Optimistic Imprecise Shortest Watchtower in 1.5D and 2.5D

TL;DR

This paper introduces a linear-time exact algorithm for the 1.5D optimistic shortest watchtower problem and an approximation scheme for the 2.5D case, optimizing watchtower placement in imprecise terrains.

Contribution

It provides the first linear-time solution for the 1.5D case and an efficient approximation scheme for the 2.5D case with bounded error.

Findings

Linear time algorithm for 1.5D case

Additive approximation scheme for 2.5D case

Solution within ε of optimal with polynomial complexity

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 ${OPT}$.