Face-guarding polyhedra

Viglietta, Giovanni

doi:10.1016/j.comgeo.2014.04.009

Cited by 5 publications

(6 citation statements)

References 8 publications

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“…If R is a collar, then b = b−1, m = m, and r = r−4 (see Figure 1). Plugging these into (12) and combining the result with (11) immediately yields (10), as claimed.…”

Section: Bounds In Terms Of Mmentioning

confidence: 74%

“…In this paper we study the Art Gallery problem for 2-reflex orthogonal polyhedra, which are orthogonal polyhedra having reflex edges oriented in just two directions, as opposed to three. These polyhedra have been introduced by the author and investigated in the context of face guards in [11]. In Section 2 we motivate the study of this special class of polyhedra, and we establish some terminology.…”

Section: Our Contributionsmentioning

confidence: 99%

“…Face guards have also been explored, first by Souvaine et al in [7], and then by the author in [11].…”

Section: Introduction Backgroundmentioning

confidence: 99%

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Optimally guarding 2-reflex orthogonal polyhedra by reflex edge guards

Viglietta

2020

Computational Geometry

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We study the problem of guarding an orthogonal polyhedron having reflex edges in just two directions (as opposed to three) by placing guards on reflex edges only.We show that r − g 2 + 1 reflex edge guards are sufficient, where r is the number of reflex edges in a given polyhedron and g is its genus. This bound is tight for g = 0. We thereby generalize a classic planar Art Gallery theorem of O'Rourke, which states that the same upper bound holds for vertex guards in an orthogonal polygon with r reflex vertices and g holes. Then we give a similar upper bound in terms of m, the total number of edges in the polyhedron. We prove that m − 4 8 + g reflex edge guards are sufficient, whereas the previous best known bound was 11m/72 + g/6 − 1 edge guards (not necessarily reflex). We also discuss the setting in which guards are open (i.e., they are segments without the endpoints), proving that the same results hold even in this more challenging case.Finally, we show how to compute guard locations in O(n log n) time.

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“…If R is a collar, then b = b−1, m = m, and r = r−4 (see Figure 1). Plugging these into (12) and combining the result with (11) immediately yields (10), as claimed.…”

Section: Bounds In Terms Of Mmentioning

confidence: 74%

Section: Our Contributionsmentioning

confidence: 99%

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Optimally guarding 2-reflex orthogonal polyhedra by reflex edge guards

Viglietta

2020

Computational Geometry

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“…One step up from terrains, we may consider simple polyhedra (surfaces of genus 0), or full 3D scenes. Visibility has been studied in full 3D as well [30][31][32]. To our knowledge, minimum-link paths in higher dimensions have not been studied before (with the exception of [33] that considered rectilinear paths).…”

Section: Domainsmentioning

confidence: 99%

On the complexity of minimum-link path problems

Kostitsyna,

Löffler,

Polishchuk

et al. 2016

Preprint

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We revisit the minimum-link path problem: Given a polyhedral domain and two points in it, connect the points by a polygonal path with minimum number of edges. We consider settings where the vertices and/or the edges of the path are restricted to lie on the boundary of the domain, or can be in its interior. Our results include bit complexity bounds, a novel general hardness construction, and a polynomial-time approximation scheme. We fully characterize the situation in 2 dimensions, and provide first results in dimensions 3 and higher for several variants of the problem.Concretely, our results resolve several open problems. We prove that computing the minimum-link diffuse reflection path, motivated by ray tracing in computer graphics, is NP-hard, even for two-dimensional polygonal domains with holes. This has remained an open problem [1] despite a large body of work on the topic. We also resolve the open problem from [2] mentioned in the handbook [3] (see Chapter 27.5, Open problem 3) and The Open Problems Project [4] (see Problem 22): "What is the complexity of the minimum-link path problem in 3-space?" Our results imply that the problem is NP-hard even on terrains (and hence, due to discreteness of the answer, there is no FPTAS unless P=NP), but admits a PTAS.

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“…As noted above, covering a 3-dimensional polyhedron by vertex guards may be unfeasible. There are some initial results on the minimum number of edge guards [2,5,13,27] and face guards [14,15,24,26], under the notion of weak visibility: An edge or a face f sees a point p if some point s ∈ f sees the point p. The problem formulation for edge and face guards can be further refined depending on whether (topologically) open or closed edges and faces are allowed. However, none of the current bounds is known to be tight.…”

Section: Introductionmentioning

confidence: 99%