Variations of largest rectangle recognition amidst a bichromatic point set

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 = P-r boolean OR P-b in R-2 and R-3, where P-r and P-b represent a set of n red points and a set of m blue points respectively, and the objective is to compute a monochromatic object of a 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 root n + m log m n log n)) and O(m(3) root 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, that 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 orthogonal range counting query using O(1) extra workspace, and with O(n(1-1/k)) query time complexity. The construction time of this data structure is O(n log n). Both the construction and query algorithms are non-recursive in nature that do not need O(logn) 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 = P-r boolean OR P-b, where each point in P-b (resp. P-r) 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. (C) 2019 Elsevier B.V. All rights reserved.