arXiv subjects

Chan-Su Shin

Publications and source records attributed to Chan-Su Shin.

11 recordsLinked to original sources

arXiv2022

Reachability of turn sequences

A turn sequence of left and right turns is realized as a simple rectilinear chain of integral segments whose turns at its bends are the same as the turn sequence. The chain starts from the origin and ends at some point which we call a reachable point of the turn sequence. We investigate the combinatorial and geometric properties of the set of reachable points of a given turn sequence such as the shape, connectedness, and sufficient and necessary conditions on the reachability to the four signed axes. We also prove the upper and lower bounds on the maximum distance from the origin to the closest reachable point on signed axes for a turn sequence. The bounds are expressed in terms of the difference between the number of left and right turns in the sequence as well as, in certain cases, the length of the maximal monotone prefix or suffix of the turn sequence. The bounds are exactly matched or tight within additive constants for some signed axes.

cs.CG↗

arXiv2020

Representing Graphs and Hypergraphs by Touching Polygons in 3D

Contact representations of graphs have a long history. Most research has focused on problems in 2D, but 3D contact representations have also been investigated, mostly concerning fully-dimensional geometric objects such as spheres or cubes. In this paper we study contact representations with convex polygons in 3D. We show that every graph admits such a representation. Since our representations use super-polynomial coordinates, we also construct representations on grids of polynomial size for specific graph classes (bipartite, subcubic). For hypergraphs, we represent their duals, that is, each vertex is represented by a point and each edge by a polygon. We show that even regular and quite small hypergraphs do not admit such representations. On the other hand, the two smallest Steiner triple systems can be represented.

cs.CG↗

arXiv2016

Minimum Rectilinear Polygons for Given Angle Sequences

A rectilinear polygon is a polygon whose edges are axis-aligned. Walking counterclockwise on the boundary of such a polygon yields a sequence of left turns and right turns. The number of left turns always equals the number of right turns plus 4. It is known that any such sequence can be realized by a rectilinear polygon. In this paper, we consider the problem of finding realizations that minimize the perimeter or the area of the polygon or the area of the bounding box of the polygon. We show that all three problems are NP-hard in general. This answers an open question of Patrignani [CGTA 2001], who showed that it is NP-hard to minimize the area of the bounding box of an orthogonal drawing of a given planar graph. We also show that realizing polylines with minimum bounding box area is NP-hard. Then we consider the special cases of $x$-monotone and $xy$-monotone rectilinear polygons. For these, we can optimize the three objectives efficiently.

cs.CG↗

arXiv2015

Improved Bounds for Beacon-Based Coverage and Routing in Simple Rectilinear Polygons

We establish tight bounds for beacon-based coverage problems, and improve the bounds for beacon-based routing problems in simple rectilinear polygons. Specifically, we show that $\lfloor \frac{n}{6} \rfloor$ beacons are always sufficient and sometimes necessary to cover a simple rectilinear polygon $P$ with $n$ vertices. We also prove tight bounds for the case where $P$ is monotone, and we present an optimal linear-time algorithm that computes the beacon-based kernel of $P$. For the routing problem, we show that $\lfloor \frac{3n-4}{8} \rfloor - 1$ beacons are always sufficient, and $\lceil \frac{n}{4}\rceil-1$ beacons are sometimes necessary to route between all pairs of points in $P$.

cs.CG↗

arXiv2013

Local Event Boundary Detection with Unreliable Sensors: Analysis of the Majority Vote Scheme

In this paper we study the identification of an event region $X$ within a larger region $Y$, in which the sensors are distributed by a Poisson process of density $λ$ to detect this event region, i.e., its boundary. The model of sensor is a 0-1 sensor that decides whether it lies in $X$ or not, and which might be incorrect with probability $p$. It also collects information on the 0-1 values of the neighbors within some distance $r$ and revises its decision by the majority vote of these neighbors. In the most general setting, we analyze this simple majority vote scheme and derive some upper and lower bounds on the expected number of misclassified sensors. These bounds depend on several sensing parameters of $p$, $r$, and some geometric parameters of the event region $X$. By making some assumptions on the shape of $X$, we prove a significantly improved upper bound on the expected number of misclassified sensors; especially for convex regions with sufficiently round boundary, and we find that the majority vote scheme performs well in the simulation rather than its theoretical upper bound.

cs.CG↗

arXiv2012

A Note on Minimum-Sum Coverage by Aligned Disks

In this paper, we consider a facility location problem to find a minimum-sum coverage of n points by disks centered at a fixed line. The cost of a disk with radius r has a form of a non-decreasing function f(r) = r^a for any a >= 1. The goal is to find a set of disks under Lp metric such that the disks are centered on the x-axis, their union covers n points, and the sum of the cost of the disks is minimized. Alt et al. [1] presented an algorithm in O(n^4 log n) time for any a > 1 under any Lp metric. We present a faster algorithm for this problem in O(n^2 log n) time for any a > 1 and any Lp metric.

cs.CG↗

arXiv2012

Covering and Piercing Disks with Two Centers

We give exact and approximation algorithms for two-center problems when the input is a set $\mathcal{D}$ of disks in the plane. We first study the problem of finding two smallest congruent disks such that each disk in $\mathcal{D}$ intersects one of these two disks. Then we study the problem of covering the set $\mathcal{D}$ by two smallest congruent disks.

cs.CG↗

arXiv2009

Computing k-Centers On a Line

In this paper we consider several instances of the k-center on a line problem where the goal is, given a set of points S in the plane and a parameter k >= 1, to find k disks with centers on a line l such that their union covers S and the maximum radius of the disks is minimized. This problem is a constraint version of the well-known k-center problem in which the centers are constrained to lie in a particular region such as a segment, a line, and a polygon. We first consider the simplest version of the problem where the line l is given in advance; we can solve this problem in O(n log^2 n) time. We then investigate the cases where only the orientation of the line l is fixed and where the line l can be arbitrary. We can solve these problems in O(n^2 log^2 n) time and in O(n^4 log^2 n) expected time, respectively. For the last two problems, we present (1 + e)-approximation algorithms, which run in O((1/e) n log^2 n) time and O((1/e^2) n log^2 n) time, respectively.

cs.CG↗

arXiv2009

Untangling a Planar Graph

A straight-line drawing $δ$ of a planar graph $G$ need not be plane, but can be made so by \emph{untangling} it, that is, by moving some of the vertices of $G$. Let shift$(G,δ)$ denote the minimum number of vertices that need to be moved to untangle $δ$. We show that shift$(G,δ)$ is NP-hard to compute and to approximate. Our hardness results extend to a version of \textsc{1BendPointSetEmbeddability}, a well-known graph-drawing problem. Further we define fix$(G,δ)=n-shift(G,δ)$ to be the maximum number of vertices of a planar $n$-vertex graph $G$ that can be fixed when untangling $δ$. We give an algorithm that fixes at least $\sqrt{((\log n)-1)/\log \log n}$ vertices when untangling a drawing of an $n$-vertex graph $G$. If $G$ is outerplanar, the same algorithm fixes at least $\sqrt{n/2}$ vertices. On the other hand we construct, for arbitrarily large $n$, an $n$-vertex planar graph $G$ and a drawing $δ_G$ of $G$ with fix$(G,δ_G) \le \sqrt{n-2}+1$ and an $n$-vertex outerplanar graph $H$ and a drawing $δ_H$ of $H$ with fix$(H,δ_H) \le 2 \sqrt{n-1}+1$. Thus our algorithm is asymptotically worst-case optimal for outerplanar graphs.

cs.CG↗

arXiv2008

Moving Vertices to Make Drawings Plane

A straight-line drawing $δ$ of a planar graph $G$ need not be plane, but can be made so by moving some of the vertices. Let shift$(G,δ)$ denote the minimum number of vertices that need to be moved to turn $δ$ into a plane drawing of $G$. We show that shift$(G,δ)$ is NP-hard to compute and to approximate, and we give explicit bounds on shift$(G,δ)$ when $G$ is a tree or a general planar graph. Our hardness results extend to 1BendPointSetEmbeddability, a well-known graph-drawing problem.

cs.CG↗

arXiv2007

Constructing Optimal Highways

For two points $p$ and $q$ in the plane, a straight line $h$, called a highway, and a real $v>1$, we define the \emph{travel time} (also known as the \emph{city distance}) from $p$ and $q$ to be the time needed to traverse a quickest path from $p$ to $q$, where the distance is measured with speed $v$ on $h$ and with speed 1 in the underlying metric elsewhere. Given a set $S$ of $n$ points in the plane and a highway speed $v$, we consider the problem of finding a \emph{highway} that minimizes the maximum travel time over all pairs of points in $S$. If the orientation of the highway is fixed, the optimal highway can be computed in linear time, both for the $L_1$- and the Euclidean metric as the underlying metric. If arbitrary orientations are allowed, then the optimal highway can be computed in $O(n^{2} \log n)$ time. We also consider the problem of computing an optimal pair of highways, one being horizontal, one vertical.

cs.CG↗