## The zig-zag path of a pseudo-triangulation (2003)

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Venue: | IN PROC. 8TH INTERNATIONAL WORKSHOP ON ALGORITHMS AND DATA STRUCTURES (WADS |

Citations: | 16 - 5 self |

### BibTeX

@INPROCEEDINGS{Aichholzer03thezig-zag,

author = {Oswin Aichholzer and Bettina Speckmann and Ileana Streinu},

title = {The zig-zag path of a pseudo-triangulation},

booktitle = {IN PROC. 8TH INTERNATIONAL WORKSHOP ON ALGORITHMS AND DATA STRUCTURES (WADS},

year = {2003},

pages = {377--388},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

We define the path of a pseudo-triangulation, a data structure generalizing the path of a triangulation of a point set. This structure allows us to use divide-and-conquer type of approaches for suitable (i.e. decomposable) problems on pseudo-triangulations. We illustrate this method by presenting a novel algorithm that counts the number of pseudo-triangulations of a point set.

### Citations

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Citation Context ...y supported by the Deutsche Forschungsgemeinschaft (DFG) under grant RO 2338/2-1. ⋆⋆ Research supported by NSF grant CCR-0105507.sdescription method or the reverse-search technique of Avis and Fukuda =-=[4]-=- are two methods for vertex enumeration which have been implemented [3, 11]. We propose a different scheme for solving counting and optimization problems for pseudo-triangulations, inspired by an anal... |

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Constructing arrangements of lines and hyperplanes with applications
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Citation Context ...cessing of the point set which takes O(n 2 ) time and storage. In the preprocessing phase we compute and store the circular order of the edges from each point to all other points of S in O(n 2 ) time =-=[10]-=-. At this stage, we can already eliminate edges which do not lie insideP . The next edge which is added to a partial path must fulfill Properties 2 (pointedness) and 3 (empty area) of Definition 2, th... |

100 | A combinatorial approach to planar non-colliding robot arm motion planning
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Citation Context ... applications. They were originally introduced in the context of visibility complexes [15] and ray shooting [8, 12], but in the last few years they also found application in robot arm motion planning =-=[18]-=-, kinetic collision detection [1, 13], and guarding [17]. In particular the so-called minimum or pointed pseudo-triangulations introduced by Streinu [18] exhibit many fascinating properties that initi... |

96 |
Ray shooting in polygons using geodesic triangulations
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Citation Context ...unlike triangulations, only recently emerged as a promising data structure with a variety of applications. They were originally introduced in the context of visibility complexes [15] and ray shooting =-=[8, 12]-=-, but in the last few years they also found application in robot arm motion planning [18], kinetic collision detection [1, 13], and guarding [17]. In particular the so-called minimum or pointed pseudo... |

85 | Topologically sweeping visibility complexes via pseudotriangulations
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Citation Context ...seudo-triangulations, unlike triangulations, only recently emerged as a promising data structure with a variety of applications. They were originally introduced in the context of visibility complexes =-=[15]-=- and ray shooting [8, 12], but in the last few years they also found application in robot arm motion planning [18], kinetic collision detection [1, 13], and guarding [17]. In particular the so-called ... |

77 | L.: Deformable free space tilings for kinetic collision detection
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Citation Context ...y introduced in the context of visibility complexes [15] and ray shooting [8, 12], but in the last few years they also found application in robot arm motion planning [18], kinetic collision detection =-=[1, 13]-=-, and guarding [17]. In particular the so-called minimum or pointed pseudo-triangulations introduced by Streinu [18] exhibit many fascinating properties that initiated a growing interest in their geom... |

65 |
B.: Kinetic collision detection for simple polygons
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- 2002
(Show Context)
Citation Context ...y introduced in the context of visibility complexes [15] and ray shooting [8, 12], but in the last few years they also found application in robot arm motion planning [18], kinetic collision detection =-=[1, 13]-=-, and guarding [17]. In particular the so-called minimum or pointed pseudo-triangulations introduced by Streinu [18] exhibit many fascinating properties that initiated a growing interest in their geom... |

45 | Expansive motions and the polytope of pointed pseudo-triangulations
- Rote, Santos, et al.
- 2003
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Citation Context ... time complexity of this algorithm is not known, but it requires O(n 2 ) space. A third possibility is to apply some vertex enumeration algorithm to the polytope of pseudo-triangulations developed in =-=[14, 16]-=-. For example, Motzkin’s double ⋆ Research partly supported by the Deutsche Forschungsgemeinschaft (DFG) under grant RO 2338/2-1. ⋆⋆ Research supported by NSF grant CCR-0105507.sdescription method or ... |

38 | Dynamic ray shooting and shortest paths in planar subdivisions via balanced geodesic triangulations
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Citation Context ...unlike triangulations, only recently emerged as a promising data structure with a variety of applications. They were originally introduced in the context of visibility complexes [15] and ray shooting =-=[8, 12]-=-, but in the last few years they also found application in robot arm motion planning [18], kinetic collision detection [1, 13], and guarding [17]. In particular the so-called minimum or pointed pseudo... |

27 | Arc triangulations
- Aichholzer, Aigner, et al.
(Show Context)
Citation Context ...me for solving counting and optimization problems for pseudo-triangulations, inspired by an analogous approach developed for triangulations. The “path of a triangulation” was introduced by Aichholzer =-=[2]-=- in order to count the triangulations of a planar point set in a divide-and-conquer like manner. This concept can be used to attack any decomposable problem on triangulations. Dumitrescu et al. [9] pr... |

15 | Counting and enumerating pseudotriangulations with the greedy flip algorithm
- Brönnimann, Kettner, et al.
- 2001
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Citation Context ... enumerating triangulations [6], defines a lexicographical order on pseudo-triangulations which he uses to enumerate pseudo-triangulations in O(log n) time per pseudo-triangulation. Brönnimann et al. =-=[7]-=- implemented an ad-hoc technique of Pocchiola based on a greedy strategy for generating edges of pseudo-triangulations. Unfortunately the time complexity of this algorithm is not known, but it require... |

14 | An efficient algorithm for enumeration of triangulations
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(Show Context)
Citation Context ... geometric and combinatorial nature. There exist already several algorithms to enumerate pseudo-triangulations of sets of n points. Bespamyatnikh [5], extending his work on enumerating triangulations =-=[6]-=-, defines a lexicographical order on pseudo-triangulations which he uses to enumerate pseudo-triangulations in O(log n) time per pseudo-triangulation. Brönnimann et al. [7] implemented an ad-hoc techn... |

14 |
Allocating vertex π-guards in simple polygons via pseudotriangulations
- Speckmann, Tóth
- 2003
(Show Context)
Citation Context ...ntext of visibility complexes [15] and ray shooting [8, 12], but in the last few years they also found application in robot arm motion planning [18], kinetic collision detection [1, 13], and guarding =-=[17]-=-. In particular the so-called minimum or pointed pseudo-triangulations introduced by Streinu [18] exhibit many fascinating properties that initiated a growing interest in their geometric and combinato... |

11 | The polytope of non-crossing graphs on a planar point set, preprint
- Orden, Santos
- 2003
(Show Context)
Citation Context ... time complexity of this algorithm is not known, but it requires O(n 2 ) space. A third possibility is to apply some vertex enumeration algorithm to the polytope of pseudo-triangulations developed in =-=[14, 16]-=-. For example, Motzkin’s double ⋆ Research partly supported by the Deutsche Forschungsgemeinschaft (DFG) under grant RO 2338/2-1. ⋆⋆ Research supported by NSF grant CCR-0105507.sdescription method or ... |

6 | Enumerating pseudo-triangulations in the plane
- Bespamyatnikh
- 2002
(Show Context)
Citation Context ... properties that initiated a growing interest in their geometric and combinatorial nature. There exist already several algorithms to enumerate pseudo-triangulations of sets of n points. Bespamyatnikh =-=[5]-=-, extending his work on enumerating triangulations [6], defines a lexicographical order on pseudo-triangulations which he uses to enumerate pseudo-triangulations in O(log n) time per pseudo-triangulat... |

5 | Allocating vertex #- guards in simple polygons via pseudo-triangulations - Toth - 2003 |

3 | Enumerating triangulation paths
- Dumitrescu, Gartner, et al.
- 2000
(Show Context)
Citation Context ...er [2] in order to count the triangulations of a planar point set in a divide-and-conquer like manner. This concept can be used to attack any decomposable problem on triangulations. Dumitrescu et al. =-=[9]-=- provided an algorithm that enumerates all triangulation paths (of all triangulations of a given point set with respect to a given line) in O(n 3 log n) time per path and O(n) space. In this paper we ... |

2 |
cdd and cddplus by
- Fukuda, ver
(Show Context)
Citation Context ... 2338/2-1. ⋆⋆ Research supported by NSF grant CCR-0105507.sdescription method or the reverse-search technique of Avis and Fukuda [4] are two methods for vertex enumeration which have been implemented =-=[3, 11]-=-. We propose a different scheme for solving counting and optimization problems for pseudo-triangulations, inspired by an analogous approach developed for triangulations. The “path of a triangulation” ... |

1 |
lrslib Software: Reverse search algorithm for vertex enumeration/convex hull problems. http://cgm.cs.mcgill.ca/~avis/C/lrs.html
- Avis
(Show Context)
Citation Context ... 2338/2-1. ⋆⋆ Research supported by NSF grant CCR-0105507.sdescription method or the reverse-search technique of Avis and Fukuda [4] are two methods for vertex enumeration which have been implemented =-=[3, 11]-=-. We propose a different scheme for solving counting and optimization problems for pseudo-triangulations, inspired by an analogous approach developed for triangulations. The “path of a triangulation” ... |