Online Competitive Searching for Rays in the Half-plane

TL;DR

This work studies online ray searching in the half-plane, an extension of the classic cow-path problem. It introduces a cow-path–like strategy with exponential X-progression and a slope parameter, and analyzes the resulting ratio function to bound worst-case performance. The authors establish an upper bound of 9.12725 on the competitive ratio and a lower bound of 9.06357, narrowing the gap to under 0.064, and show how terrain-modeling (1.5D terrain) adaptations can only help or maintain these guarantees. They further reveal that the terrain's worst case reduces to simple vertical barriers, clarifying the core difficulty of the terrain-search variant and suggesting directions for tightening bounds.

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.

Figures (25)

• Figure 1: The cow is searching for the unknown hole in the fence and moves forth and back along the fence. Any reasonable search strategy can be represented by a sequence $X=(x_1,x_2,x_3,x_4,\ldots)$. The cow runs into a local worst case situation in relative comparison to the unknown shortest path to the target, when the target is distance $|st|=x_k+\epsilon$ away from the start and a full turn on the other side up to distance $x_{k+1}$ was done before (here for $k=5$).
• Figure 2: Searching for an unknown ray in the half-plane and competing with a search path $\Pi$ against the shortest path to the ray. If the goal set is restricted to rays perpendicular to $l$, we will simply get back to the cow-path problem.
• Figure 3: An example subset of rays we are looking for. For any such ray $r$ the shortest path from $s$ to the ray, i.e., a segment $st'$ with $t'\in r$ that runs perpendicular to $r$, also has to lie in the upper half-plane. Here the rays $r_1$ and $r_2$ are not in our focus. A reasonable search path, $\Pi$, might be given by a generalized cow-path that also increases its height successively. Note that we compete against any possible target ray with the given property.
• Figure 4: Searching for an unknown target $t$ in a 1.5D terrain $T$. By entering the visibility region, $\hbox{Vis}_T(t)$, of $t$ for the first time a ray $r_t$ that points toward the start will be visited first. Visibility rays pointing away from the start need not be considered. The shortest path for visiting the ray $r_t$ from $s$ that avoids the terrain, $\hbox{OPT}_T(s,r_t)$, need not be a single line segment.
• Figure 5: (i) Detecting unknown targets $t$ on the terrain by a search path $\Pi$ means entering the visibility region $\hbox{Vis}_T(t)$ at a first visibility ray $r_t$. All such rays point toward the target $s$. (ii) We can now shift the target points to the line $l$ that runs through $s$ in parallel to the ground line $G$ of the terrain. (iii) Considering the corresponding rays for the new targets and omitting the overall terrain itself results in an instance of our (pure) ray-search problem. The shortest path to the target ray is a line segment and can only decrease. We compete against a better offline solution. (iv) Any such ray-search scenario with a fixed set of rays can be easily transformed to a simple terrain search problem by making use of small pikes close to the target points.