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}.
The pure expansion of a polynomial $\poly$ is formed by computing all product of sums occurring in $\poly$, without combining like monomials. The pure expansion of $\poly$ generalizes ~\Cref{def:smb} by allowing monomials $m_i = m_j$ for $i \neq j$.
The logical view of \revision{$\expansion{\circuit}$} is a list of tuples $(\monom, \coef)$, where $\monom$ is a set of variables and $\coef$ is in $\reals$.
\revision{$\expansion{\circuit}$} has the following recursive definition ($\circ$ is list concatenation).
Note that similar in spirit to ~\Cref{def:reduced-bi-poly}, $\expansion{\circuit}$ reduces all variable exponents $e > 1$ to $e =1$, though ~\Cref{def:reduced-bi-poly} is more general.
$\expansion{\circuit}$ encodes the \emph{reduced} form of $\polyf\inparen{\circuit}$, decoupling each monomial into a set of variables $\monom$ and a real coefficient $\coef$.
Note, however, that unlike $\rpoly$, $\expansion{\circuit}$ does not need to be in SOP form.
{\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 $|\ell.\vari{value}|$.
Using the same factorization from ~\Cref{example:expr-tree-T}, $\polyf(\abs{\circuit})=(X +2Y)(2X + Y)=2X^2+XY +4XY +2Y^2=2X^2+5XY +2Y^2$. Note that this \textit{is not} the same as the polynomial from~\Cref{eq:poly-eg}.
Given an expression tree $\circuit$ and a valuation $\vct{a}\in\mathbb{R}^\numvar$, we define the evaluation of $\circuit$ on $\vct{a}$ as $\circuit(\vct{a})=\polyf(\circuit)(\vct{a})$.
A subcircuit of a circuit $\circuit$ is a circuit \subcircuit such that \subcircuit is a DAG \textit{subgraph} of the DAG representing \circuit. The sink of \subcircuit has exactly one gate \gate.
\noindent The proof of~\Cref{lem:approx-alg} (which relies on ~\Cref{lem:one-pass} and ~\Cref{lem:sample}) can be found in~\Cref{sec:proof-lem-approx-alg}. The proofs for the referenced lemmas are also found in ~\Cref{sec:proof-one-pass} and ~\Cref{sec:proof-sample-monom}.
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 $O_k\left(\frac1{\inparen{\error'}^2}\cdot\size(\circuit)\cdot\log{\frac{1}{\conf}}\right)$.
The restriction on $\gamma$ is satisfied by any \ti (where $\gamma=0$) as well as for all three queries of the PDBench \bi benchmark (\Cref{app:subsec:experiment} shows experimentally that $\gamma$ is negligible in practice for these queries).
We also observe that (i) tuple presence is independent across blocks, so the corresponding probabilities (and hence $\prob_0$) are independent of the number of blocks, and (ii) \bis model uncertain attributes, so block size (and hence $\gamma$) is a function of the ``messiness'' of a dataset, rather than its size.
The algorithm 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 can exactly represent $\rpoly(\vct{X})$ as follows:
Given the above, the algorithm is a sampling based algorithm for the above sum: we sample $(\monom,\coef)\in\expansion{\circuit}$ with probability proportional\footnote{We could have also uniformly sampled from $\expansion{\circuit}$ but this gives better parameters.} to $\abs{\coef}$ and compute $Y=\indicator{\monom\mod{\mathcal{B}}\not\equiv0}\cdot\prod_{X_i\in\var\inparen{\monom}} p_i$. Taking $\numsamp$ samples and computing the average of $Y$ gives us our final estimate.
To summarize, \approxq modifies \circuit with a call to \onepass. It then samples from \circuit$\numsamp$ times and uses that information to approximate $\rpoly$.
we first state the lemmas that summarize the relevant properties of $\onepass$ and $\sampmon$, the auxiliary algorithms on which ~\Cref{alg:mon-sam} relies.
The $\onepass$ function completes in $O(size(\circuit)\cdot\frac{\log{\abs{\circuit(1\ldots, 1)}}}{\log{N}})$ time, where $N =\size(\circuit)$.\footnote{In the appendix we give a sufficient condition when $\abs{\circuit}(1,\ldots, 1)$ is indeed $O(1)$ in arithmetic computations. Most notably, WCOJ and FAQ results are not affected by the general runtime of arithmetic computations, a point which we also address in the appendix.}$\onepass$ 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 ($\circuit_{\vari{mod}}$), $\circuit_{\vari{mod}}$, $\circuit_{\vari{mod}}.\vari{partial}=\abs{\circuit}(1,\dots,1)$.
The function $\sampmon$ completes in $O(\log{k}\cdot k \cdot\depth(\circuit)\cdot\frac{\log{\abs{\circuit}(1,\ldots, 1)}}{\log{\size(\circuit)}})$ time\footnote{Note that the same sufficient condition on \circuit to guarentee $O(1)$ arithmetic computations applies here, and when this condition is met, the runtime loses the $\frac{\log{\abs{\circuit}(1,\ldots, 1)}}{\log{\size(\circuit)}}$ factor}, where $k =\degree(\circuit)$. Upon completion, every $\left(\monom, sign(\coef)\right)\in\expansion{\abs{\circuit}}$ is returned 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
The evaluation of $\abs{\circuit}(1,\ldots, 1)$ can be defined recursively, as follows (where $\circuit_\linput$ and $\circuit_\rinput$ are the `left' and `right' inputs of $\circuit$ if they exist):
\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 also proved in~\Cref{sec:proofs-approx-alg}.
\State$\circuit_{\vari{samp}}\gets$ Sample from left input ($\circuit_{\linput}$) and right input ($\circuit_{\rinput}$) w.p. $\circuit.\vari{Lweight}$ and $\circuit.\vari{Rweight}$. \label{alg:sample-plus-bsamp}\Comment{Each call to \sampmon uses fresh randomness}
% We conducted an experiment running modified TPCH queries over uncertain data generated by pdbench~\cite{pdbench}, both of which (data and queries) represent what is typically encountered in practice. Queries were run two times, once filtering $\bi$ cancellations, and then second not filtering the cancellations. The purpose of this was to determine an indication for how many $\bi$ cancellations occur in practice. Details and results can be found in~.
