An iterative merging algorithm for soft rectangle packing and its extension for application of fixed-outline floorplanning of soft modules

We address an important variant of the rectangle packing problem, the soft rectangle packing problem, and explore its problem extension for the fixed-outline floorplanning with soft modules. For the soft rectangle packing problem with zero deadspace, we present an iterative merging packing algorithm that merges all the rectangles into a final composite rectangle in a bottom-up order by iteratively merging two rectangles with the least areas into a composite rectangle, and then shapes and places each pair of sibling rectangles based on the dimensions and position of their composite rectangle in an up-bottom order. We prove that the proposed algorithm can guarantee feasible layout under some conditions, which are weaker as compared with a well-known zero-dead-space packing algorithm. We then provide a deadspace distribution strategy, which can systematically assign deadspace to modules, to extend the iterative merging packing algorithm to deal with soft packing problem with deadspace. For the fixed-outline floorplanning with soft modules problem, we propose an iterative merging packing based hierarchical partitioning algorithm, which adopts a general hierarchical partitioning framework as proposed in the popular PATOMA floorplanner. The framework uses a recursive bipartitioning method to partition the original problem into a set of subproblems, where each subproblem is a soft rectangle packing problem and how to solve the subproblem plays a key role in the final efficiency of the floorplanner. Different from the PATOMA that adopts the zero-dead-space packing algorithm, we adopt our proposed iterative merging packing algorithm for the subproblems. Experiments on the IBM-HB benchmarks show that the proposed packing algorithm is more effective than the zero-dead-space packing algorithm, and experiments on the GSRC benchmarks show that our floorplanning algorithm outperforms three state-of-the-art floorplanners PATOMA, DeFer and UFO, reducing wirelength by 0.2%, 4.0% and 2.3%, respectively.     

Constant Approximation Algorithms for Rectangle Stabbing and Related Problems

In this paper we present constant approximation algorithms for two NP-hard rectangle stabbing problems, called the weighted rectangle stabbing (WRS) problem and the rectangle stabbing with rejecting cost (RSRC) problem. In the WRS problem a set of axis-aligned rectangles is given, with each rectangle associated with a positive weight, and a set of weighted horizontal and/or vertical stabbing lines is sought so that each rectangle is intersected by at least one stabbing line with a weight (called cost) no less than that of the rectangle and the total cost (or weight) of all stabbing lines is minimized. In the RSRC problem each rectangle is associated with an additional positive rejecting cost and is required to be either stabbed by a stabbing line or rejected by paying its rejecting cost. For the WRS problem, we present a polynomial time 2e-approximation algorithm, where e is the natural logarithmic base. Our algorithm is based on a number of interesting techniques such as rounding, randomization, and lower bounding. For the RSRC problem, we give a 3e-approximation algorithm by using a simple but powerful LP rounding technique to identify those to-be-rejected rectangles. Our techniques are quite general and can be easily applied to several related problems, such as the stochastic rectangle stabbing problem and polygon stabbing problem from fixed directions. Algorithms obtained by our techniques are relatively simple and can be easily implemented for practical purpose.     

Maximizing the Total Profit of Rectangles Packed into a Rectangle

We consider the following rectangle packing problem. Given a set of rectangles, each of which is associated with a profit, we are requested to pack a subset of the rectangles into a bigger rectangle so that the total profit of rectangles packed is maximized. The rectangles may not overlap. This problem is strongly NP-hard even for packing squares with identical profits. We first present a simple (3 + ε)-approximation algorithm. Then we consider a restricted version of the problem and show a (2 + ε)-approximation algorithm. This restricted problem includes the case where rotation by 90° is allowed (and is possible), and the case of packing squares. We apply a similar technique to the general problem, and get an improved algorithm with a worst-case ratio of at most 5/2 + ε. Finally, we devise a (2 + ε)-approximation algorithm for the general problem.     

矩形件优化排样问题的混合遗传算法求解

利用遗传算法结合剩余矩形排样法求解矩形件正交排样问题。通过遗传算法将矩形件正交排样问题转化为一个排列问题,并引入剩余矩形排样算法来惟一确定每一个排列所对应的排样图(即排样方案),两者结合用于求解矩形件排样问题。最后用此混合遗传算法对文献[1]中的两个算例进行了验证,表明了其有效性。     

Identifying rectangles in laser range data for urban scene reconstruction

In urban scenes, many of the surfaces are planar and bounded by simple shapes. In a laser scan of such a scene, these simple shapes can still be identified. We present a one-parameter algorithm that can identify point sets on a plane for which a rectangle is a fitting boundary. These rectangles have a guaranteed density: no large part of the rectangle is empty of points. We prove that our algorithm identifies all angles for which a rectangle fits the point set of size n in O(nlogn) time. We evaluate our method experimentally on 13 urban data sets and we compare the rectangles found by our algorithm to the α‐shape as a surface boundary.     

基于改进最低水平线方法与遗传算法的矩形件排样优化算法

传统的最低水平线方法用于矩形件排样时可能产生较多未被利用的空白区域,造 成不必要的材料浪费。针对此缺陷,在搜索过程中引入启发式判断,实现空白区域的填充处理, 提高板材利用率。在应用遗传算法优化矩形件排样顺序时,在进化过程中采用分阶段设置遗传 算子的方法,改善算法的搜索性能与效果。通过改进最低水平线方法与基于分阶段遗传算子的 遗传算法相结合,共同求解矩形件排样问题。排样测试数据表明,所提出的矩形件排样优化算 法能够有效改善排样效果,提高材料利用率。     

基于欧氏距离的矩形Packing问题的确定性启发式求解算法

使用拟人的策略,提出了基于欧氏距离的占角最大穴度优先的放置方法,为矩形Packing问题的快速求解提供了一种高效的启发式算法.算法的高效性通过应用于标准电路MCNC和GSRC得到了验证.     

基于蚁群优化算法求解矩形件排样问题

布局问题来源于生产实际,优秀的布局可以提高原料利用率,降低成本,提高经济效益,对许多行业有重要意义。矩形件优化排样是一类具有NP完全难度的组合优化问题。人工蚁群算法是对蚂蚁群体行为的模拟抽象,该算法具有分布计算、信息正反馈和启发式搜索等特点。本文将蚁群算法和剩余矩形法结合用于解决矩形排样问题,首先用蚁群算法将矩形件排样问题转化为一个排列问题;然后通过剩余矩形排样算法排出每一个排列所对应的排样图;最后用算法对文献[9]中的两个算例进行了验证,表明了其有效性。     

一种基于控制点自动提取的图像畸变校正算法

在机器视觉应用中,摄像机光学系统产生的图像存在不同程度的几何畸变。为了提高图像在做定量分析时的准确性,必须对这一类畸变进行修正。在基于控制点的校正方法基础之上,研究了一种将图像分片,利用模板自动提取控制点亚像素坐标,分片校正的快速算法。实际应用表明,该方法是可靠有效的。     

Fuzzy shell clustering algorithms in image processing: fuzzyC-rectangular and 2-rectangular shells

Objective function-based clustering has been generalized recently to detect contours of circles and ellipses or even hyperbolas in a set of binary data vectors. Although there are special algorithms to discover lines, the detection of rectangles needs further treatment. A simple line-detection algorithm is not sufficient for rectangles since for identifying four lines as one rectangle, additional information such as the length of the lines and whether they are parallel or meet at a right angle is necessary. In this paper, a special fuzzy shell-clustering algorithm for rectangular contours is developed. The principal idea behind it can be generalized for other polygons so we also derive an algorithm that is capable of detecting rectangles and other polygons as well as approximating circles, ellipses, and lines     

基于改进遗传算法的矩形件优化排样

论文利用遗传算法结合剩余矩形排样法求解矩形件正交排样问题。通过对排样问题已知解信息进行统计分析,并根据分析结果改进原遗传算法判断个体好坏的标准,对父代种群进行了优劣分类,针对不同的分类采用不同的遗传操作,构造出一种改进遗传算法。通过实例验证,该算法得到了排样问题的最优解,说明了其有效性。     

A simple method for fitting of bounding rectangle to closed regions

In this paper, we introduce a new approach for fitting of a bounding rectangle to closed regions. In this approach the coordinates of the vertices are computed directly using a closed-form solution. This approach is based on simple coordinate geometry and uses the boundary points of regions. Using a least-square approach we determine the directions of major and minor axes of the object, which gives the orientation of the object. The four vertexes of the bounding rectangle are computed by pair wise solving the four straight lines. Examples from synthetic data and some real-life data show that the approach is both accurate and efficient.     

基于五块模式的单一矩形件排样算法

如何在一个大矩形里排入尽可能多的单一规格小矩形件是广泛出现在制造业领域 的板材分割、物流业领域的集装箱装载中的问题。采用五块模式将大矩形划分为五个块,求解 每个块里面矩形件的排样方式。首先,采用动态规划算法一次性生成所有块中矩形件排样方式, 然后,采用隐式枚举法考虑所有可能的五块组合,选择包含矩形件个数最多的五块组合作为最 终的排样方案。使用算例对算法进行了测试,并与另外4 种单一排样算法进行了比较。实验结 果表明,该算法在排样利用率和切割工艺两方面都有效,而且计算时间合理。     

A hybrid simulated annealing metaheuristic algorithm for the two-dimensional knapsack packing problem

The rectangle knapsack packing problem is to pack a number of rectangles into a larger stock sheet such that the total value of packed rectangles is maximized. The paper first presents a fitness strategy, which is used to determine which rectangle is to be first packed into a given position. Based on this fitness strategy, a constructive heuristic algorithm is developed to generate a solution, i.e. a given sequence of rectangles for packing. Then, a greedy strategy is used to search a better solution. At last, a simulated annealing algorithm is introduced to jump out of the local optimal trap of the greedy strategy, to find a further improved solution. Computational results on 221 rectangular packing instances show that the presented algorithm outperforms some previous algorithms on average.     

结合Canny和距离变换的矩形模式特征提取方法

提出了一种图像矩形模式特征提取算法。首先利用Canny算子对图像纹理敏感特性求取图像的Canny边缘;对Canny边缘结果计算其距离变换图,可有效减小纹理复杂区域对后续矩形模式特征提取的影响;然后对距离变换图进行掩膜卷积,对其结果进行聚类,获得矩形模式特征的中心;最后利用几何约束求解出矩形模式特征的四个顶点。实验证明了该算法有效可行,特征定位精度在真实图像上可达1.3像素。     

基于主成分分析的珠宝自动定位及检测方法

针对不规则物体形状特征难以估计的问题,以实现对珠宝的自动测量技术为前提,通过引入主成分分析的概念,提出一种新的对不规则珠宝图像的自动检测方法。该算法首先利用主分量分析提取出目标珠宝图像的主轴,然后根据优化后的主轴方向计算珠宝外接矩形的四个顶点,最后定位出最优外接矩形的位姿从而完成对不规则珠宝轮廓的检测。将所提算法用于真实珠宝图像,结果表明,算法能够准确定位检测出图像中的目标。与利用重心原理结合最小二乘法的方法和以投影为基础计算能量最大值的算法相比,实验图像的主观效果和客观的误差分析都表明了该算法在准确性和鲁棒性的优势。     

An approximation algorithm for sequential rectangle placement

We present a constant factor, polynomial time approximation algorithm for the problem of scheduling a sequence of rectangles on a matrix. The approximation is on the area covered by the rectangles, and a rectangle is placed on the matrix only if all its preceding rectangles in the sequence were already placed.     

基于直线段提取及其参数化的矩形重构方法研究

文章根据矩形目标边缘二值图的特点,提出一种基于直线段提取及其参数化的矩形目标重构方法,实现矩形目标位姿参数高精度快速求取。该文提出的矩形目标重构方法主要分两步进行:首先从矩形目标边缘图像的二值图中提取出所有直线段,并将直线段参数化;其次由参数化的直线段提取出近似矩形,再由近似矩形重构出精确的目标矩形,并计算其位姿参数。该文提出的算法可应用于机器人装配及目标跟踪中。     

一种快速的有约束矩形件优化排样模型

为了有效地解决有约束的矩形件优化排样问题,提出一种快速的求解算法;通过比较待排样矩形件的不同排样模式,选择最优排样方案。算法完全基于解析计算,虽不能寻找理论最优解,但相比于各种启发式算法大大提高了排样速度。实验结果表明,算法能够在较短的计算时间内获得满意的排样效果,是一种效率较高的有约束矩形件排样算法。     

Optimal algorithms for rectangle problems on a mesh-connected computer

In this paper, Mesh-Connected Computer (MCC) algorithms for computing several properties of a set of, possibly intersecting rectangles are presented. Given a set of n iso-oriented rectangles, we describe MCC algorithms for determining the following properties: (i) the area of the logic “OR” of these rectangles (i.e., the area of the region covered by at least one rectangle); (ii) the area of the union of pairwise “AND” of the rectangles (i.e., the area of the region covered by two or more rectangles); (iii) the largest number of rectangles that overlap (this solves the fixed-size rectangle placement problem, i.e., given a set of planar points and a rectangle, find a placement of the rectangle in the plane so that the number of points covered by the rectangle is maximal); (iv) the minimum separation between any pair of a set of nonoverlapping rectangles. All these algorithms can be implemented on a 2√n × 2√n MCC in O(√n) time which is optimal. The algorithms compare favorably with the known sequential algorithms that have O(n log n) time complexity.