We study the problem of searching for a target at some unknown location in $\mathbb{R}^d$ when additional information regarding the position of the target is available in the form of predictions. In our setting, predictions come as approximate distances to the target: for each point $p\in \mathbb{R}^d$ that the searcher visits, we obtain a value $\lambda(p)$ such that $|p\mathbf{t}|\le \lambda(p) \le c\cdot |p\mathbf{t}|$, where $c\ge 1$ is a fixed constant, $\mathbf{t}$ is the position of the target, and $|p\mathbf{t}|$ is the Euclidean distance of $p$ to $\mathbf{t}$. The cost of the search is the length of the path followed by the searcher. Our main positive result is a strategy that achieves $(12c)^{d+1}$-competitive ratio, even when the constant $c$ is unknown. We also give a lower bound of roughly $(c/16)^{d-1}$ on the competitive ratio of any search strategy in $\mathbb{R}^d$.

翻译：本研究探讨在获得目标位置预测信息的情况下，于$\mathbb{R}^d$空间中搜索未知位置目标的问题。在我们的设定中，预测信息以近似距离的形式提供：对于搜索者访问的每个点$p\in \mathbb{R}^d$，我们获得一个满足$|p\mathbf{t}|\le \lambda(p) \le c\cdot |p\mathbf{t}|$的数值$\lambda(p)$，其中$c\ge 1$为固定常数，$\mathbf{t}$表示目标位置，$|p\mathbf{t}|$表示点$p$到目标$\mathbf{t}$的欧几里得距离。搜索成本由搜索路径的长度决定。我们的主要正向结果提出了一种搜索策略，即使在不已知常数$c$的情况下，仍能实现$(12c)^{d+1}$的竞争比。同时，我们证明了在$\mathbb{R}^d$空间中，任何搜索策略的竞争比下界约为$(c/16)^{d-1}$。