Rectangle covers revisited computationally

Information and Control, 1984

Decomposing a polygon into simple shapes is a basic problem in computational geometry, with applications in pattern recognition and integrated circuit manufacture. Here we examine the special case of covering a rectilinear polygon (or polyomino) with the minimum number of rectangles, with overlapping allowed. The problem is NP-hard. However, we give here an O(v z) algorithm for constructing a minimum rectangle cover, when the polygon is vertically convex. (Here v is the number of vertices.) The problem is first reduced to a 1-dimensional interval "basis" problem. In showing our algorithm produces an optimal cover we give a new proof of a minimum basis-maximum independent set duality theorem first proved by E.

Algorithmica, 1997

1997

Abstract The rectilinear polygon cover problem is one in which a certain class of features of a rectilinear polygon of n vertices has to be covered with the minimum number of rectangles included in the polygon. In particular, one can consider covering the entire interior, the boundary and the set of corners of the polygon. These problems have important applications in, for example, storing images and in the manufacture of integrated circuits. In this paper we consider covering the corners of the polygons, also known as the corner-cover problem.

Lecture Notes in Computer Science, 2009

In wireless communication networks, optimal use of the directional antenna is very important. The directional antenna coverage (DAC) problem is to cover all clients with the smallest number of directional antennas. In this paper, we consider the variable-size rectangle covering (VSRC) problem, which is a transformation of the DAC problem. There are n points above the base line y = 0 of the two-dimensional plane. The target is to cover all these points by minimum number of rectangles, such that the dimension of each rectangle is not fixed but the area is at most 1, and the bottom edge of each rectangle is on the base line y = 0. In some applications, the number of rectangles covering any position in the two-dimensional plane is bounded, so we also consider the variation when each position in the plane is covered by no more than two rectangles. We give two polynomial time algorithms for finding the optimal covering. Further, we propose two 2-approximation algorithms that use less running time (O(n log n) and O(n)).

Proceedings of the sixteenth annual ACM symposium on Theory of computing - STOC '84, 1984

We provide an algorithm which solves the following problem: given a polygon with edges parallel to the x and y axes, which is convex in the y direction, find a minimum size collection of rectangles, which cover the polygon and are contained within it. The algorithm is quadratic in the number of vertices of the polygon. Our method also yields a new proof of a recent duality theorem equating minimum size rectangle covers to maximum size sets of independent points in the polygon.

2020

We provide the solution for a fundamental problem of geometric optimization by giving a complete characterization of worst-case optimal disk coverings of rectangles: For any $\lambda\geq 1$, the critical covering area $A^*(\lambda)$ is the minimum value for which any set of disks with total area at least $A^*(\lambda)$ can cover a rectangle of dimensions $\lambda\times 1$. We show that there is a threshold value $\lambda_2 = \sqrt{\sqrt{7}/2 - 1/4} \approx 1.035797\ldots$, such that for $\lambda<\lambda_2$ the critical covering area $A^*(\lambda)$ is $A^*(\lambda)=3\pi\left(\frac{\lambda^2}{16} +\frac{5}{32} + \frac{9}{256\lambda^2}\right)$, and for $\lambda\geq \lambda_2$, the critical area is $A^*(\lambda)=\pi(\lambda^2+2)/4$; these values are tight. For the special case $\lambda=1$, i.e., for covering a unit square, the critical covering area is $\frac{195\pi}{256}\approx 2.39301\ldots$. The proof uses a careful combination of manual and automatic analysis, demonstrating the p...

International Conference on Computing: Theory and Applications, 2007

We consider the problem of nding two parallel rectangles, in arbitrary orientation, covering a given set of n points in a plane, such that the area of the larger rectangle is minimized. We give a simple algorithm that solves the problem in O(n3) time using O(n2) space. Without altering the complexity, the algorithm can be modied to solve another optimization

Proceedings of the …, 2001

Computational Geometry, 1996

A Tk guard G in a rectilinear polygon P is a tree of diameter k completely contained in P. The guard G is said to cover a point x if x is visible or rectangularly visible from some point c o n tained in G. W e i n vestigate the function rn h k, which i s t h e largest number of Tk guards necessary to cover any rectilinear polygon with h holes and n vertices. The aim of this paper is to prove n e w l o wer and upper bounds on parts of this function. In particular, we s h o w the following bounds: 1. rn 0 k n k+4 , with equality for even k 2. rn h 1 = 3n+4h+4 16 3. rn h 2 n 6. These bounds, along with other lower bounds that we establish, suggest that the presence of holes reduces the number of guards required, if k 1. In the course of proving the upper bounds, new results on partitioning are obtained.

Discrete Applied Mathematics, 2007

Given a rectangle R with area and a set of n positive reals A = {a 1 , a 2 ,. .. , a n } with a i ∈A a i = , we consider the problem of dissecting R into n rectangles r i with area a i (i = 1, 2,. .. , n) so that the set R of resulting rectangles minimizes an objective function such as the sum of the perimeters of the rectangles in R, the maximum perimeter of the rectangles in R, and the maximum aspect ratio of the rectangles in R, where we call the problems with these objective functions PERI-SUM, PERI-MAX and ASPECT-RATIO, respectively. We propose an O(n log n) time algorithm that finds a dissection R of R that is a 1.25-approximate solution to PERI-SUM, a 2 √ 3-approximate solution to PERI-MAX, and has an aspect ratio at most max{ (R), 3, 1 + max i=1,...,n−1 a i+1 a i }, where (R) denotes the aspect ratio of R.