Maximum Area Rectangle Separating Red and Blue Points

Armaselu, Bogdan; Daescu, Ovidiu

doi:10.48550/arxiv.1706.03268

“…We first improve the result in [3] for the outlier version. Specifically, we first give a slight improvement that runs in O(k 7 m + m log m + n) time for k outliers, and then a further improvement to O(k 3 m + km log m + n) time (which works when k > (log m) 1 4 ). We also solve the circles version and provide an algorithm that runs in O(m 2 + n) time.…”

Section: Our Resultsmentioning

confidence: 99%

“…The problem of computing an MBSR was considered by Armaselu and Daescu. For the case when the target rectangle has to be axisaligned, the algorithm runs in O(m log m + n) time [2,4]. When the target rectangle is allowed to be arbitrarily oriented, an O(m 3 + n log n) time algorithm is given, and they also provide an algorithm to find the largest separating box in 3D in O(m 2 (m + n)) time [2].…”

Section: Related Workmentioning

confidence: 99%

“…We also consider all arcs pq of circles that are part of pairs in Case D (p to the west of q), the vertical line l p through p and the horizontal line l q through q, and add l p ∩ l q to B . We then find the largest rectangle S * enclosing R and containing the fewest points in B using the algorithm in [4] in O(m + n) time. It is easy to check that S * is an optimal solution for Case 1, since any circle containing points in B intersects a circle in B.…”

Section: Finding An Optimal Solution In Each Casementioning

confidence: 99%

“…The MBSR problem, introduced in [2] (see also [4]), is stated as follows. Given a set of red points R and a set of blue points B in the plane, with |R| = n, |B| = m, compute the axis-aligned rectangle S having the following properties:…”

Section: Introductionmentioning

confidence: 99%

“…

In this paper, we study two extensions of the maximum bichromatic separating rectangle (MBSR) problem introduced in [2,4]. One of the extensions, introduced in [3], is called MBSR with outliers or MBSR-O, and is a more general version of the MBSR problem in which the optimal rectangle is allowed to contain up to k outliers, where k is given as part of the input.

…”

mentioning

confidence: 99%

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Extensions of the Maximum Bichromatic Separating Rectangle Problem

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2021

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In this paper, we study two extensions of the maximum bichromatic separating rectangle (MBSR) problem introduced in [2,4]. One of the extensions, introduced in [3], is called MBSR with outliers or MBSR-O, and is a more general version of the MBSR problem in which the optimal rectangle is allowed to contain up to k outliers, where k is given as part of the input. For MBSR-O, we improve the previous known running time bounds ofThe other extension is called MBSR among circles or MBSR-C and asks for the largest axisaligned rectangle separating red points from blue unit circles. For MBSR-C, we provide an algorithm that runs in O(m 2 + n) time.Keywords extensions • bichromatic • separating rectangle • outliers • circles

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

Alegrı́a

¹

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Orden

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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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Geometric Heuristics for Transfer Learning in Decision Trees

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¹

,

Rzepecki

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Nicholson

³

et al. 2021

Proceedings of the 30th ACM International Conference on Information &Amp; Knowledge Management

1

0

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Motivated by a network fault detection problem, we study how recall can be boosted in a decision tree classifier, without sacrificing too much precision. This problem is relevant and novel in the context of transfer learning (TL), in which few target domain training samples are available. We define a geometric optimization problem for boosting the recall of a decision tree classifier, and show it is NP-hard. To solve it efficiently, we propose several near-linear time heuristics, and experimentally validate these heuristics in the context of TL. Our evaluation includes 7 public datasets, as well as 6 network fault datasets, and we compare our heuristics with several existing TL algorithms, as well as exact mixed integer linear programming (MILP) solutions to our optimization problem. We find that our heuristics boost recall in a manner similar to optimal MILP solutions, yet require several orders of magnitude less compute time. In many cases the 𝐹 1 score of our approach is competitive, and often better, than other TL algorithms. Moreover, our approach can be used as a building block to apply transfer learning to more powerful ensemble methods, such as random forests. CCS CONCEPTS• Computing methodologies → Classification and regression trees; Transfer learning.

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Maximum Area Rectangle Separating Red and Blue Points

Cited by 5 publications

References 17 publications

Extensions of the Maximum Bichromatic Separating Rectangle Problem

Extensions of the Maximum Bichromatic Separating Rectangle Problem

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

Geometric Heuristics for Transfer Learning in Decision Trees

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