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Skyline query computes all the “best” elements which are
not dominated by any other elements and thus is very important
for decision-making applications. Recently, it is generalized
to

Skyline queries [

By using the common hotel example in the literature, assuming that each hotel has the information of its distance from the beach and its price, and that one prefers the hotels which are cheap and close to the beach, Figure

Skyline versus 1-skyband.

Recently, the database research community witnessed a paradigm shift to continuous queries, and much attention has been put on sliding-window skyline queries [

Monitoring sliding-window skybands needs to extract all skyband elements from the live elements in the window and continuously report skyband changes as the window slides. In this paper, we first introduce the

The rest of this paper is organized as follows. Section

Many algorithms have been proposed for computing static skylines, including the non-index-based algorithms [

Under the assumptions of statistical independence across dimensions, no duplicate values over each dimension, and dimension domains being all totally ordered, the problem of estimating the number of the skyline elements, that is, the skyline cardinality, has been addressed in the works [

As stated before, continuous skyline queries over sliding windows in data streams [

In this section, we present some preliminary results that will be used in the next section. In addition, we also describe a data structure called

Suppose that

Suppose that

By Theorem

In a

Now we are able to describe a data structure called

Figure

The architecture of the skyband sketch.

In this section, we present our robust approaches for estimating the space usage of sliding-window skybands under the assumption of statistical independence across dimensions based on the preliminary results in the previous section.

Here, we give our theoretical analysis for the space usage of sliding-window skybands over data which is distribution constrained, that is, there are no duplicate values over each dimension. By mapping the problem of evaluating the number of the elements in a finite set which satisfy no more than

Suppose that

We map

Under assumptions of statistical independence across dimensions, no duplicate values over each dimension, and domains being all totally-ordered, an element has a

Suppose that there are

By Lemma

Theorem

Under assumptions of statistical independence across dimensions, no duplicate values over each dimension, and domains being all totally-ordered, the expected number of the skyband elements in case of monitoring a

By Lemma

In this subsection, based on the theoretical analysis proposed in the above subsection, we propose an efficient dynamic programming algorithm to estimate the space usage. Since there exist inherent correlations among the expected number of the skyband elements, the expected number of the potential-skyband elements, and the expected number of the elements stored by the skyband sketch, we only consider how to estimate the number of the skyband elements.

Estimating the number of the skyband elements using (

Algorithm

In this section, we verify our theoretical results on space usage estimation of the k-skyband operator monitoring skybands over sliding windows in the stream environment by extensive experiments. The algorithms have been implemented by the C++ programming language and run on a 2.0 GHz Intel CPU with 2 GB of memory, and the data over each dimension is generated by the (GNU Scientific Library GSL:

Figures

Space performance of monitoring skylines over sliding windows in the stream environment in a 4-dimensional space where the data over each dimension is continuously distributed.

Theoretical results

Experimental results

Experimental results

Experimental results

Theoretical results

Experimental results

Experimental results

Experimental results

Space performance of monitoring skylines over sliding windows in the stream environment in a 8-dimensional space where the data over each dimension is continuously distributed.

Theoretical results

Experimental results

Experimental results

Experimental results

Theoretical results

Experimental results

Experimental results

Experimental results

Skyband query is of great importance for multi-criteria decision-making applications. To support skyband query in the stream engine, the problem of effective space usage estimation must be solved, which is important for extending the query optimizers cost model. In this paper, under the assumption of statistically independent [

This work is partially supported by China “863” Hi-tech Program (Grant no. 2007AA01Z153), Zhejiang Provincial NSF (Grant no. Y1090096), and the National Natural Science Foundation of China (NSFC) under Grant no. 60573125 and 60873264.