Online Hitting Set for Axis-Aligned Squares

TL;DR

The paper studies online hitting sets for geometric ranges in the plane, focusing on finite point sets $P$ and a sequence of axis-aligned squares arriving online. It introduces a deterministic $O(\log n)$-competitive algorithm for squares of arbitrary sizes, using a Balanced Box Decomposition (BBD) tree and a carefully defined set of extremal points to ensure monotone growth of the hitting set. The approach extends to axis-aligned rectangles with bounded aspect ratio, achieving $O(\varrho\log n)$-competitiveness, and further generalizes to positive homothets of polygons with $k\ge3$ vertices, achieving $O(k^2\log n)$. This work advances the state of online geometric hitting sets by breaking the $O(\log^2 n)$ barrier and raises open questions about disks and higher-dimensional extensions.

Abstract

We are given a set $P$ of $n$ points in the plane, and a sequence of axis-aligned squares that arrive in an online fashion. The online hitting set problem consists of maintaining, by adding new points if necessary, a set $H\subseteq P$ that contains at least one point in each input square. We present an $O(\log n)$-competitive deterministic algorithm for this problem. The competitive ratio is the best possible, apart from constant factors. In fact, this is the first $O(\log n)$-competitive algorithm for the online hitting set problem that works for geometric objects of arbitrary sizes (i.e., arbitrary scaling factors) in the plane. We further generalize this result to positive homothets of a polygon with $k\geq 3$ vertices in the plane and provide an $O(k^2\log n)$-competitive algorithm.

Figures (8)

• Figure 1: The rectangle $r_{\rm in}$ is not sticky for the rectangle $r_{\rm out}$ in (a) and (b), and sticky in (c) and (d).
• Figure 2: In both (a) and (b), the rectangle $R$ crosses the cell $C_v$, but $R'$ does not.
• Figure 3: The extremal points of the cell $C_v$, i.e., ${\rm Ext}_v$ are colored red, when (a) $r_{\rm in}(v)=\emptyset$; (b) $r_{\rm in}(v)\neq\emptyset$.
• Figure 4: Illustration for (a) \ref{['lem_line']}; (b) \ref{['lem:halfplane']}.
• Figure 5: Illustration for \ref{['lem:strip']}.

Theorems & Definitions (19)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof