## Computing the visibility graph of points within a polygon (2004)

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Venue: | In Symposium on Computational Geometry |

Citations: | 9 - 4 self |

### BibTeX

@INPROCEEDINGS{Ben-moshe04computingthe,

author = {Boaz Ben-moshe},

title = {Computing the visibility graph of points within a polygon},

booktitle = {In Symposium on Computational Geometry},

year = {2004},

pages = {27--35},

publisher = {ACM Press}

}

### OpenURL

### Abstract

We study the problem of computing the visibility graph defined by a set P of n points inside a polygon Q: two points p, q ∈ P are joined by an edge if the segment pq ⊂ Q. Efficient output-sensitive algorithms are known for the case in which P is the set of all vertices of Q. We examine the general case in which P is an arbitrary set of points, interior or on the boundary of Q and study a variety of algorithmic questions. We give an output-sensitive algorithm, which is nearly optimal, when Q is a simple polygon. We introduce a notion of “fat ” or “robust ” visibility, and give a nearly optimal algorithm for computing visibility graphs according to it, in polygons Q that may have holes. Other results include an algorithm to detect if there are any visible pairs among P, and algorithms for output-sensitive computation of visibility graphs with distance restrictions, invisibility graphs, and rectangle visibility graphs.

### Citations

289 | Triangulating a Simple Polygon in Linear Time - Chazelle - 1991 |

251 | Geometric range searching and its relatives
- Agrawal, Erickson
- 1999
(Show Context)
Citation Context ...The query ranges associated with point si ∈ P will be sectors of radius ds i , each of which is the intersection of two halfplanes and a disk. See Figure 4. Thus, we can use known results (see, e.g., =-=[2, 4]-=-) in multi-level range search data structures to answer queries in time (roughly) output size plus O(m 1/2 ): the first two levels of the structure apply halfspace range searching to be able to report... |

144 | Voronoi diagrams
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- 2000
(Show Context)
Citation Context ...e Figure 6. This can be done in O(log n) time per p, after preprocessing Q in time O(n) (if Q is simple [12]) or O(n log n) (if Q has holes). For this step, we compute the Voronoi diagram (see, e.g., =-=[6, 11]-=-) of Q according to the convex distance function defined by a rectangle of aspect ratio δ with origin at the focus point, and we preprocess the diagram for point location queries. Then, for a given p ... |

93 |
An output sensitive algorithm for computing visibility graphs
- Ghosh, Mount
- 1987
(Show Context)
Citation Context ... to compute VGQ(V ) for a simple polygon Q. For general polygons (with holes), Overmars and Welzl [21] obtained a relatively simple O(k log n)-time method, requiring O(n) space. Then, Ghosh and Mount =-=[13]-=- obtained an O(k + n log n)-time algorithm, using O(k) storage. Pocchiola and Vegter [23] have improved the space bound to optimal using the concept of the visibility complex; their algorithm requires... |

86 | Art gallery and illumination problems
- Urrutia
- 2000
(Show Context)
Citation Context ...in Q). See Figure 1 for an example. In this paper, we present efficient algorithms to compute VGQ(P). There has been considerable study of visibility graphs and algorithms to compute them (see, e.g., =-=[14]-=-). Optimal algorithms are known for computing the visibility graph of all vertices of Q (i.e., the case that P = V is the vertex set of Q): one can compute VGQ(V ) in time O(k+n log n), where Copyrigh... |

80 |
Optimal shortest path queries in a simple polygon
- Guibas, Hershberger
(Show Context)
Citation Context ...(log m) that multiplies the output size, k, in the running time, which is caused by over-reporting the visible pairs. Our improved algorithm makes use of the query structure of Guibas and Hershberger =-=[14]-=-, modified in order to account for the presence of sites in the polygon. We first summarize their construction, which was developed to answer shortest path queries in a polygon, and then describe how ... |

71 |
Voronoi diagrams based on convex distance functions
- Chew, Drysdale
- 1985
(Show Context)
Citation Context ...e Figure 6. This can be done in O(log n) time per p, after preprocessing Q in time O(n) (if Q is simple [12]) or O(n log n) (if Q has holes). For this step, we compute the Voronoi diagram (see, e.g., =-=[6, 11]-=-) of Q according to the convex distance function defined by a rectangle of aspect ratio δ with origin at the focus point, and we preprocess the diagram for point location queries. Then, for a given p ... |

69 | Range searching
- Agarwal
- 1997
(Show Context)
Citation Context ...The query ranges associated with point si ∈ P will be sectors of radius ds i , each of which is the intersection of two halfplanes and a disk. See Figure 4. Thus, we can use known results (see, e.g., =-=[2, 4]-=-) in multi-level range search data structures to answer queries in time (roughly) output size plus O(m 1/2 ): the first two levels of the structure apply halfspace range searching to be able to report... |

68 | Finding the medial axis of a simple polygon in linear time
- Chin, Snoeyink, et al.
- 1999
(Show Context)
Citation Context ...maximal rectangle Rp ⊆ Q of orientation ρ, with p at its focus point and with aspect ratio δ. See Figure 6. This can be done in O(log n) time per p, after preprocessing Q in time O(n) (if Q is simple =-=[12]-=-) or O(n log n) (if Q has holes). For this step, we compute the Voronoi diagram (see, e.g., [6, 11]) of Q according to the convex distance function defined by a rectangle of aspect ratio δ with origin... |

60 | Approximating polygons and subdivisions with minimum link paths
- Guibas, Hershberger, et al.
- 1993
(Show Context)
Citation Context ...case of polygons with h holes, at a cost of a factor of O(h) in the time complexity. The extension utilizes a decomposition of the polygon with holes into O(h) corridors and junction triangles (as in =-=[15, 19]-=-). 3. DISTANCE-RESTRICTED VISIBILITY GRAPHS Consider now the case in which each point p ∈ P has an associated range of sight, dp > 0, such that an observer at point p can see only up to distance dp. T... |

51 | Dynamic planar convex hull
- Brodal, Jacob
- 2002
(Show Context)
Citation Context ...S. Assuming w lies on l (and is on or above T ), then the reported halfplanes correspond to the points in PS that see w. The best known bounds for the first data structure are due to Brodal and Jacob =-=[9]-=-, who present a linear-size dynamic data structure that supports some basic queries on the convex hull of a set of points in the plane. Both the update time and the query time of their data structure ... |

44 |
Constructing the visibility graph for n line segments
- Welzl
- 1985
(Show Context)
Citation Context ...given as the set V of all vertices of the polygon Q. The first algorithms to construct the visibility graph VGQ(V ) required time O(n 2 log n) [10], using a radial sweep about each vertex of Q. Welzl =-=[15]-=- and Asano et al. [2] improved the time bound to O(n 2 ), which is worstcase optimal but not output-sensitive. Hershberger [9] gave an optimal O(n + k) output-sensitive algorithm to compute VGQ(V ) fo... |

38 |
H.: Visibility of disjoint polygons
- Asano, Asano, et al.
- 1986
(Show Context)
Citation Context ...all vertices of the polygon Q. The first algorithms to construct the visibility graph VGQ(V ) required time O(n 2 log n) [20], using a radial sweep about each vertex of Q. Welzl [25] and Asano et al. =-=[5]-=- improved the time bound to O(n 2 ), which is worst-case optimal but not output-sensitive. Hershberger [18] gave an optimal O(n+k) output-sensitive algorithm to compute VGQ(V ) for a simple polygon Q.... |

35 | New Methods for Computing Visibility Graphs
- Overmars, Welzl
- 1988
(Show Context)
Citation Context ... optimal but not output-sensitive. Hershberger [9] gave an optimal O(n + k) output-sensitive algorithm to compute VGQ(V ) for a simple polygon Q. For general polygons (with holes), Overmars and Welzl =-=[11]-=- obtained a relatively simple O(k log n)-time method, requiring O(n) space. Then, Ghosh and Mount [7] obtained an O(k+n log n)-time algorithm, using O(k) storage. Pocchiola and Vegter [12] and Rivière... |

34 | Dynamic planar convex hull operations in near-logarithmic amortized time
- Chan
- 1999
(Show Context)
Citation Context ...is a connected set and that the union of the blue segments is also connected; in such a case, the K red-blue intersections can be reported in time Õ(K + r + b), for r red segments and b blue segments =-=[7, 10, 17]-=-. (If the unions are not known to be connected, then the intersections can be computed in time O((r + b) 4/3 log(r + b) + K) [1, 16].) Instead of computing only the red-blue segment intersections, tho... |

31 | Computing the visibility graph via pseudotriangulations
- Pocchiola, Vegter
- 1996
(Show Context)
Citation Context ... ′ may be Ω(n 2 ), with n >> m, while the size of VGQ(P) may be very small (e.g., O(m), or O(1)). Computing the relative convex hull, C, of P with respect to Q, and then computing the free bitangents =-=[12]-=- within C reduces the number of superfluous edges computed, but there may still be a substantial discrepency between the Ω(m 2 ) edges that may be reported and the output size (possibly O(1)). (2) We ... |

24 |
Partitioning arrangements of lines
- Agarwal
- 1990
(Show Context)
Citation Context ...ed in time Õ(K + r + b), for r red segments and b blue segments [7, 10, 17]. (If the unions are not known to be connected, then the intersections can be computed in time O((r + b) 4/3 log(r + b) + K) =-=[1, 16]-=-.) Instead of computing only the red-blue segment intersections, though, we can compute all intersections among the dual segments L ∗ i , regardless of which subpolygon contains the corresponding site... |

24 |
Proximi[y and Reachability in the Plane
- Lee
(Show Context)
Citation Context ...dressed only the special case in which Q the sites P are given as the set V of all vertices of the polygon Q. The first algorithms to construct the visibility graph VGQ(V ) required time O(n 2 log n) =-=[10]-=-, using a radial sweep about each vertex of Q. Welzl [15] and Asano et al. [2] improved the time bound to O(n 2 ), which is worstcase optimal but not output-sensitive. Hershberger [9] gave an optimal ... |

22 |
An Optimal Visibility Graph Algorithm for Triangulated Simple Polygons, Algorithmica 4
- Hershberger
- 1989
(Show Context)
Citation Context ...GQ(P). The specialization of this result to the case in which the sites are exactly the vertices of Q (P = V ) yields an O(n log 2 n+k) algorithm that nearly matches the optimal O(n + k) algorithm of =-=[9]-=-. (2) For a simple polygon Q and a set P of m sites in Q each having an associated range (distance) of vision, we show that the range-restricted visibility graph can be computed in time Õ(nlog n + m3/... |

20 |
Can visibility graphs be represented compactly? Discrete and Computational Geometry
- Agarwal, Alon, et al.
- 1994
(Show Context)
Citation Context ... , there are situations in which the visibility graph is 2 particularly dense and it would be advantageous to compute the invisibility graph in output-sensitive time, depending on ¯k. (Agarwal et al. =-=[3]-=- have studied the related problem of compact representations of a visibility graph, as a union of a small number of complete subgraphs.) We observe that the algorithms described in Section 3 (for comp... |

20 | Reporting red-blue intersections between two sets of connected line segments
- Basch, Guibas, et al.
- 1996
(Show Context)
Citation Context ...is a connected set and that the union of the blue segments is also connected; in such a case, the K red-blue intersections can be reported in time Õ(K + r + b), for r red segments and b blue segments =-=[7, 10, 17]-=-. (If the unions are not known to be connected, then the intersections can be computed in time O((r + b) 4/3 log(r + b) + K) [1, 16].) Instead of computing only the red-blue segment intersections, tho... |

14 | Visibility preserving terrain simplification- an experimental study
- Ben-Moshe, Katz, et al.
(Show Context)
Citation Context ...roduce a witness visible pair. This problem of detecting “visibility independence” of a point set arises in applications of sensor coverage (guarding) and visibilitypreserving terrain simplifications =-=[8]-=-. For a simple polygon Q visibility detection can be done by applying the simple version of the algorithm of Section 2, just applying a line segment intersection detection algorithm (for the dual segm... |

10 |
Intersecting line segments, ray shooting, and other applications of geometric partitioning techniques
- Guibas, Overmars, et al.
- 1988
(Show Context)
Citation Context ...ed in time Õ(K + r + b), for r red segments and b blue segments [7, 10, 17]. (If the unions are not known to be connected, then the intersections can be computed in time O((r + b) 4/3 log(r + b) + K) =-=[1, 16]-=-.) Instead of computing only the red-blue segment intersections, though, we can compute all intersections among the dual segments L ∗ i , regardless of which subpolygon contains the corresponding site... |

10 |
Topologically sweeping the visibility complex of polygonal scenes
- Rivière
- 1995
(Show Context)
Citation Context ...obtained a relatively simple O(k log n)-time method, requiring O(n) space. Then, Ghosh and Mount [7] obtained an O(k+n log n)-time algorithm, using O(k) storage. Pocchiola and Vegter [12] and Rivière =-=[13]-=- have improved the space bound to optimal using the concept of the visibility complex; their algorithm requires time O(k + nlog n) and uses O(n) space. Preliminaries. Let Q be a polygon in the plane h... |

4 | Partitioning arrangements of lines, II: Applications, Discrete and Computational Geometry 5 - Agarwal - 1990 |

1 | Visibility preserving terrain simplification - An experimental study - Brodal, Jacob - 2002 |