Guarding Path Polygons with Orthogonal Visibility

Abstract

We are interested in the problem of guarding simple orthogonal polygons with the minimum number of $ r $-guards. The interior point $ p $ belongs an orthogonal polygon $ P $ is visible from $ r $-guard $ g $, if the minimum area rectangle contained $ p $ and $ q $ lies within $ P $. A set of point guards in polygon $ P $ is named guard set (as denoted $ G $) if the union of visibility areas of these point guards be equal to polygon $ P $ i.e. every point in $ P $ be visible from at least one point guards in $ G $. For an orthogonal polygon, if dual graph of vertical decomposition is a path, it is named path polygon. In this paper, we show that the problem of finding the minimum number of $ r $-guards (minimum guard set) becomes linear-time solvable in orthogonal path polygons. The path polygon may have dent edges in every four orientations. For this class of orthogonal polygon, the problem has been considered by Worman and Keil who described an algorithm running in $ O(n^{17} poly\log n) $-time where $ n $ is the size of the input polygon. The problem of finding minimum number of guards for simple polygon with general visibility is NP-hard, even if polygon be orthogonal. Our algorithm is purely geometric and presents a new strategy for $ r $-guarding orthogonal polygons and guards can be placed everywhere in the interior and boundary of polygon.