28 lines
2.4 KiB
TeX
28 lines
2.4 KiB
TeX
%!TEX root=./main.tex
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\section{Conclusions and Future Work}\label{sec:concl-future-work}
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We have studied the problem of calculating the expected multiplicity of a query result tuple, %expectation of lineage polynomials over BIDBs. %random integer variables.
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a problem that has a practical application in probabilistic databases over multisets. %, where it corresponds to calculating the expected multiplicity of a query result tuple.
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% It has been studied extensively for sets (lineage formulas), but the bag settings has not received much attention.
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%While the expectation of a polynomial can be calculated in linear time for % in the size of
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% polynomials % that are
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%in SOP form, the problem is \sharpwonehard for factorized polynomials (proven through a reduction from the problem of counting k-matchings).
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%We have proven this claim through a reduction from the problem of counting k-matchings.
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We show that under various parameterized complexity hardness results/conjectures computing the expected multiplicities exactly is not possible in time linear in the corresponding deterministic query processing time.
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We prove that it is possible to approximate the expectation of a lineage polynomial in linear time
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% When only considering polynomials for result tuples of
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in the deterministic query processing over TIDBs and BIDBs (assuming that there are few cancellations).
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Interesting directions for future work include development of a dichotomy for bag \abbrPDB\xplural. While we can handle higher moments (this follows fairly easily from our existing results-- see \Cref{sec:momemts}), more general approximations are an interesting area for exploration, including those for more general data models. % beyond what we consider in this paper.
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% Furthermore, it would be interesting to see whether our approximation algorithm can be extended to support queries with negations, perhaps using circuits with monus as a representation system.
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% \BG{I am not sure what interesting future work is here. Some wild guesses, if anybody agrees I'll try to flesh them out:
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% \textbullet{More queries: what happens with negation can circuits with monus be used?}
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% \textbullet{More databases: can we push beyond BIDBs? E.g., C-tables / aggregate semimodules or just TIDBs where each input tuple is a random variable over $\mathbb{N}$?}
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% \textbullet{Other results: can we extend the work to approximate $P(R(t) = n)$}
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% }
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