We formalize our claim from \Cref{sec:intro} that a linear approximation algorithm for our problem implies that PDB queries (under bag semantics) can be answered (approximately) in the same runtime as deterministic queries under reasonable assumptions.
%\subsection{Cost Model, Query Plans, and Runtime}
%As in the introduction, we could consider polynomials to be represented as an expression tree.
%However, they do not capture many of the compressed polynomial representations that we can get from query processing algorithms on bags, including the recent work on worst-case optimal join algorithms~\cite{ngo-survey,skew}, factorized databases~\cite{factorized-db}, and FAQ~\cite{DBLP:conf/pods/KhamisNR16}. Intuitively, the main reason is that an expression tree does not allow for `sharing' of intermediate results, which is crucial for these algorithms (and other query processing methods as well).
We now show that this model corresponds to the behavior of a deterministic database by proving that for any \raPlus query $\query$, we can construct a compressed circuit for $\poly$ and \bi$\pxdb$ of size and runtime linear in that of a general class of query processing algorithms for the same query $\query$ on a deterministic database $\db$.
\footnote{This is a reasonable assumption because each block of a \bi represents entities with uncertain attributes.
In practice there is often a limited number of alternatives for each block (e.g., which of five conflicting data sources to trust). Note that all \tis trivially fulfill this condition (i.e., $c =1$).}
%That is for \bis that fulfill this restriction approximating the expectation of results of SPJU queries is only has a constant factor overhead over deterministic query processing (using one of the algorithms for which we prove the claim).
Under this model a query $Q$ evaluated over database $D$ has runtime $O(\qruntime{Q,D})$.
We assume that full table scans are used for every base relation access. We can model index scans by treating an index scan query $\sigma_\theta(R)$ as a base relation.
It can be verified that worst-case optimal join algorithms~\cite{skew,ngo-survey}, as well as query evaluation via factorized databases~\cite{factorized-db}\AR{See my comment on element on whether we should include this ref or not.} (and work on FAQs~\cite{DBLP:conf/pods/KhamisNR16}) can be modeled as select-union-project-join queries (though the size of these queries is data dependent).\footnote{This claim can be verified by e.g. simply looking at the {\em Generic-Join} algorithm in~\cite{skew} and {\em factorize} algorithm in~\cite{factorized-db}.} It can be verified that the above cost model on the corresponding SPJU join queries correctly captures their runtime.
Given an SPJU query $Q$ over a \ti$\pxdb$ and let $\db_{max}$ denote the world containing all tuples of $\pxdb$, we can compute a $(1\pm\eps)$-approximation of the expectation for each output tuple in $\query(\pxdb)$ with probability at least $1-\delta$ in time
This follows from \Cref{lem:circuits-model-runtime} (\Cref{sec:circuit-runtime}) and \Cref{cor:approx-algo-const-p} (where the latter is used with $\delta$ being substituted\footnote{Recall that \Cref{cor:approx-algo-const-p} is stated for a single output tuple so to get the required guarantee for all (at most $n^k$) output tuples of $Q$ we get at most $\frac\delta{n^k}$ probability of failure for each output tuple and then just a union bound over all output tuples. } with $\frac\delta{n^k}$).
For any positive integer $m$ we can compute the $m$-th moment of the multiplicities, allowing us to e.g. use the Chebyschev inequality or other high moment based probability bounds on the events we might be interested in.