Guarding Terrains with Guards on a Line

2025-05-05

AI Summary

This paper studies the minimum-height coverage problem of deploying $k$ collinear horizontal guards above an $x$-monotone polygonal chain $T$: find a horizontal line $L$ with minimal $y$-coordinate admitting $k$ point guards on $L$ such that every point on $T$ is visible from at least one guard. A variant partitions $T$ into $k$ subchains, each fully visible to and paired with a distinct guard. We present the first exact optimal algorithm for the primary problem, running in $O(k^2 lambda_{k-1}(n) log n)$ time for even $k$, or $O(k^2 lambda_{k-2}(n) log n)$ for odd $k$, where $lambda_s(cdot)$ denotes the Davenport–Schinzel sequence function. For the variant, we devise near-linear algorithms: $O(n)$ for fixed $k$, or $O(kn)$ when the guard line is fixed. Our approach integrates visibility graph modeling, sweep-line techniques, and divide-and-conquer optimization. These results significantly advance the theoretical foundations and computational efficiency of collinear guard deployment in terrain surveillance.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Applications

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $O(k^2lambda_{k-1}(n)log n)$ time for even numbers $kge 2$, and in $O(k^2lambda_{k-2}(n)log n)$ 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.

Problem

• Find optimal guard positions on a line to cover a polygonal chain.
• Minimize the y-coordinate of the guard line for efficiency.
• Partition the chain into subchains each guarded by one guard.

Innovation

• Optimized guard placement on horizontal line
• Efficient algorithm for visibility coverage
• Partitioning terrain into guard-specific subchains