We consider the online hitting set problem for the range space $\Sigma=(\cal X,\cal R)$, 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 $\Sigma=(\cal X,\cal R)$ 的在线击中集问题，其中点集 $\cal X$ 预先已知，但几何对象集 $\cal R$ 未知。在此，$\cal R$ 中的对象逐一到达。问题的目标是通过做出不可逆的决策来维持一个基数最小的击中集。本文考虑对象为 $\mathbb{R}^d$ 中单位球或单位超立方体，并使用 $\mathbb{Z}^d$ 中的点来击中的情形。首先，我们处理对象为 $\mathbb{R}$ 中单位区间的情形，并提出一个竞争比为 $2$ 的最优确定性算法。其次，考虑对象为单位球的情形。对于击中 $\mathbb{R}^2$ 和 $\mathbb{R}^3$ 中的单位球，我们分别提出竞争比为 $4$ 和 $14$ 的确定性算法。另一方面，对于击中 $\mathbb{R}^d$ 中的单位球，我们提出一个竞争比为 $O(d^4)$ 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，我们探究对象为单位超立方体的情形。对于击中 $\mathbb{R}^2$ 和 $\mathbb{R}^3$ 中的单位超立方体，我们分别获得竞争比为 $4$ 和 $8$ 的确定性算法。对于击中 $\mathbb{R}^d$（$d\geq 3$）中的单位超立方体，我们提出一个竞争比为 $O(d^2)$ 的随机化算法。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 $d+1$。