On fuzzy approaches for enlarging skyline query results
Introduction
Preference queries have gained much attention in the Database field in the recent years. Skyline queries [1] are a good example of SQL extensions that allow users to express their preferences in queries. Based on Pareto dominance relationship, skyline queries select all non-dominated points based on a multi-criteria comparison. Let be a set of -dimensional points, a skyline query returns, the skyline , set of points of that are not dominated by any other point of . A point dominates, in the sense of Pareto, another point iff is better than or equal to in all dimensions and strictly better than in at least one dimension. One can observe that skyline points are incomparable.
It is worthy to note that skyline queries have benefits many types of database applications since its introduction in the database field. They are widely used in non-trivial real applications, including multi-criteria decision making applications [2], [3], Web services such as hotel recommender [4], restaurant finder [5], and peer-to-peer network database [6]. With this widen use of skyline queries over several real life database applications, a lot of research studies have been devoted to efficient computing skyline and introducing multiple variants of skyline queries [7], [8], [9], [3], [10].
However, querying -dimensional datasets using a skyline operator may lead to two possible scenarios: (i) a large number of skyline points returned, which could be less informative for users requirements, (ii) a small number of skyline points returned, which could be insufficient for users needs. To solve the problem stemmed from the first scenario, various approaches have been proposed to refine the skyline, therefore reducing its size [11], [12], [13], [14], [15], [16], [17], [2], [18]. While for the second scenario only very few works exist to relax the skyline in order to increase the number of skyline results [13], [19], [20]. In this paper, we address the problem of low skyline and propose advanced fuzzy-set-based solutions to enlarge it with a set of particular interesting (non-skyline) points. Such solutions exhibit a cooperative behavior in the sense that they assist the users to obtain the desired results to their skyline queries. Users’ preferences and controlling are two key elements of our solutions. The former is leveraged to choose some specific skyline relaxation parameters and the latter allows ending the relaxation process when the results are satisfactory. In summary, the new main contributions made in this paper1 are as follows:
- 1.
We address the skyline relaxation problem by proposing two efficient fuzzy approaches 22 and 2.3 The former relies on a novel fuzzy dominance relationship which makes more demanding the dominance between two points. As for the latter, it leverages an appropriate fuzzy closeness relation to retrieve non skyline points that are fuzzily close to skyline points. Both approaches allow adding new points to the skyline result.
- 2.
For each approach, the semantic basis for the relaxed variant of skyline are discussed in depth. Then, optimized algorithms are developed to efficiently compute each variant of skyline. A theoretical complexity analysis of the proposed algorithms is also investigated.
- 3.
We conduct a set of thorough experiments to study and analyze the relevance and effectiveness of the proposed approaches. A comparative study between these approaches and the Gonclaves and Tineo approah [19] is performed as well.
The paper is structured as follows: In Section 2, we introduce some basic notions about fuzzy set theory and skyline queries. Section 3 describes the approaches 2 and 2 for relaxing the skyline and discusses the semantic basis of each of them. The computation part of the relaxed variants of skyline is presented in Section 4. In Section 5, we provide an overview of existing works. Section 6 is devoted to the experimental study. Finally, we conclude and discuss some perspectives in Section 7.
Section snippets
Fuzzy sets
The concept of fuzzy sets has been developed by Zadeh [23] in 1965 to represent classes or sets whose limits are imprecise. They can describe gradual transitions between total belonging and rejection. Typical examples of these fuzzy classes are those described with adjectives or adverbs natural language, as not expensive, fast and very close.
Formally, a fuzzy set on the universe is described by a membership function , where represents the degree of membership of in . By
Fuzzy skyline relaxation
We discuss here our fuzzy approaches to skyline relaxation. Let and be the relaxed skyline returned respectively by the approaches 2 and 2 described formally in Sections 3.1 , 3.2 .
Both approaches rely on the main idea that consists of computing the extent to which a point, discarded by the Pareto-dominance relationship, may belong to the relaxed skyline. To this end, and as it will be illustrated further, we associate with each skyline attribute a pair of
and Computation
To compute the two relaxed variants of Skyline, i.e. and , we propose a two-steps procedure (see Fig. 8): (i) the skyline computation step; and (ii) the skyline relaxation step.
In the first step, we calculate the regular skyline using a slightly improved algorithm version, called LIBNL (see algorithm 1), of algorithm BNL proposed in [1]. The LIBNL algorithm uses a function named SkylineCompare( ) to evaluate the dominance , in the sense of Pareto, between and on all
Related work
Our study can be related to the previous works on skyline computation and controlling the skyline size. In this section, we review the major existing approaches on these two topics.
Experimental study
In this section, we present the experimental study that we have conducted. The aim of this study is to prove and demonstrate the effectiveness of the proposed approaches (2 and 2) and their ability to relax small skyline with the most interesting points. In addition, this study allows us to develop a comparative assessment on the quantitative and qualitative aspects of the relaxation process between our two approaches and the approach proposed by Goncalves and Tineo (denoted GT approach)
Conclusion and perspectives
In this paper, we have addressed the problem of skyline relaxation in a controlled way. The basic idea is to make the skyline more permissive by adding points that strictly speaking do not belong to skyline, but are not far from belonging to it. We have explored two strategies: first, we propose to recover the much interesting points among those discriminated by Pareto dominance relationship by leveraging a novel fuzzy dominance relationship Much Preferred (MP). Then, we advocate to enlarge the
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