## Parametric search made practical (2002)

### Cached

### Download Links

- [igitur-archive.library.uu.nl]
- [archive.cs.uu.nl]
- [www.cs.uu.nl]
- DBLP

### Other Repositories/Bibliography

Venue: | SoCG: 18th Symposium on Computational Geometry |

Citations: | 10 - 1 self |

### BibTeX

@INPROCEEDINGS{Vanoostrum02parametricsearch,

author = {René Vanoostrum},

title = {Parametric search made practical},

booktitle = {SoCG: 18th Symposium on Computational Geometry},

year = {2002},

pages = {1--9},

publisher = {ACM Press}

}

### OpenURL

### Abstract

In this paper we show that in sorting-based applications of parametric search, Quicksort can replace the parallel sorting algorithms that are usually advocated, and we argue that Cole’s optimization of certain parametric-search algorithms may be unnecessary under realistic assumptions about the input. Furthermore, we present a generic, flexible, and easyto-use framework that greatly simplifies the implementation of algorithms based on parametric search. We use our framework to implement an algorithm that solves the Fréchetdistance problem. The implementation based on parametric search is faster than the binary-search approach that is often suggested as a practical replacement for the parametricsearch technique.

### Citations

8530 |
Introduction to Algorithms
- Cormen, Leiserson, et al.
- 1990
(Show Context)
Citation Context ...ion suffices to be able to batch comparisons. For a complete description and analysis of Quicksort, see any introductory textbook on algorithms, for instance the one by Cormen, Leiserson, and Rivest. =-=[11]-=-. In short, Quicksort sorts an input array A with elements numbered 1 ...n as follows: 1. If A has less than two elements, return; 2. Otherwise, choose A[1] as the pivot element; 3. Partition A into t... |

501 | Sorting networks and their applications - Batcher - 1968 |

286 | Parallel Merge Sort - Cole - 1988 |

232 | Applying parallel computation algorithms in the design of serial algorithms
- Megiddo
- 1983
(Show Context)
Citation Context ...ory Keywords Framework for Parametric Search, Implementation 1. INTRODUCTION Since the late 1980s, parametric search, the optimization technique developed by Megiddo in the late 1970s and early 1980s =-=[14, 15]-=-, has become an important tool for solving many geometric optimization queries efficiently. The main principle of parametric search is to compute a value λ ∗ that ∗ This research was supported by the ... |

129 |
Computing the Fréchet distance between two polygonal curves
- Alt, Godau
- 1995
(Show Context)
Citation Context ...sign of an efficient parallel algorithm. Fortunately, the generic algorithm does not necessarily have to solve the same problem as the concrete decision; in several cases, sorting can be used instead =-=[3, 7, 10, 12, 15]-=-. However, the existing parallel sorting algorithms that have good worst-case time bounds are not easily implemented, and in some cases the hidden constants in the asymptotic running times are enormou... |

121 |
Combinatorial Optimization with Rational Objective Functions
- Megiddo
- 1979
(Show Context)
Citation Context ...ory Keywords Framework for Parametric Search, Implementation 1. INTRODUCTION Since the late 1980s, parametric search, the optimization technique developed by Megiddo in the late 1970s and early 1980s =-=[14, 15]-=-, has become an important tool for solving many geometric optimization queries efficiently. The main principle of parametric search is to compute a value λ ∗ that ∗ This research was supported by the ... |

112 |
Sorting in c log n parallel steps
- Ajtai, Komlos, et al.
- 1983
(Show Context)
Citation Context ...However, the existing parallel sorting algorithms that have good worst-case time bounds are not easily implemented, and in some cases the hidden constants in the asymptotic running times are enormous =-=[2]-=-. Cole [8] shows how sorting-based parametric search can be optimized even further, but the optimization comes at the expense of making the technique even more complicated than it already is. In this ... |

105 |
Slowing down sorting networks to obtain faster sorting algorithms
- Cole
- 1987
(Show Context)
Citation Context ...he existing parallel sorting algorithms that have good worst-case time bounds are not easily implemented, and in some cases the hidden constants in the asymptotic running times are enormous [2]. Cole =-=[8]-=- shows how sorting-based parametric search can be optimized even further, but the optimization comes at the expense of making the technique even more complicated than it already is. In this paper we s... |

105 |
Parallelism in comparison problems
- Valiant
- 1975
(Show Context)
Citation Context ...rather complex and the constants hidden in the O-notation are very large. For practical use it is oftenssuggested to use a sub-optimal parallel sorting algorithm instead, such as Valiant’s merge sort =-=[20]-=-, which runs in O(log n log log n) parallel steps using O(n) processors. Less commonly known are the two sorting algorithms by Cole [9] that have the same time bounds as the AKS-network, and also use ... |

93 | Efficient algorithms for geometric optimization - Agarwal, Sharir - 1998 |

72 | Geometric Pattern Matching under Euclidean Motion
- Chew, Goodrich, et al.
- 1993
(Show Context)
Citation Context ...sign of an efficient parallel algorithm. Fortunately, the generic algorithm does not necessarily have to solve the same problem as the concrete decision; in several cases, sorting can be used instead =-=[3, 7, 10, 12, 15]-=-. However, the existing parallel sorting algorithms that have good worst-case time bounds are not easily implemented, and in some cases the hidden constants in the asymptotic running times are enormou... |

52 |
An optimal-time algorithm for slope selection
- Cole, Salowe, et al.
- 1989
(Show Context)
Citation Context ...sign of an efficient parallel algorithm. Fortunately, the generic algorithm does not necessarily have to solve the same problem as the concrete decision; in several cases, sorting can be used instead =-=[3, 7, 10, 12, 15]-=-. However, the existing parallel sorting algorithms that have good worst-case time bounds are not easily implemented, and in some cases the hidden constants in the asymptotic running times are enormou... |

49 | Geometric applications of a randomized optimization technique - Chan - 1999 |

35 | Itai A.: Improvements on bottleneck matching and related problems using geometry
- Efrat
- 1996
(Show Context)
Citation Context |

23 |
New parallel sorting schemes
- Preparata
- 1978
(Show Context)
Citation Context ...w, sorting-based parametric search has always relied on parallel sorting algorithms for the batching of comparisons. Megiddo [15] suggests the parallel sorting algorithms of Valiant [20] or Preparata =-=[17]-=-. Other authors propose using the AKS-network [2] to derive running times that are asymptotically faster. Cole [8] originally applies his opti3 mization technique to the AKS-network. In a later paper ... |

17 | M.: Fast algorithms for collision and proximity problems involving moving geometric objects
- Gupta, Janardan, et al.
- 1996
(Show Context)
Citation Context ...e selection problem” [10]. The second implementation that we know of is by Schwerdt et al. [18], who implement the algorithm for finding the minimum diameter of a set of moving points by Gupta et al. =-=[13]-=-. The second reason why parametric search is hardly ever used in practice is that it is generally assumed that the overhead involved (i.e., the hidden constants in the Onotation) are too large to be o... |

16 |
E cient algorithms for geometric optimization
- Agarwal, Sharir
- 1998
(Show Context)
Citation Context ...d by a log-factor, thus improving the running time to O(PTp + TpTs). We will explain his idea and comment on it in Section 4. One of the drawbacks of parametric search mentioned by Agarwal and Sharir =-=[1]-=- is that it requires the design of an efficient parallel algorithm for the generic version of the decision problem, which is not always easy. However, it is instructive to point out that the generic a... |

16 | The LEDA class real number
- Burnikel, Mehlhorn, et al.
- 1996
(Show Context)
Citation Context ...e encounter λ ∗ ; however, difference between the smallest and the highest number of iterations was less than 2 percent. We also tested the Fréchet-distance algorithms using the number type leda_real =-=[6]-=-. This number type provides exact relational operators (=, �=,<,≤,>,≥) and is closed under addition, subtraction, multiplication, division, and computation of k-th roots. Exact comparisons with leda_r... |

15 | Applications of the generic programming paradigm in the design of CGAL
- Brönnimann, Kettner, et al.
- 2000
(Show Context)
Citation Context ...to use Bitonic Sort or Quicksort, and decide on the number type to be used for the calculations. The parametric-search framework and the two sorting algorithms have the same design philosophy as CGAL =-=[5]-=-, the library of algorithms for computational geometry. Our framework doesn’t depend on CGAL, but the two cooperate very well. We consider this important, since it is our goal to use the framework for... |

6 | Extremal polygon containment problems and other issues in parametric searching
- Toledo
- 1991
(Show Context)
Citation Context ...ric search is a complicated technique, it is believed that implementing it must also be complicated and hard. To our knowledge, the technique has been implemented only twice; the first time by Toledo =-=[19]-=-, who applies it to the “median-oflines”-problem that Megiddo gave as a simple illustration of parametric search [15], and to the “slope selection problem” [10]. The second implementation that we know... |

6 | Sorting in clogn parallel steps - Ajtai, Komlós, et al. - 1983 |

4 | Labeling Points with Rectangles of Various Shapes
- Koike, Nakano, et al.
- 2002
(Show Context)
Citation Context ...Hausdorff distance, under rotation and rotation, between two sets of points, lines, or polygons [7]. In several other cases, the generic algorithm consists of several steps, sorting being one of them =-=[16, 18]-=-. For sorting-based parametric search we have several parallel sorting algorithms at our disposal. The first sorting algorithm that sorts in O(log n) parallel steps using O(n) processors is the sortin... |

3 | Computing the minimum diameter for moving points: an exact implementation using parametric search
- Schwerdt, Smid, et al.
- 1997
(Show Context)
Citation Context ...Hausdorff distance, under rotation and rotation, between two sets of points, lines, or polygons [7]. In several other cases, the generic algorithm consists of several steps, sorting being one of them =-=[16, 18]-=-. For sorting-based parametric search we have several parallel sorting algorithms at our disposal. The first sorting algorithm that sorts in O(log n) parallel steps using O(n) processors is the sortin... |