88 lines
6.7 KiB
TeX
88 lines
6.7 KiB
TeX
%!TEX root=./main.tex
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\section{More on Circuits and Moments}\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 $\approxq$ 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 $\pxdb$ of size and runtime linear in that of a general class of query processing algorithms for the same query $\query$ on a deterministic database $\db$.
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We assume a linear relationship between input sizes $|\pxdb|$ and $|\db|$ (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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We adopt a minimalistic compute-bound model of query evaluation drawn from the worst-case optimal join literature~\cite{skew,ngo-survey}.
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\newcommand{\qruntime}[1]{\textbf{cost}(#1)}
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%
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\noindent\resizebox{1\linewidth}{!}{
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\begin{minipage}{1.0\linewidth}
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\begin{align*}
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\qruntime{R,D} & = |R| &
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\qruntime{\sigma Q, D} & = \qruntime{Q,D} &
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\qruntime{\pi Q, D} & = \qruntime{Q,D} + \abs{Q(D)}
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\end{align*}\\[-15mm]
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\begin{align*}
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\qruntime{Q \cup Q', D} & = \qruntime{Q, D} + \qruntime{Q', D} +\abs{Q(D)}+\abs{Q'(D)} \\
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\qruntime{Q_1 \bowtie \ldots \bowtie Q_n, D} & = \qruntime{Q_1, D} + \ldots + \qruntime{Q_n,D} + \abs{Q_1(D) \bowtie \ldots \bowtie Q_n(D)}
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\end{align*}
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\end{minipage}
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}\\
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Under this model a query $Q$ evaluated over database $D$ has runtime $O(\qruntime{Q,D})$.
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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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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 SPJU join queries correctly captures their runtime.
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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 $\qruntime{Q}$ 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 from \Cref{sec:intro}:
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{Corollary}
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Given an SPJU query $Q$ over a \ti $\pxdb$ and let $\db_{max}$ denote the world containing all tuples of $\pxdb$, we can compute a $(1\pm\eps)$-approximation of the expectation for each output tuple in $\query(\pxdb)$ with probability at least $1-\delta$ in time
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%
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\[
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O_k\left(\frac 1{\eps^2}\cdot\qruntime{Q,\db_{max}}\cdot \log{\frac{1}{\conf}}\cdot \log(n)\right)
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\]
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\end{Corollary}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{proof}
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This follows from \Cref{lem:circuits-model-runtime} (\Cref{sec:circuit-runtime}) and \Cref{cor:approx-algo-const-p} (where the latter is used with $\delta$ being substituted\footnote{Recall that \Cref{cor:approx-algo-const-p} is stated for a single output tuple so to get the required guarantee for all (at most $n^k$) output tuples of $Q$ we get at most $\frac \delta{n^k}$ probability of failure for each output tuple and then just a union bound over all output tuples. } with $\frac \delta{n^k}$).
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\end{proof}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mypar{Higher Moments}
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%\label{sec:momemts}
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%
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We make a simple observation to conclude the presentation of our results.
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So far we have only focused on the expectation of $\poly$.
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In addition, we could e.g. prove bounds of the probability of a tuple's multiplicity being at least $1$.
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Progress can be made on this as follows:
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For any positive integer $m$ we can compute the $m$-th moment of the multiplicities, allowing us to e.g. use the Chebyschev inequality or other high moment based probability bounds on the events we might be interested in.
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We leave further investigations for future work.
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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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