In~\Cref{sec:hard}, we showed that computing the expected multiplicity of a compressed representation of a bag polynomial for \ti (even just based on project-join queries) 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}.
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}.
We now introduce useful definitions and notation related to circuits and polynomials. Kindly note that all proofs and pseudocode can be found in \cref{sec:proofs-approx-alg}.
{\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 $|\ell.\vari{value}|$.
\begin{Definition}[$\degree(\cdot)$]\footnote{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}[$\multc{\cdot}{\cdot}$]\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 $\expansion{\circuit}$ to be `canceled' when it is modded with $\mathcal{B}$ (\Cref{def:mod-set-polys}).
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 (Please 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:
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 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.
\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 ($\circuit_{\vari{mod}}$), $\circuit_{\vari{mod}}.\vari{partial}=\abs{\circuit}(1,\dots,1)$.
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
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.
\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}.
