Fast parallel algorithms for the Maximum Empty Rectangle problem

doi:10.1007/BF02811344

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Fast parallel algorithms for the Maximum Empty Rectangle problem

Journal Article•DOI•

Fast parallel algorithms for the Maximum Empty Rectangle problem

Amitava Datta¹, R. Srikant¹, G. D. S. Ramkumar¹, Kamala Krithivasan¹•Institutions (1)

Indian Institute of Technology Madras¹

01 Mar 1992-Sadhana-academy Proceedings in Engineering Sciences (Springer India)-Vol. 17, Iss: 1, pp 221-236

TL;DR: This paper presents efficient parallel algorithms for the maximum empty rectangle problem on crew pram and anO(logn) time algorithm on a mesh-of-trees architecture.

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Abstract: We present efficient parallel algorithms for the maximum empty rectangle problem in this paper. On crew pram, we solve the area version of this problem inO(log 2 n) time usingO(nlogn) processors. The perimeter version of this problem is solved inO(logn) time usingO(nlog 2 n) processors. On erew pram, we solve both the problems inO(logn) time usingO(n 2/logn) processors. We also present anO(logn) time algorithm on a mesh-of-trees architecture.

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Journal Article•DOI•

Determining the maximal inscribed rectangle of an irregularly shaped stone using machine vision

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Syh-Shiuh Yeh¹•Institutions (1)

National Taipei University of Technology¹

05 Jan 2022-International Journal of Computer Integrated Manufacturing

TL;DR: In this article , a rotation histogram method was developed to automatically determine the maximal inscribed rectangle of an irregularly shaped stone, which is applicable to irregularly-shaped stones with internal voids, and the area utilization of the stone will not be influenced by the placement orientation.

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Abstract: This study uses machine vision approaches to develop a rotation histogram method with better area utilization and shorter execution time that automatically determines the maximal inscribed rectangle of an irregularly shaped stone. This image processing procedure is designed to enhance the stone image, reduce the contour noise, determine the stone contour and the calculation range of the computational algorithm, and speed up the determination of the maximal inscribed rectangle of irregularly shaped stones. The rotation histogram method transforms to a histogram the stone image after-image processing, and then uses the stack method to determine the maximal rectangle of the histogram. Finally, the stone image is rotated to determine the maximal inscribed rectangle and angle of the irregularly shaped stone. The rotation histogram method is applicable to irregularly shaped stones with internal voids, and the area utilization of the stone will not be influenced by the placement orientation. The experimental results show that the rotation histogram method, as designed in this study, not only increases the area utilization for achieving the average value of 3.27%, but it also significantly shortens the execution time to achieve an average value of 64.83% compared with the existing traversal center diffusion method.

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1 citations

Journal Article•DOI•

Determining the maximal inscribed rectangle of an irregularly shaped stone using machine vision

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Yue-Bei Luo, Ching Fang Chen, Syh-Shiuh Yeh

05 Jan 2022-International Journal of Computer Integrated Manufacturing

TL;DR: In this paper , a rotation histogram method was proposed to automatically determine the maximal inscribed rectangle of an irregularly shaped stone with internal voids, which can increase the area utilization of the irregularly-shaped stones.

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Abstract: ABSTRACT This study uses machine vision approaches to develop a rotation histogram method with better area utilization and shorter execution time that automatically determines the maximal inscribed rectangle of an irregularly shaped stone. This image processing procedure is designed to enhance the stone image, reduce the contour noise, determine the stone contour and the calculation range of the computational algorithm, and speed up the determination of the maximal inscribed rectangle of irregularly shaped stones. The rotation histogram method transforms to a histogram the stone image after-image processing, and then uses the stack method to determine the maximal rectangle of the histogram. Finally, the stone image is rotated to determine the maximal inscribed rectangle and angle of the irregularly shaped stone. The rotation histogram method is applicable to irregularly shaped stones with internal voids, and the area utilization of the stone will not be influenced by the placement orientation. The experimental results show that the rotation histogram method, as designed in this study, not only increases the area utilization for achieving the average value of 3.27%, but it also significantly shortens the execution time to achieve an average value of 64.83% compared with the existing traversal center diffusion method.

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Parallel algorithm for maximum empty L-shaped polygon

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A. J. Preetham¹, Kamala Krithivasan, C. Pandu Rangan¹, Jop F. Sibeyn¹•Institutions (1)

Max Planck Society¹

01 Jan 1996

TL;DR: This paper presents a O(log n) time algorithm using O(n 3 = log n) processors on an EREW PRAM, using the idea of parallel preex to achieve optimal speed-up over the best known sequential algorithm.

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Abstract: For a given rectangle containing N points, an important problem is to nd the largest L-shaped polygon with sides parallel to those of the original rectangle, which contains none of the points. An L-shaped polygon is an orthogonal polygon with exactly one reeex angle. This paper presents a O(log n) time algorithm using O(n 3 = log n) processors on an EREW PRAM, using the idea of parallel preex. Herewith we achieve optimal speed-up over the best known sequential algorithm. We also present a O(logn) time algorithm on a mesh of trees architecture with O(n 3) processors.

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Cites methods from "Fast parallel algorithms for the Ma..."

...References[1] A. Datta, The Maximum Empty Rectangle Problem and its Variations, PhDThesis, Indian Institute of Technology, Madras, 1992[2] A. Datta, R. Srikant, G.D.S. Ramkumar, K. Krithivasan, Fast Parallel Algorithmsfor the Maximum Empty Rectangle Problem, Sadhana, Vol. 17, Part 1, pp. 221{236,1992....
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...We apply a parallel pre x idea, similar to the one used by Datta [2]....
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...It has been studied intensively by Amitava Datta [1] in a sequential context....
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...We apply a parallel pre x idea, similar tothe one used by Datta [2]....
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References

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Book•

Computational Aspects of Vlsi

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Jeffrey D Ullma

01 Jan 1984

862 citations

"Fast parallel algorithms for the Ma..." refers background in this paper

...Since the lower bound for solving any problem on a mesh-of-trees is fl(logn) ( Ullman 1984; Lodi & Pagli 1985), we state the following....
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...There are n rows and n columns in this grid and n is assumed to be a power of 2. Each row and each column is connected as a complete binary tree. For details see Ullman (1984) ....
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...The mesh of trees architecture ( Ullman 1984 ) is a square grid of n 2 processors without any interconnection between them....
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...This takes O(logn) time ( Ullman 1984; Lodi & Pagli 1985)....
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Journal Article•DOI•

Parallel merge sort

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Richard Cole¹•Institutions (1)

New York University¹

01 Aug 1988-SIAM Journal on Computing

TL;DR: A parallel implementation of merge sort on a CREW PRAM that uses n processors and O(logn) time; the constant in the running time is small.

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Abstract: We give a parallel implementation of merge sort on a CREW PRAM that uses n processors and $O(\log n)$ time; the constant in the running time is small. We also give a more complex version of the algorithm for the EREW PRAM; it also uses n processors and $O(\log n)$ time. The constant in the running time is still moderate, though not as small.

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847 citations

"Fast parallel algorithms for the Ma..." refers background or methods in this paper

...First we sort the points according to X coordinate in O(Iogn) time using O(n) processors (Cole 1988)....
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...We sort the points of CL by increasing the Y coordinate using O(n) processors and O(logn) time (Cole 1988)....
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...This takes O(n) processors and O(logn) time (Cole 1988)....
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...Notice that, sorting can be done in O(logn) time using O(n) processors (Cole 1988) and parallel prefix of n elements can be found in O (logn) time using O (n/logn) processors (Kruskal et al 1985) on an EREW PRAM....
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...The points in the set P can be sorted in the Y order in O(logn) time using O(n) processors on a CREW PRAM (Cole 1988)....
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Journal Article•DOI•

Geometric applications of a matrix-searching algorithm

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Alok Aggarwal¹, Maria Klawe¹, Shlomo Moran¹, Peter W. Shor², Robert E. Wilber¹ - Show less +1 more•Institutions (2)

IBM¹, Mathematical Sciences Research Institute²

01 Nov 1987-Algorithmica

TL;DR: The Θ(m) bound on finding the maxima of wide totally monotone matrices is used to speed up several geometric algorithms by a factor of logn.

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Abstract: LetA be a matrix with real entries and letj(i) be the index of the leftmost column containing the maximum value in rowi ofA.A is said to bemonotone ifi 1 >i 2 implies thatj(i 1) ≥J(i 2).A istotally monotone if all of its submatrices are monotone. We show that finding the maximum entry in each row of an arbitraryn xm monotone matrix requires Θ(m logn) time, whereas if the matrix is totally monotone the time is Θ(m) whenm≥n and is Θ(m(1 + log(n/m))) whenm

...read moreread less

506 citations

Book•

Efficient Parallel Algorithms

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Alan Gibbons¹, Wojciech Rytter²•Institutions (2)

University of Warwick¹, University of Warsaw²

01 Jan 1988

TL;DR: The emphasis of the book is on designing algorithms within the timeless and abstracted context of a high-level programming language rather than depending on highly specific computer architectures.

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Abstract: From the Publisher: This text is an introduction to the field of efficient parallel algorithms and to techniques for efficient parallelisation. It is largely self-contained and presumes no special knowledge of parallel computers or particular mathematics. The emphasis of the book is on designing algorithms within the timeless and abstracted context of a high-level programming language rather than depending on highly specific computer architectures. This approach concentrates on the essence of algorithmic theory, and on determining and taking advantage of the inherently parallel nature of certain types of problem. The authors present regularly used techniques and a range of algorithms which includes some of the more celebrated. The text is targeted at non-specialists who are considering entering the field of parallel algorithms. It will be particularly useful for courses aimed at advanced undergraduate or new postgraduate students of computer science and mathematics.

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466 citations

"Fast parallel algorithms for the Ma..." refers background or methods in this paper

...According to Brent's theorem ( Gibbons & Rytter 1988 ), the ith stage can be executed in upper(CJnloan) time with O(nloon) processors, where upper(x) is the lowest integer greater than or equal to the real number x. Hence, the total time taken is,...
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...See the book by Gibbons & Rytter (1988) for more on PRAM algorithms....
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...Lemma 2.3. ( Berkman et al 1988 ) The nearest smallers problem can be soloed on a CReW PRAM in O(logn) time using O(n/logn) processors....
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...It is easy to see that top(p~) is the first point in the array after p~ with an X coordinate greater than that of Pi. The problem of finding top(pl) is analogous to the nearest smallers problem as defined in § 2. This problem has been solved by Berkman et al (1988) taking O(logn) time using O(n/logn) processors on a CREW PRAM....
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...Proof. The function next (/) can be found in O(logn) time using O(nflogn) processors by converting it into a nearest smallers problem ( Berkman et al 1988 )....
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