Competitive Searching over Terrains

TL;DR

This paper studies competitive searching for an unknown target on known terrains when the searcher can fly above the surface. It delivers a tight lower bound and a nearly optimal strategy for 1.5D terrains, and shows that 2.5D terrains admit a lower bound that scales as $\Omega(\sqrt{\lambda})$ with the terrain’s maximum slope $\lambda$, matched by a practical $O(\sqrt{\lambda})$ strategy. The 1.5D results extend to unknown terrains, while 2.5D results rely on the slope parameter, revealing unbounded competitiveness without a slope bound. Overall, the work provides foundational limits and constructive strategies for visibility-based drone search on layered terrains with precise dependence on geometric parameters.

Abstract

We study a variant of the searching problem where the environment consists of a known terrain and the goal is to obtain visibility of an unknown target point on the surface of the terrain. The searcher starts on the surface of the terrain and is allowed to fly above the terrain. The goal is to devise a searching strategy that minimizes the competitive ratio, that is, the worst-case ratio between the distance traveled by the searching strategy and the minimum travel distance needed to detect the target. For $1.5$D terrains we show that any searching strategy has a competitive ratio of at least $\sqrt{82}$ and we present a nearly-optimal searching strategy that achieves a competitive ratio of $3\sqrt{19/2} \approx \sqrt{82} + 0.19$. This strategy extends directly to the case where the searcher has no knowledge of the terrain beforehand. For $2.5$D terrains we show that the optimal competitive ratio depends on the maximum slope $λ$ of the terrain, and is hence unbounded in general. Specifically, we provide a lower bound on the competitive ratio of $Ω(\sqrtλ)$. Finally, we complement the lower bound with a searching strategy based on the maximum slope of the known terrain, which achieves a competitive ratio of $O(\sqrtλ)$.

Figures (13)

• Figure 1: Our searching strategy starts from $p_0$ and then follows the blue searching path. When the searcher reaches $p_t\xspace$, it can see the target $t$.
• Figure 2: The lower bound construction for the 1.5D case.
• Figure 3: The strategy $P$ (blue) and the projected path $P^*$ (gray). The goal is seen at $p_t\xspace$.
• Figure 4: By moving $r$ to the left until we are longer on a right segment with slope $s$, we can either hit a left subpath (case on the left), or a right segment with slope greater than $s$ (case on the right).
• Figure 5: A flat visibility ray $r$ that intersects a left subpath.

Theorems & Definitions (34)

• theorem thmcountertheorem
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof