Guarding Terrains with Guards on a Line
Byeonguk Kang, Hwi Kim, Hee-Kap Ahn

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 -monotone polygonal chain with vertices, and an integer , we consider the problem of finding the lowest horizontal line lying above with point guards lying on , so that every point on the chain is \emph{visible} from some guard. A natural optimization is to minimize the -coordinate of . We present an algorithm for finding the optimal placements of and point guards for in time for even numbers , and in time for odd numbers , where is the length of the longest -Davenport-Schinzel sequence. We also study a variant with an additional requirement that is partitioned into subchains, each subchain is paired with exactly one guard, and every point on a subchain is visible from its paired guard. When is fixed, we can place the…
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Taxonomy
TopicsTree Root and Stability Studies · Soil Mechanics and Vehicle Dynamics
