Done till proof of Thm 4.18

app_approx-alg-analysis.tex

@@ -6,9 +6,9 @@ Before proving~\Cref{lem:mon-samp}, we use it to argue our main result,~\Cref{le
 
 Set $\mathcal{E}=\approxq(\revision{\circuit}, (\prob_1,\dots,\prob_\numvar),$ $\conf, \error')$, where
 \[\error' = \error \cdot \frac{\rpoly(\prob_1,\ldots, \prob_\numvar)\cdot (1 - \gamma)}{\abs{\revision{\circuit}}(1,\ldots, 1)},\]
- which achieves the claimed accuracy bound on $\mathcal{E}$.
+ which achieves the claimed accuracy bound on $\mathcal{E}$ due to~\Cref{lem:mon-samp}.
 
-The claim on the runtime follows since
+The claim on the runtime follows from~\Cref{lem:mon-samp} since
 \begin{align*}
 \frac 1{\inparen{\error'}^2}\cdot \log\inparen{\frac 1\conf}=&\frac{\log{\frac{1}{\conf}}}{\error^2 \left(\frac{\rpoly(\prob_1,\ldots, \prob_N)}{\abs{\revision{\circuit}}(1,\ldots, 1)}\right)^2}\\
 = &\frac{\log{\frac{1}{\conf}}\cdot \abs{\revision{\circuit}}^2(1,\ldots, 1)}{\error^2 \cdot \rpoly^2(\prob_1,\ldots, \prob_\numvar)},
@@ -46,10 +46,10 @@ Using Hoeffding's inequality, we then get:
 \end{equation*}
 where the last inequality follows from our choice of $\samplesize$ in~\Cref{alg:mon-sam-global2}.
 
-This concludes the proof for the first claim of theorem ~\ref{lem:mon-samp}.
+This concludes the proof for the first claim of theorem ~\ref{lem:mon-samp}. We prove the claim  on the runtime next.
 
-\paragraph{Run-time Analysis}
-The runtime of the algorithm is dominated by~\Cref{alg:mon-sam-onepass} (which by~\Cref{lem:one-pass} takes time $O(\revision{\size(\circuit)})$) and the $\samplesize$ iterations of the loop in~\Cref{alg:sampling-loop}. Each iteration's run time is dominated by the call to~\Cref{alg:mon-sam-sample} (which by~\Cref{lem:sample} takes $O(\log{k} \cdot k \cdot \revision{\depth(\circuit)})$
+\paragraph*{Run-time Analysis}
+The runtime of the algorithm is dominated by~\Cref{alg:mon-sam-onepass} (which by~\Cref{lem:one-pass} takes time $O\left({\size(\circuit)}\cdot \multc{\log\left(\abs{\circuit}^2(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)$) and the $\samplesize$ iterations of the loop in~\Cref{alg:sampling-loop}. Each iteration's run time is dominated by the call to~\Cref{alg:mon-sam-sample} (which by~\Cref{lem:sample} takes $O\left(\log{k} \cdot k \cdot {\depth(\circuit)}\cdot \multc{\log\left(\abs{\circuit}^2(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)$
 ) and~\Cref{alg:check-duplicate-block}, which by the subsequent argument takes $O(k\log{k})$ time. We sort the $O(k)$ variables by their block IDs and then check if there is a duplicate block ID or not. Adding up all the times discussed here gives us the desired overall runtime.
 
 \subsection{Proof of~\Cref{cor:approx-algo-const-p}}
@@ -58,4 +58,4 @@ The result follows by first noting that by definition of $\gamma$, we have
 Further, since each $\prob_i\ge \prob_0$ and $\poly(\vct{X})$ (and hence $\rpoly(\vct{X})$) has degree at most $k$, we have that
 \[ \rpoly(1,\dots,1) \ge \prob_0^k\cdot \rpoly(1,\dots,1).\]
 The above two inequalities implies $\rpoly(1,\dots,1) \ge \prob_0^k\cdot (1-\gamma)\cdot \abs{\revision{\circuit}}(1,\dots,1)$.
-Applying this bound in the runtime bound in~\Cref{lem:approx-alg} gives the first claimed runtime. The final runtime of $O_k\left(\frac 1{\eps^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)$ follows by noting that $depth(\revision{\circuit})\le \size(\revision{\circuit})$ and absorbing all factors that just depend on $k$.
+Applying this bound in the runtime bound in~\Cref{lem:approx-alg} gives the first claimed runtime. The final runtime of $O_k\left(\frac 1{\eps^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)$ follows by noting that $depth(\revision{\circuit})\le \size(\revision{\circuit})$ and absorbing all factors that just depend on $k$.