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Journal ArticleDOI

On the maximum empty rectangle problem

01 Jul 1984-Discrete Applied Mathematics (North-Holland)-Vol. 8, Iss: 3, pp 267-277
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.
About: This article is published in Discrete Applied Mathematics.The article was published on 1984-07-01 and is currently open access. It has received 111 citations till now. The article focuses on the topics: Largest empty rectangle & Rectangle method.
Citations
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Journal ArticleDOI
TL;DR: The state of the art of computational geometry is surveyed, a discipline that deals with the complexity of geometric problems within the framework of the analysis of algorithms.
Abstract: We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis of algorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computer-aided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areas—convex hulls, intersections, searching, proximity, and combinatorial optimizations—are discussed. Seven algorithmic techniques—incremental construction, plane-sweep, locus, divide-and-conquer, geometric transformation, prune-and-search, and dynamization—are each illustrated with an example. A collection of problem transformations to establish lower bounds for geo-metric problems in the algebraic computation/decision model is also included.

271 citations

Proceedings ArticleDOI
16 Jun 1990
TL;DR: A technique for image segmentation using shape-directed covers is described and applied to the fully automatic analysis of complex printed-page layouts, which for some tasks is superior to strategies currently emphasized in the literature, including bottom-up and top-down.
Abstract: A technique for image segmentation using shape-directed covers is described and applied to the fully automatic analysis of complex printed-page layouts. The structure of the background (white space) is analyzed, assisted by an enumeration of all maximal white rectangles. For this enumeration, the most computationally expensive step, an algorithm has been developed that, aside from a sort, achieves an expected runtime linear in the number of black connected components. The crucial engineering decision is the specification of a partial order on white rectangles to express domain-specific knowledge of preferred shapes and sizes. This order determines a sequence of partial covers of the background, and thus, a sequence of nested page segmentations. In experimental trials on Manhattan layouts, good segmentations often occur early in this sequence, using a simple and uniform shape-direction rule. This is a global-to-local strategy, which for some tasks is superior to strategies currently emphasized in the literature, including bottom-up and top-down. >

133 citations

Journal ArticleDOI
TL;DR: The sorting network described by Ajtaiet al. was the first to achieve a depth ofO(logn), and the networks introduced here are simplifications and improvements based strongly on their work.
Abstract: The sorting network described by Ajtaiet al was the first to achieve a depth ofO(logn) The networks introduced here are simplifications and improvements based strongly on their work While the constants obtained for the depth bound still prevent the construction being of practical value, the structure of the presentation offers a convenient basis for further development

112 citations

Journal ArticleDOI
TL;DR: A new simple algorithm for the so-called largest empty rectangle problem, i.e., the problem of finding a maximum area rectangle contained inA and not containing any point ofS in its interior, is presented.
Abstract: A rectangleA and a setS ofn points inA are given. We present a new simple algorithm for the so-called largest empty rectangle problem, i.e., the problem of finding a maximum area rectangle contained inA and not containing any point ofS in its interior. The computational complexity of the presented algorithm isO(n logn + s), where s is the number of possible restricted rectangles considered. Moreover, the expected performance isO(n · logn).

92 citations

Proceedings ArticleDOI
31 May 2009
TL;DR: Improved approximation factors are obtained for the hitting set or the set cover problems associated with the corresponding range spaces for ε-nets of size O(1/ε log log log 1/ε) for planar point sets and axis-parallel rectangular ranges.
Abstract: We show the existence of e-nets of size O(1/e log log 1/e) for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane with "fat" triangular ranges, and for point sets in reals3 and axis-parallel boxes; these are the first known non-trivial bounds for these range spaces. Our technique also yields improved bounds on the size of e-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of e-nets of sizeO(1/e log log log 1/e) for the dual range space of "fat" regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Bronnimann and Goodrich, we obtain improved approximation factors (computable in randomized polynomial time) for the hitting set or the set cover problems associated with the corresponding range spaces.

87 citations

References
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Journal ArticleDOI
F. K. Hwang1
TL;DR: A mmmaal spammuag tree for P can be derived from a Voronot diagram for P m hnear tmae by using prewously known results.
Abstract: Let P be a set of pomts m the plane with rectdmear distance An O(n log n) a lgonthm for the construeUon o f a Voronot diagram for P ts gwen By using prewously known results, a mmmaal spammuag tree for P can be derived from a Voronot diagram for P m hnear tmae.

134 citations

Journal ArticleDOI
TL;DR: It is shown in this paper that there exists a natural isometry between the L_1 and L_\infty metrics, which implies the existence of a polynomial time algorithm for the OPP in one metric and the existence for the same problem in the other metric.
Abstract: In this paper we study the problem of scheduling the read/write head movement to handle a batch of $nI/O$ requests in a 2-dimensional secondary storage device in minimum time. Two models of storage systems are assumed in which the access time of a record (being proportional to the “distance” between the position of the record and that of the read/write head) is measured in terms of $L_1 $ and $L_\infty $ metrics, respectively. The scheduling problem, referred to as the Open Path Problem (OPP), is equivalent to finding a shortest Hamiltonian path with a specified end point in a complete graph with n vertices. We first show in this paper that there exists a natural isometry between the $L_1 $ and $L_\infty $ metrics. Consequently, the existence of a polynomial time algorithm for the OPP in one metric implies the existence of a polynomial time algorithm for the same problem in the other metric. Based on a result by Garey, Graham and Johnson, it is easy to show that the OPP in $L_1 $ (hence in $L_\infty $) me...

122 citations

01 Jan 1981
TL;DR: A data structure called "generalized heap," which is designed primarily to maintain the solutions of a certain group of problems while the data set is changing over time, is introduced.
Abstract: In the complexity theory of algorithms one encounters many problems for which efficient algorithms are known, but one needs to resolve the problems from scratch in order to reflect even a minor change in the data set. This thesis introduces a data structure called "generalized heap," which is designed primarily to maintain the solutions of a certain group of problems while the data set is changing over time. Applications of generalized heaps, among other things, include: (1) Maintaining the Voronoi Diagram of a set of points in the plane in linear time. (2) Improving the running time of Dinic's algorithm for the maximum flow problem from O(V('2)E) to O(VE lg('2)V). (3) Improving the running time of Shamos' algorithm for the 2 minimum spanning circles problem from O(n('3) lg n) to O(n('3)). (4) Finding the maximum empty rectangle defined by a set of n points in the plane in O(n 1g('2) n) time (average case). The variety of applications suggests that generalized heaps can also be useful in areas other than graph theory and computational geometry.

2 citations