A regret minimizing set Q is a small size representation of a much larger database P so that user queries executed on Q return answers whose scores are not much worse than those on the full dataset. In particular, a k-regret minimizing set has the property that the regret ratio between the score of the top-1 item in Q and the score of the top-k item in P is minimized, where the score of an item is the inner product of the item's attributes with a user's weight (preference) vector. The problem is challenging because we want to find a single representative set Q whose regret ratio is small with respect to all possible user weight vectors. We show that k-regret minimization is NP-Complete for all dimensions d >= 3. This settles an open problem from [VLDB 2014], and resolves the complexity status of the problem for all d: the problem is known to have polynomial-time solution for d <= 2. In addition, we propose two new approximation schemes for regret minimization, both with provable guarantees, one based on coresets and another based on hitting sets. We also carry out extensive experimental evaluation, and show that our schemes compute regret-minimizing sets comparable in size to the greedy algorithm proposed in [VLDB 14] but our schemes are significantly faster and scalable to large data sets.
relatedWork The work on regret minimization was inspired by preference top- @math and skyline queries. Both of these research topics try to help a user find the best objects'' from a database. Top- @math queries assign scores to objects by some method, and return the objects with the topmost @math scores while the skyline query finds the objects such that no other object can be strictly better. Efficiently answering top- @math queries has seen a long line of work, see e.g. @cite_38 @cite_7 @cite_11 @cite_1 @cite_32 @cite_42 @cite_8 @cite_15 @cite_46 @cite_36 @cite_26 and the survey @cite_23 . In earlier work, the ranking of points was done by weight, i.e., ranking criterion was fixed. Recent work has considered the specification of the ranking as part of the query. Typically, this is specified as a preference vector @math and the ranking of the points is by linear projection on @math see e.g. @cite_38 @cite_19 @cite_2 . Another ranking criterion is based on the distance from a given query point in a metric space i.e., the top- @math query is a @math -nearest neighbor query @cite_41 .
@cite_38 @cite_26 @cite_7 @cite_8 @cite_46 @cite_36 @cite_41 @cite_42 @cite_1 @cite_32 @cite_19 @cite_23 @cite_2 @cite_15 @cite_11
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A top-k query retrieves the k highest scoring tuples from a data set with respect to a scoring function defined on the attributes of a tuple. The efficient evaluation of top-k queries has been an active research topic and many different instantiations of the problem, in a variety of settings, have been studied. However, techniques developed for conventional, centralized or distributed databases are not directly applicable to highly dynamic environments and on-line applications, like data streams. Recently, techniques supporting top-k queries on data streams have been introduced. Such techniques are restrictive however, as they can only efficiently report top-k answers with respect to a pre-specified (as opposed to ad-hoc) set of queries. In this paper we introduce a novel geometric representation for the top-k query problem that allows us to raise this restriction. Utilizing notions of geometric arrangements, we design and analyze algorithms for incrementally maintaining a data set organized in an arrangement representation under streaming updates. We introduce query evaluation strategies that operate on top of an arrangement data structure that are able to guarantee efficient evaluation for ad-hoc queries. The performance of our core technique is augmented by incorporating tuple pruning strategies, minimizing the number of tuples that need to be stored and manipulated. This results in a main memory indexing technique supporting both efficient incremental updates and the evaluation of ad-hoc top-k queries. A thorough experimental study evaluates the efficiency of the proposed technique. Top-k dominating queries combine the natural idea of selecting the k best items with a comprehensive “goodness” criterion based on dominance. A point p1 dominates p2 if p1 is as good as p2 in all attributes and is strictly better in at least one. Existing works address the problem in settings where data objects are multidimensional points. However, there are domains where we only have access to the distance between two objects. In cases like these, attributes reflect distances from a set of input objects and are dynamically generated as the input objects change. Consequently, prior works from the literature cannot be applied, despite the fact that the dominance relation is still meaningful and valid. For this reason, in this work, we present the first study for processing top-k dominating queries over distance-based dynamic attribute vectors, defined over a metric space. We propose four progressive algorithms that utilize the properties of the underlying metric space to efficiently solve the problem and present an extensive, comparative evaluation on both synthetic and real-world datasets. A query to a web search engine usually consists of a list of keywords, to which the search engine responds with the best or "top" k pages for the query. This top-k query model is prevalent over multimedia collections in general, but also over plain relational data for certain applications. For example, consider a relation with information on available restaurants, including their location, price range for one diner, and overall food rating. A user who queries such a relation might simply specify the user's location and target price range, and expect in return the best 10 restaurants in terms of some combination of proximity to the user, closeness of match to the target price range, and overall food rating. Processing top-k queries efficiently is challenging for a number of reasons. One critical such reason is that, in many web applications, the relation attributes might not be available other than through external web-accessible form interfaces, which we will have to query repeatedly for a potentially large set of candidate objects. In this article, we study how to process top-k queries efficiently in this setting, where the attributes for which users specify target values might be handled by external, autonomous sources with a variety of access interfaces. We present a sequential algorithm for processing such queries, but observe that any sequential top-k query processing strategy is bound to require unnecessarily long query processing times, since web accesses exhibit high and variable latency. Fortunately, web sources can be probed in parallel, and each source can typically process concurrent requests, although sources may impose some restrictions on the type and number of probes that they are willing to accept. We adapt our sequential query processing technique and introduce an efficient algorithm that maximizes source-access parallelism to minimize query response time, while satisfying source-access constraints. We evaluate our techniques experimentally using both synthetic and real web-accessible data and show that parallel algorithms can be significantly more efficient than their sequential counterparts. Top-k query asks for k tuples ordered according to a specific ranking function that combines the values from multiple participating attributes. The combined score function is usually linear. To efficiently answer top-k queries, preprocessing and indexing the data have been used to speed up the run time performance. Many indexing methods allow the online query algorithms progressively retrieve the data and stop at a certain point. However, in many cases, the number of data accesses is sensitive to the query parameters (i.e., linear weights in the score functions).In this paper, we study the sequentially layered indexing problem where tuples are put into multiple consecutive layers and any top-k query can be answered by at most k layers of tuples. We propose a new criterion for building the layered index. A layered index is robust if for any k, the number of tuples in the top k layers is minimal in comparison with all the other alternatives. The robust index guarantees the worst case performance for arbitrary query parameters. We derive a necessary and sufficient condition for robust index. The problem is shown solvable within O(ndlog n) (where d is the number of dimensions, and n is the number of tuples). To reduce the high complexity of the exact solution, we develop an approximate approach, which has time complexity O(2d n(log n)r(d)-1), where r(d) = ⌈d 2⌉ + ⌊d 2⌋ ⌈d 2⌉. Our experimental results show that our proposed method outperforms the best known previous methods. Efficient processing of top-k queries is a crucial requirement in many interactive environments that involve massive amounts of data. In particular, efficient top-k processing in domains such as the Web, multimedia search, and distributed systems has shown a great impact on performance. In this survey, we describe and classify top-k processing techniques in relational databases. We discuss different design dimensions in the current techniques including query models, data access methods, implementation levels, data and query certainty, and supported scoring functions. We show the implications of each dimension on the design of the underlying techniques. We also discuss top-k queries in XML domain, and show their connections to relational approaches. Given a set of objects, each with multiple numeric attributes, a (preference) top-k query retrieves the k objects with the highest scores according to a user preference, defined as a linear combination of attribute values. We consider the problem of processing a large number of continuous top-k queries, each with its own preference. When objects or user preferences change, the query results must be updated. We present a dynamic index that supports the reverse top k query, which is of independent interest. Combining this index with another one for top-k queries, we develop a scalable solution for processing many continuous top-k queries that exploits the clusteredness in user preferences. We also define an approximate version of the problem and present a solution significantly more efficient than the exact one with little loss in accuracy.