Worst-Case Optimal Covering of Rectangles by Disks

SIAM Journal on Computing, 2011

Let P be a simple polygon, and let Q be a set of points in P . We present an almost-linear time algorithm for computing a minimum cover of Q by disks that are contained in P . We generalize the algorithm above, so that it can compute a minimum cover of Q by homothets of a fixed compact convex set of constant description complexity O that are contained in P . This improves previous results of Katz and Morgenstern . We also consider the disk-cover problem when Q is contained in a (not too wide) annulus, and present an O(|Q| log |Q|) algorithm for this case.

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

We flnd minimal enclosures by rectangles for two and three regions of given areas. We show that each minimizer has connected regions and has shape depending on ratio of areas.

SIAM Journal on Computing, 1988

We solve the problem of decomposing a rectangle R into p rectangles of equal area so that the maximum rectangle perimeter is as small as possible. This work has applications in areas such as flexible object packing and data allocation. Our solution requires only a constant number of arithmetic operations and integer square roots to characterize the decomposition, and linear time to print the decomposition. The discrete analogue of the problem in which the rectangle R is replaced by a rectangular array of lattice points is also considered, and three heuristic methods of solution are given. All of the heuristic methods operate by finding a discrete approximation to our optimal decomposition of R, but with different tradeoffs between the accuracy of the approximation and running time.

Discrete Applied Mathematics, 1987

This paper solves the problem of subdividing a unit square into p rectangles of area 1/p in such a way that the maximal perimeter of a rectangle is as small as possible. The correctness of the solution is proved using the well-known theorems of Menger and Dilworth. Square decomposition In this work we consider the following geometric decomposition problem. Square decomposition. Given a unit square D and a positive integer p, subdivide D into p rectangles of area 1/p (all having edges parallel to those of D) in such a way that the maximum of their perimeters is minimized.

2019

Given a set D of n unit disks in the plane and an integer k ≤ n, the maximum area connected subset problem asks for a set D′ ⊆ D of size k that maximizes the area of the union of disks, under the constraint that this union is connected. This problem is motivated by wireless router deployment and is a special case of maximizing a submodular function under a connectivity constraint. We prove that the problem is NP-hard and analyze a greedy algorithm, proving that it is a 2 approximation. We then give a polynomial-time approximation scheme (PTAS) for this problem with resource augmentation, i.e., allowing an additional set of εk disks that are not drawn from the input. Additionally, for two special cases of the problem we design a PTAS without resource augmentation. 2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms

Computational Geometry, 2011

We consider the Minimum Vertex Cover problem in intersection graphs of axisparallel rectangles on the plane. We present two algorithms: The first is an EPTAS for non-crossing rectangle families, rectangle families R where R 1 \ R 2 is connected for every pair of rectangles R 1 , R 2 ∈ R. This algorithm extends to intersection graphs of pseudodisks. The second algorithm achieves a factor of (1.5 + ε) in general rectangle families, for any fixed ε > 0, and works also for the weighted variant of the problem. Both algorithms exploit the plane properties of axis-parallel rectangles in a non-trivial way.

Computational Geometry, 2008

We study the following problem: Given a set of red points and a set of blue points on the plane, find two unit disks C R and C B with disjoint interiors such that the number of red points covered by C R plus the number of blue points covered by C B is maximized. We give an algorithm to solve this problem in O(n 8/3 log 2 n) time, where n denotes the total number of points. We also show that the analogous problem of finding two axis-aligned unit squares S R and S B instead of unit disks can be solved in O(n log n) time, which is optimal. If we do not restrict ourselves to axis-aligned squares, but require that both squares have a common orientation, we give a solution using O(n 3 log n) time.

Discrete & Computational Geometry, 2014

Given a convex disk K (a convex compact planar set with nonempty interior), let δ L (K) and θ L (K) denote the lattice packing density and the lattice covering density of K, respectively. We prove that for every centrally-symmetric convex disk K we have that 1 ≤ δ L (K)θ L (K) ≤ 1.17225. .. The left inequality is tight and it improves a 10-year old result. Keywords Arrangements of convex disks • Packing density • Covering density 1 Introduction In this paper, we consider arrangements of convex disks in the Euclidean plane. A convex disk is a compact convex set with nonempty interior; its area will be denoted by A(K). An arrangement of congruent copies (translates) of a convex disk K is a family A of convex disks, each of which is congruent to (is a translate of) K. The arrangement is a packing if its members' interiors are mutually disjoint, and it is a covering if the union of its members is the whole plane.

AIP Conference Proceedings, 2019

We consider the problem of covering a square with exactly 6 identical circles of minimal radius. In the literature, a covering is presented by Melissen and Schuur, and conjectured to be optimal. We adress the problem proposing a mathematical programming formulation and solving it to global optimality. We prove that the conjectured optimal covering is indeed the global optimum.

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.

International Journal of Computational Geometry & Applications, 2011

Given a set S of n points in the plane, the disjoint two-rectangle covering problem is to find a pair of disjoint rectangles such that their union contains S and the area of the larger rectangle is minimized. In this paper we consider two variants of this optimization problem: (1) the rectangles are allowed to be reoriented freely while restricting them to be parallel to each other, and (2) one rectangle is restricted to be axis-parallel but the other rectangle is allowed to be reoriented freely. For both of the problems, we present O(n2 log n)-time algorithms using O(n) space.

We present filling as a new type of spatial subdivision problem that is related to covering and packing. Filling addresses the optimal placement of overlapping objects lying entirely inside an arbitrary shape so as to cover the most interior volume. In n-dimensional space, if the objects are polydisperse n-balls, we show that solutions correspond to sets of maximal n-balls and the solution space can reduced to the medial axis of a shape. We examine the structure of the solution space in two dimensions. For the filling of polygons, we provide detailed descriptions of a heuristic and a genetic algorithm for finding solutions of maximal discs. We also consider the properties of ideal distributions of N discs in polygons as N approaches infinity.

Theoretical Computer Science, 2014

We give an improved approximation algorithm for the unique unit-disk coverage problem: Given a set of points and a set of unit disks, both in the plane, we wish to find a subset of disks that maximizes the number of points contained in exactly one disk in the subset. Erlebach and van Leeuwen (2008) introduced this problem as the geometric version of the unique coverage problem, and gave a polynomial-time 18-approximation algorithm. In this paper, we improve this approximation ratio 18 to 2 + 4/ √ 3 + ε (< 4.3095 + ε) for any fixed constant ε > 0. Our algorithm runs in polynomial time which depends exponentially on 1/ε. The algorithm can be generalized to the budgeted unique unit-disk coverage problem in which each point has a profit, each disk has a cost, and we wish to maximize the total profit of the uniquely covered points under the condition that the total cost is at most a given bound.