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 scalar problem, the optimal competitive ratio is known to lie in the interval @math @cite_26 @cite_3 @cite_15 @cite_36 , which is substantially worse than what can be done by a simple streaming @math -approximation in space @math . Interestingly, for , the algorithm by Im al @cite_11 with ratio @math actually works in the online setting as well and needs space @math only during its execution (if the solution itself is not stored), which makes it possible to implement it in the streaming setting. This online ratio cannot be improved as there is a lower bound of @math @cite_11 @cite_16 , whereas in the streaming setting we can achieve a @math -approximation with a reasonable memory (or even @math for a fixed @math ). If all jobs have sufficiently small size, we improve the analysis in @cite_11 and show that the online algorithm achieves @math -approximation; see .
@cite_26 @cite_36 @cite_3 @cite_15 @cite_16 @cite_11
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We study the online stochastic bipartite matching problem, in a form motivated by display ad allocation on the Internet. In the online, but adversarial case, the celebrated result of Karp, Vazirani and Vazirani gives an approximation ratio of @math , a very familiar bound that holds for many online problems; further, the bound is tight in this case. In the online, stochastic case when nodes are drawn repeatedly from a known distribution, the greedy algorithm matches this approximation ratio, but still, no algorithm is known that beats the @math bound.Our main result is a @math -approximation online algorithm for stochastic bipartite matching, breaking this @math barrier. Furthermore, we show that no online algorithm can produce a @math approximation for an arbitrarily small @math for this problem. Our algorithms are based on computing an optimal offline solution to the expected instance, and using this solution as a guideline in the process of online allocation. We employ a novel application of the idea of the power of two choices from load balancing: we compute two disjoint solutions to the expected instance, and use both of them in the online algorithm in a prescribed preference order. To identify these two disjoint solutions, we solve a max flow problem in a boosted flow graph, and then carefully decompose this maximum flow to two edge-disjoint (near-) matchings. In addition to guiding the online decision making, these two offline solutions are used to characterize an upper bound for the optimum in any scenario. This is done by identifying a cut whose value we can bound under the arrival distribution. At the end, we discuss extensions of our results to more general bipartite allocations that are important in a display ad application. We first consider online speed scaling algorithms to minimize the energy used subject to the constraint that every job finishes by its deadline. We assume that the power required to run at speed s is P(s) = s sup spl alpha . We provide a tight spl alpha sup spl alpha bound on the competitive ratio of the previously proposed optimal available algorithm. This improves the best known competitive ratio by a factor of 2 sup spl alpha . We then introduce an online algorithm, and show that this algorithm's competitive ratio is at most 2( spl alpha ( spl alpha - 1)) sup spl alpha e sup spl alpha . This competitive ratio is significantly better and is approximately 2e sup spl alpha +1 for large spl alpha . Our result is essentially tight for large spl alpha . In particular, as spl alpha approaches infinity, we show that any algorithm must have competitive ratio e sup spl alpha (up to lower order terms). We then turn to the problem of dynamic speed scaling to minimize the maximum temperature that the device ever reaches, again subject to the constraint that all jobs finish by their deadlines. We assume that the device cools according to Fourier's law. We show how to solve this problem in polynomial time, within any error bound, using the ellipsoid 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. We exhibit a poly(log k)-competitive randomized algorithm for the k-server problem on any metric space. The best previous result independent of the geometry of the underlying metric space is the 2k–1 competitive ratio established for the deterministic work function algorithm by Koutsoupias and Papadimitriou (1995). Even for the special case when the underlying metric space is the real line, the best known competitive ratio was k. Since deterministic algorithms can do no better than k on any metric space with at least k+1 points, this establishes that for every metric space on which the problem is non-trivial, randomized algorithms give an exponential improvement over deterministic algorithms. Our algorithm maintains an approximation of the underlying metric space by a distribution over HSTs. The granularity and accuracy of the approximation is adjusted dynamically according to the aggregate behavior of the HST algorithms. In short: We try to obtain more accurate approximations at the locations and scales where the gactionh is happening. Thus a crucial component of our approach is the O((log k)^2)-competitive randomized algorithm for HSTs obtained in our previous work with Bubeck, Cohen, Lee, and Ma.dry, and its "multiscale information theory" perspective.