Inapproximability Results for Guarding Polygons and Terrains

Eidenbenz, Stephan; Stamm, Christoph; Widmayer, Peter

doi:10.1007/s00453-001-0040-8

Cited by 117 publications

(93 citation statements)

References 17 publications

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“…The so-called art gallery problem and its variants, widely studied in computational geometry and robotics, ask for covering P by a set of sensors so that every point in P is visible from at least one sensor. The art-gallery problem is known to be APX-hard [15], i.e., constant-factor approximation cannot be computed in polynomial time unless P = N P . If there is no restriction on the placements of sensors, no approximation algorithm is known.…”

Section: Introductionmentioning

confidence: 99%

Efficient Sensor Placement for Surveillance Problems

Agarwal

Ezra

Ganjugunte

2009

Distributed Computing in Sensor Systems

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Abstract. We study the problem of covering a two-dimensional spatial region P , cluttered with occluders, by sensors. A sensor placed at a location p covers a point x in P if x lies within sensing radius r from p and x is visible from p, i.e., the segment px does not intersect any occluder. The goal is to compute a placement of the minimum number of sensors that cover P . We propose a landmark-based approach for covering P . Suppose P has ς holes, and it can be covered by h sensors. Given a small parameter ε > 0, let λ := λ(h, ε) = (h/ε) log ς. We prove that one can compute a set L of O(λ log λ log (1/ε)) landmarks so that if a set S of sensors covers L, then S covers at least (1 − ε)-fraction of P . It is surprising that so few landmarks are needed, and that the number does not depend on the number of vertices in P . We then present efficient randomized algorithms, based on the greedy approach, that, with high probability, compute O(h log λ) sensor locations to cover L; hereh ≤ h is the number sensors needed to cover L. We propose various extensions of our approach, including: (i) a weight function over P is given and S should cover at least (1 − ε) of the weighted area of P , and (ii) each point of P is covered by at least t sensors, for a given parameter t ≥ 1.

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Section: Introductionmentioning

confidence: 99%

Efficient Sensor Placement for Surveillance Problems

Agarwal

Ezra

Ganjugunte

2009

Distributed Computing in Sensor Systems

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“…Recently, Ben-Moshe, Katz, and Mitchell [3], and, independently, Clarkson and Varadarajan [8], discovered constant-factor approximation algorithms for the problem. Eidenbenz et al [12] show that the related problem of guarding a simple polygon with a minimum number of guards is APX-hard. So for the problem of guarding polygons there is an ε such that it is NP-hard to obtain a (1 + ε)-approximation for the minimum number of guards.…”

Section: Introductionmentioning

confidence: 99%

Guarding a Terrain by Two Watchtowers

et al. 2009

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Given a polyhedral terrain T with n vertices, the two-watchtower problem for T asks to find two vertical segments, called watchtowers, of smallest common height, whose bottom endpoints (bases) lie on T , and whose top endpoints guard T , in the sense that each point on T is visible from at least one of them. There are three versions of the problem, discrete, semi-discrete, and continuous, depending on whether two, one, or none of the two bases are restricted to be among the vertices of T , respectively.In this paper we present the following results for the two-watchtower problem in R 2 and R 3 : (1) We show that the discrete two-watchtowers problem in R 2 can be solved in O(n 2 log 4 n) time, significantly improving previous solutions. The algorithm works, without increasing its asymptotic running time, for the semi-continuous version, where one of the towers is allowed to be placed anywhere on T . (2) We show that the continuous two-watchtower problem in R 2 can be solved in O(n 3 α(n) log 3 n) time, again significantly improving previous results. (3) Still in R 2 , we show that the continuous version of the problem of guarding a finite set P ⊂ T of m points by two watchtowers of smallest common height can be solved in O(mn log 4 n) time. (4) We show that the discrete version of the two-watchtower problem in R 3 can be solved in O(n 11/3 polylog(n)) time; this is the first nontrivial result for this problem in R 3 .

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“…Recently, Eidenbenz et al [7] proved inapproximability results for several versions of the art gallery problem. They show that the art gallery problem for the case of art gallery without holes is APX-hard, i.e., no polynomial time algorithm can achieve an approximation ratio of 1+6 for a constant 6 > 0 for the art gallery without holes unless P = NP.…”

Section: Related Workmentioning

confidence: 99%

“…The recent interest in the problem stems from applications in the area of wireless communications [7,12]. The signal coverage of an antenna is modeled as a sphere [7].…”

Section: Applicationsmentioning

confidence: 99%