## On approximate range counting and depth (2007)

Venue: | In Proc. 23rd Annu. ACM Sympos. Comput. Geom |

Citations: | 22 - 1 self |

### BibTeX

@INPROCEEDINGS{Afshani07onapproximate,

author = {Peyman Afshani},

title = {On approximate range counting and depth},

booktitle = {In Proc. 23rd Annu. ACM Sympos. Comput. Geom},

year = {2007},

pages = {337--343}

}

### Years of Citing Articles

### OpenURL

### Abstract

ABSTRACT We improve the previous results by Aronov and Har-Peled (SODA'05) and Kaplan and Sharir (SODA'06) and present a randomized data structure of O(n) expected size which can answer 3D approximate halfspace range counting queries in O(log n k) expected time, where k is the actual value of the count. This is the first optimal method for the problem in the standard decision tree model; moreover, unlike previous methods, the new method is Las Vegas instead of Monte Carlo. In addition, we describe new results for several related problems, including approximate Tukey depth queries in 3D, approximate regression depth queries in 2D, and approximate linear programming with violations in low dimensions. Categories and Subject Descriptors F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems--geometrical problems and computations

### Citations

1762 |
Computational Geometry: An Introduction
- Preparata, Shamos
- 1985
(Show Context)
Citation Context ...at least as hard as range emptiness (deciding whether h contains any point). The existence of efficient halfspace range emptiness data structures in 2D and 3D (with O(n) space and O(log n) query time =-=[14, 24]-=-) suggests that efficient approximate halfspace range counting structures might be possible in the same dimensions. Indeed, that is the case, as was shown in two recent SODA papers. (Both papers are o... |

260 | Epsilon-nets and simplex range queries
- Haussler, Welzl
- 1987
(Show Context)
Citation Context ...dditive error of "n is tolerable, then the problem can be solved with constant space and query time in any fixed dimension by simply working with a sample (so-called "-approximation) of constant size =-=[17, 23]-=-. In particular, when the count k is close to n, we can get low relative error easily. So, the main challenge is in getting low relative error when k is small. In particular, for k = 0, we do not tole... |

251 | Geometric range searching and its relatives
- Agrawal, Erickson
- 1999
(Show Context)
Citation Context ...f Chazelle et al. [11] uses linear space and has O(log n + k) query time; and in 3D, a data structure of Chan [8] (see also Ramos [25]) uses O(n log log n) space and has O(log n + k) query time. (See =-=[1]-=- for further background.) Since it is generally believed that one cannot beat the query time of \Omega (n 2=3 ) with linear space for counting in 3D, researchers have turned to the approximate version... |

151 |
Applications of random sampling
- Clarkson, Shor
- 1989
(Show Context)
Citation Context ...n exact levels. In 3D, the best upper bound on the complexity of the exact k-level currently is O(nk 3=2 ) [27]. However, the total complexity of the k 0 -level for all k 0 = 0; : : : ; k is O(nk 2 ) =-=[12]-=-, which means that the average complexity of a k 0 -level with (1 \Gammas")k ^ k 0 ^ (1 + ")k is O"(nk). This is still too large for our purposes. We show that a form of approximate klevel exists wit... |

114 | Size-estimation framework with applications to transitive closure and reachability
- Cohen
- 1997
(Show Context)
Citation Context ...he second paper, by Kaplan and Sharir [18], improved the query time to O(log 2 n) with the same O(n log n) space bound, by using a different strategy that combined an approximation technique of Cohen =-=[13]-=- with a new combinatorial lemma about overlaying lower envelopes over all prefixes of a randomly permuted sequence of planes. This query algorithm is also Monte Carlo. Subsequently, in an updated vers... |

66 | Output-sensitive results on convex hulls, extreme points, and related problems, Discrete Comput
- Chan
- 1996
(Show Context)
Citation Context ... only by a constant factor. In d dimensions, Matous^ek [22] gave an algorithm for LP with violations with running time O(nk d+1 opt ). For small kopt ! n ff for a constant ff ? 0 depending on d, Chan =-=[5, 6]-=- showed how to improve the running time to T small (n; kopt) = O(n log kopt) by using data structures for linear programming queries. The above theorem then implies an algorithm with running time O"(T... |

62 | On approximating the depth and related problems
- Aronov, Har-Peled
- 2005
(Show Context)
Citation Context ... that is the case, as was shown in two recent SODA papers. (Both papers are of particular relevance to us here, as some of the techniques used are related to ours.) The first, by Aronov and Har-Peled =-=[2]-=-, described a black-box reduction from approximate halfspace range counting to range emptiness, with polylogarithmic increase in space and query time. In 3D, their resulting data structure needs O(n ... |

45 | Low-dimensional linear programming with violations
- Chan
(Show Context)
Citation Context ... halfspaces have a nonempty intersection. (Equivalently, we want the minimum depth, in another sense of the word, in an arrangement of halfspaces.) This problem was studied by Matous^ek [22] and Chan =-=[7]-=-. In 2D and 3D, the current best running time, by Chan, is O(n log k + k 2 log n) and O(n log k + k 11=4 n 1=4 log O(1) n) respectively; in higher dimensions d, the time bound is slightly less than O(... |

45 | An optimal randomized algorithm for maximum tukey depth
- Chan
- 2004
(Show Context)
Citation Context ...han building data structures to estimate the depth of a query point/line. Points of maximum Tukey depth and lines of maximum regression depth in 2D can be computed in optimal O(n log n) expected time =-=[10, 19]-=-, but it appears difficult to find data structures with nontrivial worst-case performance that can compute the exact depth of an arbitrary query point/line. The maximum depth in either definition is \... |

38 |
Constructing Belts in twodimensional arrangements with applications
- Edelsbrunner, Welzl
- 1986
(Show Context)
Citation Context ...ted by applications in computational statistics (see below). Variants of approximate halfspace range counting were actually considered early on, for example, in a 1986 paper by Edelsbrunner and Welzl =-=[15]-=-, who studied the 2D problem with additive instead of relative error (and called the problem "halfplanar range estimation"). If an additive error of "n is tolerable, then the problem can be solved wit... |

24 | On range reporting, ray shooting and k-level construction, in
- Ramos
(Show Context)
Citation Context ...dmits efficient algorithms in low dimension: in 2D, a data structure of Chazelle et al. [11] uses linear space and has O(log n + k) query time; and in 3D, a data structure of Chan [8] (see also Ramos =-=[25]-=-) uses O(n log log n) space and has O(log n + k) query time. (See [1] for further background.) Since it is generally believed that one cannot beat the query time of \Omega (n 2=3 ) with linear space f... |

19 | Randomized incremental constructions of three-dimensional convex hulls and planar Voronoi diagrams, and approximate range counting
- Kaplan, Sharir
- 2006
(Show Context)
Citation Context ...e answer with high probability, but is Monte Carlo in the sense that the query algorithm does not know if the returned answer is a correct approximation or not. The second paper, by Kaplan and Sharir =-=[18]-=-, improved the query time to O(log 2 n) with the same O(n log n) space bound, by using a different strategy that combined an approximation technique of Cohen [13] with a new combinatorial lemma about ... |

1 |
Relative "-approximations in geometry. http://valis.cs.uiuc.edu/~sariel/ research/papers/06/relative/, 2006. Also with B. Aronov, to appear
- Har-Peled, Sharir
- 2007
(Show Context)
Citation Context ...mputational geometry. The Tukey depth (also called halfspace depth) of a query point q with respect to a point set P 1 After this work was submitted, we learn of a third paper by Har-Peled and Sharir =-=[16]-=-, which contains among other results an improvement of the query time to O(log n log log n) but with a larger space bound of O(n log O(1) n) for the 3D problem; as we will see, our result is better. A... |