## A Simple and Fast Incremental Randomized Algorithm for Computing Trapezoidal Decompositions and for Triangulating Polygons (1991)

Venue: | Comput. Geom. Theory Appl |

Citations: | 104 - 2 self |

### BibTeX

@ARTICLE{Seidel91asimple,

author = {Raimund Seidel},

title = {A Simple and Fast Incremental Randomized Algorithm for Computing Trapezoidal Decompositions and for Triangulating Polygons},

journal = {Comput. Geom. Theory Appl},

year = {1991},

volume = {1},

pages = {51--64}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper presents a very simple incremental randomized algorithm for computing the trapezoidal decomposition induced by a set S of n line segments in the plane. If S is given as a simple polygonal chain the expected running time of the algorithm is O(n log n). This leads to a simple algorithm of the same complexity for triangulating polygons. More generally, if S is presented as a plane graph with k connected components, then the expected running time of the algorithm is O(n log n k log n). As a by-product our algorithm creates a search structure of expected linear size that allows point location queries in the resulting trapezoidation in logarithmic expected time. The analysis of the expected performance is elementary and straightforward. All expectations are with respect to "coinflips" generated by the algorithm and are not based on assumptions about the geometric distribution of the input. Large Portions of the research reported here were conducted while the author visit...

### Citations

293 |
Triangulating a Simple Polygon in Linear Time
- Chazelle
- 1991
(Show Context)
Citation Context ...ars later. In the mean time Clarkson et al. had published a randomized algorithm with O(n log n) expected running time [CTV]. Finally in 1990 Chazelle discovered a linear time deterministic algorithm =-=[C90]-=-, which settles the question about the intrinsic computational complexity of triangulating once and for all. This paper presents another randomized algorithm with O(n log n) expected running time. Its... |

78 |
A theorem on polygon cutting with applications
- Chazelle
- 1982
(Show Context)
Citation Context ...A brief history: Garey et al. [GJPT] were the first to publish an O(n log n) algorithm based on sweeping in 1978. Four years later another algorithm with the same complexity was published by Chazelle =-=[C82]. The O(n -=-log n) bound was then improved to bounds of the form O(n log C P ), where C P is a "shape" parameter no bigger than n that depends on the polygon P to be triangulated (for instance the numbe... |

71 |
Triangulating a simple polygon
- Garey, Johnson, et al.
- 1978
(Show Context)
Citation Context ... importance in various application areas; and finally, the actual computational complexity of the problem remained unresolved for more than a decade until very recently. A brief history: Garey et al. =-=[GJPT]-=- were the first to publish an O(n log n) algorithm based on sweeping in 1978. Four years later another algorithm with the same complexity was published by Chazelle [C82]. The O(n log n) bound was then... |

59 |
Triangulating simple polygons and equivalent problems
- Fournier, DY
- 1984
(Show Context)
Citation Context ... then improved to bounds of the form O(n log C P ), where C P is a "shape" parameter no bigger than n that depends on the polygon P to be triangulated (for instance the number of reflex vert=-=ices [HM],[FM], or the &-=-quot;sinuosity" of P [CI]). On a different front an ever-increasing class of polygons were shown to be triangulatable in linear time [TA82], [T88]. After a false start, Tarjan and Van Wyk [TV] ma... |

45 | Triangulation and shape-complexity
- Chazelle, Incerpi
- 1984
(Show Context)
Citation Context ...e form O(n log C P ), where C P is a "shape" parameter no bigger than n that depends on the polygon P to be triangulated (for instance the number of reflex vertices [HM],[FM], or the "s=-=inuosity" of P [CI]-=-). On a different front an ever-increasing class of polygons were shown to be triangulatable in linear time [TA82], [T88]. After a false start, Tarjan and Van Wyk [TV] made a major breakthrough with a... |

43 |
Fast triangulation of simple polygons
- Hertel, Mehlhorn
- 1983
(Show Context)
Citation Context ...d was then improved to bounds of the form O(n log C P ), where C P is a "shape" parameter no bigger than n that depends on the polygon P to be triangulated (for instance the number of reflex=-= vertices [HM],[FM], or -=-the "sinuosity" of P [CI]). On a different front an ever-increasing class of polygons were shown to be triangulatable in linear time [TA82], [T88]. After a false start, Tarjan and Van Wyk [T... |

38 | An O(n log log n)-time algorithm for triangulating simple polygons
- Tarjan, Wyk
- 1988
(Show Context)
Citation Context ...M],[FM], or the "sinuosity" of P [CI]). On a different front an ever-increasing class of polygons were shown to be triangulatable in linear time [TA82], [T88]. After a false start, Tarjan an=-=d Van Wyk [TV]-=- made a major breakthrough with an O(n log log n) algorithm in 1986. This time bound was matched by a different but simpler algorithm by Kirkpatrick et al. [KKT] three years later. In the mean time Cl... |

28 |
Sorting Jordan sequences in linear time using level-linked search trees
- Hoffman, Mehlhorn, et al.
- 1986
(Show Context)
Citation Context ...l. This paper presents another randomized algorithm with O(n log n) expected running time. Its virtues lie in its simplicity. It uses no divide-and-conquer or recursion, and no "Jordan-sorting&qu=-=ot; [CTV],[HMRT]-=-. Its expected performance admits a very straightforward and self-contained analysis. Finally, it is practical and relatively simple to implement, a property that very few, if any, of the algorithms m... |

27 |
A fast Las Vegas algorithm for triangulating a simple polygon
- Clarkson, Tarjan, et al.
- 1989
(Show Context)
Citation Context ...matched by a different but simpler algorithm by Kirkpatrick et al. [KKT] three years later. In the mean time Clarkson et al. had published a randomized algorithm with O(n log n) expected running time =-=[CTV]-=-. Finally in 1990 Chazelle discovered a linear time deterministic algorithm [C90], which settles the question about the intrinsic computational complexity of triangulating once and for all. This paper... |

25 | Randomized parallel algorithms for trapezoidal diagrams
- Clarkson, Cole, et al.
- 1992
(Show Context)
Citation Context ...ed cost for this over the entire algorithm is O(log n) per component. 4 Remarks An algorithm somewhat similar to the one described in this paper has also been discovered by Clarkson, Cole, and Tarjan =-=[CCT]-=-. However, their approach is based on divide-and-conquer and the main thrust of their approach is towards a fast parallel trapezoidation algorithm. Our algorithm can be viewed as a holistic version of... |

18 |
On a convex hull algorithm for polygons and its application to triangulation problems
- Toussaint, Avis
- 1982
(Show Context)
Citation Context ...ngulated (for instance the number of reflex vertices [HM],[FM], or the "sinuosity" of P [CI]). On a different front an ever-increasing class of polygons were shown to be triangulatable in li=-=near time [TA82]-=-, [T88]. After a false start, Tarjan and Van Wyk [TV] made a major breakthrough with an O(n log log n) algorithm in 1986. This time bound was matched by a different but simpler algorithm by Kirkpatric... |

8 |
An output-complexity-sensitive polygon triangulation algorithm
- Toussaint
- 1988
(Show Context)
Citation Context ... (for instance the number of reflex vertices [HM],[FM], or the "sinuosity" of P [CI]). On a different front an ever-increasing class of polygons were shown to be triangulatable in linear tim=-=e [TA82], [T88]-=-. After a false start, Tarjan and Van Wyk [TV] made a major breakthrough with an O(n log log n) algorithm in 1986. This time bound was matched by a different but simpler algorithm by Kirkpatrick et al... |

4 |
O(n log log n) polygon triangulation with simple data structures
- Kirkpatrick, Klawe, et al.
- 1990
(Show Context)
Citation Context ...After a false start, Tarjan and Van Wyk [TV] made a major breakthrough with an O(n log log n) algorithm in 1986. This time bound was matched by a different but simpler algorithm by Kirkpatrick et al. =-=[KKT]-=- three years later. In the mean time Clarkson et al. had published a randomized algorithm with O(n log n) expected running time [CTV]. Finally in 1990 Chazelle discovered a linear time deterministic a... |