A Coreset for Approximate Furthest-Neighbor Queries in a Simple Polygon

TL;DR

The paper addresses approximate furthest-neighbor queries for a set of points $P$ inside a simple polygon $\mathcal{P}$ under geodesic distance. It introduces an $\varepsilon$-coreset of size $O(1/\varepsilon^2)$, $C\subset P$, such that for every query point $q\in\mathcal{P}$, the geodesic distance to the furthest neighbor in $C$ approximates the true furthest distance within a factor $1-\varepsilon$, independently of $|P|$ and the polygon complexity. The construction relies on reducing to a diameter path $\Gamma=\pi(p_1,p_2)$, a cone-based decomposition, pockets along $\Gamma$, and a careful handling of points inside and outside the relative convex hull, with a total time of $O\left(\frac{1}{\varepsilon} \left( n\log(1/\varepsilon) + (n+m)\log(n+m)\right) \right)$ for building the coreset. This enables a data structure whose storage is $O(m+1/\varepsilon^2)$ for answering $\varepsilon$-approximate furthest-neighbor queries, significantly reducing dependence on $n$ and $m$ while preserving fast query times. The results have implications for geometric optimization tasks in polygons and open avenues for further tightening the core-set size and extending to polygons with holes.

Abstract

Let $\mathcal{P}$ be a simple polygon with $m$ vertices and let $P$ be a set of $n$ points inside $\mathcal{P}$. We prove that there exists, for any $\varepsilon>0$, a set $\mathcal{C} \subset P$ of size $O(1/\varepsilon^2)$ such that the following holds: for any query point $q$ inside the polygon $\mathcal{P}$, the geodesic distance from $q$ to its furthest neighbor in $\mathcal{C}$ is at least $1-\varepsilon$ times the geodesic distance to its further neighbor in $P$. Thus the set $\mathcal{C}$ can be used for answering $\varepsilon$-approximate furthest-neighbor queries with a data structure whose storage requirement is independent of the size of $P$. The coreset can be constructed in $O\left(\frac{1}{\varepsilon} \left( n\log(1/\varepsilon) + (n+m)\log(n+m)\right) \right)$ time.

Figures (7)

• Figure 1: (i) The boundary of $\hbox{\sc rch}(P)$, the relative convex hull, is shown in green. The points $p_1,p_2$ define $\mathrm{diam}(P)$, so $\Gamma=\pi(p_1,p_2)$. The extension of $\Gamma$ splits $\mathcal{P}$ into two pieces: $\mathcal{P}_1(\Gamma^*)$, shown in pink, and $\mathcal{P}_2(\Gamma^*)$, shown in white. (ii) The 1-sphere of directions $\mathbb{S}^1$.
• Figure 2: (i) A geodesic triangle $\Delta_{\pi}p'q'r'$ and the pseudo-triangle $\Delta_{\pi}pqr$ defined by the points where the paths meet. The extension of $\mathrm{last}(\pi(p,u))$ hits $\pi(q,r)$. (ii) Illustration for Lemma \ref{['lem:geo-triangle']}.
• Figure 4: Pockets in $\mathcal{P}_1(\Gamma^*)$, which is the region below $\Gamma^*$, are shown in blue and pockets in $\mathcal{P}_2(\Gamma^*)$ are shown in green. (i) The point $p$ is reachable from $x$ in direction of the cones $\Phi_j$, which is shown in pink. (ii) the edge $e$ is an intermediate edge.
• Figure 5: Illustration for the proof of Lemma \ref{['lem:cross-case']}.
• Figure 9: The segments in $B$ are shown in red. Segment $s_1$ intersects $\partial\sigma$, so besides $s_1\cap\sigma$, the segments forming the connected component of $\partial\sigma \cap \mathcal{P}$ containing $x_1$ are added to $B$ as well. Since $s_2$ lies fully inside $\sigma$, no segments are added to $B$ for $s_2$, except $s_2$ itself.

Theorems & Definitions (9)

• Theorem 1
• Lemma 2
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 9