Coverage with k-Transmitters in the Presence of Obstacles

Ballinger, Brad; Benbernou, Nadia M.; Bose, Prosenjit; Damian, Mirela; Demaine, Erik D.; Dujmović, Vida; Flatland, Robin; Hurtado, Ferrán; Iacono, John; Lubiw, Anna; Morin, Pat; Sacristán, Vera; Souvaine, Diane L.; Uehara, Ryuhei

doi:10.1007/978-3-642-17461-2_1

Cited by 10 publications

(16 citation statements)

References 13 publications

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“…Our focus is on finding covers of lower power transmitters, that is, mainly 2-transmitters. This is in line with the work of Ballinger et al [3] and is motivated both by practical applications and by virtue of being the natural extension of classical Art Gallery results, that is, results for k = 0.…”

Section: Always Sufficientsupporting

confidence: 86%

“…Aichholzer et al [2] improved the bounds on monotone polygons to a tight value of n−2 2k+3 . In addition, they gave tight bounds for monotone orthogonal polygons for all values of k. Other publications explored k-transmitter coverage of regions other than simple polygons, such as coverage of the plane in the presence of line or line segment obstacles [3,11]. For example, Ballinger et al established that for disjoint segments in the plane, where each segment has one of two slopes and the entire plane is to be covered, 1 2 ( 5 6 log(k+1) n + 1) k-transmitters are always sufficient, and n+1 2k+2 k-transmitters are sometimes necessary.…”

Section: Introductionmentioning

confidence: 99%

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Combinatorics and complexity of guarding polygons with edge and point 2-transmitters

Cannon

Fai

Iwerks³

et al. 2018

Computational Geometry

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We consider a generalization of the classical Art Gallery Problem, where instead of a light source, the guards, called k-transmitters, model a wireless device with a signal that can pass through at most k walls. We show it is NP-hard to compute a minimum cover of point 2transmitters, point k-transmitters, and edge 2-transmitters in a simple polygon. The point 2transmitter result extends to orthogonal polygons. In addition, we give necessity and sufficiency results for the number of edge 2-transmitters in general, monotone, orthogonal monotone, and orthogonal polygons. * Abstracts of part of this work appeared in the informal workshops FWCG [7] and EuroCG [8] †

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Section: Always Sufficientsupporting

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Combinatorics and complexity of guarding polygons with edge and point 2-transmitters

Cannon

Fai

Iwerks³

et al. 2018

Computational Geometry

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“…The notion of k-visibility has previously been considered in the context of art-gallerystyle questions [5,13,16,22] and in the definition of certain geometric graphs [11,15,18]. While the 0-visibility region is always connected, the k-visibility region may have several components.…”

Section: Introductionmentioning

confidence: 99%

A time–space trade-off for computing the k-visibility region of a point in a polygon

Bahoo

Banyassady

Bose

et al. 2019

Theoretical Computer Science

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Let P be a simple polygon with n vertices, and let q ∈ P be a point in P . Let k ∈ {0, . . . , n − 1}. A point p ∈ P is k-visible from q if and only if the line segment pq crosses the boundary of P at most k times. The k-visibility region of q in P is the set of all points that are k-visible from q. We study the problem of computing the k-visibility region in the limited workspace model, where the input resides in a random-access read-only memory of O(n) words, each with Ω(log n) bits. The algorithm can read and write O(s) additional words of workspace, where s ∈ N is a parameter of the model. The output is written to a write-only stream.Given a simple polygon P with n vertices and a point q ∈ P , we present an algorithm that reports the k-visibility region of q in P in O(cn/s + c log s + min{ k/s n, n log log s n}) expected time using O(s) words of workspace. Here, c ∈ {1, . . . , n} is the number of critical vertices of P for q where the k-visibility region of q may change. We generalize this result for polygons with holes and for sets of non-crossing line segments.

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“…Over the last few decades, numerous variations of the above art gallery problem have been studied, including mobile guards, guards with limited visibility or mobility, guarding for special classes of polygons, etc. ; see O'Rourke's monograph [30], the survey articles [27,34,42], and recent papers [1,2,13,39].…”

Section: Introductionmentioning

confidence: 99%

Watchman routes for lines and line segments

Dumitrescu

Mitchell

Żyliński

2014

Computational Geometry

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Given a set L of non-parallel lines in the plane, a watchman route (tour) for L is a closed curve contained in the union of the lines in L such that every line is visited (intersected) by the route; we similarly define a watchman route (tour) for a connected set S of line segments. The watchman route problem for a given set of lines or line segments is to find a shortest watchman route for the input set, and these problems are natural special cases of the watchman route problem in a polygon with holes (a polygonal domain).In this paper, we show that the problem of computing a shortest watchman route for a set of n non-parallel lines in the plane is polynomially tractable, while it becomes NP-hard in 3D. We give an alternative NP-hardness proof of this problem for line segments in the plane and obtain a polynomial-time approximation algorithm with ratio O(log 3 n). Additionally, we consider some special cases of the watchman route problem on line segments, for which we provide exact algorithms or improved approximations.

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Coverage with k-Transmitters in the Presence of Obstacles

Cited by 10 publications

References 13 publications

Combinatorics and complexity of guarding polygons with edge and point 2-transmitters

Combinatorics and complexity of guarding polygons with edge and point 2-transmitters

A time–space trade-off for computing the k-visibility region of a point in a polygon

Watchman routes for lines and line segments

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