363 lines
27 KiB
TeX
363 lines
27 KiB
TeX
%root: main.tex
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%!TEX root=./main.tex
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\section{Introduction}
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\label{sec:intro}
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% \AR{\textbf{Oliver/Boris:} What is missing from the intro is why would someone care about bag-PDBs in {\em practice}? This is kinda obliquely referred to in the first para but it would be good to motivate this more. The intro (rightly) focuses on the theoretical reasons to study bag PDBs but what (if any) are the practical significance of getting bag PDBs done in linear-time? Would this lead to much faster real-life PDB systems?}
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As explainability and fairness become more relevant to the data science community, it is now more critical than ever to understand how reliable a dataset is.
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Probabilistic databases (PDBs)~\cite{DBLP:series/synthesis/2011Suciu} are a compelling solution, but a major roadblock to their adoption remains:
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PDBs are orders of magnitude slower than classical (i.e., deterministic) database systems~\cite{feng:2019:sigmod:uncertainty}.
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A naive strategy might be to move from the theoretically simpler set-relational model~\cite{DBLP:series/synthesis/2011Suciu,DBLP:conf/sigmod/BoulosDMMRS05,DBLP:conf/icde/AntovaKO07a,DBLP:conf/sigmod/SinghMMPHS08} to the computationally simpler bag-relational model, mirroring a similar transition in deterministic datbases decades ago.
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However, after discarding a long-held approach to representing lineage, we prove that query processing in Bag-PDBs is \sharpwonehard.
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This finding shows that even Bag-PDB query processing has a higher complexity than deterministic query processing, and opens a rich landscape of opportunities for research on approximate algorithms.
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The fundamental challenge is lineage formulas, a key component of query processing in PDBs.
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Under standard assumptions about how these are encoded, computing typical statistics like marginal probabilities or moments is easy (at worst linear in the size of the lineage) for bags and hence, perhaps not worthy of research attention, but hard (at worst exponential in the size of the lineage) for sets and hence, interesting from a research perspective.
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However, conventional encodings of a result's lineage are typically large, and so even for Bag-PDBs, computing such statistics from lineage formulas still has a higher complexity than answering queries in a deterministic (i.e., non-probabilistic) database.
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In this paper, we formally prove this limitation of PDBs, and address it by proposing an approximation algorithm that, to the best of our knowledge, is the first $(1-\epsilon)$-approximation for expectations of counts to have a runtime within a constant factor of deterministic query processing\footnote{
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MCDB~\cite{jampani2008mcdb} is notable in that it is also a constant factor slower, but only guarantees additive rather than multiplicative bounds.
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}.
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Consider the dominant problem in Set-PDBs: Computing marginal probabilities, and the corresponding problem in Bag-PDBs: computing expectations of counts.
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In work that addresses the former problem~\cite{DBLP:series/synthesis/2011Suciu}, the lineage of a query result tuple is a Boolean formula over random variables that captures the conditions under which the tuple appears in the result.
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Computing the probability of the tuple appearing in the result is thus analogous to weighted model counting (a known \sharpphard problem).
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In the corresponding problem for Bag-PDBs~\cite{kennedy:2010:icde:pip,DBLP:conf/vldb/AgrawalBSHNSW06,feng:2019:sigmod:uncertainty,GL16}, lineage is a polynomial over random variables that captures the multiplicity of the output tuple.
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Thus, the expectation of the multiplicity is the expectation of this polynomial.
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Lineage in Set-PDBs is typically encoded in disjunctive normal form.
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This representation is significantly larger than the query result sans lineage.
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However, even with alternative encodings~\cite{DBLP:journals/vldb/FinkHO13}, the limiting factor in computing marginal probabilities remains the probability computation itself, and not the lineage formula.
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The corresponding lineage encoding for Bag-PDBs is a polynomial in sum of products (SOP) form --- a sum of `clauses', each of which is the product of a set of integer or variable atoms.
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Thanks to linearity of expectation, computing the expectation of a count query is linear in the number of clauses in the SOP polynomial.
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Unlike Set-PDBs, however, when we consider compressed representations of this polynomial, the complexity landscape becomes much more nuanced and is \textit{not} linear in general.
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Such compressed representations like Factorized Databases~\cite{10.1145/3003665.3003667,DBLP:conf/tapp/Zavodny11} or Arithmetic/Polynomial Circuits~\cite{arith-complexity}, are analogous to deterministic query optimizations (e.g. pushing down projections)~\cite{DBLP:conf/pods/KhamisNR16,10.1145/3003665.3003667}.
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Thus, measuring the performance of a PDB algorithm in terms of the size of the \emph{compressed} lineage formula allows us to more closely relate the algorithm's performance to the complexity of query evaluation in a deterministic database.
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The initial picture is not good.
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In this paper, we prove that computing expected counts is \emph{not} linear in the size of a compressed --- specifically a factorized~\cite{10.1145/3003665.3003667} --- lineage polynomial by reduction from counting $k$-matchings.
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Thus, even bag PDBs do not enjoy the same computational complexity as deterministic databases.
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This motivates our second goal, a linear time approximation algorithm for computing expected counts in a bag database, with complexity linear in the size of a factorized lineage formula.
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As we will show, the size of the factorized
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lineage formula for a query --- and by extension, our approximation algorithm --- is proportional to the complexity of evaluating the same query on a comparable deterministic database instance~\cite{DBLP:conf/pods/KhamisNR16,10.1145/3003665.3003667}.
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In other words, our approximation algorithm can estimate expected multiplicities for tuples in the result of an SPJU query with a complexity comparable to deterministic query-processing.
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\subsection{Sets vs Bags}
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%Consider an arbitrary output polynomial $\poly$. Further, consider the same polynomial, with all exponents $e > 1$ set to $1$ and call the resulting polynomial $\rpoly$.
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%Figures, etc
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%Relations for example 1
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\begin{figure}[ht]
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\begin{subfigure}{0.2\textwidth}
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\centering
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\begin{tabular}{ c | c c c}
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$\rel$ & A & $\Phi_{set}$ & $\Phi_{bag}$\\
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\hline
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& a & $W_a$ & $W_a$\\
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& b & $W_b$ & $W_b$\\
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& c & $W_c$ & $W_c$\\
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\end{tabular}
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%\caption{Atom 1 of query $\poly$ in ~\Cref{intro:ex}}
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\label{subfig:ex-atom1}
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\end{subfigure}
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\begin{subfigure}{0.24\textwidth}
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\centering
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\begin{tabular}{ c | c c c c}
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$E$ & A & B & $\Phi_{set}$ & $\Phi_{bag}$ \\
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\hline
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& a & b & $\top$ & $1$\\
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& b & c & $\top$ & $1$\\
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& c & a & $\top$ & $1$\\
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\end{tabular}
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%\caption{Atom 3 of query $\poly$ in ~\Cref{intro:ex}}
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\label{subfig:ex-atom3}
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\end{subfigure}
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% \begin{subfigure}{0.15\textwidth}
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% \centering
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% \begin{tabular}{ c | c | c}
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% $\rel$ & B & $\Phi$\\
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% \hline
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% & b & $W_b$\\
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% & c & $W_c$\\
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% & a & $W_a$\\
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% \end{tabular}
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% %\caption{Atom 2 of query $\poly$ in ~\Cref{intro:ex}}
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% \label{subfig:ex-atom2}
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% \end{subfigure}
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\caption{$\ti$ relations for $\poly$}
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\label{fig:intro-ex}
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\end{figure}
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%Graph of query output for intro example
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%\begin{figure}
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% \begin{tikzpicture}
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% \node at (1.5, 3) [tree_node](top){a};
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% \node at (0, 0) [tree_node](left){b};
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% \node at (3, 0) [tree_node](right){c};
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% \draw (top)--(left);
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% \draw (left)--(right);
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% \draw (right)--(top);
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% \end{tikzpicture}
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%\caption{Graph of tuples in table E}
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%\label{fig:intro-ex-graph}
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%\end{figure}
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\begin{Example}\label{ex:intro}
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Consider the Tuple Independent ($\ti$) Set-PDB\footnote{Our work also handles Block Independent Disjoint Databases ($\bi$)~\cite{DBLP:conf/sigmod/BoulosDMMRS05,DBLP:series/synthesis/2011Suciu} and we return to this model later.} given in \Cref{fig:intro-ex} with two input relations $R$ and $E$.
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Each input tuple is assigned an annotation (attribute $\Phi_{set}$): an independent random Boolean variable ($W_i$) or the constant $\top$.
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% Each assignment of values to variables ($\{\;W_a,W_b,W_c\;\}\mapsto \{\;\top,\bot\;\}$) \SF{Do we need to state the meaning of $\top$ and $\bot$? Also do we want to add bag annotation to Figure 1 too since we are discussing both sets and bags later?} identifies one \emph{possible world}, a deterministic database instance that contains exactly the tuples annotated by the constant $\top$ or by a variable assigned to $\top$.
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The probability of this world is the joint probability of the corresponding assignments.
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For example, let $P[W_a] = P[W_b] = P[W_c] = p$ and consider the possible world where $R = \{\;\tuple{a}, \tuple{b}\;\}$.
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The corresponding variable assignment is $\{\;W_a \mapsto \top, W_b \mapsto \top, W_c \mapsto \bot\;\}$, and the probability of this world is $P[W_a]\cdot P[W_b] \cdot P[\neg W_c] = p\cdot p\cdot (1-p)=p^2-p^3$.
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\end{Example}
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Prior efforts to generalize incomplete databases to bags~\cite{feng:2019:sigmod:uncertainty,DBLP:conf/pods/GreenKT07,DBLP:journals/sigmod/GuagliardoL17} replace the Boolean annotations with natural numbers.
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Analogously, we generalize the above model of Set-PDBs to bags by using natural-number-valued random variables (i.e., $Dom(W_i) \subseteq \mathbb N$) and positive natural number constants ($\Phi_{bag}$ in the example).
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Without loss of generality, we assume that input relations are sets (i.e. $Dom(W_i) = \{0, 1\}$), while query evaluation follows bag semantics.
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We contrast bag and set query evaluation with the following example:
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\begin{Example}\label{ex:bag-vs-set}
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Continuing the prior example, we are given the following Boolean (resp,. count) query
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$$\poly() :- R(A), E(A, B), R(B)$$
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The lineage of the result in a Set-PDB (resp., Bag-PDB) is a Boolean (resp., polynomial) formula over random variables annotating the input relations (i.e., $W_a$, $W_b$, $W_c$).
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Because the Boolean query has only a nullary relation, we write $Q(\cdot)$ to denote the function that evaluates the lineage over one specific assignment of values to the variables (i.e., the value of the lineage in the corresponding possible world):
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\begin{align*}
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\poly_{set}(W_a, W_b, W_c) &= W_aW_b \vee W_bW_c \vee W_cW_a\\
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\poly_{bag}(W_a, W_b, W_c) &= W_aW_b + W_bW_c + W_cW_a
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\end{align*}
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It is left as an exercise for the reader to show that, given assignments to $W_a$, $W_b$, $W_c$, these expressions correspond to the existence (resp., count) of the single nullary result tuple for $\poly$ applied to the database instance in \Cref{fig:intro-ex}.
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We show one possible world here, with the set assignment $\{\;W_a\mapsto \top, W_b \mapsto \top, W_c \mapsto \bot\;\}$ (and the corresponding bag assignment),
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The polynomials evaluate as:
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\begin{align*}
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&\poly_{set}(\top, \top, \bot) = \top\top \vee \top\bot \vee \top\bot = \top\\
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&\poly_{bag}(1, 1, 0) = 1 \cdot 1 + 1\cdot 0 + 0 \cdot 1 = 1
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\end{align*}
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The Set-PDB query is satisfied in this possible world, while the Bag-PDB query produces a nullary tuple with a multiplicity of 1.
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The marginal probability (resp., expected count) of this query is computed over all possible worlds:
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% \AR{What is $\mu$ below?}
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{\small
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\begin{align*}
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P[\poly_{set}] &= \hspace*{-1mm}
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\sum_{w_i \in \{\top,\bot\}} \indicator{\poly_{set}(w_a, w_b, w_c)}P[W_a = w_a,W_b = w_b,W_c = w_c]\\
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\expct[\poly_{bag}] &= \sum_{w_i \in \{0,1\}} \poly_{bag}(w_a, w_b, w_c)\cdot P[W_a = w_a,W_b = w_b,W_c = w_c]
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\end{align*}
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}
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\end{Example}
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Note that the query of \Cref{ex:bag-vs-set} in set semantics is indeed non-hierarchical~\cite{10.1145/1265530.1265571}, and thus \sharpphard.
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To see why computing this probability is hard, observe that the clauses of the disjunctive normal form Boolean lineage are neither independent nor disjoint, leading to e.g.~\cite{DBLP:journals/vldb/FinkHO13} the use of Shannon decomposition, which is at worst exponential in the size of the input.
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% \begin{equation*}
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% \expct\pbox{\poly(W_a, W_b, W_c)} = W_aW_b + W_a\overline{W_b}W_c + \overline{W_a}W_bW_c = 3\prob^2 - 2\prob^3
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% \end{equation*}
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% In general, such a computation can be exponential in the size of the database.
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%Using Shannon's Expansion,
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%\begin{align*}
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%&W_aW_b \vee W_bW_c \vee W_cW_a
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%= &W_a
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%\end{align*}
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Conversely, in Bag-PDBs, correlations between clauses of the SOP polynomial are not problematic thanks to linearity of expectation.
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The expectation computation over the output lineage is simply the sum of expectations of each clause.
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For \Cref{ex:intro}, the expectation is simply
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{\small
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\begin{align*}
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\expct\pbox{\poly(W_a, W_b, W_c)} &= \expct\pbox{W_aW_b} + \expct\pbox{W_bW_c} + \expct\pbox{W_cW_a}\\
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\intertext{\normalsize
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In this particular lineage polynomial, all variables in each product clause are independent, so we can push expectations through.
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}
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&= \expct\pbox{W_a}\expct\pbox{W_b} + \expct\pbox{W_b}\expct\pbox{W_c} + \expct\pbox{W_c}\expct\pbox{W_a}
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\end{align*}
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}
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Computing such expectations is indeed linear in the size of the SOP as the number of operations in the computation is \textit{exactly} the number of multiplication and addition operations of the polynomial.
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As a further interesting feature of this example, note that $\expct\pbox{W_i} = P[W_i = 1]$, and so taking the same polynomial over the reals:
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\begin{multline}
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\label{eqn:can-inline-probabilities-into-polynomial}
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\expct\pbox{\poly_{bag}} = P[W_a = 1]P[W_b = 1] + P[W_b = 1]P[W_c = 1]\\
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+ P[W_c = 1]P[W_a = 1]\\
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= \poly_{bag}(P[W_a=1], P[W_b=1], P[W_c=1])
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\end{multline}
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\begin{figure}[h!]
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\resizebox{\columnwidth}{!}{
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\begin{tikzpicture}[thick, level distance=0.9cm,level 1/.style={sibling distance=4.55cm}, level 2/.style={sibling distance=1.5cm}, level 3/.style={sibling distance=0.7cm}]% level/.style={sibling distance=6cm/(#1 * 1.5)}]
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\node[tree_node](root){$\boldsymbol{\times}$}
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child{node[tree_node]{$\boldsymbol{+}$}
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child{node[tree_node]{$\boldsymbol{\times}$}
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child{node[tree_node]{$W_a$}}
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child{node[tree_node]{$W_b$}}
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}
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child{node[tree_node]{$\boldsymbol{\times}$}
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child{node[tree_node]{$W_b$}}
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child{node[tree_node]{$W_c$}}
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}
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child{node[tree_node]{$\boldsymbol{\times}$}
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child{node[tree_node]{$W_c$}}
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child{node[tree_node]{$W_a$}}
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}
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}
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child{node[tree_node]{$\boldsymbol{+}$}
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child{node[tree_node]{$\boldsymbol{\times}$}
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child{node[tree_node]{$W_a$}}
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child{node[tree_node]{$W_b$}}
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}
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child{node[tree_node]{$\boldsymbol{\times}$}
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child{node[tree_node]{$W_b$}}
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child{node[tree_node]{$W_c$}}
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}
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child{node[tree_node]{$\boldsymbol{\times}$}
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child{node[tree_node]{$W_c$}}
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child{node[tree_node]{$W_a$}}
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}
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};
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\end{tikzpicture}
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}
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\caption{Expression tree for query $\poly^2$.}
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\label{fig:intro-q2-etree}
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\end{figure}
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\subsection{Superlinearity of Bag PDBs}
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Moving forward, we focus exclusively on bags and drop the subscript from $\poly_{bag}$.
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Consider the Cartesian product of $\poly$ with itself:
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\begin{equation*}
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\poly^2() := \rel(A), E(A, B), \rel(B),\; \rel(C), E(C, D), \rel(D)
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\end{equation*}
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For an arbitrary polynomial, it is known that there may exist equivalent compressed representations.
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One such compression is the factorized polynomial~\cite{10.1145/3003665.3003667}, where the polynomial is broken up into separate factors.
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For example:
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{\small
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\begin{equation*}
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\poly^2(W_a, W_b, W_c) = \left(W_aW_b + W_bW_c + W_cW_a\right) \cdot \left(W_aW_b + W_bW_c + W_cW_a\right)
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\end{equation*}
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}
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This factorized expression can be easily modeled as an expression tree, as in \Cref{fig:intro-q2-etree},
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while the equivalent SOP representation is
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\begin{equation*}
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W_a^2W_b^2 + W_b^2W_c^2 + W_c^2W_a^2 + 2W_a^2W_bW_c + 2W_aW_b^2W_c + 2W_aW_bW_c^2.
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\end{equation*}
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The expectation then is
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\begin{align*}
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&\expct\pbox{\poly^2(W_a, W_b, W_c)}\\
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&= \expct\pbox{W_a^2}\expct\pbox{W_b^2} + \expct\pbox{W_b^2}\expct\pbox{W_c^2} + \expct\pbox{W_c^2}\expct\pbox{W_a^2} +\\
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&\qquad \expct\pbox{2W_a^2}\expct\pbox{W_b}\expct\pbox{W_c} + \expct\pbox{2W_a}\expct\pbox{W_b^2}\expct\pbox{W_c} +\\
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&\qquad \expct\pbox{2W_a}\expct\pbox{W_b}\expct\pbox{W_c^2}\\
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\end{align*}
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In our original example, the lineage polynomial for $\poly$ had the nice property that the expected count could be computed by replacing each variable with its probability (i.e., \Cref{eqn:can-inline-probabilities-into-polynomial}).
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This property does not hold for $\poly^2$ (i.e., $\expct\pbox{\poly^2} \neq \poly^2(P\pbox{W_a}, P\pbox{W_b}, P\pbox{W_c})$).
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However, a similar closed form formula for the expected count might be possible.
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Under the assumption that $Dom(W_i) = \{0, 1\}$, it is generally true that for any $k$, $\expct\pbox{W_i^k} = \expct\pbox{W_i}$.
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This property leads us to consider another structure related to $\poly$.
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% \AH{I don't know if we want to include the following statement: \par \emph{ bags are only hard with self-joins }
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% \par Atri suggests a proof in the appendix regarding this claim.}
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For any polynomial $\poly(\vct{X})$, we define the \emph{reduced polynomial} $\rpoly(\vct{X})$ to be the polynomial obtained by setting all exponents $e > 1$ in $\poly(\vct{X})$ to $1$.
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With $\poly^2$ as an example, we have:
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\begin{align*}
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\rpoly^2(W_a, W_b, W_c) =&\; W_aW_b + W_bW_c + W_cW_a + 2W_aW_bW_c + 2W_aW_bW_c\\
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&+ 2W_aW_bW_c\\
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=&\; W_aW_b + W_bW_c + W_cW_a + 6W_aW_bW_c
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\end{align*}
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%\SF{Should this be like $\tilde{\poly^2}$ to avoid ambiguous?}
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Note that the reduced polynomial is a closed form of the expected count (i.e., $\expct\pbox{\poly^2} = \rpoly(P\pbox{W_a=1}, P\pbox{W_b=1}, P\pbox{W_c=1})$).
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Also note that the $\poly$ in~\Cref{ex:bag-vs-set} is already in reduced form.
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The reduced form of a polynomial can be obtained in a linear scan over the clauses of a SOP encoding of the polynomial.
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In prior work on lineage-based Bag-PDBs~\cite{kennedy:2010:icde:pip,DBLP:conf/vldb/AgrawalBSHNSW06,yang:2015:pvldb:lenses} where this encoding is implicitly assumed, computing the expected count is linear in the size of the encoding.
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In general however, compressed encodings of the polynomial can be exponentially smaller in $k$ for $k$-products --- the query $\poly^k$ obtained by taking the Cartesian product of $k$ copies of $\poly$ has a factorized encoding of size $6\cdot k$, while the SOP encoding is of size $2\cdot 3^k$.
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This leads us to the \textbf{central question of this paper}:
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\begin{quote}
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{\em
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Is it always the case that the expectation of a UCQ in a Bag-PDB can be computed in time linear in the size of the \textbf{compressed} lineage polynomial?}
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\end{quote}
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If the answer is yes, then it is possible for Bag-PDBs to achieve performance competitive with deterministic databases.
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The answer, unfortunately, is no, and an approximation algorithm is required.
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% Consider the :
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% \begin{equation*}
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% \poly^3() := \left(\rel(A), E(A, B), R(B)\right), \left(\rel(C), E(C, D), R(D)\right), \left(\rel(F), E(F, G), R(G)\right).
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% \end{equation*}
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% The factorized output polynomial consists of a product of three identical three-way summations, while the SOP encoding is exponential --- $3^3$ clauses to be precise.
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\subsection{Overview of our results and techniques}
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Concretely, in this paper:
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(i) We show that conjunctive queries over a bag-$\ti$ are hard (i.e., superlinear in the size of a compressed lineage encoding) by reduction from counting the number of $k$-matchings over an arbitrary graph;
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(ii) We present an $(1-\epsilon)$-approximation algorithm for bag-$\ti$s and show that its complexity is linear in the size of the compressed lineage encoding;
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(iii) We generalize the approximation algorithm to bag-$\bi$s, a more general model of probabilistic data;
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(iv) We further generalize our results to higher moments, polynomial circuits, and prove that for RA+ queries, the processing time in approximation is within a constant factor of the same query processed deterministically.
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Our hardness results follow by considering a suitable generalization of the lineage polynomial in Example~\ref{ex:bag-vs-set}. First it is easy to generalize the polynomial in Example~\ref{ex:bag-vs-set} to $\poly_G^k(X_1,\dots,X_n)$ that represents the edge set of a graph $G$ in $n$ vertices. Then $\inparen{\poly_G^k(X_1,\dots,X_n)}^k$ encodes as its monomials all subgraphs of $G$ with at most $k$ edges in it. This implies that the corresponding reduced polynomial $\rpoly_G^k(p,\dots,p)$ can be written as $\sum_{i=0}^{2k} c_i\cdot p^i$ and we observe that $c_{2k}$ is proportional to the number of $k$-matchings (computing which is \sharpwonehard\ ) in $G$. Thus, if we have access to $\rpoly_G^k(p_i,\dots,p_i)$ for distinct values of $p_i$ for $0\le i\le 2k$, then we can setup a system of linear equations and compute $c_{2k}$ (and hence the number of $k$-matchings in $G$). This result, however, does not rule out the possibility that computing $\rpoly_G^k(p,\dots,p)$ for a {\em single specific} value of $p$ might be easy: indeed it is easy for $p=0$ or $p=1$. However, we are able to show that for any other value of $p$, computing $\rpoly_G^k(p,\dots,p)$ exactly will most probably require super-linear time. This reduction needs more work (and we cannot yet extend our results to $k>3$). Further, we have to rely on more recent conjectures in {\em fine-grained} complexity on e.g. the complexity of counting the number of triangles in $G$ and not more standard parameterized hardness like \sharpwonehard.
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The starting point of our approximation algorithm was the simple observation that for any lineage polynomial $\poly(X_1,\dots,X_n)$, we have $\rpoly(1,\dots,1)=Q(1,\dots,1)$ and if all the coefficients of $\poly$ are constants, then $\poly(X_1,\dots,X_n)$ (which can be easily computed in linear time) is a $p^k$ approximation to the value $\rpoly(p,\dots,p)$ that we are after. If $p$ and $k=\deg(\poly)$ are constants, then this gives a constant factor approximation. We then use sampling to get a better approximation factor of $(1\pm \eps)$: we sample monomials from $\poly(X_1,\dots,X_n)$ and do an appropriate weighted sum of their coefficients. Standard tail bounds then allow us to get our desired approximation scheme. To get a linear runtime, it turns out that we need the following properties from our compressed representation of $\poly$: (i) be able to compute $\poly(X_1,\dots,X_n)$ in linear time and (ii) be able to sample monomials from $\poly(X_1,\dots,X_n)$ quickly as well. For the ease of exposition, we start off with expression trees (see~\Cref{fig:intro-q2-etree} for an example) and show that they satisfy both of these properties. Later we show that it is easy to show that these properties also extend to polynomial circuits as well (we essentially show that in the required time bound, we can simulate access to the `unrolled' expression tree by considering the polynomial circuit).
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We also formalize our claim that, since our approximation algorithm runs in time linear in the size of the polynomial circuit, we can approximate the expected output tuple multiplicities with only a $O(\log{Z})$ overhead (where $Z$ is the number of output tuples) over the runtime of a broad class of query processing algorithms. We also observe that our results trivially extend to the problem of computing higher moments of the tuple multiplicity (instead of just the expectation).
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\paragraph{Paper Organization.} We present some relevant background and setup our notation in~\Cref{sec:background}. We present our hardness results in~\Cref{sec:hard} and our approximation algorithm in~\Cref{sec:algo}. We present some (easy) generalizations of our results in~\Cref{sec:gen}. We do a quick overview of related work in~\Cref{sec:related-work} and conclude with some open questions in~\Cref{sec:concl-future-work}.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%Interesting contributions, problem definition, known results, our results, etc
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%
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%\paragraph{Problem Definition/Known Results/Our Results/Our Techniques}
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%This work addresses the problem of performing computations over the output query polynomial efficiently. We specifically focus on computing the
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%expectation over the polynomial that is the result of a query over a PDB. This is a problem where, to the best of our knowledge, there has not
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%been a lot of study. Our results show that the problem is hard (superlinear) in the general case via a reduction to known hardness results
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%in the field of graph theory. Further we introduce a linear approximation time algorithm with guaranteed confidence bounds. We then prove the
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%claimed runtime and confidence bounds. The algorithm accepts an expression tree which models the output polynomial, samples uniformly from the
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%expression tree, and then outputs an approximation within the claimed bounds in the claimed runtime.
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%
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%\paragraph{Interesting Mathematical Contributions}
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%This work shows an equivalence between the polynomial $\poly$ and $\rpoly$, where $\rpoly$ is the polynomial $\poly$ such that all
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%exponents $e > 1$ are set to $1$ across all variables over all monomials. The equivalence is realized when $\vct{X}$ is in $\{0, 1\}^\numvar$.
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%This setting then allows for yet another equivalence, where we prove that $\rpoly(\prob,\ldots, \prob)$ is indeed $\expct\pbox{\poly(\vct{X})}$.
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%This realization facilitates the building of an algorithm which approximates $\rpoly(\prob,\ldots, \prob)$ and in turn the expectation of
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%$\poly(\vct{X})$.
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%
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%Another interesting result in this work is the reduction of the computation of $\rpoly(\prob,\ldots, \prob)$ to finding the number of
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%3-paths, 3-matchings, and triangles of an arbitrary graph, a problem that is known to be superlinear in the general case, which is, by our definition
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%hard. We show in Thm 2.1 that the exact computation of $\rpoly(\prob, \ldots, \prob)$ is indeed hard. We finally propose and prove
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%an approximation algorithm of $\rpoly(\prob,\ldots, \prob)$, a linear time algorithm with guaranteed $\epsilon/\delta$ bounds. The algorithm
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%leverages the efficiency of compressed polynomial input by taking in an expression tree of the output polynomial, which allows for factorized
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%forms of the polynomial to be input and efficiently sampled from. One subtlety that comes up in the discussion of the algorithm is that the input
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%of the algorithm is the output polynomial of the query as opposed to the input DB of the query. This then implies that our results are linear
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%in the size of the output polynomial rather than the input DB of the query, a polynomial that might be greater or lesser than the input depending
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%on the structure of the query.
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%
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%\section{Outline of the rest of the paper}
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%\begin{enumerate}
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% \item Background Knowledge and Notation
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% \begin{enumerate}
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% \item Review notation for PDBs
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% \item Review the use of semirings as generating output polynomials
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% \item Review the translation of semiring operators to RA operators
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% \item Polynomial formulation and notation
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% \end{enumerate}
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% \item Reduction to hardness results in graph theory
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% \begin{enumerate}
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% \item $\rpoly$ and its equivalence to $\expct\pbox{\poly}$ when $\vct{X} \in \{0, 1\}^\numvar$
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% \item Results for SOP polynomial
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% \item Results for compressed version of polynomial
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% \item ~\Cref{lem:const-p} proof
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% \end{enumerate}
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% \item Approximation Algorithm
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% \begin{enumerate}
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% \item Description of the Algorithm
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% \item Theoretical guarantees
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% \item Will we have time to tackle BIDB?
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% \begin{enumerate}
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% \item If so, experiments on BIDBs?
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% \end{enumerate}
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% \end{enumerate}
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% \item Future Work
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% \item Conclusion
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%\end{enumerate}
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