We consider the online hitting set problem for the range space $Σ=(\cal X,\cal R)$, where the point set $\cal X$ is known beforehand, but the set $\cal R$ of geometric objects is not known in advance. Here, objects from $\cal R$ arrive one by one. The objective of the problem is to maintain a hitting set of the minimum cardinality by taking irrevocable decisions. In this paper, we consider the problem when objects are unit balls or unit hypercubes in $\mathbb{R}^d$, and the points from $\mathbb{Z}^d$ are used for hitting them. First, we address the case when objects are unit intervals in $\mathbb{R}$ and present an optimal deterministic algorithm with a competitive ratio of~$2$. Then, we consider the case when objects are unit balls. For hitting unit balls in $\mathbb{R}^2$ and $\mathbb{R}^3$, we present $4$ and $14$-competitive deterministic algorithms, respectively. On the other hand, for hitting unit balls in $\mathbb{R}^d$, we propose an $O(d^4)$-competitive deterministic algorithm, and we demonstrate that}, for $d<4$, the competitive ratio of any deterministic algorithm is at least $d+1$. In the end, we explore the case where objects are unit hypercubes. For hitting unit hypercubes in $\mathbb{R}^2$ and $\mathbb{R}^3$, we obtain $4$ and $8$-competitive deterministic algorithms, respectively. For hitting unit hypercubes in $\mathbb{R}^d$ ($d\geq 3$), we present an $O(d^2)$-competitive randomized algorithm. Furthermore, we prove that the competitive ratio of any deterministic algorithm for the problem is at least $d+1$ for any $d\in\mathbb{N}$.

翻译：我们考虑范围空间$Σ=(\cal X,\cal R)$的在线命中集问题，其中点集$\cal X$事先已知，但几何对象集$\cal R$并非预先知晓。此处，$\cal R$中的对象逐一到达。该问题的目标是通过做出不可撤销的决策，维护一个具有最小基数的命中集。在本文中，我们研究对象为$\mathbb{R}^d$中的单位球或单位超立方体，并使用$\mathbb{Z}^d$中的点进行命中的情形。首先，我们处理对象为$\mathbb{R}$中单位区间的情况，并提出一种竞争比为~$2$的最优确定性算法。接着，我们考虑对象为单位球的情形。对于命中$\mathbb{R}^2$和$\mathbb{R}^3$中的单位球，我们分别提出了竞争比为$4$和$14$的确定性算法。另一方面，对于命中$\mathbb{R}^d$中的单位球，我们提出了一种$O(d^4)$-竞争的确定性算法，并证明当$d<4$时，任何确定性算法的竞争比至少为$d+1$。最后，我们探讨对象为单位超立方体的情形。对于命中$\mathbb{R}^2$和$\mathbb{R}^3$中的单位超立方体，我们分别得到了竞争比为$4$和$8$的确定性算法。对于命中$\mathbb{R}^d$（$d\geq 3$）中的单位超立方体，我们提出了一种$O(d^2)$-竞争的随机算法。此外，我们证明对于任意$d\in\mathbb{N}$，该问题的任何确定性算法的竞争比至少为$d+1$。