TY - JOUR

T1 - Guarding orthogonal art galleries with sliding cameras

AU - Durocher, Stephane

AU - Filtser, Omrit

AU - Fraser, Robert

AU - Mehrabi, Ali D.

AU - Mehrabi, Saeed

N1 - Publisher Copyright:
© 2017 Elsevier B.V.

PY - 2017/10/1

Y1 - 2017/10/1

N2 - 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).

AB - 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).

KW - Approximation algorithms

KW - Orthogonal art galleries

KW - Sliding cameras

UR - http://www.scopus.com/inward/record.url?scp=85018412383&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2017.04.001

DO - 10.1016/j.comgeo.2017.04.001

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AN - SCOPUS:85018412383

SN - 0925-7721

VL - 65

SP - 12

EP - 26

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

ER -