Problems involving the efficient arrangement of simple objects, as captured by bin packing and makespan scheduling, are fundamental tasks in combinatorial optimization. These are well understood in the traditional online and offline cases, but have been less well-studied when the volume of the input is truly massive, and cannot even be read into memory. This is captured by the streaming model of computation, where the aim is to approximate the cost of the solution in one pass over the data, using small space. As a result, streaming algorithms produce concise input summaries that approximately preserve the optimum value. We design the first efficient streaming algorithms for these fundamental problems in combinatorial optimization. For Bin Packing, we provide a streaming asymptotic @math -approximation with @math memory, where @math hides logarithmic factors. Moreover, such a space bound is essentially optimal. Our algorithm implies a streaming @math -approximation for Vector Bin Packing in @math dimensions, running in space @math . For the related Vector Scheduling problem, we show how to construct an input summary in space @math that preserves the optimum value up to a factor of @math , where @math is the number of identical machines.
For , the best possible competitive ratio is substantially worse than what we can achieve offline or even in the streaming setting. Balogh al @cite_7 designed an asymptotically @math -competitive algorithm, while the current lower bound on the asymptotic competitive ratio is @math @cite_1 . This (relatively complicated) online algorithm is based on the algorithm @cite_20 , which for some integer @math classifies items into size groups @math . It packs each group separately by , keeping just one bin open, which is closed whenever the next item does not fit. Thus can run in memory of size @math , unlike most other online algorithms which require maintaining the levels of all bins opened so far. Its competitive ratio tends to approximately @math as @math goes to infinity. Surprisingly, this is also the best possible ratio if only a bounded number of bins is allowed to be open for an online algorithm @cite_20 , which can be seen as the intersection of online and streaming algorithms.
@cite_20 @cite_1 @cite_7
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We revisit the classic online bin packing problem studied in the half-century. In this problem, items of positive sizes no larger than 1 are presented one by one to be packed into subsets called bins of total sizes no larger than 1, such that every item is assigned to a bin before the next item is presented. We use online partitioning of items into classes based on sizes, as in previous work, but we also apply a new method where items of one class can be packed into more than two types of bins, where a bin type is defined according to the number of such items grouped together. Additionally, we allow the smallest class of items to be packed in multiple kinds of bins, and not only into their own bins. We combine this with the approach of packing of sufficiently big items according to their exact sizes. Finally, we simplify the analysis of such algorithms, allowing the analysis to be based on the most standard weight functions. This simplified analysis allows us to study the algorithm which we defined based on all these ideas. This leads us to the design and analysis of the first algorithm of asymptotic competitive ratio strictly below 1.58, specifically, we break this barrier by providing an algorithm AH (Advanced Harmonic) whose asymptotic competitive ratio does not exceed 1.57829. Our main contribution is the introduction of the simple analysis based on weight function to analyze the state of the art online algorithms for the classic online bin packing problem. The previously used analytic tool named weight system was too complicated for the community in this area to adjust it for other problems and other algorithmic tools that are needed in order to improve the current best algorithms. We show that the weight system based analysis is not needed for the analysis of the current algorithms for the classic online bin packing problem. The importance of a simple analysis is demonstrated by analyzing several new features together with all existing techniques, and by proving a better competitive ratio than the previously best one. In this paper we consider the Online Bin Packing Problem in three variants: Circles in Squares, Circles in Isosceles Right Triangles, and Spheres in Cubes. The two first ones receive an online sequence of circles (items) of different radii while the third one receive an online sequence of spheres (items) of different radii, and they want to pack the items into the minimum number of unit squares, isosceles right triangles of leg length one, and unit cubes, respectively. For Online Circle Packing in Squares, we improve the previous best-known competitive ratio for the bounded space version, when at most a constant number of bins can be open at any given time, from 2.439 to 2.3536. For Online Circle Packing in Isosceles Right Triangles and Online Sphere Packing in Cubes we show bounded space algorithms of asymptotic competitive ratios 2.5490 and 3.5316, respectively, as well as lower bounds of 2.1193 and 2.7707 on the competitive ratio of any online bounded space algorithm for these two problems. We also considered the online unbounded space variant of these three problems which admits a small reorganization of the items inside the bin after their packing, and we present algorithms of competitive ratios 2.3105, 2.5094, and 3.5146 for Circles in Squares, Circles in Isosceles Right Triangles, and Spheres in Cubes, respectively. We study randomized algorithms for the online vector bin packing and vector scheduling problems. For vector bin packing, we achieve a competitive ratio of O(d1 B), where d is the number of dimensions and B the size of a bin. This improves the previous bound of O(d1 (B−1)) by a polynomial factor, and is tight up to logarithmic factors. For vector scheduling, we show a lower bound of [EQUATION] on the competitive ratio of randomized algorithms, which is the first result for randomized algorithms and is asymptotically tight. Finally, we analyze the widely used "power of two choices' algorithm for vector scheduling, and show that its competitive ratio is [EQUATION], which is optimal up to the additive O(log log n) term that also appears in the scalar version of this algorithm. We consider the problem of searching on m current rays for a target of unknown location. If no upper bound on the distance to the target is known in advance, then the optimal competitive ratio is 1+2mm (m−1)m−1. We show that even if an upper bound of D on the distance to the target is known in advance, then the competitive ratio of any search strategy is at least 1+2mm (m−1)m−1−O(1 log2D). This is again optimal – but in a stricter sense. In particular, this result implies the same lower bound for a robot searching for a target on infinite rays and finding it at a distance of D. To show that our lower bound is, indeed, optimal we construct a search strategy that achieves this ratio. Our strategy does not need to know an upper bound on the distance to the target in advance; it achieves a competitive ratio of 1+2mm (m−1)m−1−O(1 log2D) if the target is found at distance D. Finally, we also present a linear time algorithm to compute the strategy that allows the robot to search the farthest for a given competitive ratio C.