We consider the problem of minimizing the worst-case search time for a hidden point target in the plane using multiple mobile agents of differing speeds, all starting from a common origin. The search time is normalized by the target's distance to the origin, following the standard convention in competitive analysis. The goal is to minimize the maximum such normalized time over all target locations, the search cost. As a base case, we extend the known result for a single unit-speed agent, which achieves an optimal cost of about $\mathcal{U}_1 = 17.28935$ via a logarithmic spiral, to $n$ unit-speed agents. We give a symmetric spiral-based algorithm where each agent follows a logarithmic spiral offset by equal angular phases. This yields a search cost independent of which agent finds the target. We provide a closed-form upper bound $\mathcal{U}_n$ for this setting, which we use in our general result. Our main contribution is an upper bound on the worst-case normalized search time for $n$ agents with arbitrary speeds. We give a framework that selects a subset of agents and assigns spiral-type trajectories with speed-dependent angular offsets, again making the search cost independent of which agent reaches the target. A corollary shows that $n$ multi-speed agents (fastest speed 1) can beat $k$ unit-speed agents (cost below $\mathcal{U}_k$) if the geometric mean of their speeds exceeds $\mathcal{U}_n / \mathcal{U}_k$. This means slow agents may be excluded if they lower the mean too much, motivating non-spiral algorithms. We also give new upper bounds for point search in cones and conic complements using a single unit-speed agent. These are then used to design hybrid spiral-directional strategies, which outperform the spiral-based algorithms when some agents are slow. This suggests that spiral-type trajectories may not be optimal in the general multi-speed setting.

翻译：我们研究了在平面上使用多个速度不同的移动智能体（均从共同原点出发）来最小化隐藏点目标最坏情况搜索时间的问题。搜索时间按目标到原点的距离进行归一化处理，遵循竞争分析中的标准惯例。目标是最小化所有目标位置对应的最大归一化时间，即搜索成本。作为基础情形，我们将已知的单个单位速度智能体（通过对数螺旋实现最优成本约 $\mathcal{U}_1 = 17.28935$）的结果推广至 $n$ 个单位速度智能体。我们提出一种基于对称螺旋的算法，其中每个智能体遵循具有相等角相位偏移的对数螺旋路径。这使得搜索成本与具体由哪个智能体发现目标无关。我们为此场景给出了闭形式上界 $\mathcal{U}_n$，并将其用于一般性结论中。我们的主要贡献是给出了 $n$ 个任意速度智能体在最坏情况归一化搜索时间上的上界。我们提出一个框架，通过选择智能体子集并分配具有速度相关角偏移的螺旋型轨迹，再次使搜索成本与抵达目标的智能体无关。一个推论表明：若 $n$ 个多速度智能体（最高速度为1）的速度几何平均数超过 $\mathcal{U}_n / \mathcal{U}_k$，则其性能可超越 $k$ 个单位速度智能体（成本低于 $\mathcal{U}_k$）。这意味着若低速智能体过度拉低均值，则可能被排除在搜索策略之外，这启发了非螺旋算法的设计。我们还使用单个单位速度智能体，给出了在锥形区域及其补集中进行点搜索的新上界。这些结果随后被用于设计混合螺旋-定向策略，当部分智能体速度较慢时，该策略性能优于基于螺旋的算法。这表明螺旋型轨迹在一般多速度场景中可能并非最优。