Online Hitting Sets for Disks of Bounded Radii

TL;DR

This work advances online hitting-set algorithms for planar geometric objects, achieving $O( ext{log } M ext{log } n)$ competitiveness for disks and positive homothets of convex bodies with radii or scaling in $[1,M]$, by reducing to a finite, lowest-point-property setting and employing layered tilings and line-separated reductions. It also delivers an $O( ext{log } N)$-competitive algorithm for bottomless rectangles in a lattice point setting, via canonical partitions and a structured hitting strategy. The core technique unifies a two-stage approach: reduce to line-separated problems with the lowest-point property, then extend via tilings and multi-layer decompositions to handle radii-varied objects and general convex bodies, including centrally symmetric ones. Collectively, the results push toward $O( ext{log } n)$-competitive online geometric hitting sets in 2D for a broad class of bounded-size range spaces and lay groundwork for higher-dimensional generalizations.

Abstract

We present algorithms for the online minimum hitting set problem in geometric range spaces: given a set $P$ of $n$ points in the plane and a sequence of geometric objects that arrive one-by-one, we need to maintain a hitting set at all times by making irrevocable decisions. For disks of radii in the interval $[1,M]$, we present an $O(\log M \log n)$-competitive algorithm. This result generalizes from disks to positive homothets of any convex body in the plane with scaling factors in the interval $[1,M]$. As a main technical tool, we reduce the problem to the online hitting set problem for a finite subset of integer points and geometric objects with the lowest point property, introduced in this paper, which behave similarly to bottomless rectangles. Specifically, for a given $N>1$, we present an $O(\log N)$-competitive algorithm for the variant where $P$ is a subset of an $N\times N$ section of the integer lattice, and the geometric objects have the lowest point property.

Figures (19)

• Figure 1: When the $i$th bottomless rectangle $r_i=[5,11)\times [0,c_i)$ arrives, suppose that the hitting set $H_i$ contains the points marked with hollow dots, and $r_i\cap H_i = \emptyset$. The splitting point of $[5,11)$ is 8, with canonical partitions $\mathcal{A}_i=[5,6)\cup[6,8)$ and $\mathcal{B}_i=[8,10)\cup[10,11)$, respectively. Here, $I_1=[6,8)\subset \mathcal{A}_i$ and $I_2=[8,10) \subset \mathcal{B}_i$ are the largest canonical intervals in $\mathcal{A}_i$ and $\mathcal{B}_i$, respectively. The points marked with hollow square (resp., solid square) is the lowest-point in $(I_1\times[0,N))\cap P$ (resp., $(I_2\times[0,N))\cap P$). We add both points to $H_i$.
• Figure 2: (a) Object $S$ has the lowest-point property. For example, in the yellow (shaded) strip $I\times \mathbb{R}$, two points have the minimum $y$-coordinate, and $S$ contains both. (b) Disk $S$, with center below the $x$-axis, does not have the lowest-point property: we have $s\in S$, and the interval $I\subset \mathrm{span}(S)$ contains $s_x$, but $S$ does not contain the point $p\in P$ which has the minimum $y$-coordinate in $I\times \mathbb{R}$.
• Figure 3: A point set $P$ (red) and region $\mathrm{hull}_3(P)$ (pink). The boundary $\partial \mathrm{hull}_3(P)$ is composed of horizontal lines and circular arcs.
• Figure 4: A point set $P$ (red), $\mathrm{hull}_3(P)$ (pink), and $\mathrm{hull}_1(P)$ (light blue or pink). A disk $D\in \mathcal{D}_3$ of radius 3 (dotted blue), where the intersection $D\cap (\partial \mathrm{hull}_1(P))$ has two components.
• Figure 5: Example for the bijection $\pi$. Left: $\partial \mathrm{hull}_1(P)$ is orange arcs, $\partial \mathrm{hull}_2(P)$ is dashed green arcs, and $\partial \mathrm{hull}_3(P)$ is dash-dot blue arcs. Right: the grid points $\pi(p_0),\ldots , \pi(p_9)$ corresponding to $p_0,\ldots ,p_9$.

Theorems & Definitions (44)

• Theorem 1
• proof
• Theorem 2
• proof
• Definition 1
• Lemma 1: Dumitrescu et al. DumitrescuGT22
• Lemma 2
• proof
• Lemma 3
• proof