%!TEX root=./main.tex

\subsection{More on Circuits and Moments}\label{sec:gen}
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.
Lastly, we generalize our result for expectation to other moments.

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%\revision{
%\subsection{Cost Model, Query Plans, and Runtime}
%As in the introduction, we could consider polynomials to be represented as an expression tree.
%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).
%}
%
%\label{sec:circuits}

\mypar{The cost model}
%\label{sec:cost-model}
So far our analysis of $\approxq$ has been in terms of the size of the lineage circuits.
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 $\dbbase$.  
% Note that by definition, there exists a linear relationship between input sizes $|\pxdb|$ and $|\dbbase|$ (i.e., $\exists c, \db \in \pxdb$ s.t. $\abs{\pxdb} \leq c \cdot \abs{\db})$).
% \footnote{This is a reasonable assumption because each block of a \bi represents entities with uncertain attributes.
% 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., $c = 1$).}
%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).
% 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.
We adopt a minimalistic compute-bound model of query evaluation drawn from the worst-case optimal join literature~\cite{skew,ngo-survey}.

%
\noindent\resizebox{1\linewidth}{!}{
\begin{minipage}{1.0\linewidth}
  \begin{align*}
\qruntime{R,\dbbase}                               & = |\dbbase.R|                                                        &
                                                                                                              \qruntime{\sigma Q, \dbbase}                       & = \qruntime{Q,\dbbase}                                             &
                                                                                                                                                                                                                            \qruntime{\pi Q, \dbbase}                          & = \qruntime{Q,\dbbase} + \abs{Q(D)}
  \end{align*}\\[-15mm]
\begin{align*}
\qruntime{Q \cup Q', \dbbase}                      & = \qruntime{Q, \dbbase} + \qruntime{Q', \dbbase} +\abs{Q(D)}+\abs{Q'(D)} \\
\qruntime{Q_1 \bowtie \ldots \bowtie Q_n, \dbbase} & = \qruntime{Q_1, \dbbase} + \ldots + \qruntime{Q_n,\dbbase} + \abs{Q_1(D) \bowtie \ldots \bowtie Q_n(D)}
\end{align*}
\end{minipage}
}\\

Under this model a query $Q$ evaluated over database $\dbbase$ has runtime $O(\qruntime{Q,\dbbase})$.
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.

It can be verified that worst-case optimal join algorithms~\cite{skew,ngo-survey}, as well as query evaluation via factorized databases~\cite{factorized-db}\AR{See my comment on element on whether we should include this ref or not.} (and work on FAQs~\cite{DBLP:conf/pods/KhamisNR16}) can be modeled as select-union-project-join queries (though the size of these queries is data dependent).\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 their runtime.
%
%We now make a simple observation on the above cost model:
%\begin{proposition}
%\label{prop:queries-need-to-output-tuples}
%The runtime $\qruntime{Q}$ of any query $Q$ is at least $|Q|$
%\end{proposition}
%
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%
We are now ready to formally state our claim from \Cref{sec:intro}:
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\begin{Corollary}\label{cor:cost-model}
  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
%
  \[
    O_k\left(\frac 1{\eps^2}\cdot\qruntime{Q,\dbbase}\cdot \log{\frac{1}{\conf}}\cdot \log(n)\right)
    \]
\end{Corollary}
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