arXiv subjects

Eunku Park

Publications and source records attributed to Eunku Park.

5 recordsLinked to original sources

arXiv2026

Quadratic Complexity of Voronoi Diagrams in $\mathbb{R}^3$ for Lines in a Single Ruling of a Regulus

We study nearest and farthest Voronoi diagrams of lines in $\mathbb{R}^3$ under the Euclidean metric when all $n$ lines belong to one ruling of a smooth doubly ruled real quadric. For arbitrary line sites, the combinatorial complexity of the nearest Voronoi diagram is known only to lie between $\Omega(n^2)$ and $O(n^{3+\varepsilon})$. Under general-position assumptions, we prove that both diagrams in the ruling class have at most $4n(n-3)$ vertices and $O(n^2)$ total combinatorial complexity. Conversely, for every $n \ge 4$, one ruling of a fixed non-rotational one-sheeted hyperboloid contains a general-position set of $n$ lines with at least $(n-2)(n-3)/2$ distinct regular nearest vertices, where regular means that exactly four lines support the vertex and their three defining bisectors meet transversely. Thus the worst-case complexity of the nearest Voronoi diagram in this class is $\Theta(n^2)$, while the farthest diagram has $\Theta(n^2)$ complexity for every general-position input, since it has exactly $n(n-1)$ three-dimensional cells. Under the Pl\"ucker embedding, the ruling is a conic, and the condition for a line to be tangent to a Euclidean sphere restricts to a binary quartic. At a regular vertex, the four supporting parameters exhaust its roots, and sign alternation forces two arcs of the parameter circle to be site-free. This leaves only $n(n-3)/2$ possible cyclic support types, while B\'ezout's theorem bounds the number of centers for each type by eight. The same reduction yields an exact $O(n^2)$-time algorithm that, after cyclically sorting the site parameters, enumerates all finite nearest and farthest vertices as constant-degree real univariate representations.

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arXiv2026

Coresets for Continuous $k$-Center in Hyperbolic Space

We study the continuous $k$-center problem in fixed-dimensional hyperbolic space. The input is a finite point set $P \subset \mathbb H^D$, while the centers may be placed anywhere in the ambient space. A nonempty subset $P_\varepsilon\subseteq P$ is an $\varepsilon$-coreset if every exact optimal continuous $k$-center solution for $P_\varepsilon$ is a $(1+\varepsilon)$-approximate solution for $P$. Our construction uses Gonzalez's farthest-first traversal, whose radius $R_G$ gives a constant-factor estimate of the optimum. When $R_G$ is bounded, we map each cluster to the origin of the Poincar\'e ball model and apply a local Euclidean grid. For larger $R_G$, we partition each cluster into radial shells and project them onto representative hyperbolic spheres, avoiding a discretization of the ambient ball. By the hyperbolic law of cosines, the intersection of a hyperbolic ball with each sphere corresponds to a spherical cap. A recursive $k$-cap coverage certificate then replaces each shell with a radius-independent set of input witnesses that preserves its farthest-point behavior for every set of at most $k$ centers. The resulting coreset $P_\varepsilon$ has size $|P_\varepsilon| = \left({k}/{\varepsilon}\right)^{O(kD)}$ and satisfies $\Phi_P(C_{P_\varepsilon}) \le (1+\varepsilon)\operatorname{OPT}_k(P)$ for every optimal continuous $k$-center solution $C_{P_\varepsilon}$ of $P_\varepsilon$. It can be constructed in $O\left( nkD\left({k}/{\varepsilon}\right)^{O(kD)} \right)$ time. Consequently, for fixed $D$, $k$, and $\varepsilon$, the coreset has constant size and can be computed in linear time. We complement this upper bound by showing that, for every fixed $D \ge 1$, every $k \ge 2$, and all sufficiently small $\varepsilon$, there are instances for which every $\varepsilon$-coreset has size $\Omega\left(k\varepsilon^{-(D-1)/2}\right)$.

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arXiv2026

Computing Topological Transition Sets for Line-Line-Circle Trisectors in $R^3$

Computing the Voronoi diagram of mixed geometric objects in $R^3$ is challenging due to the high cost of exact geometric predicates via Cylindrical Algebraic Decomposition (CAD). We propose an efficient exact verification framework that characterizes the parameter space connectivity by computing certified topological transition sets. We analyze the fundamental non-quadric case: the trisector of two skew lines and one circle in $R^3$. Since the bisectors of circles and lines are not quadric surfaces, the pencil-of-quadrics analysis previously used for the trisectors of three lines is no longer applicable. Our pipeline uses exact symbolic evaluations to identify transition walls. Jacobian computations certify the absence of affine singularities, while projective closure shows singular behavior is isolated at a single point at infinity, $p_{\infty}$. Tangent-cone analysis at $p_{\infty}$ yields a discriminant $\Delta_Q = 4ks^2(k-1)$, identifying $k=0,1$ as bifurcation values. Using directional blow-up coordinates, we rigorously verify that the trisector's real topology remains locally constant between these walls. Finally, we certify that $k=0,1$ are actual topological walls exhibiting reducible splitting. This work provides the exact predicates required for constructing mixed-object Voronoi diagrams beyond the quadric-only regime.

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arXiv2025

Coresets for Farthest Point Problems in Hyperbolic Space

We show how to construct in linear time coresets of constant size for farthest point problems in fixed-dimensional hyperbolic space. Our coresets provide both an arbitrarily small relative error and additive error $\varepsilon$. More precisely, we are given a set $P$ of $n$ points in the hyperbolic space $\mathbb{H}^D$, where $D=O(1)$, and an error tolerance $\varepsilon\in (0,1)$. Then we can construct in $O(n/\varepsilon^D)$ time a subset $P_\varepsilon \subset P$ of size $O(1/\varepsilon^D)$ such that for any query point $q \in \mathbb{H}^D$, there is a point $p_\varepsilon \in P_\varepsilon$ that satisfies $d_H(q,p_\varepsilon) \geq (1-\varepsilon)d_H(q,f_P(q))$ and $d_H(q,p_\varepsilon) \geq d_H(q,f_P(q))-\varepsilon$, where $d_H$ denotes the hyperbolic metric and $f_P(q)$ is the point in $P$ that is farthest from $q$ according to this metric. This coreset allows us to answer approximate farthest-point queries in time $O(1/\varepsilon^D)$ after $O(n/\varepsilon^D)$ preprocessing time. It yields efficient approximation algorithms for the diameter, the center, and the maximum spanning tree problems in hyperbolic space.

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arXiv2024

Embeddings and near-neighbor searching with constant additive error for hyperbolic spaces

We give an embedding of the Poincaré halfspace $H^D$ into a discrete metric space based on a binary tiling of $H^D$, with additive distortion $O(\log D)$. It yields the following results. We show that any subset $P$ of $n$ points in $H^D$ can be embedded into a graph-metric with $2^{O(D)}n$ vertices and edges, and with additive distortion $O(\log D)$. We also show how to construct, for any $k$, an $O(k\log D)$-purely additive spanner of $P$ with $2^{O(D)}n$ Steiner vertices and $2^{O(D)}n \cdot λ_k(n)$ edges, where $λ_k(n)$ is the $k$th-row inverse Ackermann function. Finally, we show how to construct an approximate Voronoi diagram for $P$ of size $2^{O(D)}n$. It allows us to answer approximate near-neighbor queries in $2^{O(D)}+O(\log n)$ time, with additive error $O(\log D)$. These constructions can be done in $2^{O(D)}n \log n$ time.

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