Optimistic Imprecise Shortest Watchtower in 1.5D and 2.5D

TL;DR

This work investigates the optimistic shortest watchtower problem on imprecise terrains, focusing on 1.5D and 2.5D models. It delivers a linear-time exact algorithm for 1.5D by showing the optimum realization belongs to $O(n)$ canonical forms and that the top of the watchtower lies on the shortest path $\pi$ from $t_1$ to $t_n$, with height derived from the visibility polygon $P(\pi)$. For the discrete 2.5D case, it provides an $O\left(\frac{OPT}{\varepsilon}n^3\right)$-time additive approximation scheme, validating near-optimal solutions when the watchtower base is restricted to a vertex. The paper also presents an $O(n^3)$-time algorithm to decide zero-watchtower feasibility in 3D and discusses open questions about the tractability of the general 2.5D problem and potential PTAS approaches for the discrete version. Overall, the results advance efficient guarding of imprecise terrains and delineate clear computational boundaries between 1.5D and 2.5D cases.

Abstract

A 1.5D imprecise terrain is an $x$-monotone polyline with fixed $x$-coordinates, the $y$-coordinate of each vertex is not fixed but is constrained to be in a given vertical interval. A 2.5D imprecise terrain is a triangulation with fixed $x$ and $y$-coordinates, but the $z$-coordinate of each vertex is constrained to a given vertical interval. Given an imprecise terrain with $n$ intervals, the optimistic shortest watchtower problem asks for a terrain $T$ realized by a precise point in each vertical interval such that the height of the shortest vertical line segment whose lower endpoint lies on $T$ and upper endpoint sees the entire terrain is minimized. In this paper, we present a linear time algorithm to solve the 1.5D optimistic shortest watchtower problem exactly. For the discrete version of the 2.5D case (where the watchtower must be placed on a vertex of $T$), and we give an additive approximation scheme running in $O(\frac{OPT}{\varepsilon}n^3)$ time, achieving a solution within an additive error of $\varepsilon$ from the optimal solution value ${OPT}$.

Figures (7)

• Figure 1: (\ref{['fig:1.5D_with_polygon']}) A possible realization of a 1.5D imprecise terrain. The shaded region represents the unbounded convex polygon $P$. (\ref{['fig:2.5D_imprecise']}) A possible realization of a 2.5D imprecise terrain.
• Figure 2: The quadrilateral $D$ from the proof of Lemma \ref{['lem:wings']}. The dashed diagonal is visible from the top of the watchtower.
• Figure 3: The dashed line represents the shortening of the solid realization. Shortening a path with fixed endpoints does not increase the height of a watchtower.
• Figure 4: A realization where raising the edge containing the base of the watchtower too much increases the watchtower height.
• Figure 5: (\ref{['fig:too-much-p']}) The intersection of extended edges determine the point $p$. (\ref{['fig:poly-p']}) The polygon $Q_p$.

Theorems & Definitions (23)

• lemma thmcounterlemma: Raise Wings
• proof
• lemma thmcounterlemma: Shorten
• proof
• lemma thmcounterlemma: Top of tower determined by $\pi$
• proof
• lemma thmcounterlemma: Raising the base
• proof
• theorem thmcountertheorem: Discrete Watchtower
• proof