From 684c88fd860da4393e1402b7cac15aaa06dba8e5 Mon Sep 17 00:00:00 2001 From: Atri Rudra Date: Wed, 8 Jun 2022 02:48:43 +0000 Subject: [PATCH] Update on Overleaf. --- introduction.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/introduction.tex b/introduction.tex index feb574f..bf55066 100644 --- a/introduction.tex +++ b/introduction.tex @@ -240,9 +240,9 @@ Our negative results (\Cref{tab:lbs}) indicate that \abbrCTIDB{}s (even for $\bo We adopt the two-step intensional model of query evaluation used in set-\abbrPDB\xplural, as illustrated in \Cref{fig:two-step}: (i) \termStepOne (\abbrStepOne): Given input $\tupset$ and $\query$, output every tuple $\tup$ that possibly satisfies $\query$, annotated with its lineage polynomial $\poly(\vct{X})%=\textcolor{red}{CHANGE}\apolyqdt\inparen{\vct{X}}$ $; -(ii) \termStepTwo (\abbrStepTwo): Given $\poly(\vct{X})$ for each tuple, compute an $(1\pm \eps)$-approximation $\expct_{\randWorld\sim\bpd}\pbox{\poly(\vct{\randWorld})}$. +(ii) \termStepTwo (\abbrStepTwo): Given $\poly(\vct{X})$ for each tuple, compute a $(1\pm \eps)$-approximation $\expct_{\randWorld\sim\bpd}\pbox{\poly(\vct{\randWorld})}$. Let $\timeOf{\abbrStepOne}(\query,\tupset,\circuit)$ denote the runtime of \abbrStepOne when it outputs $\circuit$ (a representation of $\poly$ as an arithmetic circuit --- more on this representation in~\Cref{sec:expression-trees}). -Denote by $\timeOf{\abbrStepTwo}(\circuit, \epsilon)$ (recall $\circuit$ is the output of \abbrStepOne) the runtime of \abbrStepTwo (when $\poly$ is input as $\circuit$). Then +Denote by $\timeOf{\abbrStepTwo}(\circuit, \epsilon)$ (recall $\circuit$ is the output of \abbrStepOne) the runtime of \abbrStepTwo (when $\poly$ is input as $\circuit$). Then to answer if we can compute %which we can leverage~\Cref{def:reduced-poly} and~\Cref{lem:tidb-reduce-poly} to address the next formal objective: \begin{Problem}[\abbrCTIDB linear time approximation]\label{prob:big-o-joint-steps}