Let P be an orthogonal polygon with n vertices. A sliding camera travels back and forth along an orthogonal line segment s⊆P corresponding to its trajectory. The camera sees a point p∈P if there is a point q∈s such that pq‾ is a line segment normal to s that is completely contained in P. In the Minimum-Cardinality Sliding Cameras (MCSC) problem, the objective is to find a set S of sliding cameras of minimum cardinality to guard P (i.e., every point in P can be seen by some sliding camera in S), while in the Minimum-Length Sliding Cameras (MLSC) problem the goal is to find such a set S so as to minimize the total length of trajectories along which the cameras in S travel. In this paper, we answer questions posed by Katz and Morgenstern (2011) by presenting the following results: (i) the MLSC problem is polynomially tractable even for orthogonal polygons with holes, (ii) the MCSC problem is NP-complete when P is allowed to have holes, and (iii) an O(n3logn)-time 2-approximation algorithm for the MCSC problem on [NE]-star-shaped orthogonal polygons with n vertices (similarly, [NW]-, [SE]-, or [SW]-star-shaped orthogonal polygons).
|Number of pages||15|
|Journal||Computational Geometry: Theory and Applications|
|State||Published - 1 Oct 2017|
Bibliographical noteFunding Information:
Work of the author is supported by the Dutch Science Foundation (NWO) under grant 612.001.118.
© 2017 Elsevier B.V.
- Approximation algorithms
- Orthogonal art galleries
- Sliding cameras