Updated definiton of deg(C)

app_approx-alg-analysis.tex

@@ -61,7 +61,7 @@ The above two inequalities implies $\rpoly(1,\dots,1) \ge \prob_0^k\cdot (1-\gam
 Applying this bound in the runtime bound in~\Cref{lem:approx-alg} gives the first claimed runtime. The final runtime of $O_k\left(\frac 1{\eps^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\cdot \multc{\log\left(\abs{\circuit}^2(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)$ follows by noting that $\depth({\circuit})\le \size({\circuit})$ and absorbing all factors that just depend on $k$.
 
 \subsection{Proof of~\Cref{lem:val-ub}}
-
+\label{app:proof-lem-val-ub}
 
 %\paragraph{Sufficient condition for $\abs{\circuit}(1,\ldots, 1)$ to be size $O(N)$}
 %For our runtime results to be relevant, it must be the case that the sum of the coefficients computed by \onepass is indeed size $O(N)$ since there are $O(\log{N})$ bits in the RAM model where $N$ is the size of the input.  The size of the input here is \size(\circuit).  We show that when \size$(\circuit_\linput) = N_\linput$, \size$(\circuit_\rinput) = N_\rinput$, where $N_\linput + N_\rinput \leq N$, this is indeed the case.

approx_alg.tex

@@ -81,8 +81,20 @@ The function \depth~ has circuit $\circuit$ as input and outputs the number of l
 \end{Definition}
 
 \begin{Definition}[$\degree(\cdot)$]
-The function $\degree(\cdot)$ takes a circuit \circuit as input and outputs the degree of $\polyf(\abs{\circuit})$.
+\revision{
+$\deg(\circuit)$ is defined recursively as follows:
+\[\deg(\circuit)=
+\begin{cases}
+\max(\deg(\circuit_\linput),\deg(\circuit_\rinput)) & \text{ if }\circuit.type=+\\
+\deg(\circuit_\linput) + \deg(\circuit_\rinput)+1 &\text{ if }\circuit.type=\times\\
+0 & \text{otherwise}.
+\end{cases}
+\]
+}
+%If $\circuit$ has no $+$ or $\times$ gate, then $\deg(\circuit)=0$. Otherwise if 
+%The function $\degree(\cdot)$ takes a circuit \circuit as input and outputs the degree of $\polyf(\abs{\circuit})$.
 \end{Definition}
+\revision{Note that the degree of $\polyf(\abs{\circuit})$ is always upper bounded by $\deg(\circuit)$ and the latter can be strictly larger (e.g. consider the case when $\circuit$ multiplies two copies of the constant $1$-- here we have $\deg(\circuit)=1$ but degree of $\polyf(\abs{\circuit})$ is $0$).}
 
 \begin{Definition}[Subcircuit]
 A subcircuit of a circuit $\circuit$ is a circuit \subcircuit such that \subcircuit is a DAG \textit{subgraph} of the DAG representing \circuit.  The sink of \subcircuit has exactly one gate \gate.
@@ -134,7 +146,7 @@ Finally, we address the $\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\l
 \begin{Lemma}
 \label{lem:val-ub}
 For any circuit $\circuit$ with $\degree(\circuit)=k$, we have
-\[\abs{\circuit}(1,\ldots, 1)\le 2^{O(k\size(\circuit))}.\]
+\[\abs{\circuit}(1,\ldots, 1)\le 2^{2^k\cdot \size(\circuit)}.\]
 Further, under the following conditions:
 \begin{enumerate}
 \item $\circuit$ is a tree,
@@ -144,7 +156,7 @@ we have
 \[\abs{\circuit}(1,\ldots, 1)\le  \size(\circuit)^{O(k)}.\]
 \end{Lemma}
 
-Note that the above implies that with the assumption $\prob_0>0$ and $\gamma<1$ are absolute constants from Corollary~\Cref{cor:approx-algo-const-p}, then the runtime there simplies to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)^2\cdot \log{\frac{1}{\conf}}\right)$ for general circuits $\circuit$ and to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)$ for the case when $\circuit$ satisfies the special conditions in~\Cref{lem:val-ub}. In Appendix\revision{Fill in ref later on} we argue that these conditions are very general and encompass many interesting scenarios.
+Note that the above implies that with the assumption $\prob_0>0$ and $\gamma<1$ are absolute constants from Corollary~\Cref{cor:approx-algo-const-p}, then the runtime there simplies to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)^2\cdot \log{\frac{1}{\conf}}\right)$ for general circuits $\circuit$ and to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)$ for the case when $\circuit$ satisfies the special conditions in~\Cref{lem:val-ub}. In~\Cref{app:proof-lem-val-ub} we argue that these conditions are very general and encompass many interesting scenarios.
 
 \subsection{Approximating $\rpoly$}