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Covering a Square by Equal Circles

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... These models, however, are not entirely new in principle, because we have already used similar mechanical models for finding (locally) optimal circle packings and coverings in other domains (e.g. polygons, 2-sphere) [35,36,37,38]. These models are different a little from those used in [2,7]. ...
... We note, if k > 3 bars are connected to a joint of the second kind, then this joint is considered as a multiple joint that should be split into k -2 joints such that, multiplying the connecting bars appropriately, 3 bars will be connected to each of the joints obtained by splitting. The idea of multiplying (splitting) was introduced independently by [2] and [37]. Multiplying the connecting bars is appropriate if either Journal of Computational Geometry jocg.org ...
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How must n equal circles of given radius be placed so that they cover as great a part of the area of the unit circle as possible? To analyse this mathematical problem, mechanical models are introduced. A generalized tensegrity structure is associated with a maximum area configuration of the n circles, whose equilibrium configuration is determined numerically with the method of dynamic relaxation, and the stability of equilibrium is investigated by means of the stiffness matrix of the tensegrity structure. In this Part I, the principles of the models are presented, while an application will be shown in the forthcoming Part II.
... Authors of [12] propose the approach to cover a square by single circles, based on the theory of temperature expansions and compressions of pivotal structures. The approach, proposed in [13], uses the Dirichlet-Voronoi scheme. ...
... This problem has several applications, such as finding the optimal broadcast tower distribution over a region or the optimal allocation of Wi-Fi routers in a building, for example. Some specific cases have been attacked and solved in the last few decades non-algorithmically, but using results from linear algebra and discrete mathematics instead; for example, T. Tarnai and Z. Gáspár in 1995 presented minimal coverings of a square with up to ten circles (in [30]). Their results were generalized for rectangles by A. Heppes and H. Melissen in 1997 (see [37]), when they found the best coverings of a rectangle with up to five equal circles and also, under some conditions, with seven circles as well. ...
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In this work, we present an Augmented Lagrangian algorithm for nonlinear semidefinite problems (NLSDPs), which is a natural extension of its consolidated counterpart in nonlinear programming. This method works with two levels of constraints; one that is penalized and other that is kept within the subproblems. This is done to allow exploiting the subproblem structure while solving it. The global convergence theory is based on recent results regarding approximate Karush–Kuhn–Tucker optimality conditions for NLSDPs, which are stronger than the usually employed Fritz John optimality conditions. Additionally, we approach the problem of covering a given object with a fixed number of balls with a minimum radius, where we employ some convex algebraic geometry tools, such as Stengle’s Positivstellensatz and its variations, which allows for a much more general model. Preliminary numerical experiments are presented.
... Two classical problems of discrete geometry have been addresses as (Tarnai and Gaspar 1995). First, packing of 'n' equal non-overlapping circles in the unit square such that diameter is as large as possible. ...
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Event critical applications demand blanket coverage. On the other hand, nodes closer to the base station are exploited as they have to spend additional energy in relaying data of far away nodes. This brings in the idea of implementing blanket coverage in heterogeneous wireless sensor networks. I-DEEC improvises distributed energy efficient clustering (DEEC) by deploying network nodes in two layers. Layer 1 strategically tessellate hexagons to deploy nodes as normal or super nodes based on distance from the base station, considering the high data requirement within hop distance around the base station. Layer 2 randomly deploys advanced nodes with condition that no two advanced nodes sense the same area. Further, it uses the sum of the ratio of node’s distance to the base station along with residual energy ratio to calculate the possibility of a node to be selected as a cluster head, followed by the selection of the optimal percentage high possibility nodes as cluster heads. I-DEEC provisions blanket coverage by extending the stability period by reducing the ratio between initial energy of different types of nodes. I-DEEC revamps DEEC protocol in terms of network lifetime, percentage area coverage, throughput, and residual energy.
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The acquisition of point cloud data from a space frame using terrestrial laser scanning is usually affected by many occluding components and site conditions and therefore needs to achieve optimal priori planning, which is handled as the planning for scanning (P4S) problem. This paper describes a three-dimensional model-based P4S approach for space frame structures, where a space modeling solution is employed to simulate the scanning target and environment. The P4S problem modeling is used to define the visibility analysis and constraints. Lastly, a two-phase optimization is proposed to solve the P4S problem and compared with a weighted greedy algorithm. Experiments were conducted on a full-scale space frame to validate the proposed approach.
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Mobile Crowdsensing (MCS), an important component of the Internet of Things (IoT), is a paradigm which utilizes people carrying smart devices, referred to as “workers”, to perform various sensing tasks. A type of such tasks is localization, where the location of a certain target or event is to be found. The recruitment of the right set of workers to perform a localization task plays a paramount role in the outcome quality in terms of localization time, energy, cost, and accuracy. The stability of workers in MCS, which is defined as their spatio-temporal availability, makes the problem of localization more complex, since such tasks are continuous. In this work, a novel Stable Data-based Recruitment System (SDRS) for localization tasks is proposed, which-a) integrates a new data-based recruitment parameter that dynamically exploits data readings to guide the recruitment system into selecting informative workers, while considering their mobility; b) presents a stable coverage assessment method that considers range-free sensors and the mobility of workers; and c) integrates a two-phase recruitment approach that is optimized using greedy and genetic methods. The testing and evaluation of the proposed approach is conducted using datasets of MCS workers and compared with existing benchmarks. The results demonstrate that the proposed approach efficiently and reliably leads to a speedy localization, with high outcome quality.
Chapter
The paper is devoted to the circle covering problem with unequal circles. The number of circles is given. Also, we know a function, which determines a relation between the radii of two neighboring circles. The circle covering problem is usually studied in the case when the distance between points is Euclidean. We assume that the distance is determined by means of some special metric, which, generally speaking, is not Euclidean. The special numerical algorithm is suggested and implemented. It based on optical-geometric approach, which is developed by the authors in recent years and previously used only for circles of the equal radius. The results of a computational experiment are presented and discussed.
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A numerical method for investigating k-coverings of a convex bounded closed set with nonempty interior with circles of two given radii is proposed. An algorithm for finding an approximate number of such circles and the arrangement of their centers is described. For certain specific cases, approximate lower bounds of the density of the k-covering of the given domain are found. Cases with constraints on the distances between the covering circle centers and problems with a variable (given) covering multiplicity are also considered. Numerical results demonstrating the effectiveness of the proposed methods are presented.
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We propose a method for determining the number of sensors, their arrangement, and approximate lower bounds for the number of sensors for the multiple covering of an arbitrary closed bounded convex area in a plane. The problem of multiple covering is considered with restrictions on the minimal possible distances between the sensors and without such restrictions. To solve these problems, some 0–1 linear programming (LP) problems are constructed.We use a heuristic solution algorithm for 0–1 LP problems of higher dimensions. The results of numerical implementation are given and for some particular cases it is obtained that the number of sensors found can not be decreased.
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