Separating bichromatic point sets by two disjoint isothetic rectangles

Moslehi, Zahra; Bagheri, Alireza

doi:10.24200/sci.2016.3891

Cited by 4 publications

(4 citation statements)

References 11 publications

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“…For n = |R| + |B|, an arbitrarily-oriented separating rectangle can be found in O(n log n) time and O(n) space [49]. Several variations have also been solved including separability by two disjoint rectangles [33], bichromatic sets of imprecise points [46], maximizing the area of the separating rectangle [2,9], and an extension where the separator is a box in three dimensions [27]. Along with the axis-aligned rectangle, two more ortho-convex separators can be found in the literature.…”

Section: Background and Related Workmentioning

confidence: 99%

“…Several separability criteria have been considered in the literature, as well as separators of different complexities. Well-known constant-complexity separators include a line or a hyperplane [10,24,27,32], a wedge or a double-wedge [1,3,25,26,44], a circle [7,8,17,37], and one or two boxes [2,16,33,50]. Typical separators of linear complexity include different types of polygonal chains (e.g.…”

Section: Introductionmentioning

confidence: 99%

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Separating bichromatic point sets in the plane by restricted orientation convex hulls

et al. 2022

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We explore the separability of point sets in the plane by a restricted-orientation convex hull, which is an orientation-dependent, possibly disconnected, and non-convex enclosing shape that generalizes the convex hull. Let R and B be two disjoint sets of red and blue points in the plane, and $$\mathcal {O}$$ O be a set of $$k\ge 2$$ k ≥ 2 lines passing through the origin. We study the problem of computing the set of orientations of the lines of $$\mathcal {O}$$ O for which the $$\mathcal {O}$$ O -convex hull of R contains no points of B. For $$k=2$$ k = 2 orthogonal lines we have the rectilinear convex hull. In optimal $$O(n\log n)$$ O ( n log n ) time and O(n) space, $$n = \vert R \vert + \vert B \vert $$ n = | R | + | B | , we compute the set of rotation angles such that, after simultaneously rotating the lines of $$\mathcal {O}$$ O around the origin in the same direction, the rectilinear convex hull of R contains no points of B. We generalize this result to the case where $$\mathcal {O}$$ O is formed by $$k \ge 2$$ k ≥ 2 lines with arbitrary orientations. In the counter-clockwise circular order of the lines of $$\mathcal {O}$$ O , let $$\alpha _i$$ α i be the angle required to clockwise rotate the ith line so it coincides with its successor. We solve the problem in this case in $$O({1}/{\Theta }\cdot N \log N)$$ O ( 1 / Θ · N log N ) time and $$O({1}/{\Theta }\cdot N)$$ O ( 1 / Θ · N ) space, where $$\Theta = \min \{ \alpha _1,\ldots ,\alpha _k \}$$ Θ = min { α 1 , … , α k } and $$N=\max \{k,\vert R \vert + \vert B \vert \}$$ N = max { k , | R | + | B | } . We finally consider the case in which $$\mathcal {O}$$ O is formed by $$k=2$$ k = 2 lines, one of the lines is fixed, and the second line rotates by an angle that goes from 0 to $$\pi $$ π . We show that this last case can also be solved in optimal $$O(n\log n)$$ O ( n log n ) time and O(n) space, where $$n = \vert R \vert + \vert B \vert $$ n = | R | + | B | .

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Section: Background and Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Separating bichromatic point sets in the plane by restricted orientation convex hulls

et al. 2022

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“…Separability of two point sets for the case where separator is an L-shape with orientation of  was studied by Sheikhi et al [17]. The problem of deciding whether the two point sets can be separated by two disjoint isothetic rectangles was solved by Moslehi and Bagheri [18]. Separability of imprecise bichromatic points was investigated by Sheikhi et al [19].…”

Section: B Q B P mentioning

confidence: 99%

Separating Bichromatic Polylines by Fixed-angle Minimal Triangles

Moghaddam

Bagheri

2022

Scientia Iranica

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Separation of desired objects from undesired ones is one of the most important problems in computational geometry. It is tended to cover the desired objects by one or a couple of geometric shapes in a way that all of the desired objects are included by the covering shapes, while the undesired objects are excluded. We study separation of polylines by minimal triangles with a given fixed angle and present () O NlogNtime algorithm, where N is the number of all the desired and undesired polylines. By a minimal triangle, we mean a triangle in which all of its edges are tangential to the convex hull of the desired polylines. The motivation for studying this separation problem stems from that we need to separate bichromatic objects that are modeled by polylines not points in real life scenarios.

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“…They solved this problem in O(n 2 log n) time using O(n) space. The problem of separating bichromatic point sets by two disjoint axis-parallel rectangles such that each of the rectangles is monochromatic, is solved in O(n log n) time by Moslehi and Bagheri[31] (if such a solution exists). If these two rectangles are of arbitrary orientation then they solved the problem in O(n 2 log n) time.…”

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Variations of largest rectangle recognition amidst a bichromatic point set

Acharyya

Nandy

et al. 2020

Discrete Applied Mathematics

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Classical separability problem involving multi-color point sets is an important area of study in computational geometry. In this paper, we study different separability problems for bichromatic point set P = Pr ∪ P b on a plane, where Pr and P b represent the set of n red points and m blue points respectively, and the objective is to compute a monochromatic object of the desired type and of maximum size. We propose in-place algorithms for computing (i) an arbitrarily oriented monochromatic rectangle of maximum size in R 2 , and (ii) an axis-parallel monochromatic cuboid of maximum size in R 3 . The time complexities of the algorithms for problems (i) and (ii) are O(m(m+n)(m √ n+m log m+n log n)) and O(m 3 √ n+m 2 n log n), respectively. As a prerequisite, we propose an in-place construction of the classic data structure the k-d tree, which was originally invented by J. L. Bentley in 1975. Our in-place variant of the k-d tree for a set of n points in R k supports both orthogonal range reporting and counting query using O(1) extra workspace, and these query time complexities are same as the classical complexities, i.e., O(n 1−1/k + µ) and O(n 1−1/k ), respectively, where µ is the output size of the reporting query. The construction time of this data structure is O(n log n). Both the construction and query algorithms are nonrecursive in nature that do not need O(log n) size recursion stack compared to the previously known construction algorithm for in-place k-d tree and query in it. We believe that this result is of independent interest. We also propose an algorithm for the problem of computing an arbitrarily oriented rectangle of maximum weight among a point set P = Pr ∪ P b , where each point in P b (resp. Pr) is associated with a negative (resp. positive) real-valued weight that runs in O(m 2 (n + m) log(n + m)) time using O(n) extra space.

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Separating bichromatic point sets by two disjoint isothetic rectangles

Cited by 4 publications

References 11 publications

Separating bichromatic point sets in the plane by restricted orientation convex hulls

Separating bichromatic point sets in the plane by restricted orientation convex hulls

Separating Bichromatic Polylines by Fixed-angle Minimal Triangles

Variations of largest rectangle recognition amidst a bichromatic point set

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