Chaeyoon Chung

Ahmad Biniaz, Prosenjit Bose, Chaeyoon Chung et al.

Let $S$ be a set of $n$ points in $\mathbb{R}^d$, where $d \geq 2$ is a constant, and let $H_1,H_2,\ldots,H_{m+1}$ be a sequence of vertical hyperplanes that are sorted by their first coordinates, such that exactly $n/m$ points of $S$ are between any two successive hyperplanes. Let $A(S,m)$ be the set of different closest pairs in the ${{m+1} \choose 2}$ vertical slabs that are bounded by $H_i$ and $H_j$, over all $1 \leq i < j \leq m+1$. We prove tight bounds for the largest possible size of $A(S,m)$, over all point sets of size $n$, and for all values of $1 \leq m \leq n$. As a result of these bounds, we obtain, for any constant $ε>0$, a data structure of size $O(n)$, such that for any vertical query slab $Q$, the closest pair in the set $Q \cap S$ can be reported in $O(n^{1/2+ε})$ time. Prior to this work, no linear space data structure with sublinear query time was known.

Chaeyoon Chung, Anil Maheshwari, Michiel Smid

Given a set $S$ of $n$ points in the plane, we study the two-line-center problem: finding two lines that minimize the maximum distance from each point in $S$ to its closest line. We present a $(1+\varepsilon)$-approximation algorithm for the two-line-center problem that runs in $O((n/\varepsilon) \log (1/\varepsilon))$ time, which improves the previously best $O(n\log n + ({n}/{\varepsilon^2}) \log ({1}/{\varepsilon}) + (1/\varepsilon^3)\log ({1}/{\varepsilon}))$-time algorithm. We also consider three variants of this problem, in which the orientations of the two lines are restricted: (1) the orientation of one of the two lines is fixed, (2) the orientations of both lines are fixed, and (3) the two lines are required to be parallel. For each of these three variants, we give the first $(1+\varepsilon)$-approximation algorithm that runs in linear time. In particular, for the variant where the orientation of one of the two lines is fixed, we also give an improved exact algorithm that runs in $O(n \log n)$ time and show that it is optimal.