## A package for exact kinetic data structures and sweepline algorithms (2007)

Citations: | 8 - 1 self |

### BibTeX

@MISC{Russel07apackage,

author = {Daniel Russel and Menelaos I. Karavelas and Leonidas J. Guibas},

title = {A package for exact kinetic data structures and sweepline algorithms},

year = {2007}

}

### OpenURL

### Abstract

In this paper we present a package for implementing exact kinetic data structures built on objects which move along polynomial trajectories. We discuss how the package design was influenced by various considerations, including extensibility, support for multiple kinetic data structures, access to existing data structures and algorithms in CGAL, as well as debugging. Due to the similarity between the operations involved, the software can also be used to compute arrangements of polynomial objects using a sweepline approach. The package consists of three main parts, the kinetic data structure framework support code, an algebraic kernel which implements the set of algebraic operations required for kinetic data structure processing, and kinetic data structures for Delaunay triangulations in one and two dimensions, and Delaunay and regular triangulations in three dimensions. The models provided for the algebraic kernel support both exact operations and inexact approximations with heuristics to improve numerical stability. 1

### Citations

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- 1999
(Show Context)
Citation Context ...inetic data structures event times are only representable by algebraic numbers, rendering computations at event times extremely expensive, requiring a number type such as CORE::Expr [7] or LEDA::real =-=[8, 9]-=-. We can often use the combinatorial invariance of the structure during the interval between events to perform operations at a rational valued time. See Section 4.3 for an example of where this equiva... |

242 | Data structures for mobile data
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(Show Context)
Citation Context ...necessary underlying algebra and allows for the addition of filtering mechanisms in the near future. 1.1 Kinetic data structures Kinetic data structures were first introduced by Basch et. al. in 1997 =-=[2]-=-. The idea stems from the observation that most, if not all, computational geometry structures are built using predicates — functions on quantities defining the geometric input (e.g., point coordinate... |

99 | Epsilon Geometry : Building robust algorithms from imprecise computations
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(Show Context)
Citation Context ...n than their exact counterparts and are close to correct most of the time. While producing stable algorithms using only inexact predicates has proved to be an extremely difficult task for static data =-=[17]-=-, we have good reason to expect it to be easier for kinetic data structures. Let us hint on some empirical evidence as to why we expect that building stable algorithms on top of inexact predicates wou... |

87 |
An empirical comparison of priority-queues and event-set implementations
- Jones
- 1986
(Show Context)
Citation Context ...el and, as such, it can be chosen by the user (however, there is a default choice — see below). In our package, we provide two different types of priority queues, a heap and a two-list priority queue =-=[12]-=-. A two-list queue is a queue in which there is a sorted front list, containing all events before some time (and after the current time) and an unsorted back list. The queue tries to maintain a 17ssma... |

70 |
Kinetic collision detection for simple polygons
- Kirkpatrick, Snoeyink, et al.
(Show Context)
Citation Context ...nterface between the geometry and the underlying algebra, we can then implement a variety of computational models. Kinetic data structures have been implemented numerous times, for example [3, 4],[5],=-=[6]-=-. However, previous implementations punted on the hard problems involved in exactly comparing failure times and handling degeneracies. The combination of inexact computation and the lack of generality... |

68 | A Core library for robust numerical and geometric computation
- Karamcheti, Li, et al.
- 1999
(Show Context)
Citation Context ...lent. In typical kinetic data structures event times are only representable by algebraic numbers, rendering computations at event times extremely expensive, requiring a number type such as CORE::Expr =-=[7]-=- or LEDA::real [8, 9]. We can often use the combinatorial invariance of the structure during the interval between events to perform operations at a rational valued time. See Section 4.3 for an example... |

33 | Exacus: Efficient and exact algorithms for curves and surfaces
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- 2005
(Show Context)
Citation Context ...e computations are implemented. The rational kernel layer provides a useful general purpose polynomial manipulation library. However, it is being superseded in CGAL by the more general EXACUS library =-=[14]-=-. Example operations provided by the RationalKernel include • computation of the sign of a polynomial at a rational number • computation of the Sturm sequence of two polynomials • computation of a pol... |

26 | An empirical comparison of techniques for updating Delaunay triangulations
- Guibas, Russel
- 2004
(Show Context)
Citation Context ...h 24srecent releases and is now almost as fast. 7 Future work Kinetic data structures are expensive compared to rebuilding in many circumstances. For an example involving Delaunay triangulations, see =-=[16]-=-. This is especially true when trying to do exact calculations: filtering is quite well developed for static computations but just in its infancy for polynomial computations. As shown in Section 6 the... |

25 |
A computational framework for handling motion
- Guibas, Karavelas, et al.
(Show Context)
Citation Context ...ch still guarantee exact results. The algorithms for handling geometric objects under motion and, in particular, kinetic data structures have not 1 A preliminary version of this paper has appeared in =-=[1]-=-. Preprint submitted to Computational Geometry:Theory & Applications3 May 2007sreached the same level of maturity, as compared to their static (i.e., nonmoving) counterparts. The implementations have ... |

21 | A practical evaluation of kinetic data structures
- Basch, Guibas, et al.
- 1997
(Show Context)
Citation Context ...nd simple interface between the geometry and the underlying algebra, we can then implement a variety of computational models. Kinetic data structures have been implemented numerous times, for example =-=[3, 4]-=-,[5],[6]. However, previous implementations punted on the hard problems involved in exactly comparing failure times and handling degeneracies. The combination of inexact computation and the lack of ge... |

16 | The LEDA class real number
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(Show Context)
Citation Context ...inetic data structures event times are only representable by algebraic numbers, rendering computations at event times extremely expensive, requiring a number type such as CORE::Expr [7] or LEDA::real =-=[8, 9]-=-. We can often use the combinatorial invariance of the structure during the interval between events to perform operations at a rational valued time. See Section 4.3 for an example of where this equiva... |

15 | splaying: An algorithm for repairing delaunay triangulations and convex hulls
- Star
- 2005
(Show Context)
Citation Context ...an one dimension), it generally does. If, instead of the normal kinetic Delaunay 25sdata structure, which maintains a triangulation at all times, we implement Shewchuk’s star splaying based algorithm =-=[18]-=-, we should be able to guarantee that kinetic Delaunay will always converge if not too many certificates are missed. Second, we have access to good, fast exact static predicates. These allow us to rel... |

14 | An approximate arrangement algorithm for semi-algebraic curves
- Milenkovic, Sacks
- 2006
(Show Context)
Citation Context ...ing shortly in the future. Testing the derivative reliably disambiguates the two cases and is essential in order to get such kinetic data structures to work properly. Recent work by Milenkovic et al. =-=[15]-=- provides more rigorous analysis for computing arrangements of algebraic curves using approximate polynomial solvers. Their algorithm is equivalent to running a kinetic data structures based sweepline... |

8 | Interval methods for kinetic simulations
- Guibas, Karavelas
- 1999
(Show Context)
Citation Context ...le interface between the geometry and the underlying algebra, we can then implement a variety of computational models. Kinetic data structures have been implemented numerous times, for example [3, 4],=-=[5]-=-,[6]. However, previous implementations punted on the hard problems involved in exactly comparing failure times and handling degeneracies. The combination of inexact computation and the lack of genera... |

2 |
The demokin library
- Basch, Guibas, et al.
- 1997
(Show Context)
Citation Context ...nd simple interface between the geometry and the underlying algebra, we can then implement a variety of computational models. Kinetic data structures have been implemented numerous times, for example =-=[3, 4]-=-,[5],[6]. However, previous implementations punted on the hard problems involved in exactly comparing failure times and handling degeneracies. The combination of inexact computation and the lack of ge... |