## Dynamic planar convex hull (2002)

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Venue: | Proc. 43rd IEEE Sympos. Found. Comput. Sci |

Citations: | 52 - 1 self |

### BibTeX

@INPROCEEDINGS{Brodal02dynamicplanar,

author = {Gerth Stølting Brodal and Riko Jacob},

title = {Dynamic planar convex hull},

booktitle = { Proc. 43rd IEEE Sympos. Found. Comput. Sci},

year = {2002},

pages = {617--626},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we determine the amortized computational complexity of the dynamic convex hull problem in the planar case. We present a data structure that maintains a finite set of n points in the plane under insertion and deletion of points in amortized O(log n) time per operation. The space usage of the data structure is O(n). The data structure supports extreme point queries in a given direction, tangent queries through a given point, and queries for the neighboring points on the convex hull in O(log n) time. The extreme point queries can be used to decide whether or not a given line intersects the convex hull, and the tangent queries to determine whether a given point is inside the convex hull. We give a lower bound on the amortized asymptotic time complexity that matches the performance of this data structure.

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Citation Context ...w growing inverse of Ackerman’s function. Lower bounds For the static convex hull computation there is a well known reduction to sorting, presented for example in the textbook by Preparata and Shamos =-=[19]-=-. This establishes together with Ben-Or’s [2] result lower bound on the real-RAM for computing the convex hull. In the dynamic setting this implies that the sum of the running times of INSERT �Å�� � a... |

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Citation Context ...set of points in the plane is one of the most prominent objects in computational geometry. Computing the convex hull of a static set of point set can be done in optimal time, e.g., with Graham’s scan =-=[9]-=- or Andrew’s vertical sweep line variant [1] of it. Optimal output sensitive algorithms are due to Kirkpatrick and Seidel [16] and also to Chan [5], they achieve ��running time, where�denotes the numb... |

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Citation Context ...r bounds For the static convex hull computation there is a well known reduction to sorting, presented for example in the textbook by Preparata and Shamos [19]. This establishes together with Ben-Or’s =-=[2]-=- result lower bound on the real-RAM for computing the convex hull. In the dynamic setting this implies that the sum of the running times of INSERT �Å�� � and QUERY has to beÅ�. In Section 7 we prove t... |

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Citation Context ...ent(s) between two different � � convex hulls. Furthermore we might want to report (some consecutive subsequence of) the points on the convex hull or count their cardinality. Overmars and van Leeuwen =-=[17]-=- provide a solution that uses time per update operation and maintains a leaf-linked balanced search tree of the vertices on the convex hull in clockwise order. Such a tree allows all of the above ment... |

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Citation Context ... for updates (for any constant # > 0). His construction does not maintain an explicit representation of the convex hull. It is based on a general dynamization technique attributed to Bentley and Saxe =-=[BS80]-=-. Using the semidynamic deletions only data structure of Hershberger and Suri [HS92], and the right choice of parameters in a finite number of bootstrapping steps, the construction achieves update tim... |

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Citation Context ...tic set of point set can be done in optimal time, e.g., with Graham’s scan [9] or Andrew’s vertical sweep line variant [1] of it. Optimal output sensitive algorithms are due to Kirkpatrick and Seidel =-=[16]-=- and also to Chan [5], they achieve ��running time, where�denotes the number of vertices on the convex hull. In the dynamic setting we consider a data structure: Given a set of points in the plane tha... |

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Citation Context ... only. For the insertiononly problem Preparata [18] gives an �worst-case time algorithm that maintains the convex hull in a search tree. The deletion-only problem is solved by Hershberger and Suri in =-=[12]-=-, where initializing the data structure (build) with points and up to deletions are accomplished in overall �� �time. Hershberger and Suri in [13] consider the off-line variant of the problem, where b... |

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Citation Context ...an be done in optimal time, e.g., with Graham’s scan [9] or Andrew’s vertical sweep line variant [1] of it. Optimal output sensitive algorithms are due to Kirkpatrick and Seidel [16] and also to Chan =-=[5]-=-, they achieve ��running time, where�denotes the number of vertices on the convex hull. In the dynamic setting we consider a data structure: Given a set of points in the plane that is changed by inser... |

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Citation Context ...rominent objects in computational geometry. Computing the convex hull of a static point set can be done in optimal O(n log n) time with Graham 's scan [Gra72] (or Andrew's vertical sweep line variant =-=[And79]-=- of it). An optimal output sensitive algorithm is due to Kirkpatrick and Seidel [KS86] achieving O(n log h) running time, where h denotes the number of vertices on the convex hull. The dynamic planar ... |

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Citation Context ...vel of a set of lines. If we have segments on the�-level (the output size), the resulting algorithm completes in time. This improves over the fastest deterministic algorithms, (Edelsbrunner and Welzl =-=[8]-=-, using Chan’s data structure achieving ��time). It is faster aÅ than the �ex pected running time �of the randomized algorithm of Har-Peled and Sharir [11]. Here is the slow growing inverse of Ackerma... |

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Citation Context ...icit representation of the convex hull in terms of a � search tree with the points on the convex hull. The space usage can be reduced to if the queries are also part of the off-line information. Chan =-=[7]-=- gives a construction for the fully dynamic problem with ��amortized time for updates (for any constant��), and time for extreme point queries. His construction does not maintain an explicit represent... |

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Citation Context ...ion-only problem is solved by Hershberger and Suri in [12], where initializing the data structure (build) with points and up to deletions are accomplished in overall �� �time. Hershberger and Suri in =-=[13]-=- consider the off-line variant of the problem, where both insertions and deletions are allowed, but the times (and by this the order) of all insertions and deletions are known a priori. The algorithm ... |

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Citation Context ...tioned queries in time. Semidynamic variants of the problems have been considered. There updates are restricted to be either insertions only or deletions only. For the insertiononly problem Preparata =-=[18]-=- gives an �worst-case time algorithm that maintains the convex hull in a search tree. The deletion-only problem is solved by Hershberger and Suri in [12], where initializing the data structure (build)... |

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Citation Context ...nt��. The construction uses an augmented variant of an interval tree to store the convex hulls of the semidynamic deletion only data structures. This achieves �time extreme point queries. The authors =-=[4]-=- and independently Kaplan, Tarjan and Tsioutsiouliklis [10] improve the amortized update time to ��. The improved update time in [4] is achieved by constructing a semidynamic data structure that is ad... |

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Citation Context ...rministic algorithms, (Edelsbrunner and Welzl [8], using Chan’s data structure achieving ��time). It is faster aÅ than the �ex pected running time �of the randomized algorithm of Har-Peled and Sharir =-=[11]-=-. Here is the slow growing inverse of Ackerman’s function. Lower bounds For the static convex hull computation there is a well known reduction to sorting, presented for example in the textbook by Prep... |

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Citation Context ...erval tree to store the convex hulls of the semidynamic deletion only data structures. This achieves �time extreme point queries. The authors [4] and independently Kaplan, Tarjan and Tsioutsiouliklis =-=[10]-=- improve the amortized update time to ��. The improved update time in [4] is achieved by constructing a semidynamic data structure that is adapted � � � better to the particular use. More precisely th... |

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Citation Context ...To achieve the O(log n) time extreme point queries, a construction based on an interval tree is used to store the convex hulls of the semidynamic deletion only data structures maintained. The authors =-=[BJ00]-=- (and independently Kaplan, Tarjan and Tsioutsiouliklis [HTK01]) improve the amortized update time to O(log n log log n). The improved update time in [BJ00] is achieved by constructing a semidynamic d... |

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