## Optimal Output-Sensitive Convex Hull Algorithms in Two and Three Dimensions (1996)

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Citations: | 47 - 6 self |

### BibTeX

@MISC{Chan96optimaloutput-sensitive,

author = {T. M. Chan},

title = {Optimal Output-Sensitive Convex Hull Algorithms in Two and Three Dimensions},

year = {1996}

}

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### Abstract

We present simple output-sensitive algorithms that construct the convex hull of a set of n points in two or three dimensions in worst-case optimal O(n log h) time and O(n) space, where h denotes the number of vertices of the convex hull.

### Citations

1764 |
Computational Geometry: An Introduction
- Preparata, Shamos
- 1985
(Show Context)
Citation Context ... problem of computing the convex hull of P , conv(P ), which is defined as the smallest convex set containing P . The convex hull problem has received considerable attention in computational geometry =-=[11, 21, 23, 25]-=-. In E 2 , an algorithm known as Graham's scan [15] achieves O(n log n) running time, and in E 3 , an algorithm by Preparata and Hong [24] has the same complexity. These algorithms are optimal in the ... |

690 |
Algorithms in combinatorial geometry
- Edelsbrunner
- 1987
(Show Context)
Citation Context ... problem of computing the convex hull of P , conv(P ), which is defined as the smallest convex set containing P . The convex hull problem has received considerable attention in computational geometry =-=[11, 21, 23, 25]-=-. In E 2 , an algorithm known as Graham's scan [15] achieves O(n log n) running time, and in E 3 , an algorithm by Preparata and Hong [24] has the same complexity. These algorithms are optimal in the ... |

504 |
Computational Geometry in C
- O'Rourke
- 1998
(Show Context)
Citation Context ... problem of computing the convex hull of P , conv(P ), which is defined as the smallest convex set containing P . The convex hull problem has received considerable attention in computational geometry =-=[11, 21, 23, 25]-=-. In E 2 , an algorithm known as Graham's scan [15] achieves O(n log n) running time, and in E 3 , an algorithm by Preparata and Hong [24] has the same complexity. These algorithms are optimal in the ... |

389 | Applications of random sampling in computational geometry
- Clarkson, Shor
- 1987
(Show Context)
Citation Context ...d and Kapur [3]. A faster but more involved algorithm in E 3 was discovered by Edelsbrunner and Shi [13], having running time O(n log 2 h). Finally, by derandomizing an algorithm of Clarkson and Shor =-=[8], Chazelle-=- and Matousek [7] succeeded in attaining optimal O(n log h) time in E 3 . These algorithms, with complexity measured as a function of both n and the "output size" h, are said to be output-se... |

284 |
Computational Geometry. An Introduction Through Randomized Algorithms
- Mulmuley
- 1994
(Show Context)
Citation Context |

277 | Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms
- Edelsbrunner, Mücke
- 1990
(Show Context)
Citation Context ...gorithms Hull2D(P; m; H) and Hull3D(P; m; H), we have assumed that the points of P are in general position. One way to cope with degenerate point sets is to apply general perturbation methods such as =-=[12, 14]-=-; however, these methods may cause the output size h to increase, as a point that is not a hull vertex but lies on the hull boundary may become a vertex after perturbation. Thus, it is better to handl... |

241 |
An efficient algorithm for determining the convex hull of a finite planar set
- Graham
- 1972
(Show Context)
Citation Context ... defined as the smallest convex set containing P . The convex hull problem has received considerable attention in computational geometry [11, 21, 23, 25]. In E 2 , an algorithm known as Graham's scan =-=[15]-=- achieves O(n log n) running time, and in E 3 , an algorithm by Preparata and Hong [24] has the same complexity. These algorithms are optimal in the worst case, but if h, the number of hull vertices, ... |

106 | D.G.: Determining the separation of preprocessed polyhedra — a unified approach
- Dobkin, Kirkpatrick
- 1990
(Show Context)
Citation Context ...ther; Dobkin-Kirkpatrick Supported by a Killam Predoctoral Fellowship and an NSERC Postgraduate Scholarship. p k p k k +1 p -1 Figure 1: Wrapping a set of dn=me convex polygons of size m. hierarchies =-=[9, 10]-=- are the only data structures used in the three-dimensional case. Our idea is to speed up Jarvis's march and the gift-wrapping method by using a very simple grouping trick. 2 An Output-Sensitive Algor... |

105 | Nonlinearity of Davenport-Schinzel sequences and of a generalized path compressioa scheme, Combinatorica 6
- Hart, Sharir
- 1986
(Show Context)
Citation Context ... for a given segment s. (Convex hulls correspond to lower envelopes of lines in the dual.) Let h be the output size, i.e., the number of edges in the envelope; it is known that h is at most O(nff(n)) =-=[16]-=-. Hershberger [17] has given a worst-case optimal algorithm that computes lower envelopes in O(n log n) time. We now describe how his algorithm can be made output-sensitive with our technique. First, ... |

100 |
The ultimate planar convex hull algorithm
- Kirkpatrick, Seidel
- 1986
(Show Context)
Citation Context ... example, in E 2 , a simple algorithm called Jarvis's march [19] can construct the convex hull in O(nh) time. This bound was later improved to O(n log h) by an algorithm due to Kirkpatrick and Seidel =-=[20]-=-, who also provided a matching lower bound; a simplification of their algorithm has been recently reported by Chan, Snoeyink, and Yap [2]. In E 3 , one can obtain an O(nh)-time algorithm using the gif... |

97 |
Convex hulls of finite sets of points in two and three dimensions
- Preparata, Hong
- 1977
(Show Context)
Citation Context ... considerable attention in computational geometry [11, 21, 23, 25]. In E 2 , an algorithm known as Graham's scan [15] achieves O(n log n) running time, and in E 3 , an algorithm by Preparata and Hong =-=[24]-=- has the same complexity. These algorithms are optimal in the worst case, but if h, the number of hull vertices, is small, then it is possible to obtain better time bounds. For example, in E 2 , a sim... |

96 |
Ray shooting in polygons using geodesic triangulations, Algorithmica 12
- Chazelle, Edelsbrunner, et al.
- 1994
(Show Context)
Citation Context ...tition S into dn=me groups each of at most m segments and compute the lower envelope of each group by Hershberger's algorithm; this takes O(n log m) time in total. Using known data structures such as =-=[6, 18]-=-, we can perform ray shooting under each of these dn=me envelopes in O(log m) time after O(mff(m)) preprocessing (the ray shooting methods can be simplified in our case since envelopes are monotone). ... |

84 |
A pedestrian approach to ray shooting: Shoot a ray, take a walk
- Hershberger, Suri
- 1995
(Show Context)
Citation Context ...tition S into dn=me groups each of at most m segments and compute the lower envelope of each group by Hershberger's algorithm; this takes O(n log m) time in total. Using known data structures such as =-=[6, 18]-=-, we can perform ray shooting under each of these dn=me envelopes in O(log m) time after O(mff(m)) preprocessing (the ray shooting methods can be simplified in our case since envelopes are monotone). ... |

77 |
Fast detection of polyhedral intersection
- Dobkin, Kirkpatrick
- 1983
(Show Context)
Citation Context ...ther; Dobkin-Kirkpatrick Supported by a Killam Predoctoral Fellowship and an NSERC Postgraduate Scholarship. p k p k k +1 p -1 Figure 1: Wrapping a set of dn=me convex polygons of size m. hierarchies =-=[9, 10]-=- are the only data structures used in the three-dimensional case. Our idea is to speed up Jarvis's march and the gift-wrapping method by using a very simple grouping trick. 2 An Output-Sensitive Algor... |

66 | Output-sensitive results on convex hulls, extreme points, and related problems, Discrete Comput
- Chan
- 1996
(Show Context)
Citation Context ... appropriate group size m and guessing the output size h give us an optimal output-sensitive O(n log h) algorithm for computing the lower envelope. Other applications of our technique can be found in =-=[1]-=-, including the output-sensitive construction of higher-dimensional convex hulls and k-levels. In many cases, our grouping idea, combined with appropriate data structures, can be used to obtain optima... |

64 |
Finding the upper envelope of n line segments in O(n log n) time
- Hershberger
- 1989
(Show Context)
Citation Context ...nt s. (Convex hulls correspond to lower envelopes of lines in the dual.) Let h be the output size, i.e., the number of edges in the envelope; it is known that h is at most O(nff(n)) [16]. Hershberger =-=[17]-=- has given a worst-case optimal algorithm that computes lower envelopes in O(n log n) time. We now describe how his algorithm can be made output-sensitive with our technique. First, observe that we ca... |

61 | An optimal algorithm for intersecting three-dimensional convex polyhedra
- Chazelle
- 1992
(Show Context)
Citation Context ...lls in line 3 can be constructed in O(n log(m=m 0 )) rather than O(n log m) time. The same can be said for the three-dimensional case, but merging two convex polyhedra, though possible in linear time =-=[4]-=-, is more complicated. Idea 4. In Hull2D(P ), we use the sequence of group sizes m = 2 2 t , t = 1; 2; : : :, to guess h. The improvements from Ideas 2 and 3 in fact permit us to choose slower growing... |

60 |
An algorithm for convex polytopes
- Chand, Kapur
- 1970
(Show Context)
Citation Context ...has been recently reported by Chan, Snoeyink, and Yap [2]. In E 3 , one can obtain an O(nh)-time algorithm using the gift-wrapping method, an extension of Jarvis's march originated by Chand and Kapur =-=[3]-=-. A faster but more involved algorithm in E 3 was discovered by Edelsbrunner and Shi [13], having running time O(n log 2 h). Finally, by derandomizing an algorithm of Clarkson and Shor [8], Chazelle a... |

60 |
On the identification of the convex hull of a finite set of points in the plane
- Jarvis
- 1973
(Show Context)
Citation Context ...rithms are optimal in the worst case, but if h, the number of hull vertices, is small, then it is possible to obtain better time bounds. For example, in E 2 , a simple algorithm called Jarvis's march =-=[19]-=- can construct the convex hull in O(nh) time. This bound was later improved to O(n log h) by an algorithm due to Kirkpatrick and Seidel [20], who also provided a matching lower bound; a simplification... |

49 | Intersection of convex objects in two and three dimensions
- Chazelle, Dobkin
- 1987
(Show Context)
Citation Context ...ing tangents or supporting lines of the polygons through the current vertex p k , as shown in Figure 1. Since tangent finding takes logarithmic time for a convex polygon by binary or Fibonacci search =-=[5, 25]-=- (the dual problem is to intersect a convex polygon with a ray), the time required for a wrapping step is then O( n m log m). As h wrapping steps are needed to compute the hull, the total time of the ... |

37 |
A new linear algorithm for intersecting convex polygons, Computer Graphics and Image Processing 19
- O’Rourke, Chien, et al.
- 1982
(Show Context)
Citation Context ...ygons with a total of n vertices can be computed in O(n+hp) time by gift-wrapping; the two-polygon (p = 2) version of the algorithm is in fact the dual of an intersection algorithm by O'Rourke et al. =-=[22]-=- (see also [23, 25]). The total cost of Hull2D(P; m; H) can then be reduced to O(n log m+H(n=m)) time, which is a log m factor saving in the second term. Although the overall constant factor is unaffe... |

35 | An efficient approach to removing geometric degeneracies
- Emiris, Canny
- 1992
(Show Context)
Citation Context ...gorithms Hull2D(P; m; H) and Hull3D(P; m; H), we have assumed that the points of P are in general position. One way to cope with degenerate point sets is to apply general perturbation methods such as =-=[12, 14]-=-; however, these methods may cause the output size h to increase, as a point that is not a hull vertex but lies on the hull boundary may become a vertex after perturbation. Thus, it is better to handl... |

33 |
Output-sensitive construction of polytopes in four dimensions T.M. Chan and clipped Voronoi diagrams in three
- Chan, Snoeyink, et al.
(Show Context)
Citation Context ...d to O(n log h) by an algorithm due to Kirkpatrick and Seidel [20], who also provided a matching lower bound; a simplification of their algorithm has been recently reported by Chan, Snoeyink, and Yap =-=[2]-=-. In E 3 , one can obtain an O(nh)-time algorithm using the gift-wrapping method, an extension of Jarvis's march originated by Chand and Kapur [3]. A faster but more involved algorithm in E 3 was disc... |

31 | Finding the convex hull facet by facet - Swart - 1985 |

26 |
Derandomizing an output-sensitive convex hull algorithm in three dimensions
- Chazelle, Matouˇsek
- 1995
(Show Context)
Citation Context ...ut more involved algorithm in E 3 was discovered by Edelsbrunner and Shi [13], having running time O(n log 2 h). Finally, by derandomizing an algorithm of Clarkson and Shor [8], Chazelle and Matousek =-=[7] succeeded-=- in attaining optimal O(n log h) time in E 3 . These algorithms, with complexity measured as a function of both n and the "output size" h, are said to be output-sensitive. In this note, we p... |

13 |
An O(nlog2h) time algorithm for the threedimensional convex hull problem
- Edelsbrunner, Shi
(Show Context)
Citation Context ...nh)-time algorithm using the gift-wrapping method, an extension of Jarvis's march originated by Chand and Kapur [3]. A faster but more involved algorithm in E 3 was discovered by Edelsbrunner and Shi =-=[13]-=-, having running time O(n log 2 h). Finally, by derandomizing an algorithm of Clarkson and Shor [8], Chazelle and Matousek [7] succeeded in attaining optimal O(n log h) time in E 3 . These algorithms,... |