ODIn/paper-BagRelationalPDBsAreHard
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karp-luby discussion

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Oliver Kennedy 2021-09-20 21:56:06 -04:00
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@ -357,7 +357,7 @@ $$\probOf\inparen{\vct{W}_1 \vee \ldots \vee\vct{W}_\ell} = \probOf\inparen{\vct
The Karp-Luby estimator is employed on the \abbrSOP representation of $\circuit$, where each $W_i$ represents the event that one monomial is true.
By simple inspection, if there are $\ell$ monomials, this estimator has runtime $\Omega(\ell)$. Further, a minimum of $\lceil\frac{3\cdot n\cdot \log(\frac{2}{\delta})}{\epsilon^2}\rceil$ invocations of the estimator are required to achieve $\epsilon-\delta$ bound~\cite{DBLP:conf/icde/OlteanuHK10}, entailing a runtime at least quadratic in $\ell$.
As an arbitrary lineage circuit $\circuit$ may encode $O()$ monomials, the total runtime is at least $\Omega(|\circuit|^{2k})$.
As an arbitrary lineage circuit $\circuit$ may encode $O(|\circuit|^k)$ monomials, the total runtime is at least $\Omega(|\circuit|^{2k})$.