Linear probing-based hash tables offer high data locality and are considered among the fastest in real-world applications. However, they come with an inherent tradeoff between space efficiency and speed, i.e. when the hash table approaches full capacity, its performance tends to decline considerably due to an effect known as primary clustering. As a result they are only used at low load factors. Tombstones (markers for deleted elements) can help mitigate the effect of primary clustering in linear probing hash tables. However, tombstones require periodic redistribution, which, in turn, requires a complete halt of regular operations. This makes linear probing not suitable in practical applications where periodic halts are unacceptable. In this paper, we present a solution to forestall primary clustering in linear probing hash tables, ensuring high data locality and consistent performance even at high load factors. Our approach redistributes tombstones within small windows, deamortizing the cost of mitigating primary clustering and eliminating the need for periodic halts. We provide theoretical guarantees that our deamortization method is asymptotically optimal in efficiency and cost. We also design an efficient implementation within dominant linear-probing hash tables and show performance improvements. We introduce Zombie hashing in two variants: ordered (compact) and unordered (vectorized) linear probing hash tables. Both variants achieve consistent, high throughput and lowest variance in operation latency compared to other state-of-the-art hash tables across numerous churn cycles, while maintaining 95% space efficiency without downtime. Our results show that Zombie hashing overcomes the limitations of linear probing while preserving high data locality.

Ranking is at the core of many artificial intelligence (AI) applications, including search engines, recommender systems, etc. Modern ranking systems are often constructed with learning-to-rank (LTR) models built from user behavior signals. While previous studies have demonstrated the effectiveness of using user behavior signals (e.g., clicks) as both features and labels of LTR algorithms, we argue that existing LTR algorithms that indiscriminately treat behavior and non-behavior signals in input features could lead to suboptimal performance in practice. Because user behavior signals often have strong correlations with the ranking objective and can only be collected on items that have already been shown to users, directly using behavior signals in LTR could create an exploitation bias that hurts the system performance in the long run.

Time series discords are a useful primitive for time series anomaly detection, and the matrix profile is capable of capturing discord effectively. There exist many research efforts to improve the scalability of discord discovery with respect to the length of time series. However, there is surprisingly little work focused on reducing the time complexity of matrix profile computation associated with dimensionality of a multidimensional time series. In this work, we propose a sketch for discord mining among multi-dimensional time series. After an initial pre-processing of the sketch as fast as reading the data, the discord mining has runtime independent of the dimensionality of the original data. On several real world examples from water treatment and transportation, the proposed algorithm improves the throughput by at least an order of magnitude (50X) and only has minimal impact on the quality of the approximated solution. Additionally, the proposed method can handle the dynamic addition or deletion of dimensions inconsequential overhead. This allows a data analyst to consider “what-if” scenarios in real time while exploring the data.

A Content-based Time Series Retrieval (CTSR) system is an information retrieval system for users to interact with time series emerged from multiple domains, such as finance, healthcare, and manufacturing. For example, users seeking to learn more about the source of a time series can submit the time series as a query to the CTSR system and retrieve a list of relevant time series with associated metadata. By analyzing the retrieved metadata, users can gather more information about the source of the time series. Because the CTSR system is required to work with time series data from diverse domains, it needs a high-capacity model to effectively measure the similarity between different time series. On top of that, the model within the CTSR system has to compute the similarity scores in an efficient manner as the users interact with the system in real-time. In this paper, we propose an effective and efficient CTSR model that outperforms alternative models, while still providing reasonable inference runtimes. To demonstrate the capability of the proposed method in solving business problems, we compare it against alternative models using our in-house transaction data. Our findings reveal that the proposed model is the most suitable solution compared to others for our transaction data problem.

Oblivious RAMs (ORAMs) are proposed to completely hide access patterns. However, most ORAM constructions are expensive and not suitable to deploy in a database for supporting query processing over large data. In this work, we design a practical oblivious query processing framework to enable efficient query processing over a cloud database. In particular, we focus on processing multiple range and kNN queries asynchronously and concurrently with high throughput. The key idea is to integrate indices into ORAM which leverages a suite of optimization techniques (e.g., oblivious batch processing and caching). Our construction shows an order of magnitude speedup in comparison with other baselines over large datasets.

Increasingly, individuals and companies adopt a cloud service provider as a primary data and IT infrastructure platform. The remote access of the data inevitably brings the issue of trust. Data encryption is necessary to keep sensitive information secure and private on the cloud. Yet adversaries can still learn valuable information regarding encrypted data by observing data access patterns. To solve such problem, Oblivious RAMs (ORAMs) are proposed to completely hide access patterns. However, most ORAM constructions are expensive and not suitable to deploy in a database for supporting query processing over large data. Furthermore, an ORAM processes queriessynchronously, hence, does not provide high throughput forconcurrent query processing. In this article, we design a practicaloblivious query processing frameworkto enable efficient query processing over a cloud database. In particular, we focus on processing multiple range and$k$NN queriesasynchronously and concurrently with high throughput. The key idea is to integrate indices into ORAM which leverages a suite of optimization techniques (e.g., oblivious batch processing and caching). The effectiveness and efficiency of our oblivious query processing framework is demonstrated through extensive evaluations over large datasets. Our construction shows an order of magnitude speedup in comparison with other baselines.

Embeddings are one of the fundamental building blocks for data analysis tasks. Embeddings are already essential tools for large language models and image analysis, and their use is being extended to many other research domains. The generation of these distributed representations is often a data-and computation-expensive process; yet the holistic analysis and adjustment of them after they have been created is still a developing area. In this paper, we first propose a very general quantitatively measure for the presence of features in the embedding data based on if it can be learned. We then devise a method to remove or alleviate undesired features in the embedding while retaining the essential structure of the data. We use a Domain Adversarial Network (DAN) to generate a non-affine transformation, but we add constraints to ensure the essential structure of the embedding is preserved. Our empirical results demonstrate that the proposed algorithm significantly outperforms the state-of-art unsupervised algorithm on several data sets, including novel applications from the industry.

Word vector embeddings have been shown to contain and amplify biases in data they are extracted from. Consequently, many techniques have been proposed to identify, mitigate, and attenuate these biases in word representations. In this tutorial, we will review a collection of state-of-the-art debiasing techniques. To aid this, we provide an open source web-based visualization tool and offer hands-on experience in exploring the effects of these debiasing techniques on the geometry of high-dimensional word vectors. To help understand how various debiasing techniques change the underlying geometry, we decompose each technique into interpretable sequences of primitive operations, and study their effect on the word vectors using dimensionality reduction and interactive visual exploration.

Thanks to the wide adoption of GPS-equipped devices, the volume of collected spatial data is exploding. To achieve interactive exploration and analysis over big spatial data, people are willing to trade off accuracy for performance through approximation. As a foundation in many approximate algorithms, data sampling now requires more flexibility and better performance. In this paper, we study the spatial independent range sampling (SIRS) problem aiming at retrieving random samples with independence over points residing in a query region. Specifically, we have designed concise index structures with careful data layout based on various space decomposition strategies. Moreover, we propose novel algorithms for both uniform and weighted SIRS queries with low theoretical cost and complexity as well as excellent practical performance. Last but not least, we demonstrate how to support data updates and trade-offs between different sampling methods in practice. According to comprehensive evaluations conducted on real-world datasets, our methods achieve orders of magnitude performance improvement against baselines derived by existing works.

In the era of big data, more and more applications require the information of historical data to support rich analytics, learning, and mining operations. In these cases, it is highly desirable to retrieve information of previous versions of data. Traditionally, multi-version databases can be used to store all historical values of the data in order to support historical queries. However, storing all the historical data can be impractical due to its large space consumption. In this paper, we propose the concept of at-the-time persistent (ATTP) and back-in-time persistent (BITP) sketches, which are sketches that approximately answer queries on previous versions of data with small space. We then provide several implementations of ATTP/BITP sketches which are shown to be more efficient compared to existing state-of-the-art solutions in our empirical studies.

We develop a family of techniques to align word embeddings which are derived from different source datasets or created using different mechanisms (e.g., GloVe or word2vec). Our methods are simple and have a closed form to optimally rotate, translate, and scale to minimize root mean squared errors or maximize the average cosine similarity between two embeddings of the same vocabulary into the same dimensional space. Our methods extend approaches known as Absolute Orientation, which are popular for aligning objects in three-dimensions, and generalize an approach by Smith et al. (ICLR 2017). We prove new results for optimal scaling and for maximizing cosine similarity. Then we demonstrate how to evaluate the similarity of embeddings from different sources or mechanisms, and that certain properties like synonyms and analogies are preserved across the embeddings and can be enhanced by simply aligning and averaging ensembles of embeddings.

Abstract The available spatial data are rapidly growing and also diversifying. One may obtain in large quantities information such as annotated point/place of interest (POIs), check-in comments on those POIs, geo-tagged microblog comments, and demarked regions of interest (ROI). All sources interplay with each other, and together build a more complete picture of the spatial and social dynamics at play in a region. However, building a single fused representation of these data entries has been mainly rudimentary, such as allowing spatial joins. In this paper, we extend the concept of semantic embedding for POIs (points of interests) and devise the first semantic embedding of ROIs, and in particular ones that captures both its spatial and its semantic components. To accomplish this, we develop a multipart network model capturing the relationships between the diverse components, and through random-walk-based approaches, use this to embed the ROIs. We demonstrate the effectiveness of this embedding at simultaneously capturing both the spatial and semantic relationships between ROIs through extensive experiments. Applications like popularity region prediction demonstrate the benefit of using ROI embedding as features in comparison with baselines.

We define several new models for how to define anomalous regions among enormous sets of trajectories. These are based on spatial scan statistics, and identify a geometric region which captures a subset of trajectories which are significantly different in a measured characteristic from the background population. The model definition depends on how much a geometric region is contributed to by some overlapping trajectory. This contribution can be the full trajectory, proportional to the length within the spatial region, or dependent on the flux across the boundary of that spatial region. Our methods are based on and significantly extend a recent two-level sampling approach which provides high accuracy at enormous scales of data. We support these new models and algorithms with extensive experiments on millions of trajectories and also theoretical guarantees.

Kulldorff's (1997) seminal paper on spatial scan statistics (SSS) has led to many methods considering different regions of interest, different statistical models, and different approximations while also having numerous applications in epidemiology, environmental monitoring, and homeland security. SSS provides a way to rigorously test for the existence of an anomaly and provide statistical guarantees as to how "anomalous" that anomaly is. However, these methods rely on defining specific regions where the spatial information a point contributes is limited to binary 0 or 1, of either inside or outside the region, while in reality anomalies will tend to follow smooth distributions with decaying density further from an epicenter. In this work, we propose a method that addresses this shortcoming through a continuous scan statistic that generalizes SSS by allowing the point contribution to be defined by a kernel. We provide extensive experimental and theoretical results that shows our methods can be computed efficiently while providing high statistical power for detecting anomalous regions.

We develop a new class of distances for trajectories, based on the distance to a set of landmarks. These distances easily and interpretably map objects to a Euclidean space, are simple to compute, and perform well in data analysis tasks. For trajectories, they match and in some cases significantly out-perform all state-of-the-art other metrics, can effortlessly be used in k-means clustering, and directly plugged into approximate nearest neighbor approaches which immediately out-perform the best recent advances in trajectory similarity search by several orders of magnitude. These distances do not require complicated alignment (common in trajectory case). We show reasonable and often simple conditions under which these distances are metrics.

With the popularity of portable wireless devices it is important to model and predict how information or contagions spread by natural human mobility - for understanding the spreading of deadly infectious diseases and for improving delay tolerant communication schemes. Formally, we model this problem by considering M moving agents, where each agent initially carries a distinct bit of information. When two agents are at the same location or in close proximity to one another, they share all their information with each other. We would like to know the time it takes until all bits of information reach all agents, called the flood time, and how it depends on the way agents move, the size and shape of the network and the number of agents moving in the network.

Kernel regression is an essential and ubiquitous tool for non-parametric data analysis, particularly popular among time series and spatial data. However, the central operation which is performed many times, evaluating a kernel on the data set, takes linear time. This is impractical for modern large data sets.

Mobile and sensing devices have already become ubiquitous. They have made tracking moving objects an easy task. As a result, mobile applications like Uber and many IoT projects have generated massive amounts of trajectory data that can no longer be processed by a single machine efficiently. Among the typical query operations over trajectories, similarity search is a common yet expensive operator in querying trajectory data. It is useful for applications in different domains such as traffic and transportation optimizations, weather forecast and modeling, and sports analytics. It is also a fundamental operator for many important mining operations such as clustering and classification of trajectories. In this paper, we propose a distributed query framework to process trajectory similarity search over a large set of trajectories. We have implemented the proposed framework in Spark, a popular distributed data processing engine, by carefully considering different design choices. Our query framework supports both the Hausdorff distance the Fréchet distance. Extensive experiments have demonstrated the excellent scalability and query efficiency achieved by our design, compared to other methods and design alternatives.

Finding anomalous regions within spatial data sets is a central task for biosurveillance, homeland security, policy making, and many other important areas. These communities have mainly settled on spatial scan statistics as a rigorous way to discover regions where a measured quantity (e.g., crime) is statistically significant in its difference from a baseline population. However, most common approaches are inefficient and thus, can only be run with very modest data sizes (a few thousand data points) or make assumptions on the geographic distributions of the data.

This paper describes Sparse Frequent Directions, a variant of Frequent Directions for sketching sparse matrices. It resembles the original algorithm in many ways: both receive the rows of an input matrix An x d one by one in the streaming setting and compute a small sketch B ∈ Rl x d. Both share the same strong (provably optimal) asymptotic guarantees with respect to the space-accuracy tradeoff in the streaming setting. However, unlike Frequent Directions which runs in O(ndl) time regardless of the sparsity of the input matrix A, Sparse Frequent Directions runs in Õ(nnz(A)l + nl2) time. Our analysis loosens the dependence on computing the Singular Value Decomposition (SVD) as a black box within the Frequent Directions algorithm. Our bounds require recent results on the properties of fast approximate SVD computations. Finally, we empirically demonstrate that these asymptotic improvements are practical and significant on real and synthetic data.

Matrices have become essential data representations for many large-scale problems in data analytics, and hence matrix sketching is a critical task. Although much research has focused on improving the error/size tradeoff under various sketching paradigms, the many forms of error bounds make these approaches hard to compare in theory and in practice. This paper attempts to categorize and compare the most known methods under row-wise streaming updates with provable guarantees, and then to tweak some of these methods to gain practical improvements while retaining guarantees. For instance, we observe that a simple heuristic iSVD, with no guarantees, tends to outperform all known approaches in terms of size/error trade-off. We modify the best performing method with guarantees, FREQUENTDIRECTIONS, under the size/error trade-off to match the performance of iSVD and retain its guarantees. We also demonstrate some adversarial datasets where iSVD performs quite poorly. In comparing techniques in the time/error trade-off, techniques based on hashing or sampling tend to perform better. In this setting, we modify the most studied sampling regime to retain error guarantee but obtain dramatic improvements in the time/error trade-off. Finally, we provide easy replication of our studies on APT, a new testbed which makes available not only code and datasets, but also a computing platform with fixed environmental settings.

Presents the introductory editorial for this issue of the publication.

Kernel density estimates are a robust way to reconstruct a continuous distribution from a discrete point set. Typically their effectiveness is measured either in L1 or L2 error. In this paper we investigate the challenges in using L∞ (or worst case) error, a stronger measure than L1 or L2. We present efficient solutions to two linked challenges: how to evaluate the L∞ error between two kernel density estimates and how to choose the bandwidth parameter for a kernel density estimate built on a subsample of a large data set.

Tracking and approximating data matrices in streaming fashion is a fundamental challenge. The problem requires more care and attention when data comes from multiple distributed sites, each receiving a stream of data. This paper considers the problem of "tracking approximations to a matrix" in the distributed streaming model. In this model, there are m distributed sites each observing a distinct stream of data (where each element is a row of a distributed matrix) and has a communication channel with a coordinator, and the goal is to track an ε-approximation to the norm of the matrix along any direction. To that end, we present novel algorithms to address the matrix approximation problem. Our algorithms maintain a smaller matrix B , as an approximation to a distributed streaming matrix A , such that for any unit vector x : ||| Ax || 2 − || Bx || 2 | ≤ ε|| A || 2 F . Our algorithms work in streaming fashion and incur small communication, which is critical for distributed computation. Our best method is deterministic and uses only O (( m /ε) log(β N )) communication, where N is the size of stream (at the time of the query) and β is an upperbound on the squared norm of any row of the matrix. In addition to proving all algorithmic properties theoretically, extensive experiments with real large datasets demonstrate the efficiency of these protocols.

Nearest-neighbor (NN) search, which returns the nearest neighbor of a query point in a set of points, is an important and widely studied problem in many fields, and it has wide range of applications. In many of them, such as sensor databases, location-based services, face recognition, and mobile data, the location of data is imprecise. We therefore study nearest neighbor queries in a probabilistic framework in which the location of each input point is specified as a probability distribution function. We present efficient algorithms for (i) computing all points that are nearest neighbors of a query point with nonzero probability; (ii) estimating, within a specified additive error, the probability of a point being the nearest neighbor of a query point; (iii) using it to return the point that maximizes the probability being the nearest neighbor, or all the points with probabilities greater than some threshold to be the NN. We also present some experimental results to demonstrate the effectiveness of our approach.

Kernel density estimates are important for a broad variety of applications. Their construction has been well-studied, but existing techniques are expensive on massive datasets and/or only provide heuristic approximations without theoretical guarantees. We propose randomized and deterministic algorithms with quality guarantees which are orders of magnitude more efficient than previous algorithms. Our algorithms do not require knowledge of the kernel or its bandwidth parameter and are easily parallelizable. We demonstrate how to implement our ideas in a centralized setting and in MapReduce, although our algorithms are applicable to any large-scale data processing framework. Extensive experiments on large real datasets demonstrate the quality, efficiency, and scalability of our techniques.

We study the mergeability of data summaries. Informally speaking, mergeability requires that, given two summaries on two datasets, there is a way to merge the two summaries into a single summary on the two datasets combined together, while preserving the error and size guarantees. This property means that the summaries can be merged in a way akin to other algebraic operators such as sum and max, which is especially useful for computing summaries on massive distributed data. Several data summaries are trivially mergeable by construction, most notably all the sketches that are linear functions of the datasets. But some other fundamental ones, like those for heavy hitters and quantiles, are not (known to be) mergeable. In this article, we demonstrate that these summaries are indeed mergeable or can be made mergeable after appropriate modifications. Specifically, we show that for ε-approximate heavy hitters, there is a deterministic mergeable summary of size O (1/ε); for ε-approximate quantiles, there is a deterministic summary of size O ((1/ε) log(ε n )) that has a restricted form of mergeability, and a randomized one of size O ((1/ε) log 3/2 (1/ε)) with full mergeability. We also extend our results to geometric summaries such as ε-approximations which permit approximate multidimensional range counting queries. While most of the results in this article are theoretical in nature, some of the algorithms are actually very simple and even perform better than the previously best known algorithms, which we demonstrate through experiments in a simulated sensor network. We also achieve two results of independent interest: (1) we provide the best known randomized streaming bound for ε-approximate quantiles that depends only on ε, of size O ((1/ε) log 3/2 (1/ε)), and (2) we demonstrate that the MG and the SpaceSaving summaries for heavy hitters are isomorphic.

In distributed data management, a primary concern is monitoring the distributed data and generating an alarm when a user specified constraint is violated. A particular useful instance is the threshold based constraint, which is commonly known as the distributed threshold monitoring problem [4], [16], [19], [29]. This work extends this useful and fundamental study to distributed probabilistic data that emerge in a lot of applications, where uncertainty naturally exists when massive amounts of data are produced at multiple sources in distributed, networked locations. Examples include distributed observing stations, large sensor fields, geographically separate scientific institutes/units and many more. When dealing with probabilistic data, there are two thresholds involved, the score and the probability thresholds. One must monitor both simultaneously, as such, techniques developed for deterministic data are no longer directly applicable. This work presents a comprehensive study to this problem. Our algorithms have significantly outperformed the baseline method in terms of both the communication cost (number of messages and bytes) and the running time, as shown by an extensive experimental evaluation using several, real large datasets.

We study the mergeability of data summaries. Informally speaking, mergeability requires that, given two summaries on two data sets, there is a way to merge the two summaries into a single summary on the union of the two data sets, while preserving the error and size guarantees. This property means that the summaries can be merged in a way like other algebraic operators such as sum and max, which is especially useful for computing summaries on massive distributed data. Several data summaries are trivially mergeable by construction, most notably all the sketches that are linear functions of the data sets. But some other fundamental ones like those for heavy hitters and quantiles, are not (known to be) mergeable. In this paper, we demonstrate that these summaries are indeed mergeable or can be made mergeable after appropriate modifications. Specifically, we show that for ε-approximate heavy hitters, there is a deterministic mergeable summary of size O(1/ε) for ε-approximate quantiles, there is a deterministic summary of size O(1 over ε log(εn))that has a restricted form of mergeability, and a randomized one of size O(1 over ε log 3/21 over ε) with full mergeability. We also extend our results to geometric summaries such as ε-approximations and εkernels.

Ranking temporal data has not been studied until recently, even though ranking is an important operator (being promoted as a first-class citizen) in database systems. However, only the instant top- k queries on temporal data were studied in, where objects with the k highest scores at a query time instance t are to be retrieved. The instant top- k definition clearly comes with limitations (sensitive to outliers, difficult to choose a meaningful query time t ). A more flexible and general ranking operation is to rank objects based on the aggregation of their scores in a query interval, which we dub the aggregate top- k query on temporal data. For example, return the top-10 weather stations having the highest average temperature from 10/01/2010 to 10/07/2010; find the top-20 stocks having the largest total transaction volumes from 02/05/2011 to 02/07/2011. This work presents a comprehensive study to this problem by designing both exact and approximate methods (with approximation quality guarantees). We also provide theoretical analysis on the construction cost, the index size, the update and the query costs of each approach. Extensive experiments on large real datasets clearly demonstrate the efficiency, the effectiveness, and the scalability of our methods compared to the baseline methods.

Given a set of points with uncertain locations, we consider the problem of computing the probability of each point lying on the skyline, that is, the probability that it is not dominated by any other input point. If each point's uncertainty is described as a probability distribution over a discrete set of locations, we improve the best known exact solution. We also suggest why we believe our solution might be optimal. Next, we describe simple, near-linear time approximation algorithms for computing the probability of each point lying on the skyline. In addition, some of our methods can be adapted to construct data structures that can efficiently determine the probability of a query point lying on the skyline.

This paper proposes a new distance metric between clusterings that incorporates information about the spatial distribution of points and clusters. Our approach builds on the idea of a Hilbert space-based representation of clusters as a combination of the representations of their constituent points. We use this representation and the underlying metric to design a spatially-aware consensus clustering procedure. This consensus procedure is implemented via a novel reduction to Euclidean clustering, and is both simple and efficient. All of our results apply to both soft and hard clusterings. We accompany these algorithms with a detailed experimental evaluation that demonstrates the efficiency and quality of our techniques.

In this paper, we propose a unified algorithmic framework for solving many known variants of MDS. Our algorithm is a simple iterative scheme with guaranteed convergence, and is modular; by changing the internals of a single subroutine in the algorithm, we can switch cost functions and target spaces easily. In addition to the formal guarantees of convergence, our algorithms are accurate; in most cases, they converge to better quality solutions than existing methods in comparable time. Moreover, they have a small memory footprint and scale effectively for large data sets. We expect that this framework will be useful for a number of MDS variants that have not yet been studied.

In this paper, we present a measure associated with detection and inference of statistically anomalous clusters of a graph based on the likelihood test of observed and expected edges in a subgraph.This measure is adapted from spatial scan statistics for point sets and provides quantitative assessment for clusters.We discuss some important properties of this statistic and its relation to modularity and Bregman divergences.We apply a simple clustering algorithm to find clusters with large values of this measure in a variety of real-world data sets, and we illustrate its ability to identify statistically significant clusters of selected granularity.

Spatial scan statistics are used to determine hotspots in spatial data, and are widely used in epidemiology and biosurveillance. In recent years, there has been much effort invested in designing efficient algorithms for finding such "high discrepancy" regions, with methods ranging from fast heuristics for special cases, to general grid-based methods, and to efficient approximation algorithms with provable guarantees on performance and quality.In this paper, we make a number of contributions to the computational study of spatial scan statistics. First, we describe a simple exact algorithm for finding the largest discrepancy region in a domain. Second, we propose a new approximation algorithm for a large class of discrepancy functions (including the Kulldorff scan statistic) that improves the approximation versus run time trade-off of prior methods. Third, we extend our simple exact and our approximation algorithms to data sets which lie naturally on a grid or are accumulated onto a grid. Fourth, we conduct a detailed experimental comparison of these methods with a number of known methods, demonstrating that our approximation algorithm has far superior performance in practice to prior methods, and exhibits a good performance-accuracy trade-off.All extant methods (including those in this paper) are suitable for data sets that are modestly sized; if data sets are of the order of millions of data points, none of these methods scale well. For such massive data settings, it is natural to examine whether small-space streaming algorithms might yield accurate answers. Here, we provide some negative results, showing that any streaming algorithms that even provide approximately optimal answers to the discrepancy maximization problem must use space linear in the input.