49 lines
4.9 KiB
TeX
49 lines
4.9 KiB
TeX
%!TEX root=./main.tex
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\section{Related Work}\label{sec:related-work}
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%\subsection{Probabilistic Databases}\label{sec:prob-datab}
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\textbf{Probabilistic Databases} (PDBs) have been studied predominantly for set semantics.
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Many data models have been proposed for encoding PDBs more compactly than as sets of possible worlds.
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These include tuple-independent databases~\cite{VS17} (\tis), block-independent databases (\bis)~\cite{RS07}, and \emph{PC-tables}~\cite{GT06} pair a C-table % ~\cite{IL84a}
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with probability distribution over its variables.
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This is similar to our $\semNX$-PDBs, with Boolean expressions instead of polynomials.
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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
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% \emph{PC-tables}~\cite{GT06} pair a C-table % ~\cite{IL84a}
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% 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 marginal probabilities for 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)
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and then the probability of the lineage formula.
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In this paper we focus on intensional query evaluation with polynomials.
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It has been shown that computing the marginal probability of a tuple is \sharpphard~\cite{valiant-79-cenrp} (by reduction from weighted model counting).
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The second category, \emph{extensional} query evaluation, % avoids calculating the lineage.
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% This approach
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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):
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for any UCQ the probabilistic query evaluation problem is either \sharpphard (requires extensional evaluation) or \ptime (permits 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}, relying on Monte Carlo sampling, e.g.,~\cite{DS07}, or a branch-and-bound paradigm~\cite{DBLP:conf/icde/OlteanuHK10}.
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Our approximation algorithm is also 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.
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% \cite{FH12} identifies a tractable class of queries involving aggregation.
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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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\noindent \textbf{Compressed Encodings} are used for Boolean formulas (e.g, various types of circuits including OBDDs~\cite{jha-12-pdwm}) and polynomials (e.g., factorizations~\cite{factorized-db}) some of which have been utilized for probabilistic query processing, e.g.,~\cite{jha-12-pdwm}.
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Compact representations for which probabilities can be computed in linear time include OBDDs, SDDs, d-DNNF, and FBDD.
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\cite{DM14c} studies circuits for absorptive semirings while~\cite{S18a} studies circuits that include negation (expressed as the monus operation). Algebraic Decision Diagrams~\cite{bahar-93-al} (ADDs) generalize BDDs to variables with more than two values. Chen et al.~\cite{chen-10-cswssr} introduced the generalized disjunctive normal form.
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\noindent \Cref{sec:param-compl} covers more related work on fine-grained complexity.
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