Update on Overleaf.

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Atri Rudra 2022-06-08 02:48:58 +00:00 committed by node
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@ -242,7 +242,7 @@ We adopt the two-step intensional model of query evaluation used in set-\abbrPDB
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(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 to answer if we can compute a $(1\pm \eps)$-approximation to the expected multiplicity
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 a $(1\pm \eps)$-approximation to the expected multiplicity, it is enough to answer the following:
%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}