No abstract

We study the problem of assigning non-overlapping geometric objects centered at a given set of points such that the sum of area covered by them is maximized. If the points are placed on a straight-line and the objects are disks, then the problem is solvable in polynomial time. However, we show that the problem is NP-hard even for simplest objects like disks or squares in R 2 . Eppstein [CCCG, pages 260-265, 2016] proposed a polynomial time algorithm for maximizing the sum of radii (or perimeter) of non-overlapping balls or disks when the points are arbitrarily placed on a plane. We show that Eppstein's algorithm for maximizing sum of perimeter of the disks in R 2 gives a 2-approximation solution for the sum of area maximization problem. We propose a PTAS for our problem. These approximation results are extendible to higher dimensions. All these approximation results hold for the area maximization problem by regular convex polygons with even number of edges centered at the given points.

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.

We study several variations of line segment covering problem with axis-parallel unit squares in IR 2 . A set S of n line segments is given. The objective is to find the minimum number of axis-parallel unit squares which cover at least one end-point of each segment. The variations depend on the orientation and length of the input segments. We prove some of these problems to be NP-complete, and give constant factor approximation algorithms for those problems. For some variations, we have polynomial time exact algorithms. For the general version of the problem, where the segments are of arbitrary length and orientation, and the squares are given as input, we propose a factor 16 approximation result based on multilevel linear programming relaxation technique, which may be useful for solving some other problems. Further, we show that our problems have connections with the problems studied by Arkin et al.[2] on conflict-free covering problem. Our NP-completeness results hold for more simplified types of objects than those of Arkin et al. [2].

We study several variations of line segment covering problem with axis-parallel unit squares in IR 2 . A set S of n line segments is given. The objective is to find the minimum number of axis-parallel unit squares which cover at least one end-point of each segment. The variations depend on the orientation and length of the input segments. We prove some of these problems to be NP-complete, and give constant factor approximation algorithms for those problems. For some variations, we have polynomial time exact algorithms. For the general version of the problem, where the segments are of arbitrary length and orientation, and the squares are given as input, we propose a factor 16 approximation result based on multilevel linear programming relaxation technique, which may be useful for solving some other problems. Further, we show that our problems have connections with the problems studied by Arkin et al.[2] on conflict-free covering problem. Our NP-completeness results hold for more simplified types of objects than those of Arkin et al. [2].