approx algo desc.

app_approx-alg-analysis.tex

@@ -26,7 +26,7 @@ We prove \Cref{lem:approx-alg} constructively by presenting an algorithm \approx
 
 \begin{Lemma}\label{lem:one-pass}
 The $\onepass$ function completes in time:
-$$O\left(\size(\circuit) \cdot \multc{\log\left(\abs{\circuit(1\ldots, 1)}\right)}{\log{\size(\circuit}}\right)$$
+$$O\left(\size(\circuit) \cdot \multc{\log\left(\abs{\circuit(1\ldots, 1)}\right)}{\log{\size(\circuit)}}\right)$$
   $\onepass$ guarantees two post-conditions:  First, for each subcircuit $\vari{S}$ of $\circuit$, we have that $\vari{S}.\vari{partial}$ is set to $\abs{\vari{S}}(1,\ldots, 1)$.  Second, when $\vari{S}.\type  = \circplus$, \subcircuit.\lwght $= \frac{\abs{\subcircuit_\linput}(1,\ldots, 1)}{\abs{\subcircuit}(1,\ldots, 1)}$ and likewise for \subcircuit.\rwght.
 \end{Lemma}
 To prove correctness of \Cref{alg:mon-sam}, we only use the following fact that follows from the above lemma: for the modified circuit ($\circuit_{\vari{mod}}$) output by \onepass, $\circuit_{\vari{mod}}.\vari{partial}=\abs{\circuit}(1,\dots,1)$.