Variations of Enclosing Problem Using Axis Parallel Square(s): A General Approach

Priya Ranjan Sinha Mahapatra

doi:10.4236/ajcm.2014.43016

Abstract

Let P be a set of n points in two dimensional plane. For each point , we locate an axis- parallel unit square having one particular side passing through p and enclosing the maximum number of points from P. Considering all points , such n squares can be reported in O(nlogn) time. We show that this result can be used to (i) locate m>(2) axis-parallel unit squares which are pairwise disjoint and they together enclose the maximum number of points from P (if exists) and (ii) find the smallest axis-parallel square enclosing at least k points of P , .

Highlights

Given a set points in a plane, enclosing problem in computational geometry is concerned with finding the smallest geometrical object of a given type that encloses all the points of P
The objective is to compute a smallest region of given type that encloses at least k points of P . k -enclosing problems using rectangles and squares are studied [6]-[11] are studied extensively
We assume there exists m -sliceable axis-parallel squares and propose an algorithm to locate three axis-parallel unit squares which are pairwise disjoint and they together enclose the maximum number of points from P

Summary

Introduction

Points in a plane, enclosing problem in computational geometry is concerned with finding the smallest geometrical object of a given type that encloses all the points of P. Mahapatra maximize the number of points enclosed by the given region(s) of fixed size and shape This type of problem has similar applications as the problems mentioned above. In the context of bichromatic planar point set, Díaz-Báñez et al [16] proposed algorithms for maximal enclosing by two disjoint ( ) ( ) axis-parallel unit squares and circles in O n2 and O n3 log n time respectively. It is shown that this algorithm can be used to compute a placement of one or more axis-parallel squares enclosing the maximum number of points from P if such a placement exists We use this result to construct an efficient algorithm for finding the smallest axis-parallel square enclosing at least k points of P for large values of k

Maximal Enclosing Problem

Preliminaries

An Efficient Algorithm to Find k-Square for Large Values of k