arXiv subjects

Hee-Kap Ahn

Publications and source records attributed to Hee-Kap Ahn.

At least 19 recordsLinked to original sources

arXiv2026

Diameter and Length of Metric Graphs

A metric graph is a metric space obtained from a finite collection of intervals whose endpoints are identified in groups. It can also be seen as a finite, edge-weighted graph where the continuum of points along the interior of each edge is taken into consideration, and each edge is locally isometric to an interval whose length is the edge-weight. The diameter of a metric graph $G$ is the maximum distance between all pairs of points of $G$. We show that the total length of a metric graph $G$ with $\ell(G)$ leaves, cyclomatic number $cyc(G)$, and diameter $diam(G)$ is at most $(cyc(G) + max\{1, \ell(G)/2\}) \cdot diam(G)$. Furthermore, we show that his bound is tight, and we characterize the metric graphs where equality holds. As an application, we provide tight bounds in certain cases for the diameter of metric graphs obtained from a cycle or a star by the identification of a fixed number of points (pairwise or in groups).

math.CO↗

arXiv2025

Minimum Partition of Polygons under Width and Cut Constraints

We study the problem of partitioning a polygon into the minimum number of subpolygons using cuts in predetermined directions such that each resulting subpolygon satisfies a given width constraint. A polygon satisfies the unit-width constraint for a set of unit vectors if the length of the orthogonal projection of the polygon on a line parallel to a vector in the set is at most one. We analyze structural properties of the minimum partition numbers, focusing on monotonicity under polygon containment. We show that the minimum partition number of a simple polygon is at least that of any subpolygon, provided that the subpolygon satisfies a certain orientation-wise convexity with respect to the polygon. As a consequence, we prove a partition analogue of Bang's conjecture about coverings of convex regions in the plane: for any partition of a convex body in the plane, the sum of relative widths of all parts is at least one. For any convex polygon, there exists a direction along which an optimal partition is achieved by parallel cuts. Given such a direction, an optimal partition can be computed in linear time.

cs.CG↗

arXiv2025

Guarding Terrains with Guards on a Line

Given an $x$-monotone polygonal chain $T$ with $n$ vertices, and an integer $k$, we consider the problem of finding the lowest horizontal line $L$ lying above $T$ with $k$ point guards lying on $L$, so that every point on the chain is \emph{visible} from some guard. A natural optimization is to minimize the $y$-coordinate of $L$. We present an algorithm for finding the optimal placements of $L$ and $k$ point guards for $T$ in $O(k^2\lambda_{k-1}(n)\log n)$ time for even numbers $k\ge 2$, and in $O(k^2\lambda_{k-2}(n)\log n)$ time for odd numbers $k \ge 3$, where $\lambda_{s}(n)$ is the length of the longest $(n,s)$-Davenport-Schinzel sequence. We also study a variant with an additional requirement that $T$ is partitioned into $k$ subchains, each subchain is paired with exactly one guard, and every point on a subchain is visible from its paired guard. When $L$ is fixed, we can place the minimum number of guards in $O(n)$ time. When the number $k$ of guards is fixed, we can find an optimal placement of $L$ with $k$ point guards lying on $L$ in $O(kn)$ time.

cs.CG↗

arXiv2022

Universal convex covering problems under translation and discrete rotations

We consider the smallest-area universal covering of planar objects of perimeter 2 (or equivalently closed curves of length 2) allowing translation and discrete rotations. In particular, we show that the solution is an equilateral triangle of height 1 when translation and discrete rotation of $\pi$ are allowed. Our proof is purely geometric and elementary. We also give convex coverings of closed curves of length 2 under translation and discrete rotations of multiples of $\pi/2$ and $2\pi/3$. We show a minimality of the covering for discrete rotation of multiples of $\pi/2$, which is an equilateral triangle of height smaller than 1, and conjecture that the covering is the smallest-area convex covering. Finally, we give the smallest-area convex coverings of all unit segments under translation and discrete rotations $2\pi/k$ for all integers $k\ge 3$.

cs.CG↗

arXiv2022

Farthest-point Voronoi diagrams in the presence of rectangular obstacles

We present an algorithm to compute the geodesic $L_1$ farthest-point Voronoi diagram of $m$ point sites in the presence of $n$ rectangular obstacles in the plane. It takes $O(nm+n \log n + m\log m)$ construction time using $O(nm)$ space. This is the first optimal algorithm for constructing the farthest-point Voronoi diagram in the presence of obstacles. We can construct a data structure in the same construction time and space that answers a farthest-neighbor query in $O(\log(n+m))$ time.

cs.CG↗

arXiv2021

Rectangular Partitions of a Rectilinear Polygon

We investigate the problem of partitioning a rectilinear polygon $P$ with $n$ vertices and no holes % with no holes into rectangles using disjoint line segments drawn inside $P$ under two optimality criteria. In the minimum ink partition, the total length of the line segments drawn inside $P$ is minimized. We present an $O(n^3)$-time algorithm using $O(n^2)$ space that returns a minimum ink partition of $P$. In the thick partition, the minimum side length over all resulting rectangles is maximized. We present an $O(n^3 \log^2{n})$-time algorithm using $O(n^3)$ space that returns a thick partition using line segments incident to vertices of $P$, and an $O(n^6 \log^2{n})$-time algorithm using $O(n^6)$ space that returns a thick partition using line segments incident to the boundary of $P$. We also show that if the input rectilinear polygon has holes, the corresponding decision problem for the thick partition problem using line segments incident to vertices of the polygon is NP-complete. We also present an $O(m^3)$-time $3$-approximation algorithm for the minimum ink partition for a rectangle containing $m$ point holes.

cs.CG↗

arXiv2021

Minimum-Link Shortest Paths for Polygons amidst Rectilinear Obstacles

Consider two axis-aligned rectilinear simple polygons in the domain consisting of axis-aligned rectilinear obstacles in the plane such that the bounding boxes, one for each obstacle and one for each polygon, are disjoint. We present an algorithm that computes a minimum-link rectilinear shortest path connecting the two polygons in $O((N+n)\log (N+n))$ time using $O(N+n)$ space, where $n$ is the number of vertices in the domain and $N$ is the total number of vertices of the two polygons.

cs.CG↗

arXiv2021

Covering Convex Polygons by Two Congruent Disks

We consider the planar two-center problem for a convex polygon: given a convex polygon in the plane, find two congruent disks of minimum radius whose union contains the polygon. We present an $O(n\log n)$-time algorithm for the two-center problem for a convex polygon, where $n$ is the number of vertices of the polygon. This improves upon the previous best algorithm for the problem.

cs.CG↗

arXiv2020

Largest similar copies of convex polygons amidst polygonal obstacles

Given a convex polygon $P$ with $k$ vertices and a polygonal domain $Q$ consisting of polygonal obstacles with total size $n$ in the plane, we study the optimization problem of finding a largest similar copy of $P$ that can be placed in $Q$ without intersecting the obstacles. We improve the time complexity for solving the problem to $O(k^2n^2\lambda_4(k)\log{n})$. This is progress of improving the previously best known results by Chew and Kedem [SoCG89, CGTA93] and Sharir and Toledo [SoCG91, CGTA94] on the problem in more than 25 years.

cs.CG↗

arXiv2020

Largest triangles in a polygon

We study the problem of finding maximum-area triangles that can be inscribed in a polygon in the plane. We consider eight versions of the problem: we use either convex polygons or simple polygons as the container; we require the triangles to have either one corner with a fixed angle or all three corners with fixed angles; we either allow reorienting the triangle or require its orientation to be fixed. We present exact algorithms for all versions of the problem. In the case with reorientations for convex polygons with $n$ vertices, we also present $(1-\varepsilon)$-approximation algorithms.

cs.CG↗

arXiv2020

Efficient Planar Two-Center Algorithms

We consider the planar Euclidean two-center problem in which given $n$ points in the plane we are to find two congruent disks of the smallest radius covering the points. We present a deterministic $O(n \log n)$-time algorithm for the case that the centers of the two optimal disks are close to each other, that is, the overlap of the two optimal disks is a constant fraction of the disk area. We also present a deterministic $O(n\log n)$-time algorithm for the case that the input points are in convex position. Both results improve the previous best $O(n\log n\log\log n)$ bound on the problems.

cs.CG↗

arXiv2019

Computing the Center Region and Its Variants

We present an $O(n^2\log^4 n)$-time algorithm for computing the center region of a set of $n$ points in the three-dimensional Euclidean space. This improves the previously best known algorithm by Agarwal, Sharir and Welzl, which takes $O(n^{2+\epsilon})$ time for any $\epsilon > 0$. It is known that the combinatorial complexity of the center region is $\Omega(n^2)$ in the worst case, thus our algorithm is almost tight. We also consider the problem of computing a colored version of the center region in the two-dimensional Euclidean space and present an $O(n\log^4 n)$-time algorithm.

cs.CG↗

arXiv2019

Computing a Geodesic Two-Center of Points in a Simple Polygon

Given a simple polygon $P$ and a set $Q$ of points contained in $P$, we consider the geodesic $k$-center problem where we want to find $k$ points, called \emph{centers}, in $P$ to minimize the maximum geodesic distance of any point of $Q$ to its closest center. In this paper, we focus on the case for $k=2$ and present the first exact algorithm that efficiently computes an optimal $2$-center of $Q$ with respect to the geodesic distance in $P$.

cs.CG↗

arXiv2019

Maximum-Area Rectangles in a Simple Polygon

We study the problem of finding maximum-area rectangles contained in a polygon in the plane. There has been a fair amount of work for this problem when the rectangles have to be axis-aligned or when the polygon is convex. We consider this problem in a simple polygon with $n$ vertices, possibly with holes, and with no restriction on the orientation of the rectangles. We present an algorithm that computes a maximum-area rectangle in $O(n^3\log n)$ time using $O(kn^2)$ space, where $k$ is the number of reflex vertices of $P$. Our algorithm can report all maximum-area rectangles in the same time using $O(n^3)$ space. We also present a simple algorithm that finds a maximum-area rectangle contained in a convex polygon with $n$ vertices in $O(n^3)$ time using $O(n)$ space.

cs.CG↗

arXiv2019

Polygon Queries for Convex Hulls of Points

We study the following range searching problem: Preprocess a set $P$ of $n$ points in the plane with respect to a set $\mathcal{O}$ of $k$ orientations % , for a constant, in the plane so that given an $\mathcal{O}$-oriented convex polygon $Q$, the convex hull of $P\cap Q$ can be computed efficiently, where an $\mathcal{O}$-oriented polygon is a polygon whose edges have orientations in $\mathcal{O}$. We present a data structure with $O(nk^3\log^2n)$ space and $O(nk^3\log^2n)$ construction time, and an $O(h+s\log^2 n)$-time query algorithm for any query $\mathcal{O}$-oriented convex $s$-gon $Q$, where $h$ is the complexity of the convex hull. Also, we can compute the perimeter or area of the convex hull of $P\cap Q$ in $O(s\log^2n)$ time using the data structure.

cs.CG↗

arXiv2019

On Romeo and Juliet Problems: Minimizing Distance-to-Sight

We introduce a variant of the watchman route problem, which we call the quickest pair-visibility problem. Given two persons standing at points $s$ and $t$ in a simple polygon $P$ with no holes, we want to minimize the distance they travel in order to see each other in $P$. We solve two variants of this problem, one minimizing the longer distance the two persons travel (min-max) and one minimizing the total travel distance (min-sum), optimally in linear time. We also consider a query version of this problem for the min-max variant. We can preprocess a simple $n$-gon in linear time so that the minimum of the longer distance the two persons travel can be computed in $O(\log^2 n)$ time for any two query positions $s,t$ where the two persons start.

cs.CG↗

arXiv2018

Assigning Weights to Minimize the Covering Radius in the Plane

Given a set $P$ of $n$ points in the plane and a multiset $W$ of $k$ weights with $k\leq n$, we assign each weight in $W$ to a distinct point in $P$ to minimize the maximum weighted distance from the weighted center of $P$ to any point in $P$. In this paper, we give two algorithms which take $O(k^2n^2\log^3 n)$ time and $O(k^5n\log^3k+kn\log^3 n)$ time, respectively. For a constant $k$, the second algorithm takes only $O(n\log^3n)$ time, which is near linear.

cs.CG↗

arXiv2018

Point Location in Dynamic Planar Subdivisions

We study the point location problem on dynamic planar subdivisions that allows insertions and deletions of edges. In our problem, the underlying graph of a subdivision is not necessarily connected. We present a data structure of linear size for such a dynamic planar subdivision that supports sublinear-time update and polylogarithmic-time query. Precisely, the amortized update time is $O(\sqrt{n}\log n(\log\log n)^{3/2})$ and the query time is $O(\log n(\log\log n)^2)$, where $n$ is the number of edges in the subdivision. This answers a question posed by Snoeyink in the Handbook of Computational Geometry. When only deletions of edges are allowed, the update time and query time are just $O(\alpha(n))$ and $O(\log n)$, respectively.

cs.CG↗