Lower Bound of Face Guards of Polyhedral Terrains

Iwamoto, Chuzo; Junichi, Kishi; Morita, Kenichi

doi:10.2197/ipsjjip.20.435

Cited by 7 publications

(8 citation statements)

References 6 publications

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“…Notice that a polyhedral terrain has a different structure than a city with vertical buildings. The results related to guarding polyhedral terrain focus on edge and face guards [20,13,5,10,26,25].…”

Section: Polyhedral Terrain Resultsmentioning

confidence: 99%

City Guarding with Limited Field of View

Daescu,

Malik

2020

Preprint

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Drones and other small unmanned aerial vehicles are starting to get permission to fly within city limits. While video cameras are easily available in most cities, their purpose is to guard the streets at ground level. Guarding the aerial space of a city with video cameras is a problem that so far has been largely ignored.In this paper, we present bounds on the number of cameras needed to guard a city's aerial space (roofs, walls, and ground) using cameras with 180 • range of vision (the region in front of the guard), which is common for most commercial cameras. We assume all buildings are vertical and have a rectangular base. Each camera is placed at a top corner of a building.We considered the following two versions: (i) buildings have an axis-aligned ground base and, (ii) buildings have an arbitrary orientation. We give necessary and sufficient results for (i), necessary results for (ii), and conjecture sufficiency results for (ii). Specifically, for (i) we prove a sufficiency bound of 2k + ⌊ k 4 ⌋ + 4 on the number of vertex guards, while for (ii) we show that 3k + 1 vertex guards are sometimes necessary, where k is total number of buildings in the city.

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Section: Polyhedral Terrain Resultsmentioning

confidence: 99%

City Guarding with Limited Field of View

Daescu,

Malik

2020

Preprint

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“…A terrain whose bounded faces are triangles is called a triangulated terrain. These special terrains are studied in [9,10,11], where it is shown that computing the minimum number of closed face guards in a given triangulated terrain is NP-hard. Here we strengthen this result by showing that such a minimum is even NP-hard to approximate within a logarithmic factor, for both open and closed face guards, provided that terrains are not necessarily triangulated.…”

Section: Minimizing Face Guards 41 Hardness Of Approximationmentioning

confidence: 99%

Face-guarding polyhedra

Viglietta

2014

Computational Geometry

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We study the Art Gallery Problem for face guards in polyhedral environments. The problem can be informally stated as: how many (not necessarily convex) windows should we place on the external walls of a dark building, in order to completely illuminate its interior?We consider both closed and open face guards (i.e., faces with or without their boundary), and we study several classes of polyhedra, including orthogonal polyhedra, 4-oriented polyhedra, and 2-reflex orthostacks.We give upper and lower bounds on the minimum number of faces required to guard the interior of a given polyhedron in each of these classes, in terms of the total number of its faces, f . In several cases our bounds are tight: f /6 open face guards for orthogonal polyhedra and 2-reflex orthostacks, and f /4 open face guards for 4-oriented polyhedra. Additionally, for closed face guards in 2-reflex orthostacks, we give a lower bound of (f +3)/9 and an upper bound of (f +1)/7 .Then we show that it is NP-hard to approximate the minimum number of (closed or open) face guards within a factor of Ω(log f ), even for polyhedra that are orthogonal and simply connected. We also obtain the same hardness results for polyhedral terrains.Along the way we discuss some applications, arguing that face guards are not a reasonable model for guards patrolling on the surface of a polyhedron.

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“…It is known that ⌊(2n − 5)/7⌋ is the lower bound and ⌊n/3⌋ is the upper bound for the number of face guards of an n-vertex triangulated polyhedral terrain [10]. After that, both the lower and upper bounds are shown to be ⌊(n − 1)/3⌋ [11].…”

Section: Introductionmentioning

confidence: 99%

Visibility Problems for Manhattan Towers

Iwamoto

Kitagaki

2016

IEICE Trans. Inf. & Syst.

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SUMMARY A Manhattan tower is a monotone orthogonal polyhedron lying in the halfspace z ≥ 0 such that (i) its intersection with the xy-plane is a simply connected orthogonal polygon, and (ii) the horizontal cross section at higher levels is nested in that for lower levels. Here, a monotone polyhedron meets each vertical line in a single segment or not at all. We study the computational complexity of finding the minimum number of guards which can observe the side and upper surfaces of a Manhattan tower. It is shown that the vertex-guarding, edge-guarding, and face-guarding problems for Manhattan towers are NP-hard.

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Lower Bound of Face Guards of Polyhedral Terrains

Abstract: Abstract:We study the problem of determining the minimum number of face guards which cover the surface of a polyhedral terrain. We show that (2n − 5)/7 face guards are sometimes necessary to guard the surface of an n-vertex triangulated polyhedral terrain.

Cited by 7 publications

References 6 publications

City Guarding with Limited Field of View

City Guarding with Limited Field of View

Face-guarding polyhedra

Visibility Problems for Manhattan Towers

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