Online Geometric Covering and Piercing

TL;DR

This work analyzes online piercing/covering where geometric objects arrive and irrevocable piercing decisions must be made. It leverages a unit covering–piercing equivalence to extend known online results from translates of convex objects to piercing, and develops algorithms (notably Algorithm-Center and Algorithm-Vertex) that achieve dimension-dependent upper bounds for similarly sized α-fat and α-aspect∞ fat objects, including homothetic hypercubes. The paper provides lower bounds for 2D α-fat objects and higher-dimensional hypercubes, highlighting gaps in higher dimensions. Overall, the results yield nearly general upper bounds for online unit covering with convex shapes beyond balls and cubes, and point to future work on tighter bounds and randomized online approaches with practical implications for sensor networks and related geometric covering problems.

Abstract

We consider the online version of the piercing set problem, where geometric objects arrive one by one, and the online algorithm must maintain a valid piercing set for the already arrived objects by making irrevocable decisions. It is easy to observe that any deterministic algorithm solving this problem for intervals in $\mathbb{R}$ has a competitive ratio of at least $Ω(n)$. This paper considers the piercing set problem for similarly sized objects. We propose a deterministic online algorithm for similarly sized fat objects in $\mathbb{R}^d$. For homothetic hypercubes in $\mathbb{R}^d$ with side length in the range $[1,k]$, we propose a deterministic algorithm having a competitive ratio of at most~$3^d\lceil\log_2 k\rceil+2^d$. In the end, we show deterministic lower bounds of the competitive ratio for similarly sized $α$-fat objects in $\mathbb{R}^2$ and homothetic hypercubes in $\mathbb{R}^d$. Note that piercing translated copies of a convex object is equivalent to the unit covering problem, which is well-studied in the online setup. Surprisingly, no upper bound of the competitive ratio was known for the unit covering problem when the corresponding object is anything other than a ball or a hypercube. Our result yields an upper bound of the competitive ratio for the unit covering problem when the corresponding object is any convex object in $\mathbb{R}^d$.

Figures (9)

• Figure 1: Reflection of an object $\sigma$ through the point $o$.
• Figure 2: (a) A centrally symmetric convex object $C+v_{i-1}$ (colored yellow) contains points $c_1,c_2,\ldots,c_{i-1}$. Note that $(1 + \epsilon_{i-1})C + v_{i-1}$ is $\epsilon_{i-1}$-neighbourhood of $C+v_{i-1}$. The distance $d_C(q,x)<\epsilon_{i-1}$, where $q \in \mathop{\mathrm{int}}\nolimits((1 + \epsilon_{i-1})C + v_{i-1})$ and $x \in \mathop{\mathrm{int}}\nolimits(C + v_{i-1})$. The translate $C+v_{i-1}+v_{xq}$ (in dotted lines) contains $c_1,c_2,\ldots,c_{i-1}$ and $q$. (b) Cyan colored region, denotes the intersection region $A$, i.e., $\cap_{k=1}^{i-1} \sigma_k$.
• Figure 3: (a) The object colored green has two centers at $p$ and $q$. (b-c) Geometric interpretation of aspect ratio and aspect$_{\infty}$ ratio, respectively.
• Figure 4: (a) Illustration of annular region $A_{i}=H_{i}\setminus H_{i+1}$. (b) The cells of $\Delta'$ are depicted, where one of the corners of a cell $c\in\Delta'$ coincides with one of the corners of $H_i$.
• Figure 5: Points of $\Pi_d$ are drawn for (a) $d=1$, (b) $d=2$, and (c) $d=3$. Let $P_q$ be the hyperplane $\{x\in\mathbb{R}^{3}\ |\ x(x_{3})=q\}$. In Figure (c), the projection of $\Pi_3$ over planes $P_{\frac{k}{(1+\alpha)^i}}$ (colored yellow), $P_{0}$ (colored green) and $P_{\frac{-k}{(1+\alpha)^i}}$ (colored orange) over a rectangular region is depicted.

Theorems & Definitions (46)

• Theorem 1
• proof
• Definition 1: Unit Piercing Problem
• Definition 2: Unit Covering Problem
• Theorem 2
• Lemma 1
• proof
• Theorem 3
• proof
• Claim 1