Online Class Cover Problem

TL;DR

This paper investigates the online class cover problem in the plane with ${\mathcal P}_r$ red points and online-arriving blue points, using the family ${\mathcal F}$ of axis-parallel unit squares. It proves a fundamental lower bound of $\Omega(\log m)$ for deterministic online algorithms and furnishes a matching $O(\log m)$-competitive algorithm that, upon each uncovered blue point, emits at most five candidate squares derived from a staircase-based construction. The algorithm’s correctness is established via geometric invariants and a series of lemmas ensuring coverage without red points, and the competitive ratio is refined from $20\log m$ down to $10+10\log m$. The results illuminate the online behavior of geometric set covers and suggest avenues for extensions to other object families and potential randomized improvements, with practical implications for dynamic resource placement and pattern recognition tasks where red points denote forbidden regions.

Abstract

In this paper, we study the online class cover problem where a (finite or infinite) family $\cal F$ of geometric objects and a set ${\cal P}_r$ of red points in $\mathbb{R}^d$ are given a prior, and blue points from $\mathbb{R}^d$ arrives one after another. Upon the arrival of a blue point, the online algorithm must make an irreversible decision to cover it with objects from $\cal F$ that do not cover any points of ${\cal P}_r$. The objective of the problem is to place a minimum number of objects. When $\cal F$ consists of axis-parallel unit squares in $\mathbb{R}^2$, we prove that the competitive ratio of any deterministic online algorithm is $Ω(\log |{\cal P}_r|)$, and also propose an $O(\log |{\cal P}_r|)$-competitive deterministic algorithm for the problem.

Figures (15)

• Figure 1: (a) An axis-parallel unit square $R(u)=\square abcd$ centered at $u$. (b) The square $P+(v_1, 0)$ (red) is a horizontal translated copy of the square $P$ (black). (c) An $8\times 8$ grid ${\mathcal{G}}(Q)$ using $9$ points of $Q=\{u_1, u_2,\dots, u_9\}$ in a plane.
• Figure 2: (a) The point $u_2\in Q$ is nearest to the line $L$, where $Q=\{u_1, u_2,\dots, u_6\}$. (b) Here, $ST_{SW}(Q)$ is the sequence $\sigma=\{l_1, h_1, \dots,$$l_{4}, h_{4}\}$, where $Q=\{u_1, u_2,\dots, u_5\}$. (c) The square $P+(v_1, v_2)$ (red) is obtained by moving the square $P$ (black) rightwards along the staircase.
• Figure 3: Example of the lower bound of the online class cover problem using axis-parallel unit square where $m=9$. (a) $O_1$ is the red-colored square. The algorithm places the square $R_1$ (blue) to cover $b_1$. (b) $O_2$ is the red-colored square that covers both $b_1$ and $b_2$. The algorithm places the square $R_2$ (green) to cover $b_2$.
• Figure 5: Let $R(u)$ (black) be the square centered at the blue point $u$. (a) The sequence of $SW$ staircase points ${\cal Q}(u)_{SW}=(r_1,q_0,\dots,q_3,r_2)$ forms the staircase $ST_{SW}({\cal Q}(u)_{SW})$ (dark green), and $P_0,P_2,P_4$ (blue) are some staircase squares of $u$. Also, $u\in P_0,P_2,P_4$. (b) Case 1.1: $R'_{SW}$ contains no red points. Here, $R_4$ is a candidate square (blue) of $u$ obtained by moving $R'$ (black/dashed).
• Figure 6: Let $R(u)$ (black) be the square centered at a blue point $u$. The Red shaded sub-squares of $R(u)$ contain some red points.

Theorems & Definitions (34)

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