平面上的搜索与救援问题
Search and Rescue on the Plane
浏览论文内容
中文总结 AI 辅助
该研究针对平面搜索与救援问题，确定临界角$\theta^* \approx 15.6^\circ$，据此给出不同角度下智能体的最优检查点，推导竞争比闭式表达式，优化搜索与交付的竞争性能。
中文摘要 AI 辅助
我们研究搜索与救援问题的平面变体：一个智能体从平面上任意位置$P_{\theta,r} = (r\cos\theta, r\sin\theta)$出发，需定位正x轴上未知位置的物体并将其送至原点。我们的主要贡献包括：刻画任何竞争算法的最优形式、推导竞争比的闭式表达式、确定临界角$\theta^* \approx 15.6^\circ$，该角度会引发最优竞争搜索与交付的相变。对于每个角度$-\pi \leq \theta \leq \pi$，我们计算一个检查点（x轴上的着陆位置），智能体需先前往该点，再启动x轴上的搜索以优化搜索与交付的竞争比。我们证明：若$|\theta| \geq \theta^*$，检查点为原点；若$|\theta| < \theta^*$，智能体应着陆在x轴上的检查点$(r \cdot k_{|\theta|}, 0)$，其中$k_{|\theta|}$是我们给出显式公式的实数。
英文摘要
We study a planar variant of the search and rescue problem whereby an agent starting at an arbitrary position $P_{θ,r} = (r\cosθ, r\sinθ)$ in the plane must locate an object at an unknown position on the positive $x$-axis and deliver it to the origin. Our main contribution is to characterize the optimal form of any competitive algorithm, derive closed-form expressions for the competitive ratio, and identify a critical angle $θ^* \approx 15.6^\circ$ which yields a phase transition to optimal competitive search and delivery in the following sense. For each angle $-π\leq θ\leq π$ we compute a checkpoint (landing position on the $x$-axis) where the agent must go first prior to initiating a search on the $x$-axis in order to optimize the competitive ratio of search and delivery. We show that if $|θ| \geq θ^*$ then the checkpoint is at the origin, while if $|θ| < θ^*$ then the agent should land at the checkpoint $(r \cdot k_{|θ|}, 0)$ on the $x$-axis, where $k_{|θ|}$ is a real number given by an explicit formula we present.
