二维方形装箱的并行算法

Xiaofan Zhao, Hong Shen
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摘要

重点研究二维方形布局问题的并行化问题。在方形装箱问题中,需要将一列方形物品打包到最小数量的单位方形箱子中。所有正方形物品的边长都小于或等于1,这也是每个单位正方形箱子的边长。已装入一个箱子的物品的总面积不能超过1。利用调和的思想,可以在不超过边长1限制的情况下,将一些正方形放入同一个容器中。我们试图通过并行的计算处理系统将所有相应的方格同时打包到一个箱子中。给出了一种时间复杂度为n的9=4最坏情况渐近误差界算法。设OPT(I)和A(I)分别表示最优解的代价和由近似算法产生的代价,用于方形布局问题的实例I。Han等人[23]利用复杂度加权函数证明了迄今为止在线方形填充的最佳上界为2.1439。然而,我们的并行算法的上界比Han的算法差一些,我们的算法分析更简单,时间复杂度也得到了改善。Han的算法需要O(nlogn)时间,而我们的方法只需要(n)时间。
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A Parallel Algorithm for 2D Square Packing
We focus on the parallelization of two-dimensional square packing problem. In square packing problem, a list of square items need to be packed into a minimum number of unit square bins. All square items have side length smaller than or equal to 1 which is also the side length of each unit square bin. The total area of items that has been packed into one bin cannot exceed 1. Using the idea of harmonic, some squares can be put into the same bin without exceeding the bin limitation of side length 1. We try to concurrently pack all the corresponding squares into one bin by a parallel systerm of computation processing. A 9=4-worst case asymptotic error bound algorithm with time complexity (n) is showed. Let OPT(I) and A(I) denote, respectively, the cost of an optimal solution and the cost produced by an approximation algorithmA for an instance Iof the square packing problem. The best upper bound of on-line square packing to date is 2.1439 proved by Han et al. [23] by using complexity weighting functions. However the upper bound of our parallel algorithm is a litter worse than Han's algorithm, the analysis of our algorithm is more simple and the time complexity is improved. Han's algorithm needs O(nlogn) time, while our method only needs (n) time.
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