arXiv subjects

Frank Duque

Publications and source records attributed to Frank Duque.

12 recordsLinked to original sources

arXiv2024

Computing $k$-Crossing Visibility through $k$-levels

Let $\mathcal{A}$ be a set of straight lines in the plane (or planes in $\mathbb{R}^3$). The $k$-crossing visibility of a point $p$ on $\mathcal{A}$ is the set $Q$ of points in the elements of $\mathcal{A}$ such that the segment $pq$, where $q\in Q$, intersects at most $k$ elements of $\mathcal{A}$. In this paper, we present algorithms for computing the $k$-crossing visibility. Specifically, we provide $O(n\log n + kn)$ and $O(n\log n + k^2n)$ time algorithms for sets of $n$ lines in the plane and arrangements of $n$ planes in $\mathbb{R}^3$, which are optimal for $k=Ω(\log n)$ and $k=Ω(\sqrt{\log n})$, respectively. We also introduce an algorithm for computing $k$-crossing visibilities on polygons, which achieves the same asymptotic time complexity as the one presented by Bahoo et al. The techniques proposed in this paper can be easily adapted for computing $k$-crossing visibilities on other instances where the $(\leq k)$-level is known.

cs.CG↗

arXiv2023

RMF accessibility percolation on oriented graphs

Accessibility percolation is a new type of percolation problem inspired by evolutionary biology: a random number, called its fitness, is assigned to each vertex of a graph, then a path in the graph is accessible if fitnesses are strictly increasing through it. In the Rough Mount Fuji (RMF) model the fitness function is defined on the graph as $ω(v)=η(v)+θ\cdot d(v)$, where $θ$ is a positive number called the drift, $d$ is the distance to the source of the graph and $η(v)$ are i.i.d. random variables. In this paper we determine values of $θ$ for having RMF accessibility percolation on the hypercube and the two-dimensional lattices $\mathbb{L}^2$ and $\mathbb{L}^2_{alt}$.

math.PR↗

arXiv2020

Counting the Number of Crossings in Geometric Graphs

A geometric graph is a graph whose vertices are points in general position in the plane and its edges are straight line segments joining these points. In this paper we give an $O(n^2 \log n)$ algorithm to compute the number of pairs of edges that cross in a geometric graph on $n$ points. For layered, and convex geometric graphs the algorithm takes $O(n^2)$ time.

cs.CG↗

arXiv2020

An Ongoing Project to Improve the Rectilinear and the Pseudolinear Crossing Constants

A drawing of a graph in the plane is {\it pseudolinear} if the edges of the drawing can be extended to doubly-infinite curves that form an arrangement of pseudolines, that is, any pair of edges crosses precisely once. A special case are {\it rectilinear} drawings where the edges of the graph are drawn as straight line segments. The rectilinear (pseudolinear) crossing number of a graph is the minimum number of pairs of edges of the graph that cross in any of its rectilinear (pseudolinear) drawings. In this paper we describe an ongoing project to continuously obtain better asymptotic upper bounds on the rectilinear and pseudolinear crossing number of the complete graph $K_n$.

math.CO↗

arXiv2020

Non-crossing Monotone Paths and Binary Trees in Edge-ordered Complete Geometric Graphs

An edge-ordered graph is a graph with a total ordering of its edges. A path $P=v_1v_2\ldots v_k$ in an edge-ordered graph is called increasing if $(v_iv_{i+1}) > (v_{i+1}v_{i+2})$ for all $i = 1,\ldots,k-2$; it is called decreasing if $(v_iv_{i+1}) < (v_{i+1}v_{i+2})$ for all $i = 1,\ldots,k-2$. We say that $P$ is monotone if it is increasing or decreasing. A rooted tree $T$ in an edge-ordered graph is called monotone if either every path from the root of to a leaf is increasing or every path from the root to a leaf is decreasing. Let $G$ be a graph. In a straight-line drawing $D$ of $G$, its vertices are drawn as different points in the plane and its edges are straight line segments. Let $\overlineα(G)$ be the maximum integer such that every edge-ordered straight-line drawing of $G$ %under any edge labeling contains a monotone non-crossing path of length $\overlineα(G)$. Let $\overlineτ(G)$ be the maximum integer such that every edge-ordered straight-line drawing of $G$ %under any edge labeling contains a monotone non-crossing complete binary tree of size $\overlineτ(G)$. In this paper we show that $\overline α(K_n) = Ω(\log\log n)$, $\overline α(K_n) = O(\log n)$, $\overline τ(K_n) = Ω(\log\log \log n)$ and $\overline τ(K_n) = O(\sqrt{n \log n})$.

math.CO↗

arXiv2018

Accessiblility Percolation with Crossing Valleys on $n$-ary Trees

In this paper we study a variation of the accessibility percolation model, this is also motivated by evolutionary biology and evolutionary computation. Consider a tree whose vertices are labeled with random numbers. We study the probability of having a monotone subsequence of a path from the root to a leaf, where any $k$ consecutive vertices in the path contain at least one vertex of the subsequence. An $n$-ary tree, with height $h$, is a tree whose vertices at distance at most $h-1$ to the root have $n$ children. For the case of $n$-ary trees, we prove that, as $h$ tends to infinity the probability of having such subsequence: tends to 1, if $n$ grows significantly faster than $\sqrt[k]{h/(ek)}$; and tends to 0, if $n$ grows significantly slower than $\sqrt[k]{h/(ek)}$.

math.PR↗

arXiv2017

Updating the Number of Crossings in Rectilinear Drawings of the Complete Graph

Let $S$ be a set of $n$ points in general position in the plane. Join every pair of points in $S$ with a straight line segment. Let $\overline{cr}(S)$ be number of pairs of these edges that intersect in their interior. Suppose that this number is known. In this paper we consider the problem of computing $\overline{cr}(S')$, where $S'$ comes from adding, deleting or moving a point from $S$.

cs.CG↗

arXiv2017

The complexity of computing the cylindrical and the $t$-circle crossing number of a graph

A plane drawing of a graph is {\em cylindrical} if there exist two concentric circles that contain all the vertices of the graph, and no edge intersects (other than at its endpoints) any of these circles. The {\em cylindrical crossing number} of a graph $G$ is the minimum number of crossings in a cylindrical drawing of $G$. In his influential survey on the variants of the definition of the crossing number of a graph, Schaefer lists the complexity of computing the cylindrical crossing number of a graph as an open question. In this paper we settle this by showing that this problem is NP-complete. Moreover, we show an analogous result for the natural generalization of the cylindrical crossing number, namely the $t$-{\em circle crossing number}.

math.CO↗

arXiv2017

Point Sets with Small Integer Coordinates and with Small Convex Polygons

In 1935, Erdős and Szekeres proved that every set of $n$ points in general position in the plane contains the vertices of a convex polygon of $\frac{1}{2}\log_2(n)$ vertices. In 1961, they constructed, for every positive integer $t$, a set of $n:=2^{t-2}$ points in general position in the plane, such that every convex polygon with vertices in this set has at most $\log_2(n)+1$ vertices. In this paper we show how to realize their construction in an integer grid of size $O(n^2 \log_2(n)^3)$.

math.CO↗

arXiv2016

Drawing the Almost Convex Set in an Integer Grid of Minimum Size

In 2001, Károlyi, Pach and Tóth introduced a family of point sets to solve an Erdős-Szekeres type problem; which have been used to solve several other Edős-Szekeres type problems. In this paper we refer to these sets as nested almost convex sets. A nested almost convex set $\mathcal{X}$ has the property that the interior of every triangle determined by three points in the same convex layer of $\mathcal{X}$, contains exactly one point of $\mathcal{X}$. In this paper, we introduce a characterization of nested almost convex sets. Our characterization implies that there exists at most one (up to order type) nested almost convex set of $n$ points. We use our characterization to obtain a linear time algorithm to construct nested almost convex sets of $n$ points, with integer coordinates of absolute values at most $O(n^{\log_2 5})$. Finally, we use our characterization to obtain an $O(n\log n)$-time algorithm to determine whether a set of points is a nested almost convex set.

cs.CG↗

arXiv2015

Drawing the Horton Set in an Integer Grid of Minimum Size

In 1978 Erd\H os asked if every sufficiently large set of points in general position in the plane contains the vertices of a convex $k$-gon, with the additional property that no other point of the set lies in its interior. Shortly after, Horton provided a construction---which is now called the Horton set---with no such $7$-gon. In this paper we show that the Horton set of $n$ points can be realized with integer coordinates of absolute value at most $\frac{1}{2} n^{\frac{1}{2} \log (n/2)}$. We also show that any set of points with integer coordinates combinatorially equivalent (with the same order type) to the Horton set, contains a point with a coordinate of absolute value at least $c \cdot n^{\frac{1}{24}\log (n/2)}$, where $c$ is a positive constant.

cs.CG↗

arXiv2014

An upper bound on the k-modem illumination problem

A variation on the classical polygon illumination problem was introduced in [Aichholzer et. al. EuroCG'09]. In this variant light sources are replaced by wireless devices called k-modems, which can penetrate a fixed number k, of "walls". A point in the interior of a polygon is "illuminated" by a k-modem if the line segment joining them intersects at most k edges of the polygon. It is easy to construct polygons of n vertices where the number of k-modems required to illuminate all interior points is Omega(n/k). However, no non-trivial upper bound is known. In this paper we prove that the number of k-modems required to illuminate any polygon of n vertices is at most O(n/k). For the cases of illuminating an orthogonal polygon or a set of disjoint orthogonal segments, we give a tighter bound of 6n/k + 1. Moreover, we present an O(n log n) time algorithm to achieve this bound.

cs.CG↗