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 ${OPT}$.

翻译：1.5维不精确地形是指一条$x$坐标固定且单调的多段线，其每个顶点的$y$坐标并非固定值，而是被约束在给定的垂直区间内。2.5维不精确地形则是一个三角剖分，其顶点的$x$和$y$坐标固定，但每个顶点的$z$坐标被约束在给定的垂直区间内。给定一个包含$n$个区间的不精确地形，乐观最短瞭望塔问题要求在每个垂直区间内选取一个精确点以构建实际地形$T$，使得满足以下条件的最短垂直线段的高度最小化：该线段的底端点位于$T$上，且其顶端点能够通视整个地形。本文提出一种线性时间算法，可精确求解1.5维乐观最短瞭望塔问题。针对2.5维情况的离散版本（要求瞭望塔必须放置于$T$的顶点上），我们给出了一种加法近似方案，其时间复杂度为$O(\frac{OPT}{\varepsilon}n^3)$，所得解与最优解值${OPT}$的加法误差不超过$\varepsilon$。