We consider the problem of searching for rays (or lines) in the half-plane. The given problem turns out to be a very natural extension of the cow-path problem that is lifted into the half-plane and the problem can also directly be motivated by a 1.5-dimensional terrain search problem. We present and analyse an efficient strategy for our setting and guarantee a competitive ratio of less than 9.12725 in the worst case and also prove a lower bound of at least 9.06357 for any strategy. Thus the given strategy is almost optimal, the gap is less than 0.06368. By appropriate adjustments for the terrain search problem we can improve on former results and present geometrically motivated proof arguments. As expected, the terrain itself can only be helpful for the searcher that competes against the unknown shortest path. We somehow extract the core of the problem.

翻译：本文研究在半平面中搜索射线（或直线）的问题。该问题被证明是牛径问题在半平面中的一种自然扩展，并且可直接由一维半地形搜索问题所启发。我们提出并分析了一种适用于该场景的高效搜索策略，在最坏情况下保证竞争比小于9.12725，同时证明任何策略的竞争比下界至少为9.06357。因此所提策略近乎最优，其差距小于0.06368。通过对地形搜索问题进行适当调整，我们改进了先前的研究结果，并提出了基于几何动机的证明方法。正如预期，地形本身仅对与未知最短路径竞争的搜索者具有辅助作用。我们通过分析提取了该问题的核心本质。