## Optimal Cooperative Search In Fractional Cascaded Data Structures (1995)

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Citations: | 8 - 3 self |

### BibTeX

@MISC{Tamassia95optimalcooperative,

author = {Roberto Tamassia and Jeffrey Scott Vitter},

title = {Optimal Cooperative Search In Fractional Cascaded Data Structures},

year = {1995}

}

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### Abstract

Fractional cascading is a technique designed to allow efficient sequential search in a graph with catalogs of total size n. The search consists of locating a key in the catalogs along a path. In this paper we show how to preprocess a variety of fractional cascaded data structures whose underlying graph is a tree so that searching can be done efficiently in parallel. The preprocessing takes O(log n) time with n/log n processors on an EREW PRAM. For a balanced binary tree cooperative search along root-to-leaf paths can be done in O((logn)/logp) time using p processors on a CREW PRAM.

### Citations

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Computational Geometry: An Introduction
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- 1985
(Show Context)
Citation Context ... vertices. Find the region of S containing a given query point q = (x; y). We assume that the subdivision S is monotone and is given with a standard representation such as doubly-connected edge lists =-=[15]-=-. Nonmonotone subdivisions are handled via a preliminary triangulation that takes O(log n) time using a CREW PRAM with n processors [1, 18, 10], or a CRCW PRAM with n= log n processors [11]. Parallel ... |

453 | Computational Geometry - Preparata, Shamos - 1985 |

255 |
Optimal Search in Planar Subdivision
- Kirkpatrick
- 1983
(Show Context)
Citation Context ...k is the problem of cooperative point location search in an n-vertex planar subdivision. Dadoun and Kirkpatrick [7] show how to perform cooperative point location on their hierarchical data structure =-=[12]-=- in O((log n)= log p) time, which is optimal. Unfortunately, the preprocessing requirements are high: they use O(log 2 n) time with O(n 3 ) processors on a CREW PRAM. In [17] we show how to construct ... |

158 |
Optimal point location on a monotone subdivision
- Edelsbunner, Guibas, et al.
- 1986
(Show Context)
Citation Context ...al. Unfortunately, the preprocessing requirements are high: they use O(log 2 n) time with O(n 3 ) processors on a CREW PRAM. In [17] we show how to construct the bridged separator tree data structure =-=[13, 9]-=- for a monotone subdivision in O(log n) time using n= log n processors on an EREW PRAM. The bridged separator tree is very efficient in practice [8]. The search path used to process a point location q... |

154 | Fractional cascading: I. A data structuring technique
- Chazelle, Guibas
- 1986
(Show Context)
Citation Context ...tract N00014--91--J--4052, ARPA order 8225. 1 Introduction Fractional cascading is a preprocessing technique that allows efficient searching of the same key in a collection of catalogs (sorted lists) =-=[3, 4]-=-. More formally, there is a catalog associated with each node of a graph G. Given a search argument y and a search path in G, the goal is to find the smallest entrysy in each of the catalogs of the no... |

61 |
Location of a Point in a Planar Subdivision and its Applications
- Lee, Preparata
- 1977
(Show Context)
Citation Context ...al. Unfortunately, the preprocessing requirements are high: they use O(log 2 n) time with O(n 3 ) processors on a CREW PRAM. In [17] we show how to construct the bridged separator tree data structure =-=[13, 9]-=- for a monotone subdivision in O(log n) time using n= log n processors on an EREW PRAM. The bridged separator tree is very efficient in practice [8]. The search path used to process a point location q... |

58 |
Cascading divide-and-conquer: A technique for designing parallel algorithms
- Atallah, Cole, et al.
- 1989
(Show Context)
Citation Context ...hin the EREW PRAM model, the lower bound increases to\Omega\Gamma/29 (n=p)). Intuitively we form a preprocessed version T 0 of T by starting with the fractional cascaded data structure constructed by =-=[1]-=- and introducing certain substructures. For cooperative searches, we use p processors to traverse \Theta(log p) levels of the appropriate substructure in constant time, by simulating p-way branching. ... |

52 | Planar separators and parallel polygon triangulation
- Goodrich
- 1995
(Show Context)
Citation Context ...edge lists [15]. Nonmonotone subdivisions are handled via a preliminary triangulation that takes O(log n) time using a CREW PRAM with n processors [1, 18, 10], or a CRCW PRAM with n= log n processors =-=[11]-=-. Parallel triangulation can also be performed using a randomized CREW PRAM algorithm that runs in O(log n log log n) time and does O(n) expected work [6, 5]. The bridged separator tree [13, 9] uses O... |

43 | S.: Dynamic fractional cascading - Mehlhorn, Näher - 1990 |

41 | On parallel searching
- Snir
- 1989
(Show Context)
Citation Context ...me O((log n)= log p) for the cooperative search is optimal, since we can reduce the problem of dictionary searching to cooperative search in a binary tree with catalogs, and thus the lower bound from =-=[16]-=- applies. If cooperative search is done within the EREW PRAM model, the lower bound increases to\Omega\Gamma/29 (n=p)). Intuitively we form a preprocessed version T 0 of T by starting with the fractio... |

40 |
L.J.: Fractional cascading
- Chazelle, Guibas
- 1986
(Show Context)
Citation Context ...tract N00014--91--J--4052, ARPA order 8225. 1 Introduction Fractional cascading is a preprocessing technique that allows efficient searching of the same key in a collection of catalogs (sorted lists) =-=[3, 4]-=-. More formally, there is a catalog associated with each node of a graph G. Given a search argument y and a search path in G, the goal is to find the smallest entrysy in each of the catalogs of the no... |

26 |
How to Search in History
- Chazelle
- 1985
(Show Context)
Citation Context ...inance relation among the cells is acyclic. This problem was previously solved using an O(n)-space data structure (called canal tree) that supports sequential point location search in O(log 2 n) time =-=[2]-=-. For cooperative search we use a data structure based on separating surfaces, a three-dimensional extension of separators. This data structure can be efficiently constructed in parallel if a topologi... |

23 | Parallel transitive closure and point location in planar structures
- Tamassia, Vitter
- 1991
(Show Context)
Citation Context ...erarchical data structure [12] in O((log n)= log p) time, which is optimal. Unfortunately, the preprocessing requirements are high: they use O(log 2 n) time with O(n 3 ) processors on a CREW PRAM. In =-=[17]-=- we show how to construct the bridged separator tree data structure [13, 9] for a monotone subdivision in O(log n) time using n= log n processors on an EREW PRAM. The bridged separator tree is very ef... |

22 | Randomized Parallel Algorithms for Trapezoidal Diagrams
- Clarkson, Cole, et al.
- 1992
(Show Context)
Citation Context ...10], or a CRCW PRAM with n= log n processors [11]. Parallel triangulation can also be performed using a randomized CREW PRAM algorithm that runs in O(log n log log n) time and does O(n) expected work =-=[6, 5]-=-. The bridged separator tree [13, 9] uses O(n) space and supports point location queries in S in O(log n) time. It is a balanced binary tree T with catalogs where searches are performed implicitly. We... |

17 |
Parallel Triangulation of a Polygon in Two Calls to the Trapezoidal Map
- Yap
- 1988
(Show Context)
Citation Context ...h a standard representation such as doubly-connected edge lists [15]. Nonmonotone subdivisions are handled via a preliminary triangulation that takes O(log n) time using a CREW PRAM with n processors =-=[1, 18, 10]-=-, or a CRCW PRAM with n= log n processors [11]. Parallel triangulation can also be performed using a randomized CREW PRAM algorithm that runs in O(log n log log n) time and does O(n) expected work [6,... |

17 | A new point-location algorithm and its practical efficiency — Comparison with existing algorithms - Edahiro, Kokubo, et al. - 1984 |

14 |
Triangulating a Polygon in Parallel
- Goodrich
- 1989
(Show Context)
Citation Context ...h a standard representation such as doubly-connected edge lists [15]. Nonmonotone subdivisions are handled via a preliminary triangulation that takes O(log n) time using a CREW PRAM with n processors =-=[1, 18, 10]-=-, or a CRCW PRAM with n= log n processors [11]. Parallel triangulation can also be performed using a randomized CREW PRAM algorithm that runs in O(log n log log n) time and does O(n) expected work [6,... |

9 | Dynamization of Geometric Data Structures - Fries, Mehlhorn, et al. - 1985 |

7 | Optimal Parallel Algorithms for Transitive Closure and - Tamassia, Vitter - 1989 |

6 | Erratum: Randomized parallel algorithms for trapezoidal diagrams - Clarkson, Cole, et al. - 1992 |

5 | How to Search - Chazelle - 1985 |

4 |
A new point-location algorithm and its practical efficiency: comparison with existing algorithms
- Asano
- 1984
(Show Context)
Citation Context ...truct the bridged separator tree data structure [13, 9] for a monotone subdivision in O(log n) time using n= log n processors on an EREW PRAM. The bridged separator tree is very efficient in practice =-=[8]. The sear-=-ch path used to process a point location query is "highly" implicit, due to the spacesaving nature of the bridged separator tree, which makes cooperative search seem especially difficult. In... |

2 |
Cooperative subdivision search algorithms with applications
- Dadoun, Kirkpatrick
- 1989
(Show Context)
Citation Context ...EW PRAM. Both of these time/processor tradeoffs are optimal. One motivation for this work is the problem of cooperative point location search in an n-vertex planar subdivision. Dadoun and Kirkpatrick =-=[7]-=- show how to perform cooperative point location on their hierarchical data structure [12] in O((log n)= log p) time, which is optimal. Unfortunately, the preprocessing requirements are high: they use ... |

1 | On parallel searching - Shit - 1989 |