Approximating the Smallest $k$-Enclosing Geodesic Disc in a Simple Polygon

TL;DR

This work extends the smallest $k$-enclosing disc problem to geodesic metrics within simple polygons by introducing SKEG discs. It delivers both an exact algorithm based on higher-order geodesic Voronoi diagrams and polygon simplification, and two $2$-approximation strategies with distinct time-space tradeoffs. The main results show that a 2-SKEG disc can be found in expected time $O(n \log^2 n \log r + m)$ when $k = O(n/\log n)$, and with high probability in $O((n^2/k) \log n \log r + m)$ when $k = \omega(n/\log n)$, with all methods using $O(n+m)$ space. The paper also discusses leveraging $k$-nearest neighbour queries to potentially speed up the approximation, highlighting open questions about achieving $O(n\log r + m)$ time and tradeoffs in polygonal domains.

Abstract

We consider the problem of finding a geodesic disc of smallest radius containing at least $k$ points from a set of $n$ points in a simple polygon that has $m$ vertices, $r$ of which are reflex vertices. We refer to such a disc as a SKEG disc. We present an algorithm to compute a SKEG disc using higher-order geodesic Voronoi diagrams with worst-case time $O(k^{2} n + k^{2} r + \min(kr, r(n-k)) + m)$ ignoring polylogarithmic factors. We then present two $2$-approximation algorithms that find a geodesic disc containing at least $k$ points whose radius is at most twice that of a SKEG disc. The first algorithm computes a $2$-approximation with high probability in $O((n^{2} / k) \log n \log r + m)$ worst-case time with $O(n + m)$ space. The second algorithm runs in $O(n \log^{2} n \log r + m)$ expected time using $O(n + m)$ expected space, independent of $k$. Note that the first algorithm is faster when $k \in ω(n / \log n)$.

Figures (10)

• Figure 1: The case where the optimal disc $D^*$ contains points of $S$ from both sides of $\ell$.
• Figure 2: An example of points of $S$ (blue diamonds) and their projections onto $\ell$ (hollow black diamonds).
• Figure 3: The geodesic discs (arbitrarily red and blue) of radius $\rho$ centred on points of $S$ (blue points) intersect $\ell$. The intersection points of red (blue) disc boundaries with $\ell$ are marked by green (red) triangles. The intervals $I(u,\rho)$ for the points $u \in S$ are the intersections of $\ell$ with $D(u, \rho)$. Overlapping intervals illustrate points along $\ell$ where centering a geodesic of radius $\rho$ will contain multiple points of $S$ (i.e., the points of $S$ defining the overlapping intervals).
• Figure 4: Considering the chord $\ell$ of $P$ to be the $x$-axis, given a point $u \in S$ we refer to the dashed graph of the function $\operatorname{dist}_u(\cdot)$ as the distance function of $u$ to $\ell$. The points $f$ and $h$ on $\ell$ mark where different pieces of $\operatorname{dist}_u(\cdot)$ begin.
• Figure 5: The funnel from $u$ to the endpoints of $\ell$, including the apex $u_a$ and the projection $u_c$ of $u$ onto $\ell$. Also seen are the extensions of funnel edges (in blue) and their intersection points with $\ell$. These intersection points can be used to perform a binary search along $\ell$.

Theorems & Definitions (41)

• Theorem 1
• Remark 1
• Theorem 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 2
• proof