In \Cref{sec:hard}, we showed that computing the expected multiplicity of a compressed lineage polynomial for \ti (even just based on project-join queries), and by extension \bi (or more general \abbrPDB models) %any $\semNX$-PDB)
is unlikely to be possible in linear time (\Cref{thm:mult-p-hard-result}), even if all tuples have the same probability (\Cref{th:single-p-hard}).
Given this, we now design an approximation algorithm for our problem that runs in {\em linear time}.\footnote{For a very broad class of circuits: please see the discussion after \Cref{lem:val-ub} for more.}
The folowing approximation algorithm applies to \bi, though our bounds are more meaningful for a non-trivial subclass of \bis that contains both \tis, as well as the PDBench benchmark~\cite{pdbench}. As before, all proofs and pseudocode can be found in \Cref{sec:proofs-approx-alg}.
For a circuit $\circuit$, we define $\expansion{\circuit}$ as a list of tuples $(\monom, \coef)$, where $\monom$ is a set of variables and $\coef\in\domN$. We will denote the monomial composed of the variables in $\monom$ as $\encMon$.
{\em positive circuit}, denoted $\abs{\circuit}$, is obtained from $\circuit$ as follows. For each leaf node $\ell$ of $\circuit$ where $\ell.\type$ is $\tnum$, update $\ell.\vari{value}$ to $|\ell.\vari{value}|$.
Conveniently, $\abs{\circuit}\inparen{1,\ldots,1}$ gives us the number of terms represented in $\expansion{\circuit}$, i.e. $\sum\limits_{\inparen{\monom, \coef}\in\expansion{\circuit}}\abs{\coef}$.
%The function \depth has circuit $\circuit$ as input and outputs the number of levels in \circuit.
%\end{Definition}
%%%%%%%%%%%%%%%%%%%%%%%%%
%NEEDS to be moved to appendix
%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{Definition}[$\degree(\cdot)$]\label{def:degree}\footnote{Note that the degree of $\polyf(\abs{\circuit})$ is always upper bounded by $\degree(\circuit)$ 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 $0$).}
%$\degree(\circuit)$ is defined recursively as follows:
%\[\degree(\circuit)=
%\begin{cases}
%\max(\degree(\circuit_\linput),\degree(\circuit_\rinput)) & \text{ if }\circuit.\type=+\\
%\degree(\circuit_\linput) + \degree(\circuit_\rinput)+1 &\text{ if }\circuit.\type=\times\\
\begin{Definition}[$\multc{\cdot}{\cdot}$]\footnote{We note that when doing arithmetic operations on the RAM model for input of size $N$, we have that $\multc{O(\log{N})}{O(\log{N})}=O(1)$. More generally we have $\multc{N}{O(\log{N})}=O(N\log{N}\log\log{N})$.}
In a RAM model of word size of $W$-bits, $\multc{M}{W}$ denotes the complexity of multiplying two integers represented with $M$-bits. (We will assume that for input of size $N$, $W=O(\log{N})$.
To get linear runtime results from \Cref{lem:approx-alg}, we will need to define another parameter modeling the (weighted) number of monomials in %$\poly\inparen{\vct{X}}$
$\expansion{\circuit}$
to be `canceled' monomials with dependent variables are removed (\cref{def:reduced-bi-poly}). %def:hen it is modded with $\mathcal{B}$ (\Cref{def:mod-set-polys}).
Let $\isInd{\cdot}$ be a boolean function returning true if monomial $\encMon$ is composed of independent variables and false otherwise.
\AH{Technically, $\monom$ is a set of variables rather than a monomial. Perhaps we don't need the $\var(\cdot)$ function and can replace is with a function that returns the monomial represented by a set of variables. FIXED: need to propogate this to the appendix ($\encMon$)}
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
In particular, if $\prob_0>0$ and $\gamma<1$ are absolute constants then the above runtime simplifies to $O_k\left(\left(\frac1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot\log{\frac{1}{\conf}}\right)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)$.
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).
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:
\item$\circuit$ encodes the run of the algorithm in~\cite{DBLP:conf/pods/KhamisNR16} on an FAQ\AH{citation would help here, as a reviewer complaint on this was ``What is FAQ?'', though we do cite (I think) in the appendix.} query,
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 $O_k\left(\frac1{\inparen{\error'}^2}\cdot\size(\circuit)^2\cdot\log{\frac{1}{\conf}}\right)$ for general circuits $\circuit$ and to $O_k\left(\frac1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot\log{\frac{1}{\conf}}\right)$ for the case when $\circuit$ satisfies the specific conditions in \Cref{lem:val-ub}. In \Cref{app:proof-lem-val-ub} we argue that these conditions are very general and encompass many interesting scenarios, including query evaluation under \raPlus or FAQ.
We prove \Cref{lem:approx-alg} by developing an approximation algorithm (\approxq detailed in \Cref{alg:mon-sam}) with the desired runtime. This algorithm is based on the following observation.
% The algorithm (\approxq detailed in \Cref{alg:mon-sam}) 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 have: % can exactly represent $\rpoly(\vct{X})$ as follows:
Given the above, the algorithm is a sampling based algorithm for the above sum: we sample (via \sampmon) $(\monom,\coef)\in\expansion{\circuit}$ with probability proportional %\footnote{We could have also uniformly sampled from $\expansion{\circuit}$ but this gives better parameters.}
to $\abs{\coef}$ and compute $\vari{Y}=\indicator{\isInd{\encMon}}%\monom\mod{\mathcal{B}}\not\equiv 0}
\cdot\prod_{X_i\in\monom} p_i$. Taking $\numsamp$ samples and computing the average of $\vari{Y}$ gives us our final estimate. \onepass is used to compute the sampling probabilities needed in \sampmon(details are in \Cref{sec:proofs-approx-alg}).
%\approxq (\Cref{alg:mon-sam}) modifies \circuit with a call to \onepass. It then samples from $\circuit_{\vari{mod}}\numsamp$ times and uses that information to approximate $\rpoly$.
%In order to prove \Cref{lem:approx-alg}, we will need to argue the correctness of \approxq, which relies on the correctness of auxiliary algorithms \onepass and \sampmon.
% $\onepass$ 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.
%To prove correctness of \Cref{alg:mon-sam}, we only use the following fact that follows from the above lemma: for the modified circuit ($\circuit_{\vari{mod}}$), $\circuit_{\vari{mod}}.\vari{partial}=\abs{\circuit}(1,\dots,1)$.
%$$O(\log{k} \cdot k \cdot \depth(\circuit)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log{\size(\circuit)}})$$
% where $k = \degree(\circuit)$. The function returns every $\left(\monom, sign(\coef)\right)$ for $(\monom, \coef)\in \expansion{\circuit}$ with probability $\frac{|\coef|}{\abs{\circuit}(1,\ldots, 1)}$.
%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
%\noindent \onepass\ (Algorithm ~\ref{alg:one-pass-iter} in \Cref{sec:proofs-approx-alg}) iteratively visits each gate one time according to the topological ordering of \circuit annotating the \lwght, \rwght, and \prt variables of each node according to the definitions above. Lemma~\ref{lem:one-pass} is proved in \Cref{sec:proofs-approx-alg}.
%Instead, \Cref{alg:sample} selects a monomial from $\expansion{\circuit}$ by top-down traversal of the input \circuit. More details on the traversal can be found in \Cref{subsec:sampmon-remarks}.
