## A dynamic data structure for 3-d convex hull and 2-d nearest neighbor queries (2006)

Venue: | In: Proceedings of the seventeenth ACM-SIAM symposium on Discrete algorithm |

Citations: | 23 - 5 self |

### BibTeX

@INPROCEEDINGS{Chan06adynamic,

author = {Timothy M. Chan},

title = {A dynamic data structure for 3-d convex hull and 2-d nearest neighbor queries},

booktitle = {In: Proceedings of the seventeenth ACM-SIAM symposium on Discrete algorithm},

year = {2006},

pages = {1196--1202},

publisher = {ACM Press}

}

### OpenURL

### Abstract

We present a fully dynamic randomized data structure that can answer queries about the convex hull of a set of n points in three dimensions, where insertions take O(log 3 n) expected amortized time, deletions take O(log 6 n) expected amortized time, and extreme-point queries take O(log 2 n) worst-case time. This is the first method that guarantees polylogarithmic update and query cost for arbitrary sequences of insertions and deletions, and improves the previous O(n ε)-time method by Agarwal and Matouˇsek a decade ago. As a consequence, we obtain similar results for nearest neighbor queries in two dimensions and improved results for numerous fundamental geometric problems (such as levels in three dimensions and dynamic Euclidean minimum spanning trees in the plane). 1

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Citation Context ...s. We briefly review previous work on this dynamic 2-d problem. A straightforward sublinear method with roughly O( √ n) update and query time was known early on [3]. At FOCS’92, Agarwal and Matouˇsek =-=[2]-=- presented the first improvements. In the 2-d case, they gave two methods, one with O(log n) query time and O(n ε ) amortized update time, and another with O(n ε ) query time and O(log 2 n) amortized ... |

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Citation Context ... vertex of L between the two rays. The number of such curves is therefore at most 2n/r + 2n/r + n/r = O(n/r). We can construct the k-level in O(nβ(n) log n) expected time by an algorithm of Har-Peled =-=[22]-=-. By sweeping the vertices of L from left to right, we can keep track of the subset of curves intersecting each downward ray and therefore generate the conflict lists within the same time bound. ✷ 12... |

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Citation Context ...size. This follows as the k-level can be constructed by performing O(f) insertions, deletions, and nonvertical ray shooting queries. According to the current combinatorial bound on the k-level in 3-d =-=[36]-=-, f = O(nk 3/2 ) always. The author [7] showed how to speed up any O(T (n))-time k-level algorithm to run in O(n log n+ (n/k)T (k)) expected time. By plugging in T (n) = O(n 5/2 log 6 n), we now have ... |

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Citation Context ...evel. The conflict list of each cell would then have size close to O(n/r); extra steps are required to make it truly O(n/r). The O(n log n) expected running time was obtained by an algorithm of Ramos =-=[33]-=-. That vertical cells are sufficient for this particular case of the shallow cutting lemma was observed by the author [10, Lemma 3.1]. 3Our solution to the dynamic lower envelope problem is described... |

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Citation Context ...rected edge from each point to its nearest neighbor) in O(log 6 n) expected amortized time. This follows from a reduction of the problem to vertical ray shooting for a dynamic lower envelope noted in =-=[9]-=-. • We can maintain the width of a 2-d point set under insertions (without deletions) in O(log 7 n) expected amortized time (instead of O(n ε )). This follows from a method of Eppstein [21], which req... |

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Citation Context ...at is extreme along a given direction. (See [6, 18] for earlier results on the semi-online case.) Because these queries are not decomposable, we cannot directly apply our query algorithm. 8Matouˇsek =-=[26]-=- showed how linear programming queries reduce to ray shooting queries via a multidimensional parametric search, without any change to the data structure. Specifically, suppose that there is a parallel... |

14 |
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Citation Context ... insertions and deletions but we are given the entire sequence in advance) and the semi-online case (where during the insertion of an element, we are given the time when that element will be deleted) =-=[17]-=-. Several researchers—Clarkson, Mehlhorn, and Seidel [14], Devillers, Meiser, and Teillaud [15], Schwarzkopf [34], and Mulmuley [30]—explored the case where the update sequence is random in a certain ... |

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(Show Context)
Citation Context ...optimal method is currently not known. It might be possible to de-amortize our time bounds by known tricks [29]. Obtaining a deterministic polylogarithmic method remains open. In a recent development =-=[1]-=-, a more space-efficient data structure has been found for the static 3-d halfspace range reporting problem. By this result and the ideas from Section 8, the space bound for our dynamic 3-d data struc... |

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Citation Context ... the convex hull. Linear programming queries. For a dynamic set of halfspaces in 3-d, a linear programming query asks for a point inside the intersection that is extreme along a given direction. (See =-=[6, 18]-=- for earlier results on the semi-online case.) Because these queries are not decomposable, we cannot directly apply our query algorithm. 8Matouˇsek [26] showed how linear programming queries reduce t... |

5 |
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Citation Context ...pe noted in [9]. • We can maintain the width of a 2-d point set under insertions (without deletions) in O(log 7 n) expected amortized time (instead of O(n ε )). This follows from a method of Eppstein =-=[21]-=-, which requires an amortized O(log n) number of updates and vertical ray shooting queries for a dynamic 3-d halfspace intersection. • We can obtain a randomized algorithm for the 3-d convex layers (o... |

3 |
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Citation Context ...ard vertical rays at every (n/r)-th vertex of L. Return the cells formed by partitioning the region underneath L with these rays. (The idea of using levels to construct cuttings is not new; e.g., see =-=[24]-=-.) It is known [35] that the (≤ 2n/r)-level has O(n(n/r)β(n)) vertices in total. Thus, L has an expected O(nβ(n)) number of vertices. (If the actual number exceeds this bound by a constant factor, we ... |