On the maximum empty rectangle problem

Naamad, Amnon; Lee, D. T.; Hsu, Wen-Lian

doi:10.1016/0166-218x(84)90124-0

Cited by 96 publications

(91 citation statements)

References 2 publications

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“…Under this scenario, the MER problem has been extensively studied. The fi rst known study was by Naamad, Lee, and Hsu (1984), who described two algorithms that consider points as being randomly located within space. The fi rst algorithm needs points to be ordered and compared one with the other.…”

Section: Related Workmentioning

confidence: 99%

“…Gutiérrez et al (2014) The MER and QMER problems are formally defi ned below. Let S be a fi nite point set of size n located in a rectangle R ⊆ R d (typically d = 2) whose sides are parallel to the plane axes, and let q be a point such that q ∉S According to Naamad, Lee, and Hsu (1984), a rectangle M is said to be a restricted rectangle if it satisfi es the following three conditions. (1) M is completely contained in R, (2) M does not contain points from S in its interior, and (3) each arc of M contains a point S or coincides with the arc of R. The MER problem (Figure 1a) consists of fi nding the rectangle M with the largest area.…”

Section: Related Workmentioning

confidence: 99%

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Queries about the largest empty rectangle in large 2-dimensional datasets stored in secondary memory

et al. 2017

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Let be a set of points located in a rectangle and is a point that is not in . This article describes the design, implementation, and experimentation of different algorithms to solve the following two problems: (i) Maximum Empty Rectangle (MER), which consists in finding an empty rectangle with a maximum area contained in R and does not contain any point from and (ii) Query Maximum Empty Rectangle (QMER), which consists in finding the rectangle with the same restrictions given for the MER problem but must also contain . It is assumed that both problems have insufficient main memory to store all the objects in set . According to the literature, both problems are very practical in fields such as data mining and Geographic Information Systems (GIS). Specifically, the present study proposes two algorithms that assume that is stored in secondary memory (mainly disk) and that it is impossible to store it completely in main memory. The first algorithm solves the QMER problem and consists of decreasing the size of S by using dominance areas and then processing the points that are not eliminated using an algorithm proposed by Orlowski (1990). The second algorithm solves the MER problem and consists of dividing R into four subrectangles that generate four subsets of similar size; these are processed using an algorithm proposed in Edmons et al. (2003), and finally the partial solutions are combined to obtain a global solution. For the purpose of verifying algorithm efficiency, results are shown for a series of experiments that consider synthetic and real data.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Queries about the largest empty rectangle in large 2-dimensional datasets stored in secondary memory

et al. 2017

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“…Note that the critical cell size is within the range of (1/2l, l). The largest empty square problem can be solved by finding the largest empty circle in the Voronoi diagram constructed using the L ∞ distance metric, and has a time complexity of (n log n) [34][35][36]. Hence, we will not delve into it further.…”

Section: The Critical Grid Sizementioning

confidence: 99%

The Critical Grid Size and Transmission Radius for Local-Minimum-Free Grid Routing in Wireless Ad Hoc and Sensor Networks

Yi¹,

Wan²,

Su³

et al. 2010

The Computer Journal

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In grid routing, the plane is tessellated into equal-sized square cells. Two cells are called neighbor cells if they share a common edge, and two nodes are called routing neighbors if they are in neighbor cells and within each other's transmission range. If communication parties are in the same cell, packets can be transmitted directly; otherwise, packets are forwarded to routing neighbors that are in cells closer to destination cells. As a greedy strategy, grid routing suffers the existence of local minima at which no neighbor nodes exist for relaying packets. To guarantee deliverability, in this paper, we investigate two vital parameters of grid routing, called the grid size and the transmission radius. Assume that nodes are represented by a Poisson point process with rate n over a unit-area square, and let l denote the grid size and r the transmission radius. First, we show that if l = β ln n/n for some constant β and r = √ 5l, then β = 1 is the threshold for deliverability. In other words, there almost surely do not exist local minima if β > 1 and there almost surely exist local minima if β < 1. Next, for any given β > 1, we give sufficient and necessary conditions to determine the critical transmission radius (CTR) for deliverability. Then, we show that as β ∼ = 1.092, the CTR r ∼ = 2.09 √ ln n/n is the minimum over all β > 1. Simulation results are given to validate this theoretical work.

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“…Let F S,k be the set of all maximal k-boxes. The problem of generating all elements of F S,0 has been studied in the machine learning and computational geometry literatures (see [2,8,11,20,21]), and is motivated by the discovery of missing associations or "holes" in data mining applications (see [1,16,17]). All known algorithms that solve this problem have running time complexity which is exponential in the dimension n of the given point set.…”

Section: Maximal K-boxesmentioning

confidence: 99%

“…Such intervals are called empty boxes or empty rectangles, when k = 0. This problem has applications in computational geometry, data mining and machine learning [1,2,8,11,16,17,20,21]. …”

Section: Introductionmentioning

confidence: 99%

An Intersection Inequality for Discrete Distributions and Related Generation Problems

Boros

Elbassioni

Gurvich

et al. 2003

Automata, Languages and Programming

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Abstract. Given two finite sets of points X , Y in R n which can be separated by a nonnegative linear function, and such that the componentwise minimum of any two distinct points in X is dominated by some point in Y, we show that |X | ≤ n|Y|. As a consequence of this result, we obtain quasi-polynomial time algorithms for generating all maximal integer feasible solutions for a given monotone system of separable inequalities, for generating all p-inefficient points of a given discrete probability distribution, and for generating all maximal empty hyper-rectangles for a given set of points in R n . This provides a substantial improvement over previously known exponential algorithms for these generation problems related to Integer and Stochastic Programming, and Data Mining. Furthermore, we give an incremental polynomial time generation algorithm for monotone systems with fixed number of separable inequalities, which, for the very special case of one inequality, implies that for discrete probability distributions with independent coordinates, both p-efficient and p-inefficient points can be separately generated in incremental polynomial time.

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On the maximum empty rectangle problem

Cited by 96 publications

References 2 publications

Queries about the largest empty rectangle in large 2-dimensional datasets stored in secondary memory

Queries about the largest empty rectangle in large 2-dimensional datasets stored in secondary memory

The Critical Grid Size and Transmission Radius for Local-Minimum-Free Grid Routing in Wireless Ad Hoc and Sensor Networks

An Intersection Inequality for Discrete Distributions and Related Generation Problems

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