arXiv subjects

Sergio Cabello

Publications and source records attributed to Sergio Cabello.

At least 37 records · Page 2Linked to original sources

arXiv2023

Minimum Cuts in Geometric Intersection Graphs

Let $\mathcal{D}$ be a set of $n$ disks in the plane. The disk graph $G_\mathcal{D}$ for $\mathcal{D}$ is the undirected graph with vertex set $\mathcal{D}$ in which two disks are joined by an edge if and only if they intersect. The directed transmission graph $G^{\rightarrow}_\mathcal{D}$ for $\mathcal{D}$ is the directed graph with vertex set $\mathcal{D}$ in which there is an edge from a disk $D_1 \in \mathcal{D}$ to a disk $D_2 \in \mathcal{D}$ if and only if $D_1$ contains the center of $D_2$. Given $\mathcal{D}$ and two non-intersecting disks $s, t \in \mathcal{D}$, we show that a minimum $s$-$t$ vertex cut in $G_\mathcal{D}$ or in $G^{\rightarrow}_\mathcal{D}$ can be found in $O(n^{3/2}\text{polylog} n)$ expected time. To obtain our result, we combine an algorithm for the maximum flow problem in general graphs with dynamic geometric data structures to manipulate the disks. As an application, we consider the barrier resilience problem in a rectangular domain. In this problem, we have a vertical strip $S$ bounded by two vertical lines, $L_\ell$ and $L_r$, and a collection $\mathcal{D}$ of disks. Let $a$ be a point in $S$ above all disks of $\mathcal{D}$, and let $b$ a point in $S$ below all disks of $\mathcal{D}$. The task is to find a curve from $a$ to $b$ that lies in $S$ and that intersects as few disks of $\mathcal{D}$ as possible. Using our improved algorithm for minimum cuts in disk graphs, we can solve the barrier resilience problem in $O(n^{3/2}\text{polylog} n)$ expected time.

cs.CG↗

arXiv2023

On $k$-means for segments and polylines

We study the problem of $k$-means clustering in the space of straight-line segments in $\mathbb{R}^{2}$ under the Hausdorff distance. For this problem, we give a $(1+ε)$-approximation algorithm that, for an input of $n$ segments, for any fixed $k$, and with constant success probability, runs in time $O(n+ ε^{-O(k)} + ε^{-O(k)}\cdot \log^{O(k)} (ε^{-1}))$. The algorithm has two main ingredients. Firstly, we express the $k$-means objective in our metric space as a sum of algebraic functions and use the optimization technique of Vigneron~\cite{Vigneron14} to approximate its minimum. Secondly, we reduce the input size by computing a small size coreset using the sensitivity-based sampling framework by Feldman and Langberg~\cite{Feldman11, Feldman2020}. Our results can be extended to polylines of constant complexity with a running time of $O(n+ ε^{-O(k)})$.

cs.CG↗

arXiv2021

Computing the inverse geodesic length in planar graphs and graphs of bounded treewidth

The inverse geodesic length of a graph $G$ is the sum of the inverse of the distances between all pairs of distinct vertices of $G$. In some domains it is known as the Harary index or the global efficiency of the graph. We show that, if $G$ is planar and has $n$ vertices, then the inverse geodesic length of $G$ can be computed in roughly $O(n^{9/5})$ time. We also show that, if $G$ has $n$ vertices and treewidth at most $k$, then the inverse geodesic length of $G$ can be computed in $O(n \log^{O(k)}n)$ time. In both cases we use techniques developed for computing the sum of the distances, which does not have "inverse" component, together with batched evaluations of rational functions.

cs.DS↗

arXiv2020

Faster Distance-Based Representative Skyline and $k$-Center Along Pareto Front in the Plane

We consider the problem of computing the \emph{distance-based representative skyline} in the plane, a problem introduced by Tao, Ding, Lin and Pei [Proc. 25th IEEE International Conference on Data Engineering (ICDE), 2009] and independently considered by Dupin, Nielsen and Talbi [Optimization and Learning - Third International Conference, OLA 2020] in the context of multi-objective optimization. Given a set $P$ of $n$ points in the plane and a parameter $k$, the task is to select $k$ points of the skyline defined by $P$ (also known as Pareto front for $P$) to minimize the maximum distance from the points of the skyline to the selected points. We show that the problem can be solved in $O(n\log h)$ time, where $h$ is the number of points in the skyline of $P$. We also show that the decision problem can be solved in $O(n\log k)$ time and the optimization problem can be solved in $O(n \log k + n \operatorname{loglog} n)$ time. This improves previous algorithms and is optimal for a large range of values of $k$.

cs.CG↗

arXiv2020

The Complexity of Mixed-Connectivity

We investigate the parameterized complexity in $a$ and $b$ of determining whether a graph~$G$ has a subset of $a$ vertices and $b$ edges whose removal disconnects $G$, or disconnects two prescribed vertices $s, t \in V(G)$.

cs.DS↗

arXiv2020

The inverse Voronoi problem in graphs

We introduce the inverse Voronoi diagram problem in graphs: given a graph $G$ with positive edge-lengths and a collection $\mathbb{U}$ of subsets of vertices of $V(G)$, decide whether $\mathbb{U}$ is a Voronoi diagram in $G$ with respect to the shortest-path metric. We show that the problem is NP-hard, even for planar graphs where all the edges have unit length. We also study the parameterized complexity of the problem and show that the problem is W[1]-hard when parameterized by the number of Voronoi cells or by the pathwidth of the graph. For trees we show that the problem can be solved in $O(N+n \log^2 n)$ time, where $n$ is the number of vertices in the tree and $N=n+\sum_{U\in \mathbb{U}}|U|$ is the size of the description of the input. We also provide a lower bound of $Ω(n \log n)$ time for trees with $n$ vertices.

cs.DS↗

arXiv2020

Hardness of Minimum Barrier Shrinkage and Minimum Installation Path

In the Minimum Installation Path problem, we are given a graph $G$ with edge weights $w(.)$ and two vertices $s,t$ of $G$. We want to assign a non-negative power $p(v)$ to each vertex $v$ of $G$ so that the edges $uv$ such that $p(u)+p(v)$ is at least $w(uv)$ contain some $s$-$t$-path, and minimize the sum of assigned powers. In the Minimum Barrier Shrinkage problem, we are given a family of disks in the plane and two points $x$ and $y$ lying outside the disks. The task is to shrink the disks, each one possibly by a different amount, so that we can draw an $x$-$y$ curve that is disjoint from the interior of the shrunken disks, and the sum of the decreases in the radii is minimized. We show that the Minimum Installation Path and the Minimum Barrier Shrinkage problems (or, more precisely, the natural decision problems associated with them) are weakly NP-hard.

cs.CC↗

arXiv2019

The Inverse Kakeya Problem

We prove that the largest convex shape that can be placed inside a given convex shape $Q \subset \mathbb{R}^{d}$ in any desired orientation is the largest inscribed ball of $Q$. The statement is true both when "largest" means "largest volume" and when it means "largest surface area". The ball is the unique solution, except when maximizing the perimeter in the two-dimensional case.

math.MG↗

arXiv2019

Encoding 3SUM

We consider the following problem: given three sets of real numbers, output a word-RAM data structure from which we can efficiently recover the sign of the sum of any triple of numbers, one in each set. This is similar to a previous work by some of the authors to encode the order type of a finite set of points. While this previous work showed that it was possible to achieve slightly subquadratic space and logarithmic query time, we show here that for the simpler 3SUM problem, one can achieve an encoding that takes $\tilde{O}(N^{\frac 32})$ space for inputs sets of size $N$ and allows constant time queries in the word-RAM.

cs.DS↗

arXiv2018

Computing Shapley values in the plane

We consider the problem of computing Shapley values for points in the plane, where each point is interpreted as a player, and the value of a coalition is defined by the area of usual geometric objects, such as the convex hull or the minimum axis-parallel bounding box. For sets of $n$ points in the plane, we show how to compute in roughly $O(n^{3/2})$ time the Shapley values for the area of the minimum axis-parallel bounding box and the area of the union of the rectangles spanned by the origin and the input points. When the points form an increasing or decreasing chain, the running time can be improved to near-linear. In all these cases, we use linearity of the Shapley values and algebraic methods. We also show that Shapley values for the area and the perimeter of the convex hull or the minimum enclosing disk can be computed in $O(n^2)$ and $O(n^3)$ time, respectively. In this case the computation is closely related to the model of stochastic point sets considered in computational geometry, but here we have to consider random insertion orders of the points instead of a probabilistic existence of points.

cs.CG↗

arXiv2018

On the Minimum Consistent Subset Problem

Let $P$ be a set of $n$ colored points in the plane. Introduced by Hart (1968), a consistent subset of $P$, is a set $S\subseteq P$ such that for every point $p$ in $P\setminus S$, the closest point of $p$ in $S$ has the same color as $p$. The consistent subset problem is to find a consistent subset of $P$ with minimum cardinality. This problem is known to be NP-complete even for two-colored point sets. Since the initial presentation of this problem, aside from the hardness results, there has not been a significant progress from the algorithmic point of view. In this paper we present the following algorithmic results: 1. The first subexponential-time algorithm for the consistent subset problem. 2. An $O(n\log n)$-time algorithm that finds a consistent subset of size two in two-colored point sets (if such a subset exists). Towards our proof of this running time we present a deterministic $O(n \log n)$-time algorithm for computing a variant of the compact Voronoi diagram; this improves the previously claimed expected running time. 3. An $O(n\log^2 n)$-time algorithm that finds a minimum consistent subset in two-colored point sets where one color class contains exactly one point; this improves the previous best known $O(n^2)$ running time which is due to Wilfong (SoCG 1991). 4. An $O(n)$-time algorithm for the consistent subset problem on collinear points; this improves the previous best known $O(n^2)$ running time. 5. A non-trivial $O(n^6)$-time dynamic programming algorithm for the consistent subset problem on points arranged on two parallel lines. To obtain these results, we combine tools from planar separators, additively-weighted Voronoi diagrams with respect to convex distance functions, point location in farthest-point Voronoi diagrams, range trees, paraboloid lifting, minimum covering of a circle with arcs, and several geometric transformations.

cs.CG↗

arXiv2018

Minimum Shared-Power Edge Cut

We introduce a problem called the Minimum Shared-Power Edge Cut (MSPEC). The input to the problem is an undirected edge-weighted graph with distinguished vertices s and t, and the goal is to find an s-t cut by assigning "powers" at the vertices and removing an edge if the sum of the powers at its endpoints is at least its weight. The objective is to minimize the sum of the assigned powers. MSPEC is a graph generalization of a barrier coverage problem in a wireless sensor network: given a set of unit disks with centers in a rectangle, what is the minimum total amount by which we must shrink the disks to permit an intruder to cross the rectangle undetected, i.e. without entering any disc. This is a more sophisticated measure of barrier coverage than the minimum number of disks whose removal breaks the barrier. We develop a fully polynomial time approximation scheme (FPTAS) for MSPEC. We give polynomial time algorithms for the special cases where the edge weights are uniform, or the power values are restricted to a bounded set. Although MSPEC is related to network flow and matching problems, its computational complexity (in P or NP-hard) remains open.

cs.DS↗

arXiv2018

Covering many points with a small-area box

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 a value $α>0$, covering as many points of $P$ as possible with an axis-parallel rectangle of area at most $α$. For this problem we give a probabilistic $(1-\varepsilon)$-approximation that works in near-linear time: In $O((n/\varepsilon^4)\log^3 n \log (1/\varepsilon))$ time we find an axis-parallel rectangle of area at most $α$ that, with high probability, covers at least $(1-\varepsilon)\mathrm{κ^*}$ points, where $\mathrm{κ^*}$ is the maximum possible number of points that could be covered.

cs.CG↗

arXiv2018

Subquadratic Algorithms for the Diameter and the Sum of Pairwise Distances in Planar Graphs

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 $O(n^{15/8}{\rm polylog}(n))$ expected time we can also compute the number of pairs of vertices at distance smaller than a given threshold. These are the first algorithms for these problems using time $O(n^c)$ for some constant $c<2$, even when restricted to undirected, unweighted planar graphs.

cs.DS↗

arXiv2018

Maximum Volume Subset Selection for Anchored Boxes

Let $B$ be a set of $n$ axis-parallel boxes in $\mathbb{R}^d$ such that each box has a corner at the origin and the other corner in the positive quadrant of $\mathbb{R}^d$, and let $k$ be a positive integer. We study the problem of selecting $k$ boxes in $B$ that maximize the volume of the union of the selected boxes. This research is motivated by applications in skyline queries for databases and in multicriteria optimization, where the problem is known as the hypervolume subset selection problem. It is known that the problem can be solved in polynomial time in the plane, while the best known running time in any dimension $d \ge 3$ is $Ω\big(\binom{n}{k}\big)$. We show that: - The problem is NP-hard already in 3 dimensions. - In 3 dimensions, we break the bound $Ω\big(\binom{n}{k}\big)$, by providing an $n^{O(\sqrt{k})}$ algorithm. - For any constant dimension $d$, we present an efficient polynomial-time approximation scheme.

cs.CG↗

arXiv2018

The parameterized complexity of finding a 2-sphere in a simplicial complex

We consider the problem of finding a subcomplex K' of a simplicial complex K such that K' is homeomorphic to the 2-dimensional sphere, S^2. We study two variants of this problem. The first asks if there exists such a K' with at most k triangles, and we show that this variant is W[1]-hard and, assuming ETH, admits no O(n^{o(sqrt(k))}) time algorithm. We also give an algorithm that is tight with regards to this lower bound. The second problem is the dual of the first, and asks if K' can be found by removing at most k triangles from K. This variant has an immediate O(3^k poly(|K|)) time algorithm, and we show that it admits a polynomial kernelization to O(k^2) triangles, as well as a polynomial compression to a weighted version with bit-size O(k log k).

cs.DS↗

arXiv2017

Peeling potatoes near-optimally in near-linear time

We consider the following geometric optimization problem: find a convex polygon of maximum area contained in a given simple polygon $P$ with $n$ vertices. We give a randomized near-linear-time $(1-\varepsilon)$-approximation algorithm for this problem: in $O(n( \log^2 n + (1/\varepsilon^3) \log n + 1/\varepsilon^4))$ time we find a convex polygon contained in $P$ that, with probability at least $2/3$, has area at least $(1-\varepsilon)$ times the area of an optimal solution. We also obtain similar results for the variant of computing a convex polygon inside $P$ with maximum perimeter. To achieve these results we provide new results in geometric probability. The first result is a bound relating the probability that two points chosen uniformly at random inside $P$ are mutually visible and the area of the largest convex body inside $P$. The second result is a bound on the expected value of the difference between the perimeter of any planar convex body $K$ and the perimeter of the convex hull of a uniform random sample inside $K$.

cs.CG↗

arXiv2017

Two Optimization Problems for Unit Disks

We present an implementation of a recent algorithm to compute shortest-path trees in unit disk graphs in $O(n\log n)$ worst-case time, where $n$ is the number of disks. In the minimum-separation problem, we are given $n$ unit disks and two points $s$ and $t$, not contained in any of the disks, and we want to compute the minimum number of disks one needs to retain so that any curve connecting $s$ to $t$ intersects some of the retained disks. We present a new algorithm solving this problem in $O(n^2\log^3 n)$ worst-case time and its implementation.

cs.CG↗