## Scalable Parallel Computational Geometry for Coarse Grained Multicomputers (1994)

Venue: | International Journal on Computational Geometry |

Citations: | 74 - 14 self |

### BibTeX

@ARTICLE{Dehne94scalableparallel,

author = {Frank Dehne and Andreas Fabri and Andrew Rau-chaplin},

title = {Scalable Parallel Computational Geometry for Coarse Grained Multicomputers},

journal = {International Journal on Computational Geometry},

year = {1994},

volume = {6},

pages = {298--307}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study scalable parallel computational geometry algorithms for the coarse grained multicomputer model: p processors solving a problem on n data items, were each processor has O( n p ) AE O(1) local memory and all processors are connected via some arbitrary interconnection network (e.g. mesh, hypercube, fat tree). We present O( Tsequential p + T s (n; p)) time scalable parallel algorithms for several computational geometry problems. T s (n; p) refers to the time of a global sort operation. Our results are independent of the multicomputer's interconnection network. Their time complexities become optimal when Tsequential p dominates T s (n; p) or when T s (n; p) is optimal. This is the case for several standard architectures, including meshes and hypercubes, and a wide range of ratios n p that include many of the currently available machine configurations. Our methods also have some important practical advantages: For interprocessor communication, they use only a small fixed numb...

### Citations

1775 |
Computational Geometry: An Introduction
- Preparata, Shamos
- 1985
(Show Context)
Citation Context ... Globally sort S by z-coordinate. Processor p i stores subset H i and bounding plane H i . Each processor p i computes locally 3Dmax(H i ) using the standard sequential algorithm as described e.g. in =-=[16]-=-, and removes all points dominated in H i . Each processor p i computes locally the 2D-projection H 0 i , 2Dmax(H 0 i ), and the monotone chain 2Dmax i . Using the bounding planes V 1 ; : : : V p , ea... |

1316 |
Introduction to Parallel Algorithms and Architecturis: Arrays
- Leighton
- 1992
(Show Context)
Citation Context ...eterministic algorithms exist [6], but they are not of practical use. One could also use randomized sorting [17], but in this paper we will only consider deterministic methods. We refer the reader to =-=[3, 5, 11, 13, 14, 17]-=- for a more detailed discussion of the different architectures and routing algorithms. It is interesting to study, for which ratio of n and p the global sort becomes optimal, that is T s (n; p) = O( n... |

626 |
Tsitsiklis, Parallel and Distributed Computation: Numerical Methods
- Bertsekas, N
- 1989
(Show Context)
Citation Context ...eterministic algorithms exist [6], but they are not of practical use. One could also use randomized sorting [17], but in this paper we will only consider deterministic methods. We refer the reader to =-=[3, 5, 11, 13, 14, 17]-=- for a more detailed discussion of the different architectures and routing algorithms. It is interesting to study, for which ratio of n and p the global sort becomes optimal, that is T s (n; p) = O( n... |

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

160 | Maintenance of configurations in the plane - Overmars, Leeuwen - 1981 |

121 | External-memory computational geometry
- Goodrich, Tsay, et al.
- 1993
(Show Context)
Citation Context ... be frequently swapped between the host and the systolic array, and this "I/O bottleneck" is the main factor determining the computation time. A closely related "external memory" m=-=odel was studied in [9]-=-. At the end of Section 1 we will discuss more in detail the relationship of our work to previous results in the literature. The architectures of most existing multicomputers (e.g. the Intel Paragon, ... |

120 |
A logarithmic time sort for linear size networks
- Reif, Valiant
- 1987
(Show Context)
Citation Context ...e complexities are based on [14] and [3], respectively. Note that for the hypercube better deterministic algorithms exist [6], but they are not of practical use. One could also use randomized sorting =-=[17]-=-, but in this paper we will only consider deterministic methods. We refer the reader to [3, 5, 11, 13, 14, 17] for a more detailed discussion of the different architectures and routing algorithms. It ... |

67 | Deterministic sorting in nearly logarithmic time on the hypercube and related computers
- Cypher, Plaxton
- 1990
(Show Context)
Citation Context ... + p p)) and for a hypercube T s (n; p) = O( n p (log n + log 2 p)). These time complexities are based on [14] and [3], respectively. Note that for the hypercube better deterministic algorithms exist =-=[6]-=-, but they are not of practical use. One could also use randomized sorting [17], but in this paper we will only consider deterministic methods. We refer the reader to [3, 5, 11, 13, 14, 17] for a more... |

64 |
Finding the upper envelope of n line segments in O(n log n) time
- Hershberger
- 1989
(Show Context)
Citation Context ...egments of S. Output: Each processor stores O( n p ) segment portions of LE(S). (1) Each processor p i computes sequentially LE(S i ) for its subset S i of line segments (ignoring all other segments) =-=[12]-=-. (2) Globally sort the segments in S p i=1 LE(S i ) by the x-coordinate of their right endpoints, which moves to each processor p i a new set V i of O( n p ) segments. Note that, each processor p i a... |

57 |
Solutions to Klee’s rectangle problems
- Bentley
- 1977
(Show Context)
Citation Context ...of size O( n p ), n psp, in time O( n log n p + T s (n; p)). In the remainder of this section, we study some applications of our multisearch algorithm. A segment tree, originally introduced by Bently =-=[4]-=-, is a data structure designed for storing line segments. The segment tree T for a set of n line segments is a complete binary tree with 2n leaves (for ease of description let 2n be a power of 2) repr... |

51 | Randomized routing on fat-trees
- Greenberg, Leiserson
- 1989
(Show Context)
Citation Context ...eterministic algorithms exist [6], but they are not of practical use. One could also use randomized sorting [17], but in this paper we will only consider deterministic methods. We refer the reader to =-=[3, 5, 11, 13, 14, 17]-=- for a more detailed discussion of the different architectures and routing algorithms. It is interesting to study, for which ratio of n and p the global sort becomes optimal, that is T s (n; p) = O( n... |

26 |
Sorting in constant number of row and column phases on a mesh, Algorithmica
- Marberg, Gafni
- 1988
(Show Context)
Citation Context ...time complexity of a global global sort. Note that, for a mesh T s (n; p) = \Theta( n p (log n + p p)) and for a hypercube T s (n; p) = O( n p (log n + log 2 p)). These time complexities are based on =-=[14]-=- and [3], respectively. Note that for the hypercube better deterministic algorithms exist [6], but they are not of practical use. One could also use randomized sorting [17], but in this paper we will ... |

23 |
The measure problem for rectangular ranges in d-space
- Leeuwen, Wood
- 1981
(Show Context)
Citation Context ... compute locally xcover(s) for all stripes s in the respective vertical slab. Perform a plane sweep in upwards direction in time O( n p log n), using the sequential measure of rectangles algorithm in =-=[20]-=- with minor adaptations. (3) Determine all boxes b which are contained in a rectangle r 2 R: Each processor locally builds a segment tree for L and H, each. Using these segment trees, determine for ea... |

19 |
Implementing data structures on a hypercube multiprocessor, and applications in parallel computational geometry
- Dehne, Rau-Chaplin
- 1990
(Show Context)
Citation Context ... time complexity follows from Lemma 1. 8 Parallel Tree Search and Applications Let T = (V; E) be a balanced k-ary tree of size n and height h = O(log k n), where k is a fixed constant. We recall from =-=[7]-=- the definition of the multisearch problem for T and a set Q = fq 1 ; : : : ; q m g of m = O(n) search queries on T . Each query q 2 Q has a search path, path(q) = (v 1 (q); : : : ; v h (q)), of h ver... |

18 | An implementation of a general-purpose parallel sorting algorithm
- Tridgell, Brent
- 1993
(Show Context)
Citation Context ...pe consists of log n phases which merge pairs of envelopes, starting with envelopes consisting of a single segment each. For parallel sorting we used a merge sort available as public domain code from =-=[18]-=-. The total exchange operation was implemented by using sort (see Section 2.2). Multinode broadcast was available as a cm-5 system call, but partial sum had to be re-implemented because the available ... |

15 |
Multisearch techniques for implementing data structures on a mesh-connected computer
- Atallah, Dehne, et al.
- 1991
(Show Context)
Citation Context ...duced by the m search queries. It is important to note that the m search processes may overlap arbitrarily. That is, at any time t, any node of T may be visited by an arbitrary number of queries. See =-=[1, 7]-=- for more details. Define as T 0 the subtree of T induced by the root and all nodes of T which have a distance from the root of at most log k p. Subtree T 0 has p 0sp leaves. To simplify exposition, a... |

13 |
On the parallel-decomposability of geometric problems
- Atallah, Tsay
- 1992
(Show Context)
Citation Context ...recent "Grand Challenges" report [10]. Yet, only little theoretical work has been done for designing scalable parallel algorithms for computational geometry problems. A related problem was s=-=tudied in [2, 19]-=-. The model considered there was a host machine with O(n) memory attached to a systolic array of size p with O(1) memory per processors. This model suffers however from the fact that data has to be fr... |

6 |
Performance Computing and Communications. The FY 1992 U.S. Research and Development Program. A Report by the Committee
- High
- 1992
(Show Context)
Citation Context ...scalable, that is, they must be applicable and efficient for a wide range of ratios n p . The design of such scalable algorithms is also listed as a major goal in the recent "Grand Challenges&quo=-=t; report [10]-=-. Yet, only little theoretical work has been done for designing scalable parallel algorithms for computational geometry problems. A related problem was studied in [2, 19]. The model considered there w... |

3 |
Translation separability of sets of polygons
- Dehne, Sack
- 1987
(Show Context)
Citation Context ...terconnection network and local memories of size O( n p ), n psp, in time O( n log n p + T s (n; p)). Let S be a set of r pairwise disjoint m-vertex polygons. The uni-directional separability problem =-=[8]-=- consists of determining all directions d such that S is separable by a sequence of r translations in direction d (one for each polygon). The multi-directional separability problem [8] asks if S is se... |