Guarding Terrains with Guards on a Line

TL;DR

This paper introduces algorithms for optimally placing a horizontal line and point guards to surveil an $x$-monotone polygonal chain, minimizing the line's height and ensuring full visibility with efficient computation.

Contribution

It presents new algorithms with improved time complexity for optimal guard placement and line positioning on an $x$-monotone chain, including variants with partitioned subchains.

Findings

Algorithms run in $O(k^2 ext{lambda}_{k-1}(n) ext{log} n)$ time for even $k$

Algorithms run in $O(k^2 ext{lambda}_{k-2}(n) ext{log} n)$ time for odd $k$

Optimal placement can be found in $O(n)$ time when $L$ is fixed

Abstract

Given an $x$-monotone polygonal chain $T$ with $n$ vertices, and an integer $k$, we consider the problem of finding the lowest horizontal line $L$ lying above $T$ with $k$ point guards lying on $L$, so that every point on the chain is \emph{visible} from some guard. A natural optimization is to minimize the $y$-coordinate of $L$. We present an algorithm for finding the optimal placements of $L$ and $k$ point guards for $T$ in $O(k^2\lambda_{k-1}(n)\log n)$ time for even numbers $k\ge 2$, and in $O(k^2\lambda_{k-2}(n)\log n)$ time for odd numbers $k \ge 3$, where $\lambda_{s}(n)$ is the length of the longest $(n,s)$-Davenport-Schinzel sequence. We also study a variant with an additional requirement that $T$ is partitioned into $k$ subchains, each subchain is paired with exactly one guard, and every point on a subchain is visible from its paired guard. When $L$ is fixed, we can place the minimum number of guards in $O(n)$ time. When the number $k$ of guards is fixed, we can find an optimal placement of $L$ with $k$ point guards lying on $L$ in $O(kn)$ time.