## 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...

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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... |

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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... |

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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... |

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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, ... |

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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 ... |

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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... |

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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... |

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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... |

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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 ... |

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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... |

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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... |

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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 |
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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 |
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(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 |
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(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 |
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(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... |