arXiv subjects

Sang Duk Yoon

Publications and source records attributed to Sang Duk Yoon.

3 recordsLinked to original sources

arXiv2024

Maximum-Width Rainbow-Bisecting Empty Annulus

Given a set of $n$ colored points with $k$ colors in the plane, we study the problem of computing a maximum-width rainbow-bisecting empty annulus (of objects specifically axis-parallel square, axis-parallel rectangle and circle) problem. We call a region rainbow if it contains at least one point of each color. The maximum-width rainbow-bisecting empty annulus problem asks to find an annulus $A$ of a particular shape with maximum possible width such that $A$ does not contain any input points and it bisects the input point set into two parts, each of which is a rainbow. We compute a maximum-width rainbow-bisecting empty axis-parallel square, axis-parallel rectangular and circular annulus in $O(n^3)$ time using $O(n)$ space, in $O(k^2n^2\log n)$ time using $O(n\log n)$ space and in $O(n^3)$ time using $O(n^2)$ space respectively.

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 $π$ 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 $π/2$ and $2π/3$. We show a minimality of the covering for discrete rotation of multiples of $π/2$, 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 $2π/k$ for all integers $k\ge 3$.

cs.CG↗

arXiv2019

Empty Squares in Arbitrary Orientation Among Points

This paper studies empty squares in arbitrary orientation among a set $P$ of $n$ points in the plane. We prove that the number of empty squares with four contact pairs is between $Ω(n)$ and $O(n^2)$, and that these bounds are tight, provided $P$ is in a certain general position. A contact pair of a square is a pair of a point $p\in P$ and a side $\ell$ of the square with $p\in \ell$. The upper bound $O(n^2)$ also applies to the number of empty squares with four contact points, while we construct a point set among which there is no square of four contact points. These combinatorial results are based on new observations on the $L_\infty$ Voronoi diagram with the axes rotated and its close connection to empty squares in arbitrary orientation. We then present an algorithm that maintains a combinatorial structure of the $L_\infty$ Voronoi diagram of $P$, while the axes of the plane continuously rotates by $90$ degrees, and simultaneously reports all empty squares with four contact pairs among $P$ in an output-sensitive way within $O(s\log n)$ time and $O(n)$ space, where $s$ denotes the number of reported squares. Several new algorithmic results are also obtained: a largest empty square among $P$ and a square annulus of minimum width or minimum area that encloses $P$ over all orientations can be computed in worst-case $O(n^2 \log n)$ time.

cs.CG↗