Guarding Terrains with Guards on a Line

TL;DR

The paper studies guarding an $x$-monotone terrain $T$ with $n$ vertices using $k$ point guards placed on a horizontal line $L$ above $T$, aiming to minimize the height of $L$ while ensuring visibility of every terrain point. It formalizes two problems: Altitude Terrain Cover (ATC) and Bijective Altitude Terrain Cover (BATC), and develops algorithms that leverage monotone guard-placement functions, shortest-path trees, and Davenport-Schinzel sequence bounds to achieve near-optimal runtimes. For ATC, the authors obtain $O(k^2\boldsymbol{λ}_{k-1}(n)\log n)$ time for even $k\ge2$ and $O(k^2\boldsymbol{λ}_{k-2}(n)\log n)$ for odd $k\ge3$; for BATC, they guarantee $O(n)$ time to minimize the number of guards on a fixed $L$ and $O(kn)$ time to compute the lowest $L$ with $k$ guards. The work combines parametric search, quasiconvex programming, and rational-function envelopes to handle multiple guards, offering efficient exact solutions for terrain-guarding problems with practical implications in surveillance and monitoring contexts. Overall, the results advance the algorithmic understanding of altitude-based terrain guarding and provide scalable methods for both unrestricted and bijective partition variants.

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 $O(k^2λ_{k-1}(n)\log n)$ time for even numbers $k\ge 2$, and in $O(k^2λ_{k-2}(n)\log n)$ time for odd numbers $k \ge 3$, where $λ_{s}(n)$ 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.

Figures (10)

• Figure 1: Cases for $k=2$. (a) The optimal placement of $L$ such that $T$ can be covered by two guards lying on $L$. (b) The optimal placement of $L$ such that $T$ is partitioned into two subchains: one of them is visible from the red guard and the other is visible from the blue guard.
• Figure 2: (a) The gray region is the set of points visible from every point of $e$. $g(e, h)\xspace$ (resp. $f(e, h)\xspace$) is the leftmost (resp. rightmost) point of $L(h)$ in the region. (b) $f(h)\xspace$ is the leftmost point among $f(e, h)\xspace$ for all edges $e$ of $T$.
• Figure 3: (a) Red dashed segments are edges of $\mathsf{S}_n$. For any fixed $h \in I_{0}$, (b) $f(vv', h)\xspace = f(v, h)\xspace$ if $x(\pi_n(v)) < x(\pi_n(v'))$. (c) $f(vv', h)\xspace = f(v', h)\xspace$ if $x(\pi_n(v)) \geqslant x(\pi_n(v')$.
• Figure 4: (a) Red and blue half-line represents $f(e, h)\xspace$ and $g(e, h)\xspace$ for an edge $e$ of $T$, respectively. (b) We have $W = \langle w_1, w_2, w_3, w_4, w_5 \rangle$, and $(y_0, y'_0]=(y(w_3),y(w_4)]$.
• Figure 5: (a) $e$ is an edge of $T$. $f(h)=g(e,h)$. (b) $vv'$ is an edge of $\mathsf{S}_1$. We have that $\overline{vv'} \cap e$ is contained in the interior of $e$, $q = f(h)\xspace \cap \overline{vv'}$, and $\textsf{peak}(e,q) = \{v, v'\}$ for $h = y(q)$. For arbitrary small $\epsilon > 0$, $\textsf{peak}(e,q_1) = \{v'\}$ for $h = y(q) + \epsilon$, and $\textsf{peak}(e,q_2) = \{v\}$ for $h = y(q) - \epsilon$.

Theorems & Definitions (46)

• Lemma 1
• Lemma 2
• proof
• Lemma 3
• Lemma 4
• Lemma 5
• proof
• proof
• Lemma 6
• proof