## Approximate Shape Fitting via Linearization (2001)

Venue: | In Proc. 42nd Annu. IEEE Sympos. Found. Comput. Sci |

Citations: | 14 - 7 self |

### BibTeX

@INPROCEEDINGS{Har-peled01approximateshape,

author = {Sariel Har-peled and Kasturi R. Varadarajan},

title = {Approximate Shape Fitting via Linearization},

booktitle = {In Proc. 42nd Annu. IEEE Sympos. Found. Comput. Sci},

year = {2001},

pages = {66--73}

}

### OpenURL

### Abstract

Shape fitting is a fundamental optimization problem in computer science. In this paper, we present a general and unified technique for solving a certain family of such problems. Given a point set P in R d, this technique can be used to ε-approximate: (i) the min-width annulus and shell that contains P, (ii) minimum width cylindrical shell containing P, (iii) diameter, width, minimum volume bounding box of P, and (iv) all the previous measures for the case the points are moving. The running time of the resulting algorithms is O(n + 1/ε c), where c is a constant that depends on the problem at hand. Our new general technique enable us to solve those problems without resorting to a careful and painful case by case analysis, as was previously done for those problems. Furthermore, for several of those problems our results are considerably simpler and faster than what was previously known. In particular, for the minimum width cylindrical shell problem, our solution is the first algorithm whose running time is subquadratic in n. (In fact we get running time linear in n.) 1

### Citations

1762 | Computational Geometry: An Introduction - Preparata, Shamos - 1985 |

409 | FastMap: A Fast Algorithm for Indexing, Data-Mining and
- Faloutsos, Lin
- 1995
(Show Context)
Citation Context ...le, hyperplane) that best fits the point-set. Shape fitting is a fundamental optimization problem and has numerous usages in graphics (shape simplification, collusion detection), learning, datamining =-=[FL95]-=-, databases [AWY + 99] (projective clustering), metrology, compression, and geometric optimization. A restricted variant of this problem is where the shape to be fitted to the input is defined by a (s... |

247 | Fast algorithms for projected clustering - Aggarwal, Wolf, et al. - 1999 |

231 | Data Structures for Mobile Data
- Basch, Guibas, et al.
- 1997
(Show Context)
Citation Context ...mate shape fitting for such variants. We handle both the classical static case, and also apply it to the more general (and considerably harder) case when the points are moving (i.e., kinetic settings =-=[BGH97]-=-). For example, imagine that one would like to maintain the minimum volume bounding box that contains a set of moving points. Our technique enable us to compute such a bounding box (which is also movi... |

158 | Optimal point location on a monotone subdivision - Edelsbunner, Guibas, et al. - 1986 |

89 | Applications of parametric searching in geometric optimization - Agarwal, Sharir, et al. - 1994 |

80 | Range searching with semi-algebraic sets
- Agarwal, Matousek
- 1994
(Show Context)
Citation Context ...ow in this paper that that once the problem is stated as a linearized optimization problem, it can be approximated efficiently. Linearization is quite powerful and can be applied to numerous problems =-=[AM94]-=-, and thus our approximation technique is quite broad, as it can be applied to optimization problems of this type. In particular, we show that it suffices to approximate the shape for a small subset o... |

79 | Alinear-time algorithm for computing the Voronoi diagram of a convex polygon - Aggarwal, Guibas, et al. - 1989 |

77 | Efficiently approximating the minimum-volume bounding box of a point set in three dimensions
- Barequet, Har-Peled
(Show Context)
Citation Context ...t of Computer Science, University of Iowa, kvaradar@cs.uiowa.edu 1sIf the shape we are trying to fit is convex (i.e., find a cylinder that contains the point-set), then it is implied by previous work =-=[BH99]-=- that using convex shape approximation techniques one can first approximate the given input by a small point-set, and solve the optimization problem on this sampled subset. Since convex shape approxim... |

73 |
Metric entropy of some classes of sets with differentiable boundaries
- Dudley
- 1974
(Show Context)
Citation Context ... time, a subset K ⊆ H of O(1/ε k/2 ) linear functions, such that K is an ε-approximation for H. Proof: Follows by using the algorithm of Gärtner [Gär95] together with Dudley’s approximation technique =-=[Dud74]-=- on the set generated by Theorem 2.1. The (straightforward but tedious) details are omitted and will appear in the full-version. Theorem 2.3 Given a family of k-variate linear functions H = {h1, . . .... |

60 |
Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus
- Chan
(Show Context)
Citation Context ...e in R2 , and in roughly 1 3− O(n 19 ) time in R3 . For the approximate annulus problem, we obtain an algorithm that runs in O(n+1/ε3d ) time in Rd . This is an improvement over the algorithm of Chan =-=[Cha00]-=- which runs in roughly O(n+1/εd2 /4 ) time. The running time of our algorithm can be improved further; we mention the annulus result primarily to point out that very good running times can be obtained... |

46 | A subexponential algorithm for abstract optimization problems
- Gärtner
- 1995
(Show Context)
Citation Context ...a parameter ε > 0, one can compute, in O(n + 1/ε 3k/2 ) time, a subset K ⊆ H of O(1/ε k/2 ) linear functions, such that K is an ε-approximation for H. Proof: Follows by using the algorithm of Gärtner =-=[Gär95]-=- together with Dudley’s approximation technique [Dud74] on the set generated by Theorem 2.1. The (straightforward but tedious) details are omitted and will appear in the full-version. Theorem 2.3 Give... |

46 |
M.: The spaces of convex bodies
- Gruber
- 1993
(Show Context)
Citation Context ... techniques one can first approximate the given input by a small point-set, and solve the optimization problem on this sampled subset. Since convex shape approximation is (relatively) well understood =-=[Gru93]-=-, this results in fast and efficient approximation algorithms. We state this more formally in Section 3.1, where we present a unified approach for this type of problems. However, if we are interested ... |

44 | Efficient randomized algorithms for some geometric optimization problems - Agarwal - 1995 |

41 | Computing envelopes in four dimensions with applications - Agarwal, Aronov, et al. - 1997 |

32 | Roundness algorithms using the Voronoi diagrams - Ebara, Fukuyama, et al. - 1989 |

32 | Out-of-roundness problem revisited - Le, Lee - 1991 |

31 | Maintaining approximate extent measures of moving points
- Agarwal, Har-Peled
- 2001
(Show Context)
Citation Context ...x hull of P at all times. This means that Q is an ε-approximation of P with respect to any “convex” measure. Such results were previously known only for diameter and minimumradius enclosing ball, see =-=[AH01]-=-. These results generalize to algebraic motion and to “non-convex” measures like minimum-width spherical/cylindrical shell. The paper is organized as follows: In Section 2, we introduce the basic tool... |

29 | Parametric and kinetic minimum spanning trees - Agarwal, Eppstein, et al. - 1998 |

29 | Cylindrical static and kinetic binary space partitions - Agarwal, Guibas, et al. - 1997 |

28 | Establishment of a pair of concentric circles with the minimum radial separation for assessing roundness error, Computer Aided Design 24 (3 - Roy, Zhang - 1992 |

23 | Approximation and exact algorithms for minimum – width annuli and shells - Agarwal, Aronov, et al. - 1999 |

22 | Review of dimensioning and tolerancing: Representation and processing - Roy, Liu, et al. - 1991 |

19 | Maintaining the extent of a moving point set
- Agarwal, Guibas, et al.
- 1997
(Show Context)
Citation Context ...ently. Currently, no efficient algorithms 9 p 2 i .sare known for maintaining most of those exact measures efficiently for a moving point-set. For the case of the diameter, a result of Agarwal et al. =-=[AGHV97]-=- shows that the diameter of a point set under linear motion in the plane can change quadratic number of times. Theorem 3.7 Given a point-set P with n linearly-moving points, and a parameter ε > 0, one... |

18 | A complete roundness classification procedure, in - Mehlhorn, Shermer, et al. - 1997 |

18 | T.J.: Approximation by circles - Rivlin - 1979 |

16 | Fitting a set of points by a circle - Garcia-Lopez, Ramos - 1997 |

15 | On the width and roundness of a set of points in the plane - Smid, Janardan - 1999 |

14 | Issues in the metrology of geometric tolerancing - Yap, Chang - 1997 |

13 | Exact and approximation algorithms for minimum-width cylindrical shells, Discrete Comput
- Agarwal, Aronov, et al.
(Show Context)
Citation Context ...rical shell containing P whose width is at most (1 + ε) times the width of the minimum-width cylindrical shell containing P . This problem is motivated by applications in computational metrology, see =-=[AAS00]-=-. Agarwal et al. [AAS00] present an algorithm that computes the exact minimum-width cylindrical shell for a set of n points in R3 in O(n5 ) time. Since computing the optimal shell is so expensive, the... |

13 | Simplified kinetic connectivity for rectangles and hypercubes - Hershberger, Suri - 2001 |

9 | Probing for near centers and relative roundness - Shermer, Yap - 1995 |