Spirals and Beyond: Competitive Plane Search with Multi-Speed Agents

Abstract

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.