Search and Rescue on the Plane
This study addresses an online search problem in the plane where an agent, starting from an arbitrary position and orientation, must locate an unknown target on the positive x-axis and transport it to the origin. Through competitive analysis, geometric modeling, and optimization theory, the work reveals a strategy phase transition induced by a critical angle θ* ≈ 15.6°: when the initial heading angle is below θ*, the optimal strategy involves first moving to a specific checkpoint before searching along the x-axis; otherwise, direct search is optimal. The paper provides an explicit formula for the checkpoint location as a function of the initial angle, derives a closed-form expression for the competitive ratio, and fully characterizes the structure of the optimal competitive algorithm for any initial orientation, thereby achieving theoretically optimal performance.