Online Competitive Searching for Rays in the Half-plane

TL;DR

This paper studies an online search problem for rays in the half-plane, proposing a nearly optimal strategy with a competitive ratio close to the theoretical lower bound, and extends the approach to terrain search problems.

Contribution

It introduces an efficient search strategy for rays in the half-plane with near-optimal competitive ratio and provides geometric proofs applicable to terrain search problems.

Findings

Competitive ratio less than 9.12725 achieved

Lower bound of at least 9.06357 established

Strategy is nearly optimal with less than 0.06368 gap

Abstract

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.