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

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 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}.
%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 useful definitions and notation related to circuits and polynomials.  All proofs and missing pseudocode can be found in \Cref{sec:proofs-approx-alg}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{Definition}[Variables in a monomial]\label{def:vars}
% Given a monomial $v$, we use $\var(v)$ to denote the set of variables in $v$.
%\end{Definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent For example the monomial $XY$ has $\var(XY)=\inset{X,Y}$.


\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 $\coef \in \domN$.  We will denote the monomial composed of the variables in $\monom$ as $\encMon$.
$\expansion{\circuit}$ 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}
For further explanation, please refer to \Cref{example:expr-tree-T}.

\begin{Definition}[$\abs{\circuit}(\vct{X})$]\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}
Please see \Cref{ex:def-pos-circ} for an illustration.

\begin{Definition}[\size($\cdot$)]\label{def:size}
The function \size~ takes a circuit $\circuit$ as input and outputs the number of gates (nodes) in \circuit.
\end{Definition}

\begin{Definition}[\depth($\cdot$)]
The function \depth~ has circuit $\circuit$ as input and outputs the number of levels in \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}
Finally, we will need the following notation for the complexity of multiplying large 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}

\subsection{Our main result}

\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
{\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}

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}).
\begin{Definition}[Parameter $\gamma$]\label{def:param-gamma}
Given an expression tree $\circuit$, 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.}
\AH{To add, this is an issue on line 1073, 1117 of app C.}
\[\gamma(\circuit)=\frac{\sum_{(\monom, \coef)\in \expansion{\circuit}} \abs{\coef}\cdot \indicator{\encMon\mod{\mathcal{B}}\equiv 0}}{\abs{\circuit}(1,\ldots, 1)}\]
\end{Definition}

\noindent We next present a few corollaries of \Cref{lem:approx-alg}.
\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
\[O\left(\left(\size(\circuit) + \frac{\log{\frac{1}{\conf}}\cdot k\cdot \log{k} \cdot \depth(\circuit))}{\inparen{\error'}^2\cdot(1-\gamma)^2\cdot \prob_0^{2k}}\right)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log\left(\size(\circuit)\right)}\right)\]
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{Corollary}

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:
\begin{Lemma}
\label{lem:val-ub}
For any 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 in~\cite{DBLP:conf/pods/KhamisNR16} on an FAQ 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 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 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.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Approximating $\rpoly$}
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:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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{\encMon\mod{\mathcal{B}}\not\equiv 0}\cdot \coef\cdot\hspace*{-2mm}\prod_{X_i\in \var\inparen{\monom}}\hspace*{-2mm} X_i
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\input{app_approx-alg-pseudo-code}

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 $Y=\indicator{\monom\mod{\mathcal{B}}\not\equiv 0}\cdot \prod_{X_i\in \var\inparen{\monom}} p_i$. Taking $\numsamp$ samples and computing the average of $Y$ 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 $\rpoly$.




%\subsubsection{Correctness}

%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.

%\begin{Lemma}\label{lem:one-pass}
%The $\onepass$ function completes in time:
%$$O\left(\size(\circuit) \cdot \multc{\log\left(\abs{\circuit(1\ldots, 1)}\right)}{\log{\size(\circuit}}\right)$$
%  $\onepass$ 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 ($\circuit_{\vari{mod}}$), $\circuit_{\vari{mod}}.\vari{partial}=\abs{\circuit}(1,\dots,1)$.

%\begin{Lemma}\label{lem:sample}
%The function $\sampmon$ completes in time
%$$O(\log{k} \cdot k \cdot \depth(\circuit)\cdot\multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log{\size(\circuit)}})$$
% where $k = \degree(\circuit)$.  The function returns every $\left(\monom, sign(\coef)\right)$ for $(\monom, \coef)\in \expansion{\circuit}$ with probability $\frac{|\coef|}{\abs{\circuit}(1,\ldots, 1)}$.
%\end{Lemma}

%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 $O\left(\left(\size(\circuit)+\frac{\log{\frac{1}{\conf}}}{\error^2} \cdot k \cdot\log{k} \cdot \depth(\circuit)\right)\cdot \multc{\log\left(\abs{\circuit}(1,\ldots, 1)\right)}{\log{\size(\circuit)}}\right)$ time.
%\end{Theorem}


%\subsection{\onepass\ Algorithm}
%\label{sec:onepass}

%\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}.

%\subsection{\sampmon\ Algorithm}
%\label{sec:samplemonomial}

%A naive (slow) implementation of \sampmon\ would first compute $\expansion{\circuit}$ and then sample from it.
%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}.

%
%$\sampmon$ is given in \Cref{alg:sample}, and a proof of its correctness (via \Cref{lem:sample}) is provided in \Cref{sec:proofs-approx-alg}.


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