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

2025-08-14

AI Summary

This paper studies the worst-case normalized search time minimization problem for collaborative search of a hidden point target in the plane by mobile agents with heterogeneous speeds. We propose a speed-dependent angular-offset logarithmic spiral framework, integrating conical partitioning with a hybrid spiral–directional strategy to achieve geometric coordination between task assignment and trajectory design. Theoretically, we prove that multi-speed systems can outperform same-size single-speed systems, with optimality characterized by the geometric mean of agent speeds; moreover, slow agents can be safely eliminated under specific conditions. We derive, for the first time, a closed-form upper bound (U_n) on the worst-case search time for (n) unit-speed agents, and provide a tight worst-case upper bound for arbitrary speed configurations. Experiments demonstrate that the proposed hybrid strategy significantly outperforms pure spiral methods, especially when slow agents are present.

Technology Category

• Search and Optimization: Heuristic Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Search

Application Category

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• Minimize worst-case search time for hidden target using multi-speed agents
• Extend optimal spiral search to multiple agents with equal speeds
• Develop hybrid strategies for multi-speed agents outperforming pure spiral algorithms

Innovation

• Multi-speed agents use spiral trajectories
• Speed-dependent angular offsets optimize search
• Hybrid spiral-directional strategies outperform pure spirals