Efficient Guarding of Polygons and Terrains

Ashok, Pradeesha; Reddy, Meghana M.

doi:10.1007/978-3-030-18126-0_3

Cited by 6 publications

(5 citation statements)

References 23 publications

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“…The most natural parameter for the art gallery problem is the solution size. As the art gallery problem is W [1]-hard, when parameterized by the solution size [21], research focused on other parameters [11,9,10,14,50,51,8]. Specifically, Agrawal et al [9] described an elegant FPT algorithm for the art gallery problem.…”

Section: Related Workmentioning

confidence: 99%

A Practical Algorithm with Performance Guarantees for the Art Gallery Problem

Hengeveld,

Miltzow

2024

Discrete Mathematics &Amp; Theoretical Computer Science

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Given a closed simple polygon $P$, we say two points $p,q$ see each other if the segment $pq$ is fully contained in $P$. The art gallery problem seeks a minimum size set $G\subset P$ of guards that sees $P$ completely. The only currently correct algorithm to solve the art gallery problem exactly uses algebraic methods and is attributed to Sharir. As the art gallery problem is ER-complete, it seems unlikely to avoid algebraic methods, without additional assumptions. In this paper, we introduce the notion of vision stability. In order to describe vision stability consider an enhanced guard that can see "around the corner" by an angle of $\delta$ or a diminished guard whose vision is by an angle of $\delta$ "blocked" by reflex vertices. A polygon $P$ has vision stability $\delta$ if the optimal number of enhanced guards to guard $P$ is the same as the optimal number of diminished guards to guard $P$. We will argue that most relevant polygons are vision stable. We describe a one-shot vision stable algorithm that computes an optimal guard set for visionstable polygons using polynomial time and solving one integer program. It guarantees to find the optimal solution for every vision stable polygon. We implemented an iterative visionstable algorithm and show its practical performance is slower, but comparable with other state of the art algorithms. Our iterative algorithm is inspired and follows closely the one-shot algorithm. It delays several steps and only computes them when deemed necessary. Given a chord $c$ of a polygon, we denote by $n(c)$ the number of vertices visible from $c$. The chord-width of a polygon is the maximum $n(c)$ over all possible chords $c$. The set of vision stable polygons admits an FPT algorithm when parametrized by the chord-width. Furthermore, the one-shot algorithm runs in FPT time, when parameterized by the number of reflex vertices.

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Section: Related Workmentioning

confidence: 99%

A Practical Algorithm with Performance Guarantees for the Art Gallery Problem

Hengeveld,

Miltzow

2024

Discrete Mathematics &Amp; Theoretical Computer Science

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“…Surprisingly, the art gallery problem is probably not contained in NP, as it is ∃R-complete [3]. In recent years, the art gallery problem has been studied also from various other theoretical perspectives: approximation [12,19], smoothed analysis [18,22], and parameterized complexity [6][7][8]13]. From a practical perspective, various research groups developed, implemented, and tested algorithms that are exact, reliable, and relatively fast, but theoretically may run forever [14,24,27,44].…”

Section: Art Gallery Problemmentioning

confidence: 99%

Topological Art in Simple Galleries

Bertschinger¹,

El²,

Miltzow³

et al. 2022

Symposium on Simplicity in Algorithms (SOSA)

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Let P be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in P. We say two points a, b ∈ P can see each other if the line segment seg(a, b) is contained in P. We denote by V (P) the family of all minimum guard placements. The Hausdorff distance makes V (P) a metric space and thus a topological space. We show homotopy-universality, that is, for every semi-algebraic set S there is a polygon P such that V (P) is homotopy equivalent to S. Furthermore, for various concrete topological spaces T , we describe instances I of the art gallery problem such that V (I ) is homeomorphic to T .

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“…Literature also shows a variety of multi-agent scenarios that apply to the coverage problem (Laguna and Bhattacharya 2019;Ashok and Reddy 2019). Guards in these studies are also assumed to have unlimited range and a 360 • view angle.…”

Section: Theoretical Workmentioning

confidence: 99%

Dynamic Guard Patrol in Stealth Games

Enezi

Verbrugge

2020

AIIDE

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Guard patrols are common in stealth games, both for a sense of realism and to form a basic challenge to acting out covert behaviour. Automated patrol route construction, however, is difficult, as simple approaches based on randomized movement or goal selection tend to result in unrealistic and overly repetitive guarding motion, and poor area coverage. In this work we introduce an approach based on modelling location “staleness” in a manner similar to occupancy maps, but applied to both a grid representation and a dynamically constructed coverage-visibility mesh. The latter provides a continuous abstraction of the game environment which can be efficiently used and maintained for real-time decision making. We use Unity3D and multiple game levels to compare our grid and mesh-based results, showing the advantages of the visibility mesh over the grid-based approach.

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Efficient Guarding of Polygons and Terrains

Cited by 6 publications

References 23 publications

A Practical Algorithm with Performance Guarantees for the Art Gallery Problem

A Practical Algorithm with Performance Guarantees for the Art Gallery Problem

Topological Art in Simple Galleries

Dynamic Guard Patrol in Stealth Games

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