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Small changes to Section 5 (Aaron)

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Aaron Huber 2021-04-08 12:45:35 -04:00
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circuits-model-runtime.tex
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@ -95,10 +95,10 @@ We now connect the size of a circuit (where the size of a circuit is the number
for a given SPJU query $Q$ and $\semNX$-PDB $\pxdb$ to its $\qruntime{Q,\db}$ where $\db$ is one of the possible worlds of $\pxdb$. We do this formally by showing that the size of the circuit is asymptotically no worse than the corresponding runtime of a large class of deterministic query processing algorithms.
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\begin{lemma}
\begin{Lemma}
\label{lem:circuits-model-runtime}
Given a $\semNX$-PDB $\pxdb$ and query plan $Q$, the runtime of $Q$ over $\bagdbof$ has the same or better complexity as the size of the lineage of $Q(\pxdb)$. That is, we have $\abs{V_{Q,\pxdb}} \leq (k-1)\qruntime{Q}$, where $k$ is the maximal degree of any polynomial in $Q(\pxdb)$.
\end{lemma}
Given a $\semNX$-PDB $\pxdb$ and query plan $Q$, the runtime of $Q$ over $\pxdb$ has the same or better complexity as the size of the lineage of $Q(\pxdb)$. That is, we have $\abs{V_{Q,\pxdb}} \leq (k-1)\qruntime{Q}$, where $k$ is the maximal degree of any polynomial in $Q(\pxdb)$.
\end{Lemma}
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\noindent The proof is shown in in~\Cref{app:subsec-lem-lin-vs-qplan}.
We now have all the pieces to argue that using our approximation algorithm, the expected multiplicities of a SPJU query can be computed in essentially the same runtime as deterministic query processing for the same query:
@ -112,8 +112,7 @@ We now have all the pieces to argue that using our approximation algorithm, the
\end{Corollary}
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\begin{proof}
\OK{Invalid reference}
This follows from~\Cref{lem:circuits-model-runtime} and (the circuit counterpart-- see~\Cref{sec:results-circuits})~\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 $\frac \delta{n^k}$).
This follows from~\Cref{lem:circuits-model-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 $\frac \delta{n^k}$).
\end{proof}
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