 2$ lines with arbitrary orientations. In the counter-clockwise circular order of the lines of $\mathcal{O}$, let $α_i$ be the angle required to clockwise rotate the $i$th line so it coincides with its successor. We solve the problem in this case in $O(1/Θ\cdot N \log N)$ time and $O(1/Θ\cdot N)$ space, where $Θ= \min \{ α_1,\ldots,α_k \}$ and $N=\max\{k,\vert R \vert + \vert B \vert \}$. We finally consider the case in which $\mathcal{O}$ is formed by $k=2$ lines, one of the lines is fixed, and the second line rotates by an angle that goes from $0$ to $π$. We show that this last case can also be solved in optimal $O(n\log n)$ time and $O(n)$ space, where $n = \vert R \vert + \vert B \vert$.

cs.CG↗

arXiv2022

Algorithms for the Euclidean Bipartite Edge Cover Problem

Given a graph $G=(V,E)$ with costs on its edges, the minimum-cost edge cover problem consists of finding a subset of $E$ covering all vertices in $V$ at minimum cost. If $G$ is bipartite, this problem can be solved in time $O(|V|^3)$ via a well-known reduction to a maximum-cost matching problem on $G$. If in addition $V$ is a set of points on the Euclidean line, Collanino et al. showed that the problem can be solved in time $O(|V| \log |V|)$ and asked whether it can be solved in time $o(|V|^3)$ if $V$ is a set of points on the Euclidean plane. We answer this in the affirmative, giving an $O(|V|^{2.5} \log |V|)$ algorithm based on the Hungarian method using weighted Voronoi diagrams. We also propose some 2-approximation algorithms and give experimental results of our implementations.

cs.DM↗

arXiv2021

Rainbow polygons for colored point sets in the plane

Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let $\operatorname{rb-index}(S)$ denote the smallest size of a perfect rainbow polygon for a colored point set $S$, and let $\operatorname{rb-index}(k)$ be the maximum of $\operatorname{rb-index}(S)$ over all $k$-colored point sets in general position; that is, every $k$-colored point set $S$ has a perfect rainbow polygon with at most $\operatorname{rb-index}(k)$ vertices. In this paper, we determine the values of $\operatorname{rb-index}(k)$ up to $k=7$, which is the first case where $\operatorname{rb-index}(k)\neq k$, and we prove that for $k\ge 5$, \[ \frac{40\lfloor (k-1)/2 \rfloor -8}{19} %Birgit: \leq\operatorname{rb-index}(k)\leq 10 \bigg\lfloor\frac{k}{7}\bigg\rfloor + 11. \] Furthermore, for a $k$-colored set of $n$ points in the plane in general position, a perfect rainbow polygon with at most $10 \lfloor\frac{k}{7}\rfloor + 11$ vertices can be computed in $O(n\log n)$ time.

cs.CG↗

arXiv2020

Minimizing The Maximum Distance Traveled To Form Patterns With Systems of Mobile Robots

In the pattern formation problem, robots in a system must self-coordinate to form a given pattern, regardless of translation, rotation, uniform-scaling, and/or reflection. In other words, a valid final configuration of the system is a formation that is \textit{similar} to the desired pattern. While there has been no shortage of research in the pattern formation problem under a variety of assumptions, models, and contexts, we consider the additional constraint that the maximum distance traveled among all robots in the system is minimum. Existing work in pattern formation and closely related problems are typically application-specific or not concerned with optimality (but rather feasibility). We show the necessary conditions any optimal solution must satisfy and present a solution for systems of three robots. Our work also led to an interesting result that has applications beyond pattern formation. Namely, a metric for comparing two triangles where a distance of $0$ indicates the triangles are similar, and $1$ indicates they are \emph{fully dissimilar}.

cs.CG↗

arXiv2020

A Note on Empty Balanced Tetrahedra in Two colored Point sets in $\mathbb{R}^3$

Let $S$ be a set of $n$ red and $n$ blue points in general position in $\mathbb{R}^3$. Let $τ$ be a tetrahedra with vertices on $S$. We say that $τ$ is \emph{empty} if it does not contain any point of $S$ in its interior. We say that $τ$ is \emph{balanced} if it contains two blue vertices and two red vertices. In this paper we show that $S$ spans $Ω(n^{5/2})$ empty balanced tetrahedra.

cs.CG↗

arXiv2019

Convex decompositions of point sets in the plane

Let $P$ be a set of $n$ points in general position on the plane. A set of closed convex polygons with vertices in $P$, and with pairwise disjoint interiors is called a convex decomposition of $P$ if their union is the convex hull of $P$, and no point of $P$ lies in the interior of the polygons. We show that there is a convex decomposition of $P$ with at most $\frac{4}{3}|I(P)|+\frac{1}{3}|B(P)|+1\le \frac{4}{3}|P|-2$ elements, where $B(P)\subseteq P$ is the set of points at the vertices of the convex hull of $P$, and $I(P)=P-B(P)$.

math.CO↗

arXiv2018

Balanced Islands in Two Colored Point Sets in the Plane

Let $S$ be a set of $n$ points in general position in the plane, $r$ of which are red and $b$ of which are blue. In this paper we prove that there exist: for every $α\in \left [ 0,\frac{1}{2} \right ]$, a convex set containing exactly $\lceil αr\rceil$ red points and exactly $\lceil αb \rceil$ blue points of $S$; a convex set containing exactly $\left \lceil \frac{r+1}{2}\right \rceil$ red points and exactly $\left \lceil \frac{b+1}{2}\right \rceil$ blue points of $S$. Furthermore, we present polynomial time algorithms to find these convex sets. In the first case we provide an $O(n^4)$ time algorithm and an $O(n^2\log n)$ time algorithm in the second case. Finally, if $\lceil αr\rceil+\lceil αb\rceil$ is small, that is, not much larger than $\frac{1}{3}n$, we improve the running time to $O(n \log n)$.

cs.CG↗

arXiv2018

Capturing points with a rotating polygon (and a 3D extension)

We study the problem of rotating a simple polygon to contain the maximum number of elements from a given point set in the plane. We consider variations of this problem where the rotation center is a given point or lies on a line segment, a line, or a polygonal chain. We also solve an extension to 3D where we rotate a polyhedron around a given point to contain the maximum number of elements from a set of points in the space.

cs.CG↗

arXiv2018

Colored ray configurations

We study the cyclic color sequences induced at infinity by colored rays with apices being a given balanced finite bichromatic point set. We first study the case in which the rays are required to be pairwise disjoint. We derive a lower bound on the number of color sequences that can be realized from any such fixed point set and examine color sequences that can be realized regardless of the point set, exhibiting negative examples as well. We also provide a tight upper bound on the number of configurations that can be realized from a point set, and point sets for which there are asymptotically less configurations than that number. In addition, we provide algorithms to decide whether a color sequence is realizable from a given point set in a line or in general position. We address afterwards the variant of the problem where the rays are allowed to intersect. We prove that for some configurations and point sets, the number of ray crossings must be $Θ(n^2)$ and study then configurations that can be realized by rays that pairwise cross. We show that there are point sets for which the number of configurations that can be realized by pairwise-crossing rays is asymptotically smaller than the number of configurations realizable by pairwise-disjoint rays. We provide also point sets from which any configuration can be realized by pairwise-crossing rays and show that there is no configuration that can be realized by pairwise-crossing rays from every point set.

cs.CG↗

arXiv2017

Rectilinear Convex Hull with minimum area

Let $P$ be a planar set of $n$ points in general position. We consider the problem of computing an orientation of the plane for which the Rectilinear Convex Hull of $P$ has minimum area. Bae et al. (Computational Geometry: Theory and Applications, Vol. 42, 2009) solved the problem in quadratic time and linear space. We describe an algorithm that reduces this time complexity to $Θ(n \log n)$.

cs.CG↗

arXiv2017

On the $O_β$-hull of a planar point set

We study the $O_β$-hull of a planar point set, a generalization of the Orthogonal Convex Hull where the coordinate axes form an angle $β$. Given a set $P$ of $n$ points in the plane, we show how to maintain the $O_β$-hull of $P$ while $β$ runs from $0$ to $π$ in $O(n \log n)$ time and $O(n)$ space. With the same complexity, we also find the values of $β$ that maximize the area and the perimeter of the $O_β$-hull and, furthermore, we find the value of $β$ achieving the best fitting of the point set $P$ with a two-joint chain of alternate interior angle $β$.

cs.CG↗

arXiv2017

Balanced partitions of 3-colored geometric sets in the plane

Let $S$ be a finite set of geometric objects partitioned into classes or \emph{colors}. A subset $S'\subseteq S$ is said to be \emph{balanced} if $S'$ contains the same amount of elements of $S$ from each of the colors. We study several problems on partitioning $3$-colored sets of points and lines in the plane into two balanced subsets: (a) We prove that for every 3-colored arrangement of lines there exists a segment that intersects exactly one line of each color, and that when there are $2m$ lines of each color, there is a segment intercepting $m$ lines of each color. (b) Given $n$ red points, $n$ blue points and $n$ green points on any closed Jordan curve $γ$, we show that for every integer $k$ with $0 \leq k \leq n$ there is a pair of disjoint intervals on $γ$ whose union contains exactly $k$ points of each color. (c) Given a set $S$ of $n$ red points, $n$ blue points and $n$ green points in the integer lattice satisfying certain constraints, there exist two rays with common apex, one vertical and one horizontal, whose union splits the plane into two regions, each one containing a balanced subset of $S$.

cs.CG↗