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Geometric Separability Using Orthogonal Objects

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Geometric Separability Using Orthogonal Objects Geometric Separability using Orthogonal Objects V P Abidha 1, Pradeesha Ashok International Institute of Information Technology Bangalore, India Abstract Given a bichromatic point set P = R [ B of red and blue points, a separator is an object of a certain type that separates R and B. We study the geometric separability problem when the separator is a) rectangular annulus of fixed orientation b) rectangular annulus of arbitrary orien- tation c) square annulus of fixed orientation d) orthogonal convex polygon. In this paper, we give polynomial time algorithms to construct separators of each of the above type that also optimizes a given parameter. Specifically, we give an O(n3 log n) algorithm that computes (non-uniform width) separating rectangular annulus in arbitrary orientation, of minimum possible width. Fur- ther, when the orientation is fixed, we give an O(n log n) algorithm that constructs a uniform width separating rectangular annulus of minimum possible width and an O(n log2 n) algorithm that constructs a minimum width separating concentric square annulus. We also give an opti- mal algorithm that computes a separating orthogonal convex polygon with minimum number of edges, that runs in O(n log n) time. Keywords: Geometric Separability, axis-parallel rectangular annulus, axis-parallel square annulus, orthogonal convex polygon 1. Introduction In many applications, data is represented using points in Rd. In some of them, each data point belongs to a fixed number of categories or classes. In those cases, data from different classes can be represented by points of different colors. Thus the problems involving multicolored point sets are important in applications like data mining and machine learning. In this paper, we study the separability problem for bichromatic point sets in the plane. Given a bichromatic point set P = R [ B, where R represents a set of red points and B represents a set of blue points, the separability problem asks whether there exists a geometric object of a arXiv:2010.12227v1 [cs.CG] 23 Oct 2020 particular type that separates R and B. The separability problem is closely related to the classifi- cation problem in Machine Learning and has also found applications in areas like Data mining, Computer graphics, Pattern recognition, etc. An important question studied in separability is to decide whether a separator of a particular type exists for a given bichromatic point set. For example, the linear separability question is to decide whether there exists a straight line such that Email addresses: [email protected] (V P Abidha ), [email protected] (Pradeesha Ashok) 1First author is supported by the TCS Research scholar program Preprint submitted to Information Processing Letters October 26, 2020 the blue points and red points are in different half-planes defined by the line. A generalization of this problem to higher dimensions is to check whether there exists a hyperplane that separates red and blue points. Both these problems can be solved in linear time [14]. The separability problem is also studied when the separator is a geometric object like circle, rectangle, convex polygon, etc. [9, 19, 8]. Another set of questions studied in separability is to find a separator that optimizes a given parameter. Edelsbrunner and Preparata [8] gave an algorithm to find a separating convex polygon with the minimum number of edges. However, it is NP-hard to find a separating simple polygon with the minimum number of edges [14]. O’Rourke et al. [16] gave algorithms for deciding separability using circles and to find the smallest and largest separating circles. Hurtado et al. [12] studied the separability problem for wedges and strips. Motivated by applications in urban scene reconstruction using LiDAR data, many researchers [5, 3, 4] considered the separability problem for various rectangular shapes. Rectilinear shapes are particularly important in urban scene reconstruction as many entities like roads and buidings tend to have orthogonal edges. In addition to the papers mentioned above, Sheikhi et al. [17] studied the separability problem for a family of non-convex rectilinear objects called L-shapes and gave polynomial time algorithms. We consider a more general class of these objects, namely rectangular annulus. We also consider the separability problem for a class of objects called orthogonal convex objects, which is a restriction of classical convexity to an orthogonal setting. Problems Studied and Related Work Let P = R [ B be a point set in R2 where R is a set of red points and B is a set of blue points and jRj + jBj = n, throughout the paper. For simplicity of explanation, we assume no two points in P have the same x or y coordinate. The separability problem is to find an object C such that all the blue points lie inside or on the boundary of C and all the red points lie outside or on the boundary of C. We study the separability problem for annuli. In general, an annulus refers to the region enclosed between two objects of the same type. In this paper, we study the separability problem for Rectangular and Square annuli of fixed and arbitrary orientation. We define the orientation of a line as the angle it makes with the X-axis. A rectangle/square (and a rectangular/square annulus) has orientation θ if the line containing each of its edges has orientation either θ or θ + π/2. We will now define these objects that we are considering. Rectangular Annulus: A rectangular annulus is a closed region between two rectangles D and D0 of same orientation such that D is contained in D0. For a rectangular annulus A, we shall call the larger rectangle and smaller rectangle that define A as its outer rectangle and inner rectangle respectively. The width of a rectangular annulus on a given side is the perpendicular distance between the corresponding edges of the outer and inner rectangles. If the width is uniform on all the four sides of an annulus, we call it a uniform width rectangular annulus. Note that the center of inner and outer rectangles are the same in a uniform width rectangular annulus (The converse is not always true). When the width is not uniform on all the four sides, the width of the annulus is defined as the maximum width among the four sides. Square Annulus: A square annulus is a closed region between two squares S and S 0 of same orientation such that S is contained in S 0. The definition of the width of the square annulus is similar to that of the rectangular annulus. If the center (point of intersection of the diagonals) of the outer and inner squares are the same, or equivalently the width is uniform on all the four sides, then we call it a concentric square annulus. For a given bi-chromatic point set, we aim to find separating rectangular and square annuli 2 that minimize the width. A closely related problem that arises in the monochromatic setting is to find an annulus of minimum width that encloses a given point set. Enclosing a point set using a circular annulus of minimum width is studied in [6]. Gluchshenko et al. [11] gave an O(n log n) algorithm to compute a minimum width enclosing rectilinear annulus. Mukherjee et al. [15] gave an O(n) algorithm that computes minimum width enclosing axis-parallel rectangular annu- lus and an O(n2 log n) algorithm that computes minimum width enclosing rectangular annulus in arbitrary orientation. Bae [1] gave an algorithm that runs in O(n3 log n) time that computes the minimum width enclosing square annulus. Another closely related problem in the monochro- matic setting is to find a maximum width empty annulus in a point set. Bae [2] gave algorithms that run in O(n3) and O(n2 log n) time respectively to compute the maximum width empty square and rectangular annulus. Many techniques used for solving the monochromatic problems can be extended for solving the minimum width separating annulus problem. However, the presence of red points poses certain challenges in the separability problem. For instance, it is easy to extend a nonuniform width annulus to a uniform width annulus in the monochromatic setting whereas the two cases are very different in the bichromatic setting. The algorithms presented in this paper makes non-trivial modifications to the existing techniques to solve the bichromatic separability problem. We also consider the separability problem for orthogonal convex objects. Orthogonal Convex Polygon: A polygon C is said to be orthogonal convex if all the edges of C are either horizontal or vertical and the intersection of any vertical or horizontal line with C is either null, a point or a continuous line segment [10]. An orthogonal convex polygon has alternate vertical and horizontal edges and alternate convex and reflex vertices (except for the vertices with highest and lowest x and y coordinates). A polynomial time algorithm to find a separating convex polygon with the minimum number of edges is given in [8]. We extend this result to orthogonal convexity. A summary of the results of this paper are given below. Object Orientation Parameter Running time Nonuniform width Rectangular Annulus Fixed width O(n) Uniform width Rectangular Annulus Fixed width O(n log n) Nonuniform width Rectangular Annulus Arbitrary width O(n3 log n) Uniform width Square Annulus Fixed width O(n log2 n) Orthogonal convex polygon Fixed edges O(n log n) All the above results for the separability problem using minimum width annulus can be ex- tended to the separability problem using maximum width annulus. Notations: For any point p 2 P, let x(p) and y(p) respectively denote the x-coordinate and y- coordinate values of p .
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