Guarding Monotone Art Galleries with Sliding Cameras in Linear Time

Abstract

A sliding camera in an orthogonal polygon $P$ is a point guard $g$ that travels back and forth along an orthogonal line segment $s$ inside $P$. A point $p$ in $P$ is guarded by $g$ if and only if there exists a point $q$ on $s$ such that line segment $pq$ is normal to $s$ and contained in $P$. In the minimum sliding cameras (MSC) problem, the objective is to guard $P$ with the minimum number of sliding cameras. We give a linear-time dynamic programming algorithm for the MSC problem on $x$-monotone orthogonal polygons, improving the 2-approximation algorithm of Katz and Morgenstern (2011). More generally, our algorithm can be used to solve the MSC problem in linear time on simple orthogonal polygons $P$ for which the dual graph induced by the vertical decomposition of $P$ is a path. Our results provide the first polynomial-time exact algorithms for the MSC problem on a non-trivial subclass of orthogonal polygons.

Mark de Berg—Supported by the Netherlands Organisation for Scientific Research (NWO) under project 024.002.003.

Stephane Durocher—Supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Saeed Mehrabi—Supported in part by a Manitoba Graduate Scholarship (MGS).