Counting Queries Research Articles

Space efficient data structures for dynamic orthogonal range counting

We present a linear-space data structure that maintains a dynamic set of n points with coordinates of real numbers in the plane to support orthogonal range counting in O((lgnlglgn)2) worst-case time, and insertions and deletions in O((lgnlglgn)2) amortized time. This provides faster support for updates than previous results with the same bounds on space cost and query time. We also consider the same problem for points on a U×U grid, and design the first succinct data structure for a dynamic integer sequence, S, to support range counting queries defined as follows: Given a range, [i1..i2], of indices and a range, [v1..v2], of values, count the number of entries of S[i1..i2] that store integers from [v1..v2].

A learning theory approach to noninteractive database privacy

In this article, we demonstrate that, ignoring computational constraints, it is possible to release synthetic databases that are useful for accurately answering large classes of queries while preserving differential privacy. Specifically, we give a mechanism that privately releases synthetic data useful for answering a class of queries over a discrete domain with error that grows as a function of the size of the smallest net approximately representing the answers to that class of queries. We show that this in particular implies a mechanism for counting queries that gives error guarantees that grow only with the VC-dimension of the class of queries, which itself grows at most logarithmically with the size of the query class. We also show that it is not possible to release even simple classes of queries (such as intervals and their generalizations) over continuous domains with worst-case utility guarantees while preserving differential privacy. In response to this, we consider a relaxation of the utility guarantee and give a privacy preserving polynomial time algorithm that for any halfspace query will provide an answer that is accurate for some small perturbation of the query. This algorithm does not release synthetic data, but instead another data structure capable of representing an answer for each query. We also give an efficient algorithm for releasing synthetic data for the class of interval queries and axis-aligned rectangles of constant dimension over discrete domains.

On Optimal Differentially Private Mechanisms for Count-Range Queries.

While there is a large and growing body of literature on differentially private mechanisms for answering various classes of queries, to the best of our knowledge "count-range" queries have not been studied. These are a natural class of queries that ask "is the number of rows in a relation satisfying a given predicate between two integers θ1 and θ2?" Such queries can be viewed as a simple form of SQL "having" queries. We begin by developing a provably optimal differentially private mechansim for count-range queries for a single consumer. For count queries (in contrast to count-range queries), Ghosh et al. [9] have provided a differentially private mechanism that simultaneously maximizes utility for multiple consumers. This raises the question of whether such a mechanism exists for count-range queries. We prove that the answer is no - for count range queries, no such mechanism exists. However, perhaps surprisingly, we prove that such a mechanism does exist for "threshold" queries, which are simply count-range queries for which either θ1 = 0 or θ2 = +∞. Furthermore, we prove that this mechanism is a two-approximation for general count-range queries.

Protecting count queries in study design

Today's clinical research institutions provide tools for researchers to query their data warehouses for counts of patients. To protect patient privacy, counts are perturbed before reporting; this compromises their utility for increased privacy. The goal of this study is to extend current query answer systems to guarantee a quantifiable level of privacy and allow users to tailor perturbations to maximize the usefulness according to their needs. A perturbation mechanism was designed in which users are given options with respect to scale and direction of the perturbation. The mechanism translates the true count, user preferences, and a privacy level within administrator-specified bounds into a probability distribution from which the perturbed count is drawn. Users can significantly impact the scale and direction of the count perturbation and can receive more accurate final cohort estimates. Strong and semantically meaningful differential privacy is guaranteed, providing for a unified privacy accounting system that can support role-based trust levels. This study provides an open source web-enabled tool to investigate visually and numerically the interaction between system parameters, including required privacy level and user preference settings. Quantifying privacy allows system administrators to provide users with a privacy budget and to monitor its expenditure, enabling users to control the inevitable loss of utility. While current measures of privacy are conservative, this system can take advantage of future advances in privacy measurement. The system provides new ways of trading off privacy and utility that are not provided in current study design systems.

An adaptive mechanism for accurate query answering under differential privacy

We propose a novel mechanism for answering sets of counting queries under differential privacy. Given a workload of counting queries, the mechanism automatically selects a different set of "strategy" queries to answer privately, using those answers to derive answers to the workload. The main algorithm proposed in this paper approximates the optimal strategy for any workload of linear counting queries. With no cost to the privacy guarantee, the mechanism improves significantly on prior approaches and achieves near-optimal error for many workloads, when applied under (ε, δ)-differential privacy. The result is an adaptive mechanism which can help users achieve good utility without requiring that they reason carefully about the best formulation of their task.

Universally Utility-maximizing Privacy Mechanisms

A mechanism for releasing information about a statistical database with sensitive data must resolve a trade-off between utility and privacy. Publishing fully accurate information maximizes utility while minimizing privacy, while publishing random noise accomplishes the opposite. Privacy can be rigorously quantified using the framework of differential privacy, which requires that a mechanism's output distribution is nearly the same whether a given database row is included. The goal of this paper is to formulate and provide strong and general utility guarantees, subject to differential privacy. We pursue mechanisms that guarantee near-optimal utility to every potential user, independent of its side information (modeled as a prior distribution over query results) and preferences (modeled via a symmetric and monotone loss function). Our main result is the following: for each fixed count query and differential privacy level, there is a geometric mechanism $M^*$---a discrete variant of the simple and well-studied mechanism that adds random noise from a Laplace distribution---that is simultaneously expected loss-minimizing for every possible user, subject to the differential privacy constraint. This is an extremely strong utility guarantee: every potential user $u$, no matter what its side information and preferences, derives as much utility from $M^*$ as from interacting with a differentially private mechanism $M_u$ that is optimally tailored to $u$. More precisely, for every user $u$ there is an optimal mechanism $M_u$ for it that factors into a user-independent part (the geometric mechanism $M^*$) and a user-specific postprocessing step that depends only on the output of the geometric mechanism and not on the underlying database. The first part of our proof of this result characterizes the optimal differentially private mechanism for a user as a certain basic feasible solution to a linear program with a user-specific objective function and user-independent constraints that encode differential privacy. The second part shows that all of the relevant vertices of the feasible region (ranging over all possible users) are derivable from the geometric mechanism via suitable remappings of its range.

Randomized generalization for aggregate suppression over hidden web databases

Many web databases are hidden behind restrictive form-like interfaces which allow users to execute search queries over the underlying hidden database. While it is important to support such search queries, many hidden database owners also want to maintain a certain level of privacy for aggregate information over their databases, for reasons including business secrets and homeland security. Existing work on aggregate suppression thwarts the uniform random sampling of a hidden database, but cannot address recently proposed attacks which accurately estimate SUM and COUNT queries without the need to first draw a uniform random sample. In this paper, we consider the problem of suppressing SUM and COUNT queries over a hidden database. In particular, we develop randomized generalization, a novel technique which provides rigid aggregate-suppression guarantee while maintaining the utility of individual search queries. We present theoretical analysis and extensive experiments to illustrate the effectiveness of our approach.

Differential Privacy via Wavelet Transforms

Privacy-preserving data publishing has attracted considerable research interest in recent years. Among the existing solutions, ∈-differential privacy provides the strongest privacy guarantee. Existing data publishing methods that achieve ∈-differential privacy, however, offer little data utility. In particular, if the output data set is used to answer count queries, the noise in the query answers can be proportional to the number of tuples in the data, which renders the results useless. In this paper, we develop a data publishing technique that ensures ∈-differential privacy while providing accurate answers for range-count queries, i.e., count queries where the predicate on each attribute is a range. The core of our solution is a framework that applies wavelet transforms on the data before adding noise to it. We present instantiations of the proposed framework for both ordinal and nominal data, and we provide a theoretical analysis on their privacy and utility guarantees. In an extensive experimental study on both real and synthetic data, we show the effectiveness and efficiency of our solution.

Publishing set-valued data via differential privacy

Set-valued data provides enormous opportunities for various data mining tasks. In this paper, we study the problem of publishing set-valued data for data mining tasks under the rigorous differential privacy model. All existing data publishing methods for set-valued data are based on partition-based privacy models, for example k -anonymity, which are vulnerable to privacy attacks based on background knowledge. In contrast, differential privacy provides strong privacy guarantees independent of an adversary's background knowledge and computational power. Existing data publishing approaches for differential privacy, however, are not adequate in terms of both utility and scalability in the context of set-valued data due to its high dimensionality. We demonstrate that set-valued data could be efficiently released under differential privacy with guaranteed utility with the help of context-free taxonomy trees. We propose a probabilistic top-down partitioning algorithm to generate a differentially private release, which scales linearly with the input data size. We also discuss the applicability of our idea to the context of relational data. We prove that our result is (∈, δ)-useful for the class of counting queries, the foundation of many data mining tasks. We show that our approach maintains high utility for counting queries and frequent itemset mining and scales to large datasets through extensive experiments on real-life set-valued datasets.

Private analysis of graph structure

We present efficient algorithms for releasing useful statistics about graph data while providing rigorous privacy guarantees. Our algorithms work on data sets that consist of relationships between individuals, such as social ties or email communication. The algorithms satisfy edge differential privacy , which essentially requires that the presence or absence of any particular relationship be hidden. Our algorithms output approximate answers to subgraph counting queries. Given a query graph H , e.g., a triangle, k -star or k -triangle, the goal is to return the number of edge-induced isomorphic copies of H in the input graph. The special case of triangles was considered by Nissim, Raskhodnikova and Smith (STOC 2007), and a more general investigation of arbitrary query graphs was initiated by Rastogi, Hay, Miklau and Suciu (PODS 2009). We extend the approach of [NRS] to a new class of statistics, namely, k -star queries. We also give algorithms for k -triangle queries using a different approach, based on the higher-order local sensitivity. For the specific graph statistics we consider (i.e., k -stars and k -triangles), we significantly improve on the work of [RHMS]: our algorithms satisfy a stronger notion of privacy, which does not rely on the adversary having a particular prior distribution on the data, and add less noise to the answers before releasing them. We evaluate the accuracy of our algorithms both theoretically and empirically, using a variety of real and synthetic data sets. We give explicit, simple conditions under which these algorithms add a small amount of noise. We also provide the average-case analysis in the Erdős-Rényi-Gilbert G ( n , p ) random graph model. Finally, we give hardness results indicating that the approach NRS used for triangles cannot easily be extended to k -triangles (and hence justifying our development of a new algorithmic approach).

A prediction model for web search hit counts using word frequencies

A search engine user with a well-defined information need is not interested in getting thousands of hits, but a few hits that are all highly relevant to their search. Often search words need to be refined and augmented to narrow results to more relevant pages. However, an overly specific query may lead to no hits at all, while most typical queries lead to thousands or even millions of them, both undesirable outcomes. This paper suggests a query rewriting method for generating alternative query strings and proposes a hit count prediction model for predicting the number of search engine hits for each alternative query string, based on the English language frequencies of the words in the search terms. Using the hit count prediction model, different types of search strategies, such as a lowest hit count query preference, can be utilized to improve users’ search experience. We present an evaluation experiment of the hit count prediction model for three major search engines. We also discuss and quantify how far the Google, Yahoo! and Bing search engines diverge from monotonic behaviour, considering negative and positive search terms separately.

Sampling-based lower bounds for counting queries

It is well known that the problem of computing relative approximations of weighted counting queries such as the probability of evidence in a Bayesian network, the partition function of a Markov network, and the number of solutions of a constraint satisfaction problem is NP-hard. In this paper, we settle therefore on an easier problem of computing high-confidence lower bounds. We propose to use importance sampling and Markov inequality for solving it. However, a straight-forward application of the Markov inequality often yields poor lower bounds. We therefore propose several new schemes for improving its performance in practice. Empirically, we show that our new schemes are quite powerful, often yielding substantially higher (better) lower bounds than all state-of-the-art schemes.

Towards optimal range medians

We consider the following problem: Given an unsorted array of n elements, and a sequence of intervals in the array, compute the median in each of the subarrays defined by the intervals. We describe a simple algorithm which needs O ( n log k + k log n ) time to answer k such median queries. This improves previous algorithms by a logarithmic factor and matches a comparison lower bound for k = O ( n ) . The space complexity of our simple algorithm is O ( n log n ) in the pointer machine model, and O ( n ) in the RAM model. In the latter model, a more involved O ( n ) space data structure can be constructed in O ( n log n ) time where the time per query is reduced to O ( log n / log log n ) . We also give efficient dynamic variants of both data structures, achieving O ( log 2 n ) query time using O ( n log n ) space in the comparison model and O ( ( log n / log log n ) 2 ) query time using O ( n log n / log log n ) space in the RAM model, and show that in the cell-probe model, any data structure which supports updates in O ( log O ( 1 ) n ) time must have Ω ( log n / log log n ) query time. Our approach naturally generalizes to higher-dimensional range median problems, where element positions and query ranges are multidimensional—it reduces a range median query to a logarithmic number of range counting queries.

A general approach for cache-oblivious range reporting and approximate range counting

We present cache-oblivious solutions to two important variants of range searching: range reporting and approximate range counting. Our main contribution is a general approach for constructing cache-oblivious data structures that provide relative ( 1 + ε ) -approximations for a general class of range counting queries. This class includes three-sided range counting in the plane, 3-d dominance counting, and 3-d halfspace range counting. The constructed data structures use linear space and answer queries in the optimal query bound of O ( log B ( N / K ) ) block transfers in the worst case, where K is the number of points in the query range. As a corollary, we also obtain the first approximate 3-d halfspace range counting and 3-d dominance counting data structures with a worst-case query time of O ( log ( N / K ) ) in internal memory. An easy but important consequence of our main result is the existence of O ( N log N ) -space cache-oblivious data structures with an optimal query bound of O ( log B N + K / B ) block transfers for the reporting versions of the above problems. Using standard reductions, these data structures allow us to obtain the first cache-oblivious data structures that use almost linear space and achieve the optimal query bound for circular range reporting and K-nearest neighbour searching in the plane, as well as for orthogonal range reporting in three dimensions.

Bucket‐based authentication for outsourced databases

AbstractWhen outsourced database owners delegate their data to service providers, which might be untrusted or compromised, two issues of data security emerge, including data confidentiality and data integrity. Most of the previous research focuses on only one issue and the solution to integrate two approaches is expensive. In this paper, we propose bucket‐based authentication that can keep data confidentiality and meanwhile guarantee data integrity. Specifically, we first propose a new approach based on bucket checksum, which can be used for the authentication of multiple tuples at one time. We then apply bucket checksum to the authentication of various types of queries in static scenarios, including range queries and aggregation queries, such as MIN, MAX, SUM and COUNT queries. In the authentication of aggregation queries, several pruning rules have been proposed to improve performance further. We also extend our approach to dynamic scenarios based on incremental hash. Cost analysis shows the advantages of our approach over previous ones in terms of construction and verification cost. Experimental results show that our approach is both efficient and effective. Copyright © 2010 John Wiley & Sons, Ltd.

Compact labelings for efficient first-order model-checking

We consider graph properties that can be checked from labels, i.e., bit sequences, of logarithmic length attached to vertices. We prove that there exists such a labeling for checking a first-order formula with free set variables in the graphs of every class that is \emph{nicely locally cwd-decomposable}. This notion generalizes that of a \emph{nicely locally tree-decomposable} class. The graphs of such classes can be covered by graphs of bounded \emph{clique-width} with limited overlaps. We also consider such labelings for \emph{bounded} first-order formulas on graph classes of \emph{bounded expansion}. Some of these results are extended to counting queries.

Orthogonal range searching in linear and almost-linear space

In this paper we describe space-efficient data structures for the two-dimensional range searching problem. We present a dynamic linear space data structure that supports orthogonal range reporting queries in O ( log n + k log ε n ) time, where k is the size of the answer. Our data structure also supports emptiness and one-reporting queries in O ( log n ) time and thus achieves optimal time and space for this type of queries. In the case of integer point coordinates, we describe a static and a randomized dynamic linear space data structures that support range reporting, emptiness and one-reporting queries in sub-logarithmic time. These are the first linear space data structures for these problems that achieve sub-logarithmic query time. We also present a dynamic linear space data structure for range counting queries with O ( ( log n / log log n ) 2 ) time and a dynamic O ( n log n / log log n ) space data structure for semigroup range queries with O ( ( log n / log log n ) 2 ) query time.

A Cryptographic Approach to Securely Share and Query Genomic Sequences

To support large-scale biomedical research projects, organizations need to share person-specific genomic sequences without violating the privacy of their data subjects. In the past, organizations protected subjects' identities by removing identifiers, such as name and social security number; however, recent investigations illustrate that deidentified genomic data can be "reidentified" to named individuals using simple automated methods. In this paper, we present a novel cryptographic framework that enables organizations to support genomic data mining without disclosing the raw genomic sequences. Organizations contribute encrypted genomic sequence records into a centralized repository, where the administrator can perform queries, such as frequency counts, without decrypting the data. We evaluate the efficiency of our framework with existing databases of single nucleotide polymorphism (SNP) sequences and demonstrate that the time needed to complete count queries is feasible for real world applications. For example, our experiments indicate that a count query over 40 SNPs in a database of 5000 records can be completed in approximately 30 min with off-the-shelf technology. We further show that approximation strategies can be applied to significantly speed up query execution times with minimal loss in accuracy. The framework can be implemented on top of existing information and network technologies in biomedical environments.

On k-d Range Search with Patricia Tries

Patricia tries are explored for indexing combined text and spatial data. A combined text and spatial data range search algorithm is presented for reporting all data from a set of size n intersecting a query hyperrectangle. We also use Patricia tries to answer $\epsilon$-approximate orthogonal range search on a set of n random points and hyperrectangles in k-dimensional data space. $\epsilon$-approximate orthogonal range counting queries can be answered in $O(k {\rm log} n \epsilon^{k-1})$ time, and the number of nodes visited for orthogonal range counting queries is shown to be $O({\rm log} n + k(1+2n^{1/k}\Delta)^{k-1})$ for cubical range of side length $\Delta$. Patricia tries are evaluated experimentally for both orthogonal range search and $\epsilon$-approximate orthogonal range search (for $2 \leq k \leq 14$ and n up to 1,000,000) using uniformly distributed random data. The expected range search time is determined theoretically and found to agree with experimental results.

On Approximating the Depth and Related Problems

We study the question of finding a deepest point in an arrangement of regions and provide a fast algorithm for this problem using random sampling, showing it sufficient to solve this problem when the deepest point is shallow. This implies, among other results, a fast algorithm for approximately solving linear programming problems with violations. We also use this technique to approximate the disk covering the largest number of red points, while avoiding all the blue points, given two such sets in the plane. Using similar techniques implies that approximate range counting queries have roughly the same time and space complexity as emptiness range queries.