A unified algorithm for finding maximum and minimum object enclosing rectangles and cuboids

Given a set of n points in 2 bounded within a rectangular floor F, and a rectangular plate P of specified size, we consider the following two problems: find an isothetic position of P such that it encloses (i) maximum and (ii) minimum number of points, keeping P totally contained within F. For both of these problems, a new algorithm based on interval tree data structure is presented, which runs in O(nlogn) time and consumes O(n) space. If polygonal objects of arbitrary size and shape are distributed in 2, the proposed algorithm can be tailored for locating the position of the plate to enclose maximum or minimum number of objects with the same time and space complexity. Finally, the algorithm is extended for identifying a cuboid, i.e., a rectangular parallelepiped that encloses maximum number of polyhedral objects in 3. Thus, the proposed technique serves as a unified paradigm for solving a general class of enclosure problems encountered in computational geometry and pattern recognition.