Optimistic Imprecise Shortest Watchtower in 1.5D and 2.5D

2026-01-19

AI Summary

This study addresses the problem of selecting vertex heights within prescribed vertical intervals on uncertain 1.5D and 2.5D terrains to minimize the height of the shortest watchtower that can cover the entire terrain. The work presents the first linear-time exact algorithm for the 1.5D case and introduces an approximation scheme for the discrete 2.5D setting with additive error ε, running in O((OPT/ε)·n³) time. By integrating techniques for handling interval constraints, visibility analysis, and approximation algorithm design, this research overcomes key computational bottlenecks in visibility optimization under terrain uncertainty, substantially advancing the theoretical and algorithmic foundations of the problem.

Technology Category

• Search and Optimization: Heuristic Search
• Planning, Routing, and Scheduling: Planning under Uncertainty
• Reasoning under Uncertainty: Uncertainty Representations

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 ${OPT}$.

Problem

• imprecise terrain
• shortest watchtower
• optimistic shortest watchtower
• 1.5D terrain
• 2.5D terrain

Innovation

• imprecise terrain
• shortest watchtower
• linear-time algorithm
• additive approximation scheme
• computational geometry