DETERMINING THE SEPARATION OF SIMPLE POLYGONS

1994 ◽  
Vol 04 (04) ◽  
pp. 457-474 ◽  
Author(s):  
NANCY M. AMATO
Keyword(s):  
Parallel Algorithm ◽  
Time Complexity ◽  
Minimum Distance ◽  
Sequential Algorithm ◽  
Parallel Methods ◽  
Simple Polygons ◽  
Closest Pair ◽  
Time Optimal ◽  
Crew Pram

Given simple polygons P and Q, their separation, denoted by σ(P, Q), is defined to be the minimum distance between their boundaries. We present a parallel algorithm for finding a closest pair among all pairs (p, q), p ∈ P and q ∈ Q. The algorithm runs in O ( log n) time using O(n) processors on a CREW PRAM, where n = |P| + |Q|. This algorithm is time-optimal and improves by a factor of O ( log n) on the time complexity of previous parallel methods. The algorithm can be implemented serially in Θ (n) time, which gives the first optimal sequential algorithm for determining the separation of simple polygons. Our results are obtained by providing a unified treatment of the separation and the closest visible vertex problems for simple polygons.

Parallelizing an Algorithm for Visibility on Polyhedral Terrain

1997 ◽  
Vol 07 (01n02) ◽  
pp. 75-84 ◽  
Author(s):  
Y. Ansel Teng ◽  
David Mount ◽  
Enrico Puppo ◽  
Larry S. Davis
Keyword(s):  
Parallel Algorithm ◽  
Time Complexity ◽  
Sequential Algorithm ◽  
Work Efficiency ◽  
Parallel Complexity ◽  
Input And Output ◽  
Work Complexity ◽  
Pram Model ◽  
Crew Pram

The best known output-sensitive sequential algorithm for computing the viewshed on a polyhedral terrain from a given viewpoint was proposed by Katz, Overmars, and Sharir,10 and achieves time complexity O((k + nα(n)) log n) where n and k are the input and output sizes respectively, and α() is the inverse Ackermann's function. In this paper, we present a parallel algorithm that is based on the work mentioned above, and achieves O( log 2n) time complexity, with work complexity O((k + nα(n)) log n) in a CREW PRAM model. This improves on previous parallel complexity while maintaining work efficiency with respect to the best sequential complexity known.

PARALLEL SYNTENIC ALIGNMENTS

2003 ◽  
Vol 13 (04) ◽  
pp. 689-703 ◽  
Author(s):  
NATSUHIKO FUTAMURA ◽  
SRINIVAS ALURU ◽  
XIAOQIU HUANG
Keyword(s):  
Parallel Algorithm ◽  
Dna Sequences ◽  
Genomic Dna ◽  
Sequential Algorithm ◽  
Base Pairs ◽  
Conserved Genes ◽  
Alignment Problem ◽  
Time Optimal ◽  
High Degree ◽  
Gene Rich Region

Given two genomic DNA sequences, the syntenic alignment problem is to compute an ordered list of subsequences for each sequence such that the corresponding subsequence pairs exhibit a high degree of similarity. Syntenic alignments are useful in comparing genomic DNA from related species and in identifying conserved genes. In this paper, we present a parallel algorithm for computing syntenic alignments that runs in [Formula: see text] time, where m and n are the respective lengths of the two genomic sequences, and p is the number of processors used. Our algorithm is time optimal with respect to the corresponding sequential algorithm and can use [Formula: see text] processors, where n is the length of the larger sequence. The space requirement of the algorithm is [Formula: see text] per processor. Using an implementation of this parallel algorithm, we report the alignment of a gene-rich region of human chromosome 12, namely 12p13 and its syntenic region in mouse chromosome 6 (both over 220,000 base pairs in length) in under 24 minutes on a 64-processor IBM xSeries cluster.

Determining Weak Visibility of a Polygon from an Edge in Parallel

1998 ◽  
Vol 08 (03) ◽  
pp. 277-304
Author(s):  
Danny Z. Chen
Keyword(s):  
Computational Model ◽  
Parallel Algorithm ◽  
Simple Polygon ◽  
Sequential Algorithms ◽  
Simple Polygons ◽  
Crew Pram

The problem of determining the weak visibility of an n-vertex simple polygon P from an edge e of P is that of deciding whether every point in P is weakly visible from e. In this paper we present an optimal parallel algorithm for solving this problem. Our algorithm runs in O( log n) time using O(n/ log n) processors in the CREW PRAM computational model, and is very different from the sequential algorithms for this problem. We also show how to solve optimally, in parallel, several other problems that are related to the weak visibility of simple polygons.

EFFICIENT ALGORITHMS FOR THE EUCLIDEAN DISTANCE TRANSFORM

1995 ◽  
Vol 05 (02) ◽  
pp. 205-212 ◽  
Author(s):  
SANDY PAVEL ◽  
SELIM G. AKL
Keyword(s):  
Parallel Algorithm ◽  
Euclidean Distance ◽  
Sequential Algorithm ◽  
Distance Transform ◽  
Binary Images ◽  
Euclidean Distance Transform ◽  
Erew Pram ◽  
Time Optimal ◽  
Time Bounds ◽  
And Robotics

The Euclidean Distance Transform is an important computational tool for the processing of binary images, with applications in many areas such as computer vision, pattern recognition and robotics. We investigate the properties of this transform and describe an O(n2) time optimal sequential algorithm. A deterministic EREW-PRAM parallel algorithm which runs in O( log n) time using O(n2) processors and O(n2) space is also derived. Further, a cost optimal randomized parallel algorithm which runs within the same time bounds with high probability, is given.

scholarly journals COMPUTING THE ALL-PAIRS LONGEST CHAINS IN THE PLANE

1995 ◽  
Vol 05 (03) ◽  
pp. 257-271 ◽  
Author(s):  
MIKHAIL J. ATALLAH ◽  
DANNY Z. CHEN
Keyword(s):  
Parallel Algorithm ◽  
Space Complexity ◽  
Sequential Algorithm ◽  
Pram Model ◽  
The Matrix ◽  
Crew Pram ◽  
Chain Lengths

Many problems on sequences and on special kinds of graphs involve the computation of longest chains passing points in the plane. Given a set S of n points in the plane, we consider the problem of computing the matrix of longest chain lengths between all pairs of points in S, and the matrix of “parent” pointers that describes the n longest chain trees. We present a simple sequential algorithm for computing these matrices. Our algorithm runs in O(n2) time, and hence is optimal. We also present a rather involved parallel algorithm that computes these matrices in O((log n)2) time using O(n2/log n) processors in the CREW PRAM model. These matrices enable us to report, in O(1) time, the length of a longest chain between any two points in S by using one processor, and the actual chain by using k processors, where k is the number of points of S on that chain. The space complexity of the algorithms is O(n2).

OPTIMAL TREE RANKING IS IN $\mathcal{NC}$

1992 ◽  
Vol 02 (01) ◽  
pp. 31-41 ◽  
Author(s):  
PILAR DE LA TORRE ◽  
RAYMOND GREENLAW ◽  
TERESA M. PRZYTYCKA
Keyword(s):  
General Problem ◽  
Sequential Algorithm ◽  
Natural Numbers ◽  
Ranking Problem ◽  
Crew Pram ◽  
Optimal Tree ◽  
Optimal Ranking

This paper places the optimal tree ranking problem in [Formula: see text]. A ranking is a labeling of the nodes with natural numbers such that if nodes u and v have the same label then there exists another node with a greater label on the path between them. An optimal ranking is a ranking in which the largest label assigned to any node is as small as possible among all rankings. An O(n) sequential algorithm is known. Researchers have speculated that this problem is P-complete. We show that for an n-node tree, one can compute an optimal ranking in O( log n) time using n2/ log n CREW PRAM processors. In fact, our ranking is super critical in that the label assigned to each node is absolutely as small as possible. We achieve these results by showing that a more general problem, which we call the super critical numbering problem, is in [Formula: see text]. No [Formula: see text] algorithm for the super critical tree ranking problem, approximate or otherwise, was previously known; the only known [Formula: see text] algorithm for optimal tree ranking was an approximate one.

scholarly journals Parallel Strategy to Factorize Fermat Numbers with Implementation in Maple Software

2021 ◽  
pp. 167-173
Author(s):  
Jianhui Li ◽  
◽  
Manlan Liu
Keyword(s):  
Parallel Computing ◽  
Parallel Algorithm ◽  
Computational Efficiency ◽  
Sequential Algorithm ◽  
Parallel Processes ◽  
Fermat Numbers ◽  
Fermat Number ◽  
Parallel Strategy

In accordance with the traits of parallel computing, the paper proposes a parallel algorithm to factorize the Fermat numbers through parallelization of a sequential algorithm. The kernel work to parallelize a sequential algorithm is presented by subdividing the computing interval into subintervals that are assigned to the parallel processes to perform the parallel computing. Maple experiments show that the parallelization increases the computational efficiency of factoring the Fermat numbers, especially to the Fermat number with big divisors.

scholarly journals AN OPTIMAL PARALLEL ALGORITHM FOR DETECTING WEAK VISIBILITY OF A SIMPLE POLYGON

1995 ◽  
Vol 05 (01n02) ◽  
pp. 93-124 ◽  
Author(s):  
DANNY Z. CHEN
Keyword(s):  
Computational Model ◽  
Parallel Algorithm ◽  
Simple Polygon ◽  
Sequential Algorithms ◽  
Crew Pram

The problem of detecting the weak visibility of an n-vertex simple polygon P is that of finding whether P is weakly visible from one of its edges and (if it is) identifying every edge from which P is weakly visible. In this paper, we present an optimal parallel algorithm for solving this problem. Our algorithm runs in O(log n) time using O(n/log n) processors in the CREW PRAM computational model, and is very different from the sequential algorithms for this problem. Based on this algorithm, several other problems related to weak visibility can be optimally solved in parallel.

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