A Characterization of Link-2 LR-visibility Polygons with Applications

Abstract

Two points $x, y$ inside a polygon $P$ are said to be mutually link-2 visible if there exists the third point $z \in P$ such that $z$ is visible from both $x$ and $y$. The polygon $P$ is link-2 $LR$ -visible if there are two points $s$, $t$ on the boundary of $P$ such that every point on the clockwise boundary of $P$ from $s$ to $t$ is link-2 visible from some point of the other boundary of $P$ from $t$ to $s$ and vice versa. We give a characterization of link-2 $LR$-visibility polygons by generalizing the known result on $LR$-visibility polygons. A main idea is to extend the concepts of ray-shootings and components to those under notion of link-2 visibility. Then, we develop an $O(n \log n)$ time algorithm to determine whether a given polygon is link-2 $LR$-visible.

Using the characterization of link-2 $LR$-visibility polygons, we further present an $O(n \log n)$ time algorithm for determining whether a polygonal region is searchable by a $k$-searcher, $k \ge 2$, improving upon the previous $O(n^2)$ time bound. A polygonal region is searchable by a searcher if the searcher can detect (or see) an unpredictable intruder inside the region, no matter how fast the intruder moves. A $k$ -searcher holds $k$ flashlights and can see only along the rays of the flashlights emanating from his position. Our result can also be used to simplify the existing solutions of other polygon search problems.

This work was partially supported by the Grant-in-Aid (MEXT/JSPS KAKENHI 23500024) for Scientific Research from Japan Society for the Promotion of Science and the National Natural Science Foundation of China under grant 61173034.