  We present an $O(nrG)$ time algorithm for computing and maintaining a minimum
length shortest watchman tour that sees a simple polygon under monotone
visibility in direction $\theta$, while $\theta$ varies in $[0,180^{\circ})$,
obtaining the directions for the tour to be the shortest one over all tours,
where $n$ is the number of vertices, $r$ is the number of reflex vertices, and
$G\leq r$ is the maximum number of gates of the polygon used at any time in the
algorithm.

Computational Geometry