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\section{$1 \pm \epsilon$ Approximation Algorithm}\label{sec:algo}

In \Cref{sec:hard}, we showed that the answer to \Cref{prob:intro-stmt} is no.
With this result, we now design an approximation algorithm for our problem that runs in $\bigO{\abs{\circuit}}$ for a very broad class of circuits (see the discussion after \Cref{lem:val-ub} for more).
The folowing approximation algorithm applies to \abbrBIDB lineage polynomials (over $\raPlus$ queries), though our bounds are more meaningful for a non-trivial subclass of queries over \bis that contains all queries on \tis, as well as the queries of the PDBench benchmark~\cite{pdbench}.  All proofs and pseudocode can be found in \Cref{sec:proofs-approx-alg}.
%it is then desirable to have an algorithm to approximate the multiplicity in linear time, which is what we describe next.

\subsection{Preliminaries and some more notation}

We now introduce definitions and notation related to circuits and polynomials that we will need to state our upper bound results.  

\begin{Definition}[$\expansion{\circuit}$]\label{def:expand-circuit}
For a circuit $\circuit$, we define $\expansion{\circuit}$ as a list of tuples $(\monom, \coef)$, where $\monom$ is a set of variables and $\coef \in \domN$.  
$\expansion{\circuit}$ has the following recursive definition ($\circ$ is list concatenation).
$\expansion{\circuit} =
\begin{cases}
					\expansion{\circuit_\linput} \circ \expansion{\circuit_\rinput}		&\textbf{ if }\circuit.\type = \circplus\\
					\left\{(\monom_\linput \cup \monom_\rinput, \coef_\linput \cdot \coef_\rinput) ~|~(\monom_\linput, \coef_\linput) \in \expansion{\circuit_\linput}, (\monom_\rinput, \coef_\rinput) \in \expansion{\circuit_\rinput}\right\} 		&\textbf{ if }\circuit.\type = \circmult\\
					\elist{(\emptyset, \circuit.\val)}								&\textbf{ if }\circuit.\type = \tnum\\
					\elist{(\{\circuit.\val\}, 1)}									&\textbf{ if }\circuit.\type = \var.\\
\end{cases}
$
\end{Definition}
Later on, we will denote the monomial composed of the variables in $\monom$ as $\encMon$.  As an example of $\expansion{\circuit}$, consider $\circuit$ illustrated in \Cref{fig:circuit}.  $\expansion{\circuit}$ is then $[(X, 2), (XY, -1), (XY, 4), (Y, -2)]$.

\begin{Definition}[$\abs{\circuit}$]\label{def:positive-circuit}
For any circuit $\circuit$, the corresponding
{\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 $|\ell.\vari{value}|$.
\end{Definition}
We will overload notation and use $\abs{\circuit}\inparen{\vct{X}}$ to mean $\polyf\inparen{\abs{\circuit}}$.
Conveniently, $\abs{\circuit}\inparen{1,\ldots,1}$ gives us $\sum\limits_{\inparen{\monom, \coef} \in \expansion{\circuit}}\abs{\coef}$.

\begin{Definition}[\size($\cdot$), \depth$\inparen{\cdot}$]\label{def:size-depth}
The functions \size and \depth output the number of gates and levels respectively for input \circuit.
\end{Definition}

\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:
\[\degree(\circuit)=
\begin{cases}
\max(\degree(\circuit_\linput),\degree(\circuit_\rinput)) & \text{ if }\circuit.\type=+\\
\degree(\circuit_\linput) + \degree(\circuit_\rinput)+1 &\text{ if }\circuit.\type=\times\\
1 & \text{ if }\circuit.\type = \var\\
0 & \text{otherwise}.
\end{cases}
\]
\end{Definition}

Next, we use the following notation for the complexity of multiplying integers:
\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 $N$, $W=O(\log{N})$.)
\end{Definition}

Finally, to get linear runtime results from \Cref{lem:approx-alg}, we will need to define another parameter modeling the (weighted) number of monomials in %$\poly\inparen{\vct{X}}$ 
$\expansion{\circuit}$ 
that need to be `canceled' when 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; further, let $\indicator{\theta}$ also be a boolean function returning true if $\theta$ evaluates to true.
\begin{Definition}[Parameter $\gamma$]\label{def:param-gamma}
Given a \abbrBIDB circuit $\circuit$ define
\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 ($\encMon$)}
\AH{To add, this is an issue on line 1073, 1117 of app C.}
\[\gamma(\circuit)=\frac{\sum_{(\monom, \coef)\in \expansion{\circuit}} \abs{\coef}\cdot \indicator{\neg\isInd{\encMon}} }%\encMon\mod{\mathcal{B}}\equiv 0}}
{\abs{\circuit}(1,\ldots, 1)}.\]
\end{Definition}
\subsection{Our main result}\label{sec:algo:sub:main-result}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\mypar{Algorithm Idea}
%We prove \Cref{lem:approx-alg} by developing an 
Our approximation algorithm (\approxq pseudo code in \Cref{sec:proof-lem-approx-alg}) 
%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 lineage polynomial $\poly(\vct{X})=\polyf(\circuit)$ for circuit \circuit over $\bi$, we have: % can exactly represent $\rpoly(\vct{X})$ as follows:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}
\label{eq:tilde-Q-bi}
\rpoly\inparen{X_1,\dots,X_\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.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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
 to $\abs{\coef}$ and compute $\vari{Y}=\indicator{\isInd{\encMon}}
 \cdot \prod_{X_i\in \monom} p_i$. %Taking $\ceil{\frac{2 \log{\frac{2}{\conf}}}{\error^2}}$ samples 
Repeating the sampling appropriate number of times
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}).
%%%%%%%%%%%%%%%%%%%%%%%

%The following results assume input circuit \circuit computed from an arbitrary $\raPlus$ query $\query$ and arbitrary \abbrBIDB $\pdb$.  We refer to \circuit as a \abbrBIDB circuit.
%\AH{Verify that the proof for \Cref{lem:approx-alg} doesn't rely on properties of $\raPlus$ or \abbrBIDB.}
%\begin{Theorem}\label{lem:approx-alg}
%Let \circuit be an arbitrary \abbrBIDB 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
%{\small
%\[O\left(\left(\size(\circuit) + \frac{\log{\frac{1}{\conf}}\cdot \abs{\circuit}^2(1,\ldots, 1)\cdot  k\cdot \log{k} \cdot \depth(\circuit))}{\inparen{\error}^2\cdot\rpoly^2(\prob_1,\ldots, \prob_\numvar)}\right)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)\]
%}
%such that
%\begin{equation}
%\label{eq:approx-algo-bound}
%\probOf\left(\left|\mathcal{E} - \rpoly(\prob_1,\dots,\prob_\numvar)\right|> \error \cdot \rpoly(\prob_1,\dots,\prob_\numvar)\right) \leq \conf.
%\end{equation}
%\end{Theorem}


\mypar{Runtime analysis} We can argue the following runtime for the algorithm outlined above:
% We next present a few corollaries of \Cref{lem:approx-alg}.
\begin{Theorem}
\label{cor:approx-algo-const-p}
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 $\rpoly(\prob_1,\ldots, \prob_\numvar)$ 
satisfying 
\begin{equation}
\label{eq:approx-algo-bound-main}
\probOf\left(\left|\mathcal{E} - \rpoly(\prob_1,\dots,\prob_\numvar)\right|> \error' \cdot \rpoly(\prob_1,\dots,\prob_\numvar)\right) \leq \conf
\end{equation}
 can be computed in time
\begin{equation}
\label{eq:approx-algo-runtime}
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}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right).
\end{equation}
In particular, if $\prob_0>0$ and $\gamma<1$ are absolute constants then the above runtime simplifies to $O_k\left(\left(\frac 1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot \log{\frac{1}{\conf}}\right)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)$.
\end{Theorem}

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 $\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}$). 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}
\label{lem:val-ub}
For any \abbrBIDB circuit $\circuit$ with $\degree(\circuit)=k$, we have
$\abs{\circuit}(1,\ldots, 1)\le 2^{2^k\cdot \size(\circuit)}.$
Further, under either of the following conditions:
\begin{enumerate}
\item $\circuit$ is a tree,
\item $\circuit$ encodes the run of the algorithm on a FAQ~\cite{DBLP:conf/pods/KhamisNR16}/AJAR~\cite{ajar} query,
\end{enumerate}
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 \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
\begin{Corollary}
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}
\AR{The above Corollary needs to be improved/generalized. This is a place-holder for now.}
In \Cref{app:proof-lem-val-ub} we argue that these conditions are very general and encompass many interesting scenarios, including query evaluation under FAQ/AJAR setup.
%\AH{AJAR reference.}


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