Fast sequential and parallel algorithms for finding the largest rectangle separating two sets
01 Jan 1990-International Journal of Computer Mathematics (Gordon and Breach Science Publishers)-Vol. 37, pp 49-61
TL;DR: This work considers two limiting cases of this problem when the cardinalities of set A is much greater than that of set B, and presents efficient sequential and parallel algorithms for these two problems.
Abstract: Given a bounding isothetic rectangle BR and two sets of points A and B with cardinalities n and m inside it, we have to find an isothetic rectangle containing maximum number of points from set A and no point from set B. We consider two limiting cases of this problem when the cardinalities of set A (resp. set B) is much greater than that of set B (resp. set ,A). We present efficient sequential and parallel algorithms for these two problems. Our sequential algorithms run in O((n + m)log m + m 2) and O((m+ n) log n + n 2) time respectively. The parallel algorithms in CREW PRAM run in o(log n) ando(log m 2) time using O(max(n,m 2/logm)) and O(max(m,n 2/logn)) processors respectively. Our sequential algorithms are faster than a previous algorithm under these constraints on cardinality. No previous parallel algorithm was known for this problem. We also present an optimal systolic algorithm for the original problem.
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17 Dec 2002
TL;DR: An O(log log N) bus cycles parallel algorithm for the medial axis transform of an N/spl times/N binary image on a linear array with a reconfigurable pipelined bus system using N/sup 2/ processors is provided.
Abstract: In this paper based on the advantages of both optical transmission and electronic computation, we first provide an O(log log N) bus cycles parallel algorithm for the medial axis transform of an N/spl times/N binary image on a linear array with a reconfigurable pipelined bus system using N/sup 2/ processors. By increasing the number of processors, the proposed algorithm can be modified to run in O(log log/sub q/ N) and O(1) bus cycles using qN/sup 2/ and N/sup 2+1//spl isin// processors respectively, where 1/spl les/q/spl les//spl radic/N, /spl isin/ is a constant and /spl isin//spl ges/1. These results improve on previously known algorithms developed on various parallel computation models. Key Words: Medial axis transform, image processing, image compression, computer vision, parallel algorithms, linear array with a reconfigurable pipelined bus system.
11 citations
References
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Book Chapter•
01 Apr 1979
TL;DR: Very large scale integrated (VLSI) circuit technology has made it possible to build multiprocessor hardware devices to aid in the rapid solution of sophisticated problems but an algorithms designer wishing to take full advantage of the massive parallelism offered by VLSI must address geometric issues hitherto relegated to layout artists.
Abstract: Very large scale integrated (VLSI) circuit technology has made it possible to build
multiprocessor hardware devices to aid in the rapid solution of sophisticated problems.
An algorithms designer wishing to take full advantage of the massive parallelism offered
by VLSI must address geometric issues hitherto relegated to layout artists. The reason
for this is that VLSI is a planar technology in which the interconnections among
components on a chip may cost more than the components themselves. The designer of a
multiprocessor algorithm to be implemented in this technology must consider the
complexity of the data paths between processors in evaluating the algorithm.
164 citations
01 Oct 1987
TL;DR: The first algorithm for computing the largest-area empty rectangle is optimal within a multiplicative constant and the two algorithms for computing such a rectangle can be modified to compute thelargest-perimeter rectangle in memory space.
Abstract: We provide two algorithms for solving the following problem: Given a rectangle containing n points, compute the largest-area and the largest-perimeter subrectangles with sides parallel to the given rectangle that lie within this rectangle and that do not contain any points in their interior. For finding the largest-area empty rectangle, the first algorithm takes O(n log3n) time and O(n) memory space and it simplifies the algorithm given by Chazelle, Drysdale and Lee which takes O(n log3n) time but O(n log n) storage. The second algorithm for computing the largest-area empty rectangle is more complicated but it only takes O(n log2n) time and O(n) memory space. The two algorithms for computing the largest-area rectangle can be modified to compute the largest-perimeter rectangle in O(n log2n) and O(n log n) time, respectively. Since O(n log n) is a lower bound on time for computing the largest-perimeter empty rectangle, the second algorithm for computing such a rectangle is optimal within a multiplicative constant.
134 citations
TL;DR: A divide-and-conquer approach similar to the ones used by Bentley is used and a new notion of Voronoi diagram is introduced along with a method for efficient computation of certain functions over paths of a tree.
Abstract: We consider the following problem: Given a rectangle containing N points, find the largest area subrectangle with sides parallel to those of the original rectangle which contains none of the given points. If the rectangle is a piece of fabric or sheet metal and the points are flaws, this problem is finding the largest-area rectangular piece which can be salvaged. A previously known result [13] takes $O(N^2 )$ worst-case and $O(N\log ^2 N)$ expected time. This paper presents an $O(N\log ^3 N)$ time, $O(N\log N)$ space algorithm to solve this problem. It uses a divide-and-conquer approach similar to the ones used by Bentley [1] and introduces a new notion of Voronoi diagram along with a method for efficient computation of certain functions over paths of a tree.
125 citations
TL;DR: In this article, the authors considered the problem of finding a maximum area rectangle that is fully contained in a given rectangle A and does not contain any point of S in its interior.
Abstract: Given a rectangle A and a set S of n points in A , we consider the problem, called the maximum empty rectangle problem , of finding a maximum area rectangle that is fully contained in A and does not contain any point of S in its interior. An O( n 2 ) time algorithm is presented. Furthermore, it is shown that if the points of S are drawn randomly and independently from A , the problem can be solved in O( n (log n ) 2 ) expected time.
105 citations
Journal Article•
TL;DR: It is shown that if the points of S are drawn randomly and independently from A , the problem can be solved in O( n (log n ) 2 ) expected time.
Abstract: Abstract Given a rectangle A and a set S of n points in A , we consider the problem, called the maximum empty rectangle problem , of finding a maximum area rectangle that is fully contained in A and does not contain any point of S in its interior. An O( n 2 ) time algorithm is presented. Furthermore, it is shown that if the points of S are drawn randomly and independently from A , the problem can be solved in O( n (log n ) 2 ) expected time.
105 citations
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