## On approximating the depth and related problems (2005)

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Venue: | In Proc. 16th ACM-SIAM Sympos. Discrete Algorithms |

Citations: | 63 - 11 self |

### BibTeX

@INPROCEEDINGS{Aronov05onapproximating,

author = {Boris Aronov and Sariel Har-peled},

title = {On approximating the depth and related problems},

booktitle = {In Proc. 16th ACM-SIAM Sympos. Discrete Algorithms},

year = {2005},

pages = {886--894}

}

### Years of Citing Articles

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### Abstract

We study the question of finding a deepest point in an arrangement of regions, and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for solving linear programming with violations approximately. We also use this technique to approximate the disk covering the largest number of red points, while avoiding all the blue points, given two such sets in the plane. Using similar techniques imply that approximate range counting queries have roughly the same time

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Citation Context ...ithms exist [Cha05]. See Section 4.1.) To appreciate this result, consider the case where k = √ n. A natural approach to approximate linear programming with violations is to compute a δ-approximation =-=[VC71]-=- to the set of constraints in L, and apply the exact algorithm on this sample. However, in this case δ = ε/ √ n, and the required random sample would include (almost) all the constraints of L, thus ac... |

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419 | Davenport-Schinzel Sequences and their Geometric Applications
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Citation Context ... the complexity of the arrangement formed by S be bounded by |S| O(d) . The arrangement formed by S is the decomposition of the plane (or, more generally IR d ) induced by a collection of such shapes =-=[SA95]-=-. The combinatorial complexity of an arrangement is the total number of edges, faces, and vertices in the arrangement. A k-level in an arrangement of curves is the closure of the set of points on the ... |

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Citation Context ...his point, we observe that B can be computed in time O(n log n) time (for example, by computing the convex hull of the dual points, see [Ede87]) and preprocessed into the Dobkin-Kirkpatrick hierarchy =-=[DK90]-=-, so that Cπ can be manipulated implicitly. More specifically, it is easily checked that the following operations can be performed on the resulting pseudodisks in logarithmic time: Computation of boun... |

103 |
Algorithmic Geometry
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Citation Context ...epth at most k in their arrangement A = A(S) is O(nk) [CS89, Sha03, Sha91]. (iv) Using a randomized incremental construction, one can compute all faces at depth at most k in A in time O(nk + n log n) =-=[BY98]-=-. (In fact, if we are interested in “depth thresholding” rather than construction of a subset of A, this can be done with a standard plane sweep in O(nk log n) deterministic time [dBvKOS00].) Theorem ... |

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85 | Approximate range searching
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Citation Context ...ge searching are computationally nearly equivalent. Since this circumvents the aforementioned gap, we believe it to be of independent interest. We contrast our results with the work of Arya and Mount =-=[AM00]-=- that considers a different notion of approximation for the range searching problem. They approximate the range (using a distance which depends on the diameter of the query shape) and return the exact... |

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Citation Context ...In the (somewhat reasonable) computation model assumed by Brönnimann et al. [BCP93], halfspace counting queries require Ω ∗ (n 1−2/(d+1) ) time per query, if only linear space is allowed. 1 Matouˇsek =-=[Mat93]-=- showed how to answer such queries in O(n 1−1/d ) time. On the other hand, halfspace emptiness queries can be answered in logarithmic time in two and three dimensions, and in O ∗ (n 1−1/⌊d/2⌋ ) time i... |

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Reporting points in halfspaces
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Citation Context ...r such queries in O(n 1−1/d ) time. On the other hand, halfspace emptiness queries can be answered in logarithmic time in two and three dimensions, and in O ∗ (n 1−1/⌊d/2⌋ ) time in higher dimensions =-=[Mat92]-=-, using near-linear space; this is slightly faster than the aforementioned lower bound for the counting problem. Especially interesting is the situation in dimensions two and three, where the gap is b... |

50 | Range searching with efficient hierarchical cuttings - Matousek - 1993 |

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37 |
On geometric optimization with few violated constraints, SCG ’94
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Citation Context ... at most (1 + ε)kopt constraints of L and such that f(u) ≤ f(v). The running time of the new algorithm is O(n(ε −2 log n) d+1 ). This compares favorably with the previous exact algorithm of Matouˇsek =-=[Mat95]-=- which requires O(nk d+1 ) running time. (In two and three dimensions faster exact algorithms exist [Cha05]. See Section 4.1.) To appreciate this result, consider the case where k = √ n. A natural app... |

32 | Random sampling, halfspace range reporting, and construction of (#k)-levels in three dimensions - CHAN |

28 | Shape fitting with outliers - Har-Peled, Wang - 2004 |

23 | The complexity of the union of (α, β)-covered objects - Efrat |

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18 |
Randomized optimal algorithm for slope selection
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Citation Context ...points of the lines (and it can be computed in O(n) time). Alternatively, one can compute the leftmost, rightmost, topmost and bottommost vertices in the arrangement of the lines L in O(n log n) time =-=[Mat91]-=-. Furthermore, one can compute a lower bound ∆ on the distance between a vertex of A(L) and a line of L not passing through it. Indeed, consider a line ℓ: ax + by = c, where |a|, |b|, |c| ≤ M, and a p... |

18 | The clarkson-shor technique revisited and extended - Sharir - 2001 |

17 | On approximate halfspace range counting and relative epsilonapproximations
- Aronov, Har-Peled, et al.
(Show Context)
Citation Context ...s can be improved. Notice that this argument only applies to the case where emptiness queries are treated as a “black box” operations. If this assumption is removed, certain improvements are possible =-=[AHS07]-=-. 5.3 Applications 5.3.1 Halfplane and halfspace range counting Using the data structure of Dobkin and Kirkpatrick [DK85], one can answer emptiness halfspace range searching queries in logarithmic tim... |

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14 | How hard is halfspace range searching? Discrete Comput - Brönnimann, Chazelle, et al. - 1993 |

12 | When crossings count: Approximating the minimum spanning tree - Har-Peled, Indyk - 2000 |

5 | Speeding up the incremental construction of the union of geometric objects in practice - Ezra, Halperin, et al. |

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1 | 21 + 02] [AHS07 - Agarwal, Hagerup, et al. - 2005 |

1 | The Clarkson-Shor technique revisited and extended - Geom - 1991 |