Terrain-Like Graphs: PTASs for Guarding Weakly-Visible Polygons and Terrains

Ashur, Stav; Filtser, Omrit; Katz, Matthew J.; Saban, Rachel

doi:10.1007/978-3-030-39479-0_1

Cited by 3 publications

(10 citation statements)

References 14 publications

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“…Our algorithm for vertex guarding a WV-polygon uses a solution to the problem of guarding the boundary of a WV-polygon using vertex guards. This problem admits a local-search-based PTAS (see [4,20]), which is similar to the local-search-based PTAS of Gibson et al [18] for vertex guarding the vertices of a 1.5D-terrain. The proof of both these PTASs is based on the proof scheme of Mustafa and Ray [27].…”

Section: Related Workmentioning

confidence: 99%

“…In the second part, it computes a subset G of the vertices of P of size at most |G|, such that G ∪ G guards P (boundary plus interior). Thus, if we apply the algorithm of [4] for computing G, then the approximation ratio of our algorithm is 2 + ε, since the former algorithm guarantees that |G| ≤ (1 + ε/2)OPT ∂ , where OPT ∂ is the size of a minimum-cardinality subset of the vertices of P that guards P 's boundary, and clearly OPT ∂ ≤ OPT.…”

Section: Related Workmentioning

confidence: 99%

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A Constant-Factor Approximation Algorithm for Vertex Guarding a WV-Polygon

Ashur

Filtser

Katz

2021

Approximation and Online Algorithms

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The problem of vertex guarding a simple polygon was first studied by Subir K. Ghosh (1987), who presented a polynomial-time O(log n)-approximation algorithm for placing as few guards as possible at vertices of a simple n-gon P , such that every point in P is visible to at least one of the guards. Ghosh also conjectured that this problem admits a polynomial-time algorithm with constant approximation ratio. Due to the centrality of guarding problems in the field of computational geometry, much effort has been invested throughout the years in trying to resolve this conjecture. Despite some progress (surveyed below), the conjecture remains unresolved to date. In this paper, we confirm the conjecture for the important case of weakly visible polygons, by presenting a (2 + ε)-approximation algorithm for guarding such a polygon using vertex guards. A simple polygon P is weakly visible if it has an edge e, such that every point in P is visible from some point on e. We also present a (2 + ε)-approximation algorithm for guarding a weakly visible polygon P , where guards may be placed anywhere on P 's boundary (except in the interior of the edge e). Finally, we present an O(1)-approximation algorithm for vertex guarding a polygon P that is weakly visible from a chord.Our algorithms are based on an in-depth analysis of the geometric properties of the regions that remain unguarded after placing guards at the vertices to guard the polygon's boundary. Finally, our algorithms may become useful as part of the grand attempt of Bhattacharya et al. to prove the original conjecture, as their approach is based on partitioning the underlying simple polygon into a hierarchy of weakly visible polygons. * An earlier version of this paper (excluding the proof of Theorem 16) was presented at WAOA'20 [3]. O.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

A Constant-Factor Approximation Algorithm for Vertex Guarding a WV-Polygon

Ashur

Filtser

Katz

2021

Approximation and Online Algorithms

Self Cite

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“…Regarding possible induced subgraphs, it is known that terrain visibility graphs can contain arbitrary large holes but no antiholes of size larger than five [14]. Ashur et al [5] introduced terrain-like graphs and studied approximation algorithms computing dominating sets.…”

Section: Related Workmentioning

confidence: 99%

“…However, Ameer et al [4] recently disproved the conjecture and gave a counterexample. Now, a terrain-like graph is simply defined to have a vertex ordering fulfilling the X-property [5]. Note that every induced subgraph of a terrain visibility graph is terrain-like.…”

Section: X-propertymentioning

confidence: 99%

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A Fast Shortest Path Algorithm on Terrain-like Graphs

Froese

Renken

2020

Discrete Comput Geom

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Terrain visibility graphs are a well-known graph class in computational geometry. They are closely related to polygon visibility graphs, but a precise graph-theoretical characterization is still unknown. Over the last decade, terrain visibility graphs attracted considerable attention in the context of time series analysis (there called time series visibility graphs) with various practical applications in areas such as physics, geography, and medical sciences. Computing shortest paths in visibility graphs is a common task, for example, in line-of-sight communication. For time series analysis, graph characteristics involving shortest paths lengths (such as centrality measures) have proven useful. In this paper, we devise a fast output-sensitive shortest path algorithm on a superclass of terrain visibility graphs called terrain-like graphs (including all induced subgraphs of terrain visibility graphs). Our algorithm runs in $$O(d^*\log \varDelta )$$ O ( d ∗ log Δ ) time, where $$d^*$$ d ∗ is the length (that is, the number of edges) of the shortest path and $$\varDelta $$ Δ is the maximum vertex degree. Alternatively, with an $$O(n^2)$$ O ( n 2 ) -time preprocessing our algorithm runs in $$O(d^*)$$ O ( d ∗ ) time.

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