105 lines
7.3 KiB
TeX
105 lines
7.3 KiB
TeX
%!TEX root= prob-def.tex
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\subsection{Deterministic Query Runtimes}\label{sec:gen}
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%We formalize our claim from \Cref{sec:intro} that a linear approximation algorithm for our problem implies that PDB queries (under bag semantics) can be answered (approximately) in the same runtime as deterministic queries under reasonable assumptions.
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%Lastly, we generalize our result for expectation to other moments.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\revision{
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%\subsection{Cost Model, Query Plans, and Runtime}
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%As in the introduction, we could consider polynomials to be represented as an expression tree.
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%However, they do not capture many of the compressed polynomial representations that we can get from query processing algorithms on bags, including the recent work on worst-case optimal join algorithms~\cite{ngo-survey,skew}, factorized databases~\cite{factorized-db}, and FAQ~\cite{DBLP:conf/pods/KhamisNR16}. Intuitively, the main reason is that an expression tree does not allow for `sharing' of intermediate results, which is crucial for these algorithms (and other query processing methods as well).
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%}
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%
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%\label{sec:circuits}
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%\mypar{The cost model}
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%\label{sec:cost-model}
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%So far our analysis of \Cref{prob:intro-stmt} has been in terms of the size of the lineage circuits.
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%We now show that this model corresponds to the behavior of a deterministic database by proving that for any \raPlus query $\query$, we can construct a compressed circuit for $\poly$ and \bi $\pdb$ of size and runtime linear in that of a general class of query processing algorithms for the same query $\query$ on $\pdb$'s \dbbaseName $\dbbase$.
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% Note that by definition, there exists a linear relationship between input sizes $|\pxdb|$ and $|\dbbase|$ (i.e., $\exists c, \db \in \pxdb$ s.t. $\abs{\pxdb} \leq c \cdot \abs{\db})$).
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% \footnote{This is a reasonable assumption because each block of a \bi represents entities with uncertain attributes.
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% In practice there is often a limited number of alternatives for each block (e.g., which of five conflicting data sources to trust). Note that all \tis trivially fulfill this condition (i.e., $c = 1$).}
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%That is for \bis that fulfill this restriction approximating the expectation of results of SPJU queries is only has a constant factor overhead over deterministic query processing (using one of the algorithms for which we prove the claim).
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% with the same complexity as it would take to evaluate the query on a deterministic \emph{bag} database of the same size as the input PDB.
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In~\Cref{sec:intro}, we introduced the function $T_{det}\inparen{\cdot}$ to analyze the runtime complexity of~\Cref{prob:expect-mult}.
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To decouple our results from any specific join algorithm, we first lower bound the cost of a join.
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\begin{Definition}[Join Cost]
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\label{def:join-cost}
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Denote by $\jointime{R_1, \ldots, R_m}$ the runtime of an algorithm for computing the $m$-ary join $R_1 \bowtie \ldots \bowtie R_m$.
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We require only that the algorithm must enumerate its output, i.e., that $\jointime{R_1, \ldots, R_m} \geq |R_1 \bowtie \ldots \bowtie R_m|$. %With this definition of $\jointime{\cdot}$, worst-case optimal join algorithms are handled.
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\end{Definition}
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Worst-case optimal join algorithms~\cite{skew,ngo-survey} and query evaluation via factorized databases~\cite{factorized-db} (as well as work on FAQs~\cite{DBLP:conf/pods/KhamisNR16} and AJAR~\cite{ajar}) can be modeled as $\raPlus$ queries (though the query size is data dependent).
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For these algorithms, $\jointime{R_1, \ldots, R_n}$ is linear in the {\em AGM bound}~\cite{AGM}.
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% = |R_1| + \ldots + |R_n| + |R_1(\db) \bowtie \ldots \bowtie R_n(\db)|$.
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Our cost model for general query evaluation follows:
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%\noindent\resizebox{1\linewidth}{!}{
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%\begin{minipage}{1.0\linewidth}
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{\small
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\begin{flalign*}
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&\begin{aligned}
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&\qruntimenoopt{R,\gentupset,\bound} = |\gentupset.R|
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&
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&\qquad\qquad\qruntimenoopt{\sigma \query, \gentupset,\bound} = \qruntimenoopt{\query,\gentupset,\bound}\\
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\end{aligned}&\\
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%\vspace{-.6cm}
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&\qruntimenoopt{\pi \query, \gentupset,\bound} = \qruntimenoopt{\query,\gentupset,\bound} +\abs{\query(\gentupset)}&\\
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&\qruntimenoopt{\query \cup \query', \gentupset,\bound} = \qruntimenoopt{\query, \gentupset,\bound} + \qruntimenoopt{\query', \gentupset,\bound} + \abs{\query\inparen{\gentupset}}+\abs{\query'\inparen{\gentupset}}& \\
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&\qruntimenoopt{\query_1 \bowtie \ldots \bowtie \query_m, \gentupset,\bound} = &\\
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&\qquad\qruntimenoopt{\query_1, \gentupset,\bound} + \ldots + \qruntimenoopt{\query_m,\gentupset,\bound} +
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\jointime{\query_1(\gentupset), \ldots, \query_m(\gentupset)}&\\
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\end{flalign*}
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}
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\vspace{-0.93cm}
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%\begin{align*}
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% &\qruntimenoopt{\query_1 \bowtie \ldots \bowtie \query_m, \gentupset,\bound} = \\
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% &\qquad\qruntimenoopt{\query_1, \gentupset,\bound} + \ldots + \qruntimenoopt{\query_m,\gentupset,\bound} +
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% \jointime{\query_1(\gentupset), \ldots, \query_m(\gentupset)}
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%\end{align*}\\%[-10mm]
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%\end{minipage}
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%}\\
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\noindent
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Under this model, an $\raPlus$ query $\query$ evaluated over over any deterministic database
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%\dbbaseName
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that maps each tuple in $\gentupset$ to a multiplicity in $[0,\bound]$ %database $\gentupset$
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has runtime $O\inparen{\qruntimenoopt{Q,\gentupset, \bound}}$.
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We assume that full table scans are used for every base relation access. We can model index scans by treating an index scan query $\sigma_\theta(R)$ as a base relation.
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%Observe that
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% () .\footnote{This claim can be verified by e.g. simply looking at the {\em Generic-Join} algorithm in~\cite{skew} and {\em factorize} algorithm in~\cite{factorized-db}.} It can be verified that the above cost model on the corresponding $\raPlus$ join queries correctly captures the runtime of current best known .
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\Cref{lem:circ-model-runtime} and \Cref{lem:tlc-is-the-same-as-det} show that for any $\raPlus$ query $\query$ and \dbbaseName $\tupset$, there exists a circuit $\circuit^*$ such that $\timeOf{\abbrStepOne}(Q,\tupset,\circuit^*)$ and $|\circuit^*|$ are both $O(\qruntimenoopt{\optquery{\query}, \tupset,\bound})$, as we assumed when moving from \Cref{prob:big-o-joint-steps} to \Cref{prob:intro-stmt}. Lastly, we can handle FAQs/AJAR queries and factorized databases by allowing for optimization. %, i.e. $\qruntimenoopt{\optquery{\query}, \gentupset, \bound}$.
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%
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%We now make a simple observation on the above cost model:
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%\begin{proposition}
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%\label{prop:queries-need-to-output-tuples}
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%The runtime $\qruntimenoopt{Q}$ of any query $Q$ is at least $|Q|$
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%\end{proposition}
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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%We are now ready to formally state our claim with respect to \Cref{prob:intro-stmt}:
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\begin{Corollary}\label{cor:cost-model}
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% Given an $\raPlus$ query $\query$ over a \ti $\pdb$ with \dbbaseName $\dbbase$, we can compute a $(1\pm\eps)$-approximation of the expectation for each output tuple in $\query(\pdb)$ with probability at least $1-\delta$ in time
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%
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% \[
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% O_k\left(\frac 1{\eps^2}\cdot\qruntimenoopt{Q,\dbbase}\cdot \log{\frac{1}{\conf}}\cdot \log(n)\right)
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% \]
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%\end{Corollary}
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%Atri: The above is no longer needed
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "main"
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%%% End:
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