We address the problem of finding a "best" deterministic query answer to a query over a probabilistic database. For this purpose, we propose the notion of a consensus world (or a consensus answer) which is a deterministic world (answer) that minimizes the expected distance to the possible worlds (answers). This problem can be seen as a generalization of the well-studied inconsistent information aggregation problems (e.g. rank aggregation) to probabilistic databases. We consider this problem for various types of queries including SPJ queries, queries, group-by aggregate queries, and clustering. For different distance metrics, we obtain polynomial time optimal or approximation algorithms for computing the consensus answers (or prove NP-hardness). Most of our results are for a general probabilistic database model, called and xor tree model , which significantly generalizes previous probabilistic database models like x-tuples and block-independent disjoint models, and is of independent interest.
In recent years, there has also been much work on efficiently answering different types of queries over probabilistic databases. @cite_29 first considered the problem of ranking over probabilistic databases, and proposed two ranking functions to combine the tuple scores and probabilities. @cite_35 presented improved algorithms for the same ranking functions. Zhang and Chomicki @cite_36 presented a desiderata for ranking functions and propose Global queries. Ming @cite_5 @cite_3 recently presented a different ranking function called Probabilistic threshold queries . Finally, @cite_37 also present a semantics of ranking functions and a new ranking function called expected rank . In a recent work, we proposed a parameterized ranking function, and presented general algorithms for evaluating them @cite_31 Other types of queries have also been recently considered over probabilistic databases (e.g. clustering @cite_16 , nearest neighbors @cite_41 etc.).
@cite_35 @cite_37 @cite_36 @cite_41 @cite_29 @cite_3 @cite_5 @cite_31 @cite_16
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The dramatic growth in the number of application domains that naturally generate probabilistic, uncertain data has resulted in a need for efficiently supporting complex querying and decision-making over such data. In this paper, we present a unified approach to ranking and top-k query processing in probabilistic databases by viewing it as a multi-criteria optimization problem, and by deriving a set of features that capture the key properties of a probabilistic dataset that dictate the ranked result. We contend that a single, specific ranking function may not suffice for probabilistic databases, and we instead propose two parameterized ranking functions, called PRFω and PRFe, that generalize or can approximate many of the previously proposed ranking functions. We present novel generating functions-based algorithms for efficiently ranking large datasets according to these ranking functions, even if the datasets exhibit complex correlations modeled using probabilistic and xor trees or Markov networks. We further propose that the parameters of the ranking function be learned from user preferences, and we develop an approach to learn those parameters. Finally, we present a comprehensive experimental study that illustrates the effectiveness of our parameterized ranking functions, especially PRFe, at approximating other ranking functions and the scalability of our proposed algorithms for exact or approximate ranking. When dealing with massive quantities of data, top-k queries are a powerful technique for returning only the k most relevant tuples for inspection, based on a scoring function. The problem of efficiently answering such ranking queries has been studied and analyzed extensively within traditional database settings. The importance of the top-k is perhaps even greater in probabilistic databases, where a relation can encode exponentially many possible worlds. There have been several recent attempts to propose definitions and algorithms for ranking queries over probabilistic data. However, these all lack many of the intuitive properties of a top-k over deterministic data. Specifically, we define a number of fundamental properties, including exact-k, containment, unique-rank, value-invariance, and stability, which are all satisfied by ranking queries on certain data. We argue that all these conditions should also be fulfilled by any reasonable definition for ranking uncertain data. Unfortunately, none of the existing definitions is able to achieve this. To remedy this shortcoming, this work proposes an intuitive new approach of expected rank. This uses the well-founded notion of the expected rank of each tuple across all possible worlds as the basis of the ranking. We are able to prove that, in contrast to all existing approaches, the expected rank satisfies all the required properties for a ranking query. We provide efficient solutions to compute this ranking across the major models of uncertain data, such as attribute-level and tuple-level uncertainty. For an uncertain relation of N tuples, the processing cost is O(N logN)—no worse than simply sorting the relation. In settings where there is a high cost for generating each tuple in turn, we provide pruning techniques based on probabilistic tail bounds that can terminate the search early and guarantee that the top-k has been found. Finally, a comprehensive experimental study confirms the effectiveness of our approach. In emerging applications such as location-based services, sensor monitoring and biological management systems, the values of the database items are naturally imprecise. For these uncertain databases, an important query is the Probabilistic k-Nearest-Neighbor Query (k-PNN), which computes the probabilities of sets of k objects for being the closest to a given query point. The evaluation of this query can be both computationally- and I O-expensive, since there is an exponentially large number of k object-sets, and numerical integration is required. Often a user may not be concerned about the exact probability values. For example, he may only need answers that have sufficiently high confidence. We thus propose the Probabilistic Threshold k-Nearest-Neighbor Query (T-k-PNN), which returns sets of k objects that satisfy the query with probabilities higher than some threshold T. Three steps are proposed to handle this query efficiently. In the first stage, objects that cannot constitute an answer are filtered with the aid of a spatial index. The second step, called probabilistic candidate selection, significantly prunes a number of candidate sets to be examined. The remaining sets are sent for verification, which derives the lower and upper bounds of answer probabilities, so that a candidate set can be quickly decided on whether it should be included in the answer. We also examine spatially-efficient data structures that support these methods. Our solution can be applied to uncertain data with arbitrary probability density functions. We have also performed extensive experiments to examine the effectiveness of our methods. Various approaches for keyword proximity search have been implemented in relational databases, XML and the Web. Yet, in all of them, an answer is a Q-fragment, namely, a subtree T of the given data graph G, such that T contains all the keywords of the query Q and has no proper subtree with this property. The rank of an answer is inversely proportional to its weight. Three problems are of interest: finding an optimal (i.e., top-ranked) answer, computing the top-k answers and enumerating all the answers in ranked order. It is shown that, under data complexity, an efficient algorithm for solving the first problem is sufficient for solving the other two problems with polynomial delay. Similarly, an efficient algorithm for finding a θ-approximation of the optimal answer suffices for carrying out the following two tasks with polynomial delay, under query-and-data complexity. First, enumerating in a (θ+1)-approximate order. Second, computing a (θ+1)-approximation of the top-k answers. As a corollary, this paper gives the first efficient algorithms, under data complexity, for enumerating all the answers in ranked order and for computing the top-k answers. It also gives the first efficient algorithms, under query-and-data complexity, for enumerating in a provably approximate order and for computing an approximation of the top-k answers.