Separating Bichromatic Point Sets by Minimal Triangles with a Fixed Angle

Moslehi, Zahra; Bagheri, Alireza

doi:10.1142/s0129054117500198

“…A thorough study is presented by Seara [1]. In 2017 the separation of two bichromatic point sets by a minimal triangle with a fixed angle was posed by Moslehi and Bagheri [28] and solved by an ( log ) O n n -time algorithm. We study the separation of polylines by a triangle and present the relevant algorithm of () O NlogN -time, in fact, we generalize their solution for polylines.…”

Section: Np Hard mentioning

confidence: 99%

Separating Bichromatic Polylines by Fixed-angle Minimal Triangles

Moghaddam

¹

,

Bagheri

²

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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“…monotone or with alternating constant turn) [25,39], different types of enclosing shapes (e.g. a polygon or a non-traditional convex hull) [5,19], and sets of geometric objects of the same type, such as a hyperplanes [35] and triangles [34]. These choices of separators have been used not only on points, but also on segments, circles, simple polygons, etc.…”

Section: Introductionmentioning

confidence: 99%

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

Alegrı́a

¹

,

Orden

²

,

Seara

³

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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“…As an ω-wedge rotates around P , the locus of positions of the apex of the ω-wedge describes a curve called an ω-cloud, see Figure 1c. The ω-cloud finds applications in diverse geometric algorithms [1,2,13]. The ω-cloud can be seen as a generalization of the diameter function introduced by Rao and Goldberg [14].…”

Section: Introductionmentioning

confidence: 99%

Reconstructing a Convex Polygon from Its $$\omega $$-cloud

Arseneva

¹

,

Bose

²

,

Carufel

³

et al. 2019

Computer Science – Theory and Applications

1

0

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An ω-wedge is the closed set of points contained between two rays that are emanating from a single point (the apex), and are separated by an angle ω < π. Given a convex polygon P , we place the ω-wedge such that P is inside the wedge and both rays are tangent to P . The set of apex positions of all such placements of the ω-wedge is called the ω-cloud of P . We investigate reconstructing a polygon P from its ω-cloud. Previous work on reconstructing P from probes with the ω-wedge required knowledge of the points of tangency between P and the two rays of the ω-wedge in addition to the location of the apex. Here we consider the setting where the maximal ω-cloud alone is given. We give two conditions under which it uniquely defines P : (i) when ω < π is fixed/given, or (ii) when what is known is that ω < π/2. We show that if neither of these two conditions hold, then P may not be unique. We show that, when the uniqueness conditions hold, the polygon P can be reconstructed in O(n) time with O(1) working space in addition to the input, where n is the number of arcs in the input ω-cloud. E. A. was partially supported by F.R.S.-FNRS, and by SNF Early PostDoc Mobility project P2TIP2-168563. P. B., J. C., and S. V. were partially supported by NSERC.

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Separating Bichromatic Point Sets by Minimal Triangles with a Fixed Angle

Cited by 3 publications

References 10 publications

Separating Bichromatic Polylines by Fixed-angle Minimal Triangles

Separating Bichromatic Polylines by Fixed-angle Minimal Triangles

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

Reconstructing a Convex Polygon from Its $$\omega $$-cloud

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