The optimal representation of disjoint iso-oriented rectangles in two-dimensional trees

Partitions of the plane are often used to design an efficient data structure for configurations in the plane. In this publication we will use recursive iso-oriented partitions in order to design a search tree data structure for a set S of mutually disjoint iso-oriented rectangles. We will consider two optimization criteria for the choice of the partitions: minimization of the size of the tree and minization of the search time. We will present polynomial time algorithms to find recursive iso-oriented partitions for both optimization criteria separately. We will show that for some arbitrarily large sets S there is a trade-off between the size of the tree and the height of the tree. However, we will prove that for each set S of rectangles logarithmic search time can be obtained with a tree of almost linear size. Such a tree can be found in time almost quadratic in the size of S.     

An effective recursive partitioning approach for the packing of identical rectangles in a rectangle

In this work, we deal with the problem of packing (orthogonally and without overlapping) identical rectangles in a rectangle. This problem appears in different logistics settings, such as the loading of boxes on pallets, the arrangements of pallets in trucks and the stowing of cargo in ships. We present a recursive partitioning approach combining improved versions of a recursive five-block heuristic and an L-approach for packing rectangles into larger rectangles and L-shaped pieces. The combined approach is able to rapidly find the optimal solutions of all instances of the pallet loading problem sets Cover I and II (more than 50?000 instances). It is also effective for solving the instances of problem set Cover III (almost 100?000 instances) and practical examples of a woodpulp stowage problem, if compared to other methods from the literature. Some theoretical results are also discussed and, based on them, efficient computer implementations are introduced. The computer implementation and the data sets are available for benchmarking purposes.     

Multimagic rectangles based on large sets of orthogonal arrays

Large sets of orthogonal arrays (LOAs) have been used to construct resilient functions and zigzag functions by Stinson. In this paper, an application of LOAs in constructing multimagic rectangles is given. Further, some recursive constructions for multimagic rectangles are presented, and some infinite families of bimagic rectangles are obtained.     

Nonisomorphic complete sets of F-rectangles with prime power dimensions

A technique for constructing nonisomorphic complete sets of frequency rectangles with prime power dimensions is described. This procedure is used to establish a conservative general lower bound for the number of possible nonisomorphic complete sets of frequency rectangles of prime power order. Several cases are considered in detail which improve the lower bound for those orders. The technique can also be applied to the construction of inequivalent orthogonal arrays of strength 2.     

A block-based layer building approach for the 2D guillotine strip packing problem

We examine the 2D strip packing problems with guillotine-cut constraint, where the objective is to pack all rectangles into a strip with fixed width and minimize the total height of the strip. We combine three most successful ideas for the orthogonal rectangular packing problems into a single coherent algorithm: (1) packing a block of rectangles instead of a single rectangle in each step; (2) dividing the strip into layers and pack layer by layer; and (3) unrolling and repacking the top portion of the solutions where usually wasted space occurs. Computational experiments on benchmark test sets suggest that our approach rivals existing approaches.     

Mixed-integer programming techniques for decomposing IMRT fluence maps using rectangular apertures

We consider a matrix decomposition problem arising in Intensity Modulated Radiation Therapy (IMRT). The problem input is a matrix of intensity values that are to be delivered to a patient via IMRT from some given angle, under the condition that the IMRT device can only deliver radiation in rectangular shapes. This paper studies the problem of minimizing the number of rectangles (and their associated intensities) necessary to decompose such a matrix. We propose an integer programming-based methodology for providing lower and upper bounds on the optimal solution, and demonstrate the efficacy of our approach on clinical data.     

Computation of Spatial Covariance Matrices

In an earlier article, Ghosh derived the density for the distance between two points uniformly and independently distributed in a rectangle. This article extends that work to include the case where the two points lie in two different rectangles in a lattice. This density allows one to find the expected value of certain functions of this distance between rectangles analytically or by one-dimensional numerical integration.

In the case of isotropic spatial models or spatial models with geometric anisotropy terms for agricultural experiments one can use these theoretical results to compute the covariance between the yields in different rectangular plots. As the numerical integration is one-dimensional these results are computed quickly and accurately. The types of covariance functions used come from the Matérn and power families of processes. Analytic results are derived for the de Wijs process, a member of both families and for the power models also.

Software in R is available. Examples of the code are given for fitting spatial models to the Fairfield Smith data. Other methods for the estimation of the covariance matrices are discussed and their pros and cons are outlined.     

A matching approach for replenishing rectangular stock sizes

Consider a replenishment problem in which several different rectangular sizes of material are stocked. Customers order rectangles of the material, but the rectangles ordered have a range on specified width as well as on specified length. To satisfy the customer requirements, the stock material can be cut once longitudinally in order to satisfy two customer requirements or not cut at all, that is, an entire stock piece of material is used to satisfy one customer requirement. If an exact match is impossible in the current planning period, the unused material must be returned to stock— an expensive and undesirable situation. In this paper, a nonbipartite weighted matching problem formulation will be given for determining the replenishment requirements of rectangular stock sizes. Then, a hybrid solution approach, capable of solving real applications (typically up to 3000 nodes) efficiently, will be discussed. This solution was implemented in September 1998 and has operated successfully since then.     

An algorithm for the two-dimensional assortment problem

In this paper we consider the two-dimensional assortment problem. This is the problem of choosing from a set of stock rectangles a subset which can be used for cutting into a number of smaller rectangular pieces. Constraints are imposed upon the number of such pieces which result from the cutting.A heuristic algorithm for the guillotine cutting version of the problem is developed based on a greedy procedure for generating two-dimensional cutting patterns, a linear program for choosing the cutting patterns to use and an interchange procedure to decide the best subset of stock rectangles to cut.Computational results are presented for a number of test problems which indicate that the algorithm developed produces good quality results both for assortment problems and for two-dimensional cutting problems.     

Polychromatic colorings of rectangular partitions

A rectangular partition is a partition of a plane rectangle into an arbitrary number of non-overlapping rectangles such that no four rectangles share a corner. In this note, it is proven that every rectangular partition admits a vertex coloring with four colors such that every rectangle, except possibly the outer rectangle, has all four colors on its boundary. This settles a conjecture of Dinitz et al. [Y. Dinitz, M.J. Katz, R. Krakovski, Guarding rectangular partitions, in: Abstracts 23rd Euro. Workshop Comput. Geom., 2007, pp. 30-33]. The proof is short, simple and based on 4-edge-colorability of a specific class of planar graphs.     

On rectangular cartograms

A rectangular cartogram is a type of map where every region is a rectangle. The size of the rectangles is chosen such that their areas represent a geographic variable (e.g., population). Good rectangular cartograms are hard to generate: The area specifications for each rectangle may make it impossible to realize correct adjacencies between the regions and so hamper the intuitive understanding of the map.

We present the first algorithms for rectangular cartogram construction. Our algorithms depend on a precise formalization of region adjacencies and build upon existing VLSI layout algorithms. Furthermore, we characterize a non-trivial class of rectangular subdivisions for which exact cartograms can be computed efficiently. An implementation of our algorithms and various tests show that in practice, visually pleasing rectangular cartograms with small cartographic error can be generated effectively.     

货物尺寸相同的2维装箱问题的等价类(英文)

在生产与储运领域,把小长方体货物（盒子）装入大长方体箱子是一项重要的工作.本文涉及的问题是:把相同尺寸（a×b×c）的盒子装到一个箱子X×Y×Z中,使所装入箱子的盒子数量为最大.由于某些条件的限止,有时要求货物只能按一个重力方向进行装箱,从而使装箱问题变为把尺寸相同的2维盒子（a×b）填装到一个2维箱子X×Y中.本文讨论当盒子尺寸（a×b包括 b×a）给定,箱子尺寸充分大时,在本文所给的等价意义下,共有多少种互不等价的箱子X×Y.     

Optimising the cutting of wood fibre plates in the hardboard industry

Hardboard companies transform eucalyptus trunks into rectangular wood fibre plates basically by means of processes of disintegration and reconstitution of wood fibres. Such plates called hardboards are then cut into ordered items (smaller rectangles) to satisfy customer demands. In this paper, we present approaches to generate cutting patterns that minimize the cost or waste of material, considering different particular constraints associated with longitudinal and transversal saws, head cuts, book rotation and item unloading stations of the cutting machine. The methods are based on dynamic programming recursive formulas combined with greedy constructive heuristics and the primal simplex algorithm. To illustrate the application of these approaches, a case study was carried out in a Brazilian hardboard company. The results show that the approaches are able to produce better solutions than the ones currently used by the company.     

A novel non-linear approach to minimal area rectangular packing

This paper discusses the minimal area rectangular packing problem which is to pack a given set of rectangles into a rectangular container of minimal area such that no two rectangles overlap. Current approaches for this problem rely on metaheuristics like simulated annealing, on constraint programming or on non-linear models. Difficulties arise from the non-convexity and the combinatorial complexity. We investigate different mathematical programming approaches for this and introduce a novel approach based on non-linear optimization and the “tunneling effect” achieved by a relaxation of the non-overlapping constraints. We compare our optimization algorithm to a simulated annealing and a constraint programming approach and show that our approach is competitive. Additionally, since it is easy to extend, it is also applicable to a variety of related problems.     

A best-first branch and bound algorithm for unconstrained two-dimensional cutting problems

This paper is concerned with the problem of unconstrained two-dimensional cutting of small rectangular pieces, each of which has its own profit and size, from a large rectangular plate so as to maximize the profit-sum of the pieces produced. Hifi and Zissimopoulos's recursive algorithm using G and Kang's upper bound is presently the most efficient exact algorithm for the problem. We propose a best-first branch and bound algorithm based upon the bottom-up approach that is more efficient than their recursive algorithm. The proposed algorithm uses efficient upper bound and branching strategies that can reduce the number of nodes that must be searched significantly. We demonstrate the efficiency of the proposed algorithm through computational experiments.     

Recursive formulas for multinomial probabilities with applications

Recursive formulas are provided for computing probabilities of a multinomial distribution. Firstly, a recursive formula is provided for computing rectangular probabilities which include the cumulative distribution function as a special case. These rectangular probabilities can be used to provide goodness-of-fit tests for the cell probabilities. The probability that a certain cell count is the maximum of all cell counts is also considered, which can be used to assess the probability that the maximum cell count corresponds to the cell with the maximum probability. Finally, a recursive formula is provided for computing the probability that the cell counts satisfy a certain ordering, which can be used to assess the probability that the ordering of the cell counts corresponds to the ordering of the cell probabilities. The computational intensity of these recursive formulas is linear in the number of cells, and they provide opportunities for calculating probabilities that would otherwise be computationally challenging. Some examples of the applications of these formulas are provided.     

Regular polygons in taxicab geometry

A polygon of n sides will be called regular in taxicab geometry if it has n equal angles and n sides of equal taxicab length. This paper will show that there are no regular taxicab triangles and no regular taxicab pentagons. The sets of taxicab rectangles and taxicab squares will be shown to be the same, respectively, as the sets of Euclidean rectangles and Euclidean squares. A method of construction for a regular taxicab 2n-gon for any n will be demonstrated.     

Skew chain orders and sets of rectangles

A “skew chain order” is one which can be partitioned into saturated chains all starting at rank 0. A two-part Sperner-type theorem is derived for the direct product of two skew chain orders. This theorem is used to solve an extremal problem about certain sets of integer rectangles.     

Tiling simply connected regions with rectangles

In 1995, Beauquier, Nivat, Rémila, and Robson showed that tiling of general regions with two bars is NP-complete, except for a few trivial special cases. In a different direction, in 2005, Rémila showed that for simply connected regions by two rectangles, the tileability can be solved in quadratic time (in the area). We prove that there is a finite set of at most 10⁶ rectangles for which the tileability problem of simply connected regions is NP-complete, closing the gap between positive and negative results in the field. We also prove that counting such rectangular tilings is #P-complete, a first result of this kind.     

The concentration of a wave field at the interface of elastic media

The steady symmetrical oscillations of a transversely non-uniform elastic rectangular region, consisting of three bonded uniformisotropic rectangles, are considered, where the elastic characteristics of the inner rectangle are assumed to be different from those of the external rectangles. Using methods developed previously [1,2], the dependence of the order of the singularity of the stress field at the interface on the combinations of the constants of elasticity of the joined media and on their wave impedances is investigated.