35 lines
4.3 KiB
TeX
35 lines
4.3 KiB
TeX
%!TEX root=./main.tex
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\section{Related Work}\label{sec:related-work}
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In addition to work on probabilistic databases, our work has connections to work on compact representations of polynomials and relies on past work in fine-grained complexity which we review in \Cref{sec:compr-repr-polyn} and \Cref{sec:param-compl}.
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%\subsection{Probabilistic Databases}\label{sec:prob-datab}
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Probabilistic Databases (PDBs) have been studied predominantly under set semantics.
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A multitude of data models have been proposed for encoding a PDB more compactly than as its set of possible worlds.
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Tuple-independent databases (\tis) consist of a classical database where each tuple associated with a probability and tuples are treated as independent probabilistic events.
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While unable to encode correlations directly, \tis are popular because any finite probabilistic database can be encoded as a \ti and a set of constraints that ``condition'' the \ti~\cite{VS17}.
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Block-independent databases (\bis) generalize \tis by partitioning the input into blocks of disjoint tuples, where blocks are independent~\cite{RS07,BS06}. \emph{PC-tables}~\cite{GT06} pair a C-table~\cite{IL84a} with probability distribution over its variables. This is similar to our $\semNX$-PDBs, except that we do not allow for variables as attribute values and instead of local conditions (propositional formulas that may contain comparisons), we associate tuples with polynomials $\semNX$.
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Approaches for probabilistic query processing (i.e., computing the marginal probability for query result tuples), fall into two broad categories.
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\emph{Intensional} (or \emph{grounded}) query evaluation computes the \emph{lineage} of a tuple (a Boolean formula encoding the provenance of the tuple) and then the probability of the lineage formula.
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In this paper we focus on intensional query evaluation using polynomials instead of boolean formulas.
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It is a well-known fact that computing the marginal probability of a tuple is \sharpphard (proven through a reduction from weighted model counting~\cite{provan-83-ccccptg,valiant-79-cenrp} using the fact the tuple's marginal probability is the probability of a its lineage formula).
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The second category, \emph{extensional} query evaluation, avoids calculating the lineage.
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This approach is in \ptime, but is limited to certain classes of queries.
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Dalvi et al.~\cite{DS12} proved a dichotomy for unions of conjunctive queries (UCQs): for any UCQ the probabilistic query evaluation problem is either \sharpphard (requires extensional evaluation) or \ptime (allows intensional).
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Olteanu et al.~\cite{FO16} presented dichotomies for two classes of queries with negation, R\'e et al~\cite{RS09b} present a trichotomy for HAVING queries.
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Amarilli et al. investigated tractable classes of databases for more complex queries~\cite{AB15,AB15c}.
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Another line of work, studies which structural properties of lineage formulas lead to tractable cases~\cite{kenig-13-nclexpdc,roy-11-f,sen-10-ronfqevpd}.
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Several techniques for approximating tuple probabilities have been proposed in related work~\cite{FH13,heuvel-19-anappdsd,DBLP:conf/icde/OlteanuHK10,DS07,re-07-eftqevpd}, relying on Monte Carlo sampling, e.g., \cite{DS07,re-07-eftqevpd}, or a branch-and-bound paradigm~\cite{DBLP:conf/icde/OlteanuHK10,fink-11}.
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The approximation algorithm for bag expectation we present in this work is based on sampling.
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Fink et al.~\cite{FH12} study aggregate queries over a probabilistic version of the extension of K-relations for aggregate queries proposed in~\cite{AD11d} (this data model is referred to as \emph{pvc-tables}). As an extension of K-relations, this approach supports bags. Probabilities are computed using a decomposition approach~\cite{DBLP:conf/icde/OlteanuHK10} over the symbolic expressions that are used as tuple annotations and values in pvc-tables. \cite{FH12} identifies a tractable class of queries involving aggregation. In contrast, we study a less general data model and query class, but provide a linear time approximation algorithm and provide new insights into the complexity of computing expectation (while \cite{FH12} computes probabilities for individual output annotations).
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