## Faster Core-Set Constructions and Data Stream Algorithms in Fixed Dimensions (2003)

Venue: | Comput. Geom. Theory Appl |

Citations: | 64 - 3 self |

### BibTeX

@INPROCEEDINGS{Chan03fastercore-set,

author = {Timothy M. Chan},

title = {Faster Core-Set Constructions and Data Stream Algorithms in Fixed Dimensions},

booktitle = {Comput. Geom. Theory Appl},

year = {2003},

pages = {152--159}

}

### Years of Citing Articles

### OpenURL

### Abstract

We speed up previous (1 + ")-factor approximation algorithms for a number of geometric optimization problems in xed dimensions: diameter, width, minimum-radius enclosing cylinder, minimum-width annulus, minimum-volume bounding box, minimum-width cylindrical shell, etc.

### Citations

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Citation Context ...explicitly constructing the upper envelope, which in the dual corresponds to a planar convex hull. Since thesrst coordinates are from [E] and can thus be radix-sorted in O(n + E ) time, Graham's scan =-=[10]-=- requires O(n + E ) time. The answers q[x] for all x 2 [F ] can be computed by another linear scan. Therefore, T 2 (n) = O(n + F ). The above recurrence solves to T d (n) = O(n +E d 2 F ). We can also... |

378 | Data streams: algorithms and applications
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Citation Context ...4] have also given data structures that support insertion only but with remarkably little space. The need to handle massive data has generated considerable attention recently to the data stream model =-=[21, 22, 25]-=-, where the input is too big to be stored, only one pass over the input is possible, and algorithms can maintain only a bounded amount of information at any time. Exact algorithms with sublinear space... |

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Citation Context ...4] have also given data structures that support insertion only but with remarkably little space. The need to handle massive data has generated considerable attention recently to the data stream model =-=[21, 22, 25]-=-, where the input is too big to be stored, only one pass over the input is possible, and algorithms can maintain only a bounded amount of information at any time. Exact algorithms with sublinear space... |

112 | Approximate clustering via core-sets
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Citation Context ...nce such a subset is found, we can simply return a solution to the subset by a direct algorithm, exact or approximate. This approach has proved successful also for high-dimensional geometric problems =-=[7,-=- 8], but we will focus only on the case ofsxed dimensions. Denition 1.1 Given a single-argument measure () that is monotone (i.e., Q P implies (Q) (P )), we say that a subset R P is an "-core-... |

87 | A replacement for Voronoi diagrams of near linear size
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Citation Context ... discrete Voronoi diagrams and related constructs, which can identify the (exact) nearest neighbors of a grid point set to all grid points. This notion is dierent from \approximate Voronoi diagrams&qu=-=ot; [6, 18]-=- and does not seem to have receive as much attention in computational geometry, although the problem has been considered by Breu et al. [11] (see also [28]) and has applications in image processing. T... |

73 |
Metric entropy of some classes of sets with differentiable boundaries
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Citation Context ...proximate the extent measure, using a single core-set of a smaller total size|O(1=" (d 1)=2 ) instead of O(1=" d 1 ). This gives better results for some problems. The idea is based on those =-=of Dudley [14] and -=-Bronshteyn and Ivanov [12]. The running time for the construction is O(n+ 1=" 3(d 1)=2 ). We again show how to improve the running time using Theorem 2.1/Corollary 2.2. In order to do this, we ne... |

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Citation Context ...s notion is dierent from \approximate Voronoi diagrams" [6, 18] and does not seem to have receive as much attention in computational geometry, although the problem has been considered by Breu et =-=al. [11] (see-=- also [28]) and has applications in image processing. The second part of the paper (Section 3) is mostly independent of thesrst and uses a new coreset construction that \corrects" itself as point... |

60 |
Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus
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(Show Context)
Citation Context ...There are at least two \obvious " approximation algorithms, one running in O(n=" (d 1)=2 ) time [5] (by rounding directions) and one running in O(n + 1=" 2(d 1) ) [9] (by rounding point=-=s). The author [13] has obser-=-ved that a combination of the two yields an O(n + 1=" 3(d 1)=2 ) time bound. He has also given two simple algorithms running in O(n + 1=" d 1=2 ) time; the time bound can be reduced slightly... |

41 | Linear-size approximate Voronoi diagrams
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(Show Context)
Citation Context ... discrete Voronoi diagrams and related constructs, which can identify the (exact) nearest neighbors of a grid point set to all grid points. This notion is dierent from \approximate Voronoi diagrams&qu=-=ot; [6, 18]-=- and does not seem to have receive as much attention in computational geometry, although the problem has been considered by Breu et al. [11] (see also [28]) and has applications in image processing. T... |

34 | Efficient approximation and optimization algorithms for computational metrology
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Citation Context ...d 2 ) log(1=")) if ecient planar-point-location data structures are available. 2. The width, i.e., the minimum width over all enclosing slabs (regions between two parallel hyperplanes). Duncan et=-= al. [15] have desc-=-ribed an O(n=" (d 1)=2 )-time approximation algorithm; the technique by Barequet and Har-Peled [9] can be used in combination to yield an O(n+ 1=" 3(d 1)=2 ) time bound [13]. (Slight improve... |

33 | Line transversals of balls and smallest enclosing cylinders in three dimensions
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Citation Context ...ning time can in theory be reduced to O((n + 1=" 2 ) polylog (1=")) for d = 4, and to O((n + 1=" d 2 ) log(1=")) for d > 4. 3. An enclosing cylinder of the minimum radius. Agarwal,=-= Aronov, and Sharir [1] gave an O-=-(n=" 2 )- time approximation algorithm for d = 3. The author [13] gave an O(n + 1=" 3(d 1)=2 )-time algorithm for allsxed d (omitting minor theoretical improvements). We improve the time bou... |

28 | Computing diameter in the streaming and sliding-window models
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Citation Context ... the log n factors can be removed, or more boldly, whether constant space is possible. For diameter, one of earlier algorithms [5] uses just O(1=" (d 1)=2 ) space in the data stream setting (see =-=also [16, 22]-=-). For the width, no previous algorithms yield constant-space solutions; the problem is apparently open even in the two-dimensional case, as Hershberger and Suri [22] mentioned only partial results un... |

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25 |
Finding minimal enclosing boxes
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Citation Context ...t an O(n + 1=" d 1 ) bound (which can also be marginally improved with advanced data structures). For the bounding box problem in dimension d = 3, we can use O'Rourke's exact O(jRj 3 )-time algor=-=ithm [27] to g-=-et an O(n + 1=" 3 ) bound. Other problems (items 5 and 6) can be solved by taking appropriate lifting maps (linearization), as described by Agarwal, Har-Peled, and Varadarjan [4]. For example, fo... |

24 | A practical approach for computing the diameter of a point set
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(Show Context)
Citation Context ... c ) for some absolute constant c is the best one can hope for. It would also be interesting to see how well our new diameter algorithm competes with the experimentally ecient algorithms by Har-Peled =-=[17]-=- and Malandain and Boissonnat [24] in practice. 3 Data Stream Algorithms 3.1 Warm-Up: Constant Factor for Minimum Cylinder We now consider algorithms under the data stream model. To illustrate the mai... |

19 |
On maintaining the width and diameter of a planar point-set online
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Citation Context ...independent of n. For the width and the cylinder problem, for example, the space and amortized insertion time are bounded by O([(1=") log(1=")] d 1 ). (This result even improves known data s=-=tructures [23, 26] for -=-approximating the two-dimensional width in the traditional model; these structures required linear space and logarithmic insertion time.) For the spherical-shell problem, the bound is O([(1=") lo... |

17 | Streaming Geometric Optimization Using Graphics Hardware
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(Show Context)
Citation Context ...he problem is apparently open even in the two-dimensional case, as Hershberger and Suri [22] mentioned only partial results under the assumption that the diameter-to-width ratio is bounded. (See also =-=[3]-=- for multiple-pass algorithms that exploit graphics hardware.) In the second part of the paper, we answer the question in the armative by obtaining one-pass, data stream algorithms for all the above p... |

15 |
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Citation Context ...sing a single core-set of a smaller total size|O(1=" (d 1)=2 ) instead of O(1=" d 1 ). This gives better results for some problems. The idea is based on those of Dudley [14] and Bronshteyn a=-=nd Ivanov [12]. The-=- running time for the construction is O(n+ 1=" 3(d 1)=2 ). We again show how to improve the running time using Theorem 2.1/Corollary 2.2. In order to do this, we need to simplify one step of Agar... |

14 |
Eciently approximating the minimum-volume bounding box of a point set in three dimensions
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- 2001
(Show Context)
Citation Context ...m distance over all pairs of points. There are at least two \obvious " approximation algorithms, one running in O(n=" (d 1)=2 ) time [5] (by rounding directions) and one running in O(n + 1=&=-=quot; 2(d 1) ) [9] (by round-=-ing points). The author [13] has observed that a combination of the two yields an O(n + 1=" 3(d 1)=2 ) time bound. He has also given two simple algorithms running in O(n + 1=" d 1=2 ) time; ... |

12 | Computing the diameter of a point set
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(Show Context)
Citation Context ...is the best one can hope for. It would also be interesting to see how well our new diameter algorithm competes with the experimentally ecient algorithms by Har-Peled [17] and Malandain and Boissonnat =-=[24]-=- in practice. 3 Data Stream Algorithms 3.1 Warm-Up: Constant Factor for Minimum Cylinder We now consider algorithms under the data stream model. To illustrate the main diculties, we start with the min... |

12 | Maintaining the approximate width of a set of points in the plane
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(Show Context)
Citation Context ...independent of n. For the width and the cylinder problem, for example, the space and amortized insertion time are bounded by O([(1=") log(1=")] d 1 ). (This result even improves known data s=-=tructures [23, 26] for -=-approximating the two-dimensional width in the traditional model; these structures required linear space and logarithmic insertion time.) For the spherical-shell problem, the bound is O([(1=") lo... |

11 | Convex hulls and related problems in data streams
- Hersberger, Suri
- 2003
(Show Context)
Citation Context ...4] have also given data structures that support insertion only but with remarkably little space. The need to handle massive data has generated considerable attention recently to the data stream model =-=[21, 22, 25]-=-, where the input is too big to be stored, only one pass over the input is possible, and algorithms can maintain only a bounded amount of information at any time. Exact algorithms with sublinear space... |

9 | Computing the penetration depth of two convex polytopes
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(Show Context)
Citation Context ...n to yield an O(n+ 1=" 3(d 1)=2 ) time bound [13]. (Slight improvements were known in low dimensions using more complicated data structures; for example, for d = 3, the author [13] and Agarwal et=-= al. [2] have respectively obtain-=-ed an O((n + 1=") log(1=")) and an O(n + (1=") log 2 (1=")) bound.) We give a new algorithm that runs in O(n + 1=" d 1 ) time. With advanced data structures, the running time ... |

5 |
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Citation Context ...erent from \approximate Voronoi diagrams" [6, 18] and does not seem to have receive as much attention in computational geometry, although the problem has been considered by Breu et al. [11] (see =-=also [28]) and-=- has applications in image processing. The second part of the paper (Section 3) is mostly independent of thesrst and uses a new coreset construction that \corrects" itself as points are inserted ... |

4 |
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- Bădoiu, Clarkson
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(Show Context)
Citation Context ...nce such a subset is found, we can simply return a solution to the subset by a direct algorithm, exact or approximate. This approach has proved successful also for high-dimensional geometric problems =-=[7,-=- 8], but we will focus only on the case ofsxed dimensions. Denition 1.1 Given a single-argument measure () that is monotone (i.e., Q P implies (Q) (P )), we say that a subset R P is an "-core-... |

4 |
Shape with outliers
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(Show Context)
Citation Context ...aradarajan's (see Section 2.3), as Barequet and Har-Peled's subroutine [9] is not explicitly called. (This dimension-recursion approach is perhaps closer to a core-set algorithm by Har-Peled and Wang =-=[20] when dualized.-=-) Theorem 3.4 Given a stream of points in IR d , we can maintain an "-core-set for the extent measure, in a single pass, using O([(1=") log(1=")] d 1 ) space and O(1) amortized time. Pr... |

2 |
High-dimensional shape in linear time
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- 2003
(Show Context)
Citation Context ...approximation algorithmssrst. There is a very simple constant-factor algorithm, noted by Agarwal, Aronov, and Sharir [1] (later generalized by Barequet and Har-Peled [9] and Har-Peled and Varadarajan =-=[19]-=-). The algorithm just picks an arbitrary input point, say o,snds the farthest point v from o, and returns the farthest point from the lines! ov . Let Rad(P ) denote the minimum radius of all cylinders... |

1 |
Approximating extent measures of points. http://valis.cs.uiuc.edu/~sariel/research/papers/01/fitting/, 2003. Preliminary versions appeared
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(Show Context)
Citation Context ...egion between two concentric spheres) of the minimum width. The author [13] was thesrst to obtain linear-time approximation algorithms in allsxed dimensions d 2. Agarwal, Har-Peled, and Varadarajan [=-=4] have giv-=-en the best time bound for suciently large (butsxed) d: O(n + 1=" 3d ). We improve their time bound to O(n + 1=" 2d ), which beats previous results [13] for all d 4. 6. Agarwal, Har-Peled, ... |