Variations of largest rectangle recognition amidst a bichromatic point set

Abstract

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.

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

“…The O-convex hull is relevant for research fields that require restricted-orientation enclosing shapes [18]. In the particular case where O is formed by two orthogonal lines, Oconvexity is known as orthogonal convexity 2 and the O-convex hull is known as the rectilinear convex hull. The rectilinear convex hull has been extensively studied in the context of fields as diverse as polyhedra reconstruction [14], facility location [47], and geometric optimization [29]; as well as in practical research fields such as pattern recognition [28], shape analysis [15], and VLSI circuit layout design [48].…”

“…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.…”

“…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.…”

Steganalysis for LSB inserts in low stego-payload artificial color images

“…Finding the largest empty rectangle was first considered by A. Naamad, D. Lee and V. Shu [9] with some subsequent clarification and modifications in other studies [10].…”

Separating Bichromatic Polylines by Fixed-angle Minimal Triangles

“…Misra et al [26] worked on the special case of the separation problem when the points lie on a circle and demonstrated a polynomial-time algorithm for that case. Acharyya et al [27] worked on different problems for bichromatic point set on a plane and proposed inplace algorithms for computing (a) an arbitrarily oriented monochromatic rectangle of maximum size in 2 R and (b) an axis-parallel monochromatic cuboid of maximum size in 3 R .…”