A Constant‐Factor Approximation Algorithm for Optimal 1.5D Terrain Guarding

Ben‐Moshe, Boaz; Katz, Matthew J.; Mitchell, Joseph S. B.

doi:10.1137/s0097539704446384

Cited by 60 publications

(91 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The first constant-factor approximation algorithm for the 1.5D terrain guarding problem was given by Ben-Moshe et al [3]. Their algorithm works by first placing guards to divide the terrain into independent subterrains.…”

Section: Related Workmentioning

confidence: 99%

“…The 1.5D case is also applicable, for example, if we only need to cover the path between two points on a polyhedral terrain. It has been pointed out [3] that the 1.5D terrain guarding problem can be utilized in heuristic methods for the 2.5D case.…”

Section: Motivationmentioning

confidence: 99%

“…The recent results of Ben-Moshe et al [3] and Clarkson and Varadarajan [7] showed that constant-factor approximation algorithms exist for TG. Unfortunately they do not provide a small constant guaranteed approximation factor.…”

Section: Motivationmentioning

confidence: 99%

See 2 more Smart Citations

A 4-Approximation Algorithm for Guarding 1.5-Dimensional Terrains

King

2006

LATIN 2006: Theoretical Informatics

View full text Add to dashboard Cite

Section: Related Workmentioning

confidence: 99%

Section: Motivationmentioning

confidence: 99%

See 1 more Smart Citation

A 4-Approximation Algorithm for Guarding 1.5-Dimensional Terrains

King

2006

LATIN 2006: Theoretical Informatics

View full text Add to dashboard Cite

“…In orthogonal polygons, Nilsson [13] gives an algorithm to compute O(OP T 2 ) guards, based on the constant-factor approximation for guarding 1.5D terrains [2]. Worman and Keil [20] show that a minimum guard cover in the r-visibility model for orthogonal polygons can be found in polynomial time.…”

Section: Introductionmentioning

confidence: 99%

The Art Gallery Theorem for Polyominoes

Biedl

Irfan

Iwerks

et al. 2012

Discrete Comput Geom

Self Cite

View full text Add to dashboard Cite

We explore the art gallery problem for the special case that the domain (gallery) P is an mpolyomino, a polyform whose cells are m unit squares. We study the combinatorics of guarding polyominoes in terms of the parameter m, in contrast with the traditional parameter n, the number of vertices of P . In particular, we show that ⌊ m+1 3 ⌋ point guards are always sufficient and sometimes necessary to cover an m-polyomino, possibly with holes. When m ≤ 3n 4 − 4, the sufficiency condition yields a strictly lower guard number than ⌊ n 4 ⌋, given by the art gallery theorem for orthogonal polygons.

show abstract

“…If the guards are at some fixed height, the minimum number of guards can be found in polynomial time [19]. 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.…”

Section: Introductionmentioning

confidence: 99%

Guarding a Terrain by Two Watchtowers

et al. 2009

View full text Add to dashboard Cite

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 .

show abstract

A Constant‐Factor Approximation Algorithm for Optimal 1.5D Terrain Guarding

Cited by 60 publications

References 9 publications

A 4-Approximation Algorithm for Guarding 1.5-Dimensional Terrains

A 4-Approximation Algorithm for Guarding 1.5-Dimensional Terrains

The Art Gallery Theorem for Polyominoes

Guarding a Terrain by Two Watchtowers

Contact Info

Product

Resources

About