Done with pass on Sec 4

approx_alg.tex

@@ -100,7 +100,7 @@ In the subsequent subsections we will prove the following theorem.
 \begin{Theorem}\label{lem:approx-alg}
 Let \circuit be a circuit for a UCQ over \bi and define $\poly(\vct{X})=\polyf(\circuit)$ and let $k=\degree(\circuit)$.
 Then an estimate $\mathcal{E}$ of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ can be computed in time
-\[O\left(\size(\circuit) + \frac{\log{\frac{1}{\conf}}\cdot \abs{\circuit}^2(1,\ldots, 1)\cdot  k\cdot \log{k} \cdot \depth(\circuit))}{\inparen{\error'}^2\cdot\rpoly^2(\prob_1,\ldots, \prob_\numvar)}\right)\]
+\[O\left(\left(\size(\circuit) + \frac{\log{\frac{1}{\conf}}\cdot \abs{\circuit}^2(1,\ldots, 1)\cdot  k\cdot \log{k} \cdot \depth(\circuit))}{\inparen{\error'}^2\cdot\rpoly^2(\prob_1,\ldots, \prob_\numvar)}\right)\cdot\multc{\log\left(\abs{\circuit}^2(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)\]
 such that
 \begin{equation}
 \label{eq:approx-algo-bound}
@@ -121,8 +121,8 @@ Given an expression tree $\circuit$, define
 \begin{Corollary}
 \label{cor:approx-algo-const-p}
 Let $\poly(\vct{X})$ be as in~\Cref{lem:approx-alg} and let $\gamma=\gamma(\circuit)$. Further let it be the case that $\prob_i\ge \prob_0$ for all $i\in[\numvar]$. Then an estimate $\mathcal{E}$  of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ satisfying~\Cref{eq:approx-algo-bound} can be computed in time
-\[O\left(\size(\circuit) + \frac{\log{\frac{1}{\conf}}\cdot k\cdot \log{k} \cdot \depth(\circuit))}{\inparen{\error'}^2\cdot(1-\gamma)^2\cdot \prob_0^{2k}}\right)\]
-In particular, if $\prob_0>0$ and $\gamma<1$ are absolute constants then the above runtime simplifies to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)$.
+\[O\left(\left(\size(\circuit) + \frac{\log{\frac{1}{\conf}}\cdot k\cdot \log{k} \cdot \depth(\circuit))}{\inparen{\error'}^2\cdot(1-\gamma)^2\cdot \prob_0^{2k}}\right)\cdot\multc{\log\left(\abs{\circuit}^2(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)\]
+In particular, if $\prob_0>0$ and $\gamma<1$ are absolute constants then the above runtime simplifies to $O_k\left(\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)\cdot\multc{\log\left(\abs{\circuit}^2(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)$.
 \end{Corollary}
 
 The proof for~\Cref{cor:approx-algo-const-p} can be seen in~\Cref{sec:proofs-approx-alg}.
@@ -130,6 +130,22 @@ The restriction on $\gamma$ is satisfied by any \ti (where $\gamma=0$) as well a
 We also observe that (i) tuple presence is independent across blocks, so the corresponding probabilities (and hence $\prob_0$) are independent of the number of blocks, and (ii) \bis model uncertain attributes, so block size (and hence $\gamma$) is a function of the ``messiness'' of a dataset, rather than its size.
 Thus, we expect the corollary to hold in general.
 
+Finally, we address the $\multc{\log\left(\abs{\circuit}^2(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}$ term in the runtime. In Appendix\revision{Fill in ref later on}, we show the following:
+\begin{Lemma}
+\label{lem:val-ub}
+For any circuit $\circuit$ with $\degree(\circuit)=k$, we have
+\[\abs{\circuit}^2(1,\ldots, 1)\le 2^{O(k\size(\circuit))}.\]
+Further, under the following conditions:
+\begin{enumerate}
+\item $\circuit$ is a tree,
+\item $\circuit$ is the output of a FAQ query from algorithm in~\cite{DBLP:conf/pods/KhamisNR16},
+\end{enumerate}
+we have
+\[\abs{\circuit}^2(1,\ldots, 1)\le  \size(\circuit)^{O(k)}.\]
+\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.
+
 \subsection{Approximating $\rpoly$}
 
 The algorithm to prove~\Cref{lem:approx-alg} follows from the following observation. Given a query polynomial $\poly(\vct{X})=\polyf(\circuit)$ for circuit \circuit over $\bi$, we can exactly represent $\rpoly(\vct{X})$ as follows:
@@ -181,19 +197,22 @@ we first state the lemmas that summarize the relevant properties of $\onepass$ a
 
 
 \begin{Lemma}\label{lem:one-pass}
-The $\onepass$ function completes in $O(size(\circuit) \cdot \frac{\log{\abs{\circuit(1\ldots, 1)}}}{\log{N}})$ time, where $N = \size(\circuit)$.\footnote{In the appendix we give a sufficient condition when $\abs{\circuit}(1,\ldots, 1)$ is indeed $O(1)$ in arithmetic computations.  Most notably, WCOJ and FAQ results are not affected by the general runtime of arithmetic computations, a point which we also address in the appendix.}  $\onepass$ 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.
+The $\onepass$ function completes in $O\left(size(\circuit) \cdot \multc{\log\left(\abs{\circuit(1\ldots, 1)}\right)}{\log{\size(\circuit}}\right)$ time. %, where $N = \size(\circuit)$.\footnote{In the appendix we give a sufficient condition when $\abs{\circuit}(1,\ldots, 1)$ is indeed $O(1)$ in arithmetic computations.  Most notably, WCOJ and FAQ results are not affected by the general runtime of arithmetic computations, a point which we also address in the appendix.}
+  $\onepass$ 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 ($\circuit_{\vari{mod}}$), $\circuit_{\vari{mod}}$, $\circuit_{\vari{mod}}.\vari{partial}=\abs{\circuit}(1,\dots,1)$.
 
 \begin{Lemma}\label{lem:sample}
-The function $\sampmon$ completes in $O(\log{k} \cdot k \cdot \depth(\circuit)\cdot\frac{\log{\abs{\circuit}(1,\ldots, 1)}}{\log{\size(\circuit)}})$ time\footnote{Note that the same sufficient condition on \circuit to guarentee $O(1)$ arithmetic computations applies here, and when this condition is met, the runtime loses the $\frac{\log{\abs{\circuit}(1,\ldots, 1)}}{\log{\size(\circuit)}}$ factor}, where $k = \degree(\circuit)$.  Upon completion, every $\left(\monom, sign(\coef)\right)\in \expansion{\abs{\circuit}}$ is returned with probability $\frac{|\coef|}{\abs{\circuit}(1,\ldots, 1)}$. 
+The function $\sampmon$ completes in $O(\log{k} \cdot k \cdot \depth(\circuit)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log{\size(\circuit)}})$ time,
+%\footnote{Note that the same sufficient condition on \circuit to guarentee $O(1)$ arithmetic computations applies here, and when this condition is met, the runtime loses the $\frac{\log{\abs{\circuit}(1,\ldots, 1)}}{\log{\size(\circuit)}}$ factor},
+ where $k = \degree(\circuit)$.  Upon completion, every $\left(\monom, sign(\coef)\right)\in \expansion{\abs{\circuit}}$ is returned with probability $\frac{|\coef|}{\abs{\circuit}(1,\ldots, 1)}$. 
 \end{Lemma}
 
-Armed with the above two lemmas, we are ready to argue the following result (proof in~\Cref{sec:proofs-approx-alg}):
+With the above two lemmas, we are ready to argue the following result (proof in~\Cref{sec:proofs-approx-alg}):
 \begin{Theorem}\label{lem:mon-samp}
 For any $\circuit$ with $\degree(poly(|\circuit|)) = k$, algorithm \ref{alg:mon-sam} outputs an estimate $\vari{acc}$ of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ such that
 \[\probOf\left(\left|\vari{acc} - \rpoly(\prob_1,\ldots, \prob_\numvar)\right|> \error \cdot \abs{\circuit}(1,\ldots, 1)\right) \leq \conf,\]
- in $O\left(\size(\circuit)\right.$ $+$ $\left.\left(\frac{\log{\frac{1}{\conf}}}{\error^2} \cdot k \cdot\log{k} \cdot \depth(\circuit)\right)\right)$ time.
+ in $O\left(\left(\size(\circuit)+\frac{\log{\frac{1}{\conf}}}{\error^2} \cdot k \cdot\log{k} \cdot \depth(\circuit)\right)\cdot \multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log{\size(\circuit)}}\right)$ time.
 \end{Theorem}