Online Bin Stretching with Three Bins

Online Bin Stretching is a semi-online variant of bin packing in which the algorithm has to use the same number of bins as an optimal packing, but is allowed to slightly overpack the bins. The goal is to minimize the amount of overpacking, i.e., the maximum size packed into any bin. We give an algorithm for Online Bin Stretching with a stretching factor of $11/8 = 1.375$ for three bins. Additionally, we present a lower bound of $45/33 = 1.\overline{36}$ for Online Bin Stretching on three bins and a lower bound of $19/14$ for four and five bins that were discovered using a computer search.Comment: Preprint of a journal version. See version 2 for the conference paper. Conference paper split into two journal submissions; see arXiv:1601.0811

Discovering and Certifying Lower Bounds for the Online Bin Stretching Problem

There are several problems in the theory of online computation where tight lower bounds on the competitive ratio are unknown and expected to be difficult to describe in a short form. A good example is the Online Bin Stretching problem, in which the task is to pack the incoming items online into bins while minimizing the load of the largest bin. Additionally, the optimal load of the entire instance is known in advance. The contribution of this paper is twofold. First, we provide the first non-trivial lower bounds for Online Bin Stretching with 6, 7 and 8 bins, and increase the best known lower bound for 3 bins. We describe in detail the algorithmic improvements which were necessary for the discovery of the new lower bounds, which are several orders of magnitude more complex. The lower bounds are presented in the form of directed acyclic graphs. Second, we use the Coq proof assistant to formalize the Online Bin Stretching problem and certify these large lower bound graphs. The script we propose certified as well all the previously claimed lower bounds, which until now were never formally proven. To the best of our knowledge, this is the first use of a formal verification toolkit to certify a lower bound for an online problem

Swellfish Privacy: Exploiting Time-Dependent Relevance for Continuous Differential Privacy : Technical Report

Today, continuous publishing of differentially private query results is the de-facto standard. The challenge hereby is adding enough noise to satisfy a given privacy level, and adding as little noise as necessary to keep high data utility. In this context, we observe that privacy goals of individuals vary significantly over time. For instance, one might aim to hide whether one is on vacation only during school holidays. This observation, named time-dependent relevance, implies two effects which – properly exploited – allow to tune data utility. The effects are time-variant sensitivity (TEAS) and time-variant number of affected query results (TINAR). As today’s DP frameworks, by design, cannot exploit these effects, we propose Swellfish privacy. There, with policy collections, individuals can specify combinations of time-dependent privacy goals. Then, query results are Swellfish-private, if the streams are indistinguishable with respect to such a collection.We propose two tools for designing Swellfish-private mechanisms, namely, temporal sensitivity and a composition theorem, each allowing to exploit one of the effects. In a realistic case study, we show empirically that exploiting both effects improves data utility by one to three orders of magnitude compared to state-of-the-art w-event DP mechanisms. Finally, we generalize the case study by showing how to estimate the strength of the effects for arbitrary use cases

Comparing Predictions of Object Movements

Estimating the future location of moving objects using different estimation models, such as linear or probabilistic models, has been investigated extensively. However, the location estimations of those models are generally not comparable. For instance, one model might return a position for some object, another one a Gaussian probability distribution, and a third one a uniform distribution. Similar issues arise for query answers. In this paper, we examine the question how estimations of different models can be compared. To do so, we propose a general model based on the central limit theorem. This allows handling different PDF-based approaches as well as models from the other groups (i.e., linear estimations) in a unified manner. Furthermore, we show how to inject privacy into the general model, a fundamental pre-requisite for user acceptance. Thus, we support well-known approaches like k-anonymity and spatial obfuscation. Based on our general model, we conduct a comprehensive experimental study considering a real-world road network; comparing models form different groups for the first time. Our results, for instance, reveal that estimation models based on individual velocity profiles are not necessarily better than models, which estimate the future location of objects only based on their direction. In more abstract terms, our general model allows comparison of estimation models that could not be compared before and gives way to build models that solve the privacy-accuracy challenge

Accurate Cardinality Estimation of Co-occurring Words Using Suffix Trees (Extended Version)

Estimating the cost of a query plan is one of the hardest problems in query optimization. This includes cardinality estimates of string search patterns, of multi-word strings like phrases or text snippets in particular. At first sight, suffix trees address this problem. To curb the memory usage of a suffix tree, one often prunes the tree to a certain depth. But this pruning method "takes away" more information from long strings than from short ones. This problem is particularly severe with sets of long strings, the setting studied here. In this article, we propose respective pruning techniques. Our approaches remove characters with low information value. The various variants determine a character\u27s information value in different ways, e.g., by using conditional entropy with respect to previous characters in the string. Our experiments show that, in contrast to the well-known pruned suffix tree, our technique provides significantly better estimations when the tree size is reduced by 60% or less. Due to the redundancy of natural language, our pruning techniques yield hardly any error for tree-size reductions of up to 50%