arXiv subjects

Matias Korman

Publications and source records attributed to Matias Korman.

At least 55 records · Page 3Linked to original sources

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↗

arXiv2017

Line Segment Covering of Cells in Arrangements

Given a collection $L$ of line segments, we consider its arrangement and study the problem of covering all cells with line segments of $L$. That is, we want to find a minimum-size set $L'$ of line segments such that every cell in the arrangement has a line from $L'$ defining its boundary. We show that the problem is NP-hard, even when all segments are axis-aligned. In fact, the problem is still NP-hard when we only need to cover rectangular cells of the arrangement. For the latter problem we also show that it is fixed parameter tractable with respect to the size of the optimal solution. Finally we provide a linear time algorithm for the case where cells of the arrangement are created by recursively subdividing a rectangle using horizontal and vertical cutting segments.

cs.CG↗

arXiv2017

Packing Plane Spanning Trees and Paths in Complete Geometric Graphs

We consider the following question: How many edge-disjoint plane spanning trees are contained in a complete geometric graph $GK_n$ on any set $S$ of $n$ points in general position in the plane? We show that this number is in $Ω(\sqrt{n})$. Further, we consider variants of this problem by bounding the diameter and the degree of the trees (in particular considering spanning paths).

cs.CG↗

arXiv2017

Experimental Study of Compressed Stack Algorithms in Limited Memory Environments

The {\em compressed stack} is a data structure designed by Barba {\em et al.} (Algorithmica 2015) that allows to reduce the amount of memory needed by an algorithm (at the cost of increasing its runtime). In this paper we introduce the first implementation of this data structure and make its source code publicly available. Together with the implementation we analyze the performance of the compressed stack. In our synthetic experiments, considering different test scenarios and using data sizes ranging up to $2^{30}$ elements, we compare it with the classic (uncompressed) stack, both in terms of runtime and memory used. Our experiments show that the compressed stack needs significantly less memory than the usual stack (this difference is significant for inputs containing $2000$ or more elements). Overall, with a proper choice of parameters, we can save a significant amount of space (from two to four orders of magnitude) with a small increase in the runtime ($2.32$ times slower on average than the classic stack). These results holds even in test scenarios specifically designed to be challenging for the compressed stack.

cs.DS↗

arXiv2017

On the Complexity of Barrier Resilience for Fat Regions

In the \emph {barrier resilience} problem (introduced by Kumar {\em et al.}, Wireless Networks 2007), we are given a collection of regions of the plane, acting as obstacles, and we would like to remove the minimum number of regions so that two fixed points can be connected without crossing any region. In this paper, we show that the problem is NP-hard when the collection only contains fat regions with bounded ply $Δ$ (even when they are axis-aligned rectangles of aspect ratio $1 : (1 + \varepsilon)$). We also show that the problem is fixed-parameter tractable (FPT) for unit disks and for similarly-sized $β$-fat regions with bounded ply $Δ$ and $O(1)$ pairwise boundary intersections. Furthermore, we can use our FPT algorithm to construct an $(1+\varepsilon)$-approximation algorithm that runs in $O(2^{f(Δ, \varepsilon,β)}n^5)$ time, where $f\in O(\frac{Δ^4β^8}{\varepsilon^4}\log(βΔ/\varepsilon))$.

cs.CC↗

arXiv2017

Time-Space Trade-off Algorithms for Triangulating a Simple Polygon

An $s$-workspace algorithm is an algorithm that has read-only access to the values of the input, write-only access to the output, and only uses $O(s)$ additional words of space. We present a randomized $s$-workspace algorithm for triangulating a simple polygon $P$ of $n$ vertices that runs in $O(n^2/s+n \log n \log^{5} (n/s))$ expected time using $O(s)$ variables, for any $s \leq n$. In particular, when $s \leq \frac{n}{\log n\log^{5}\log n}$ the algorithm runs in $O(n^2/s)$ expected time.

cs.CG↗

arXiv2017

On interference among moving sensors and related problems

We show that for any set of $n$ points moving along "simple" trajectories (i.e., each coordinate is described with a polynomial of bounded degree) in $\Re^d$ and any parameter $2 \le k \le n$, one can select a fixed non-empty subset of the points of size $O(k \log k)$, such that the Voronoi diagram of this subset is "balanced" at any given time (i.e., it contains $O(n/k)$ points per cell). We also show that the bound $O(k \log k)$ is near optimal even for the one dimensional case in which points move linearly in time. As applications, we show that one can assign communication radii to the sensors of a network of $n$ moving sensors so that at any given time their interference is $O(\sqrt{n\log n})$. We also show some results in kinetic approximate range counting and kinetic discrepancy. In order to obtain these results, we extend well-known results from $\varepsilon$-net theory to kinetic environments.

cs.CG↗

arXiv2017

Stabbing segments with rectilinear objects

Given a set $S$ of $n$ line segments in the plane, we say that a region $\mathcal{R}\subseteq \mathbb{R}^2$ is a {\em stabber} for $S$ if $\mathcal{R}$ contains exactly one endpoint of each segment of $S$. In this paper we provide optimal or near-optimal algorithms for reporting all combinatorially different stabbers for several shapes of stabbers. Specifically, we consider the case in which the stabber can be described as the intersection of axis-parallel halfplanes (thus the stabbers are halfplanes, strips, quadrants, $3$-sided rectangles, or rectangles). The running times are $O(n)$ (for the halfplane case), $O(n\log n)$ (for strips, quadrants, and 3-sided rectangles), and $O(n^2 \log n)$ (for rectangles).

cs.CG↗

arXiv2017

Hanabi is NP-hard, Even for Cheaters who Look at Their Cards

In this paper we study a cooperative card game called Hanabi from the viewpoint of algorithmic combinatorial game theory. In Hanabi, each card has one among $c$ colors and a number between $1$ and $n$. The aim is to make, for each color, a pile of cards of that color with all increasing numbers from $1$ to $n$. At each time during the game, each player holds $h$ cards in hand. Cards are drawn sequentially from a deck and the players should decide whether to play, discard or store them for future use. One of the features of the game is that the players can see their partners' cards but not their own and information must be shared through hints. We introduce a single-player, perfect-information model and show that the game is intractable even for this simplified version where we forego both the hidden information and the multiplayer aspect of the game, even when the player can only hold two cards in her hand. On the positive side, we show that the decision version of the problem---to decide whether or not numbers from $1$ through $n$ can be played for every color---can be solved in (almost) linear time for some restricted cases.

cs.DM↗

arXiv2017

Geometric Biplane Graphs I: Maximal Graphs

We study biplane graphs drawn on a finite planar point set $S$ in general position. This is the family of geometric graphs whose vertex set is $S$ and can be decomposed into two plane graphs. We show that two maximal biplane graphs---in the sense that no edge can be added while staying biplane---may differ in the number of edges, and we provide an efficient algorithm for adding edges to a biplane graph to make it maximal. We also study extremal properties of maximal biplane graphs such as the maximum number of edges and the largest maximum connectivity over $n$-element point sets.

cs.CG↗

arXiv2017

Geometric Biplane Graphs II: Graph Augmentation

We study biplane graphs drawn on a finite point set $S$ in the plane in general position. This is the family of geometric graphs whose vertex set is $S$ and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6-connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges.

cs.CG↗

arXiv2016

High Dimensional Consistent Digital Segments

We consider the problem of digitalizing Euclidean line segments from $\mathbb{R}^d$ to $\mathbb{Z}^d$. Christ {\em et al.} (DCG, 2012) showed how to construct a set of {\em consistent digital segment} (CDS) for $d=2$: a collection of segments connecting any two points in $\mathbb{Z}^2$ that satisfies the natural extension of the Euclidean axioms to $\mathbb{Z}^d$. In this paper we study the construction of CDSs in higher dimensions. We show that any total order can be used to create a set of {\em consistent digital rays} CDR in $\mathbb{Z}^d$ (a set of rays emanating from a fixed point $p$ that satisfies the extension of the Euclidean axioms). We fully characterize for which total orders the construction holds and study their Hausdorff distance, which in particular positively answers the question posed by Christ {\em et al.}.

cs.CG↗

arXiv2016

Computing the $L_1$ Geodesic Diameter and Center of a Polygonal Domain

For a polygonal domain with $h$ holes and a total of $n$ vertices, we present algorithms that compute the $L_1$ geodesic diameter in $O(n^2+h^4)$ time and the $L_1$ geodesic center in $O((n^4+n^2 h^4)α(n))$ time, respectively, where $α(\cdot)$ denotes the inverse Ackermann function. No algorithms were known for these problems before. For the Euclidean counterpart, the best algorithms compute the geodesic diameter in $O(n^{7.73})$ or $O(n^7(h+\log n))$ time, and compute the geodesic center in $O(n^{11}\log n)$ time. Therefore, our algorithms are significantly faster than the algorithms for the Euclidean problems. Our algorithms are based on several interesting observations on $L_1$ shortest paths in polygonal domains.

cs.CG↗

arXiv2016

Colored Spanning Graphs for Set Visualization

We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red points belong exclusively to the red set, and purple points belong to both sets. A \emph{red-blue-purple spanning graph} (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected. We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem can be solved in polynomial time using matroid techniques. In addition, we discuss more efficient algorithms for the case in which points are located on a line or a circle, and also describe a fast $(\frac 12ρ+1)$-approximation algorithm, where $ρ$ is the Steiner ratio.

cs.CG↗

arXiv2015

The Dual Diameter of Triangulations

Let $\Poly$ be a simple polygon with $n$ vertices. The \emph{dual graph} $\triang^*$ of a triangulation~$\triang$ of~$\Poly$ is the graph whose vertices correspond to the bounded faces of $\triang$ and whose edges connect those faces of~$\triang$ that share an edge. We consider triangulations of~$\Poly$ that minimize or maximize the diameter of their dual graph. We show that both triangulations can be constructed in $O(n^3\log n)$ time using dynamic programming. If $\Poly$ is convex, we show that any minimizing triangulation has dual diameter exactly $2\cdot\lceil\log_2(n/3)\rceil$ or $2\cdot\lceil\log_2(n/3)\rceil -1$, depending on~$n$. Trivially, in this case any maximizing triangulation has dual diameter $n-2$. Furthermore, we investigate the relationship between the dual diameter and the number of \emph{ears} (triangles with exactly two edges incident to the boundary of $\Poly$) in a triangulation. For convex $\Poly$, we show that there is always a triangulation that simultaneously minimizes the dual diameter and maximizes the number of ears. In contrast, we give examples of general simple polygons where every triangulation that maximizes the number of ears has dual diameter that is quadratic in the minimum possible value. We also consider the case of point sets in general position in the plane. We show that for any such set of $n$ points there are triangulations with dual diameter in~$O(\log n)$ and in~$Ω(\sqrt n)$.

cs.CG↗

arXiv2015

Computing the Geodesic Centers of a Polygonal Domain

We present an algorithm that computes the geodesic center of a given polygonal domain. The running time of our algorithm is $O(n^{12+ε})$ for any $ε>0$, where $n$ is the number of corners of the input polygonal domain. Prior to our work, only the very special case where a simple polygon is given as input has been intensively studied in the 1980s, and an $O(n \log n)$-time algorithm is known by Pollack et al. Our algorithm is the first one that can handle general polygonal domains having one or more polygonal holes.

cs.CG↗

arXiv2015

Computing the $L_1$ Geodesic Diameter and Center of a Simple Polygon in Linear Time

In this paper, we show that the $L_1$ geodesic diameter and center of a simple polygon can be computed in linear time. For the purpose, we focus on revealing basic geometric properties of the $L_1$ geodesic balls, that is, the metric balls with respect to the $L_1$ geodesic distance. More specifically, in this paper we show that any family of $L_1$ geodesic balls in any simple polygon has Helly number two, and the $L_1$ geodesic center consists of midpoints of shortest paths between diametral pairs. These properties are crucial for our linear-time algorithms, and do not hold for the Euclidean case.

cs.CG↗

arXiv2015

A linear-time algorithm for the geodesic center of a simple polygon

Given two points in a simple polygon $P$ of $n$ vertices, its geodesic distance is the length of the shortest path that connects them among all paths that stay within $P$. The geodesic center of $P$ is the unique point in $P$ that minimizes the largest geodesic distance to all other points of $P$. In 1989, Pollack, Sharir and Rote [Disc. \& Comput. Geom. 89] showed an $O(n\log n)$-time algorithm that computes the geodesic center of $P$. Since then, a longstanding question has been whether this running time can be improved (explicitly posed by Mitchell [Handbook of Computational Geometry, 2000]). In this paper we affirmatively answer this question and present a linear time algorithm to solve this problem.

cs.CG↗