Done with pass on S4

approx_alg.tex

@@ -79,9 +79,9 @@ Given a lineage polynomial $\poly(\vct{X})=\polyf(\circuit)$ for circuit \circui
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{equation}
 \label{eq:tilde-Q-bi}
-\rpoly\inparen{X_1,\dots,X_\numvar}=\hspace*{-1mm}\sum_{(\monom,\coef)\in \expansion{\circuit}} %\hspace*{-2mm}
+\rpoly\inparen{p_1,\dots,p_\numvar}=\hspace*{-1mm}\sum_{(\monom,\coef)\in \expansion{\circuit}} %\hspace*{-2mm}
 \indicator{\isInd{\encMon}%\mod{\mathcal{B}}\not\equiv 0
-}\cdot \coef\cdot\hspace*{-2mm}\prod_{X_i\in \monom}\hspace*{-2mm} X_i.
+}\cdot \coef\cdot\hspace*{-2mm}\prod_{X_i\in \monom}\hspace*{-2mm} p_i.
 \end{equation}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
@@ -116,7 +116,7 @@ and computing the average of $\vari{Y}$ gives us our final estimate. \onepass is
 Let \circuit be an arbitrary \abbrBIDB circuit %for a UCQ over \bi 
 and define $\poly(\vct{X})=\polyf(\circuit)$ and let $k=\degree(\circuit)$.
 %Let $\poly(\vct{X})$ be as in \Cref{lem:approx-alg} and 
-Let $\gamma=\gamma(\circuit)$ for \abbrBIDB circuit \circuit. Further let it be the case that $\prob_i\ge \prob_0$ for all $i\in[\numvar]$. Then an estimate $\mathcal{E}$  of $\rpoly(\prob_1,\ldots, \prob_\numvar)$ 
+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 
 \begin{equation}
 \label{eq:approx-algo-bound-main}
@@ -132,7 +132,7 @@ In particular, if $\prob_0>0$ and $\gamma<1$ are absolute constants then the abo
 
 The restriction on $\gamma$ is satisfied by any \ti (where $\gamma=0$) as well as for all three queries of the PDBench \bi benchmark (see \Cref{app:subsec:experiment} for experimental results). 
 
-We briefly connect the runtime in \Cref{eq:approx-algo-runtime} to the algorithm outline earlier (where  we ignore the dependence on $\multc{\cdot}{\cdot)}$, which is needed to handle the cost of arithmetic operations over integers). The $\size(\circuit)$ comes from the time take to run \onepass once (\onepass essentially computes $\abs{\circuit}(1,\ldots, 1)$ using the natural circuit evaluation algorithm on $\circuit$). We make $\frac{\log{\frac{1}{\conf}}}{\inparen{\error'}^2\cdot(1-\gamma)^2\cdot \prob_0^{2k}}$ many calls to \sampmon (each of which essentially traces $O(k)$ random sink to source paths in $\circuit$ all of which by definition have length at most $\depth(\circuit)$.
+We briefly connect the runtime in \Cref{eq:approx-algo-runtime} to the algorithm outline earlier (where  we ignore the dependence on $\multc{\cdot}{\cdot}$, which is needed to handle the cost of arithmetic operations over integers). The $\size(\circuit)$ comes from the time take to run \onepass once (\onepass essentially computes $\abs{\circuit}(1,\ldots, 1)$ using the natural circuit evaluation algorithm on $\circuit$). We make $\frac{\log{\frac{1}{\conf}}}{\inparen{\error'}^2\cdot(1-\gamma)^2\cdot \prob_0^{2k}}$ many calls to \sampmon (each of which essentially traces $O(k)$ random sink to source paths in $\circuit$ all of which by definition have length at most $\depth(\circuit)$).
 
 Finally, we address the $\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}$ term in the runtime. %In \Cref{susec:proof-val-up}, we show the following:
 \begin{Lemma}
@@ -149,14 +149,14 @@ we have $\abs{\circuit}(1,\ldots, 1)\le  \size(\circuit)^{O(k)}.$
 
 Note that the above implies that with the assumption $\prob_0>0$ and $\gamma<1$ are absolute constants from \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 $\circuit$. If $\circuit$ is a tree, then the runtime simplifies to $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)$, which then answers \Cref{prob:intro-stmt} is yes for such circuits.
 
-Finally, note that by \Cref{prop:circuit-depth} and \Cref{lem:circ-model-runtime} for any $\raPlus$ query $\query$, there exists a circuit $\circuit^*$ such that $\depth(\circuit^*)\le O_{|Q|}(\log{n})$ and $\size(\circuit)\le O_k\inparen{\qruntime{\query, \dbbase}}$. Using this along with \Cref{lem:val-ub}, \Cref{cor:approx-algo-const-p} and the fact that $n\le \qruntime{\query, \dbbase}$, we answer \Cref{prob:big-o-joint-steps} in the affirmative as follows:
+Finally, note that by \Cref{prop:circuit-depth} and \Cref{lem:circ-model-runtime} for any $\raPlus$ query $\query$, there exists a circuit $\circuit^*$ for $\apolyqdt$ such that $\depth(\circuit^*)\le O_{|Q|}(\log{n})$ and $\size(\circuit)\le O_k\inparen{\qruntime{\query, \dbbase}}$. Using this along with \Cref{lem:val-ub}, \Cref{cor:approx-algo-const-p} and the fact that $n\le \qruntime{\query, \dbbase}$, we answer \Cref{prob:big-o-joint-steps} in the affirmative as follows:
 \begin{Corollary}
 \label{cor:approx-algo-punchline}
 Let $\query$ be an $\raPlus$ query and $\pdb$ be an \abbrBIDB with $p_0>0$ and $\gamma<1$ (where $p_0,\gamma$ as in \Cref{cor:approx-algo-const-p}) are absolute constants. Let $\poly(\vct{X})=\apolyqdt$ for any result tuple $\tup$ with $\deg(\poly)=k$. Then one can compute an approximation satisfying \Cref{eq:approx-algo-bound-main} in time $O_{k,|Q|,\error',\conf}\inparen{\qruntime{\query, \dbbase}}$ (given $\query,\dbbase$ and $p_i$ for each $i\in [n]$ that defines $\pd$).
 %Let $\poly(\vct{X})$ be a \abbrBIDB-lineage polynomial correspoding to an \abbrBIDB circuit $\circuit$ that satisfies the specific conditions in \Cref{lem:val-ub}. Then one can compute an approximation satisfying \Cref{eq:approx-algo-bound-main} in time
 % $O_k\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)$. % for the case when $\circuit$ satisfies the specific conditions in \Cref{lem:val-ub}. 
 \end{Corollary}
-If we want to approximate the expected multiplicities of all $Z=O(n^k)$ result tuples $\tup$, simultaneously we just need to run the above result with $\conf$ replaced by $\frac \conf Z$. Note this increases the runtime by only a logarithmic factor.
+If we want to approximate the expected multiplicities of all $Z=O(n^k)$ result tuples $\tup$ simultaneously, we just need to run the above result with $\conf$ replaced by $\frac \conf Z$. Note this increases the runtime by only a logarithmic factor.
 
 
 %\AR{The above Corollary needs to be improved/generalized. This is a place-holder for now.}