% \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?}
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.
A fundamental problem in probabilistic query processing is: given a query, probabilistic database, and possible result tuple, compute the marginal probability of the tuple to appear in the result.
In the set semantics setting, it was shown that this is equivalent to computing the probability of a Boolean formula called lineage formula which records how the result tuple has been produced by combining input tuples. Given this correspondence, the problem reduces to weighted model counting which is \sharpphard, even if the formula is in DNF. A large body of work has focused on identifying tractable cases by either identifying tractable classes of queries (e.g.,~\cite{DS12}) or studying compressed representations of lineage formulas that are tractable for certain classes of input databases (e.g.,~\cite{AB15}).
The problem of computing the marginal probability of a result tuple has a natural correspondence under bag semantics: computing the expected multiplicity of a query result tuple and, analog to the case for sets, this problem can be reduced to computing the expectation of the lineage, which under bag semantics is a polynomial. This problem has received much less attention, perhaps because the problem is trivially tractable, in fact linear time in the size of the formula, for the sum of product (SOP) representation of lineage polynomials. However, there exist compressed representations of polynomials, e.g., factorizations~\cite{factorized-db}, that can be exponentially more concise than the SOP representation of a polynomial. We would like to be able to exploit such representations for query processing. However, this is only beneficial if the problem remains tractable for such compressed representations.\footnote{Note that while for sets, compressed representations are investigated in hope of finding tractable cases, for bags the compressed representation can increase computational complexity.}
We prove that, unfortunately, this is not the case. In fact,
we prove by reduction from counting $k$-matchings that
computing the expected count of a query result tuple is super-linear (\sharpwonehard) in the size of the compressed representation. In spite of this negative result, not everything is lost. We develop a the first linear time (in the size of the factorized lineage) $(1-\epsilon)$-approximation scheme for expected counts of SPJU query results over Bag-PDBs.
As we also show for project-join queries, that this algorithm only has a constant factor overhead over deterministic query processing over the probabilistic database. This is an important result, because it implies that approximate expectations can be computed within time competitive with deterministic query evaluation over bag databases.
% is \emph{not} linear in the size of a compressed (factorized~\cite{factorized-db}) lineage polynomial by reduction from counting $k$-matchings.
% Thus, even Bag-PDBs do not enjoy the same time complexity as deterministic databases.
% This motivates our second goal, a linear time (in the size of the factorized lineage) approximation of expected counts for SPJU results in Bag-PDBs.
% As we also show, this complexity is proportional to the same query on a deterministic database.
% In this paper, we prove this
% limitation of PDBs and address it by proposing an algorithm that, to 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{
% MCDB~\cite{jampani2008mcdb} is also a constant factor slower, but only guarantees additive bounds.
% }
% The fundamental challenge is lineage formulas, a key component of query processing in PDBs.
% Using the standard (i.e., DNF) encoding of lineage, computing typical statistics like marginal probabilities or moments is easy (i.e., $O(|\text{lineage}|)$) for bags and hence, perhaps not worthy of research attention, but hard (i.e., $O(2^{|\text{lineage}|})$) for sets and hence, interesting from a research perspective.
% However, the standard encoding is unnecessarily large, and so even for Bag-PDBs, computing such statistics still has a higher complexity than answering queries in a deterministic (i.e., non-probabilistic) database.
% A naive strategy might be to move from the theoretically simpler set-relational model~\cite{DBLP:series/synthesis/2011Suciu,BD05,DBLP:conf/icde/AntovaKO07a,DBLP:conf/sigmod/SinghMMPHS08} to the computationally simpler bag-relational model, mirroring a similar transition in deterministic datbases decades ago.
% However, after discarding a long-held approach to representing lineage, we prove that query processing in Bag-PDBs is \sharpwonehard.
% 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.
% The fundamental challenge is lineage formulas, a key component of query processing in PDBs.
% Using the standard (i.e., DNF) encoding of lineage, computing typical statistics like marginal probabilities or moments is easy (i.e., $O(|\text{lineage}|)$) for bags and hence, perhaps not worthy of research attention, but hard (i.e., $O(2^{|\text{lineage}|})$) for sets and hence, interesting from a research perspective.
% However, the standard encoding is unnecessarily large, and so even for Bag-PDBs, computing such statistics still has a higher complexity than answering queries in a deterministic (i.e., non-probabilistic) database.
% In this paper, we formally prove this limitation of PDBs and address it by proposing an algorithm that, to 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{
% MCDB~\cite{jampani2008mcdb} is also a constant factor slower, but only guarantees additive bounds.
% }
% Consider the dominant problem in Set-PDBs (Computing marginal probabilities) and the corresponding problem in Bag-PDBs (computing expectations of counts).
% 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.
% Computing the probability of the tuple appearing in the result is thus analogous to weighted model counting (a known \sharpphard problem).
% In the corresponding Bag-PDB problem~\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.
% The expectation of the multiplicity is the expectation of this polynomial.
% Lineage in Set-PDBs is typically encoded in disjunctive normal form.
% This representation is significantly larger than the query result sans lineage.
% However, even with alternative encodings~\cite{FH13}, the limiting factor in computing marginal probabilities is the probability computation itself, not the lineage formula.
% The Bag-PDB analog is a polynomial in sum of products (SOP) form --- a sum of `clauses', each the product of a set of integer or variable atoms.
% Thanks to linearity of expectation, computing the expectation of a count query is linear in the number of clauses in the SOP polynomial.
% 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.
% Compressed representations like Factorized Databases~\cite{factorized-db} %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,factorized-db}.
% Thus, measuring the performance of a PDB algorithm in terms of the size of the \emph{compressed} lineage formula more closely relates the algorithm's performance to the complexity of query evaluation in a deterministic database.
% The initial picture is not good.
% We prove that computing expected counts is \emph{not} linear in the size of a compressed (factorized~\cite{factorized-db}) lineage polynomial by reduction from counting $k$-matchings.
% Thus, even Bag-PDBs do not enjoy the same time complexity as deterministic databases.
% This motivates our second goal, a linear time (in the size of the factorized lineage) approximation of expected counts for SPJU results in Bag-PDBs.
% As we also show, this complexity is proportional to the same query on a deterministic database.
% 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.
%Consider an arbitrary output polynomial $\poly$. Further, consider the same polynomial, with all exponents $e > 1$ set to $1$ and call the resulting polynomial $\rpoly$.
Consider the Tuple Independent ($\ti$) Set-PDB\footnote{Our work does also handle Block Independent Databases ($\bi$)~\cite{BD05,DBLP:series/synthesis/2011Suciu}.} given in \Cref{fig:intro-ex} with two input relations $R$ and $E$.
Each input tuple is assigned an annotation (attribute $\Phi_{set}$): an independent random Boolean variable ($W_i$) or the constant $\top$.
% 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 $\top$.
For example, let $\probOf[W_a]=\probOf[W_b]=\probOf[W_c]=\prob$ and consider the possible world where $R =\{\;\tuple{a}, \tuple{b}\;\}$.
The corresponding variable assignment is $\{\;W_a \mapsto\top, W_b \mapsto\top, W_c \mapsto\bot\;\}$, and the probability of this world is $\probOf[W_a]\cdot\probOf[W_b]\cdot\probOf[\neg W_c]=\prob\cdot\prob\cdot(1-\prob)=\prob^2-\prob^3$.
Following prior efforts~\cite{feng:2019:sigmod:uncertainty,DBLP:conf/pods/GreenKT07,GL16}, we generalize this model of Set-PDBs to bags using $\semN$-valued random variables (i.e., $Dom(W_i)\subseteq\mathbb N$) and constants (annotation $\Phi_{bag}$ in the example).
The lineage of the result in a Set-PDB (resp., Bag-PDB) is a Boolean (polynomial) formula over random variables annotating the input relations (i.e., $W_a$, $W_b$, $W_c$).
Because the query result is 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):
Given $W_a$, $W_b$, $W_c$, these functions compute the existence (resp., count) of the nullary result tuple for $\poly$ applied to the database instance in \Cref{fig:intro-ex}.
We show one possible world here, with the set assignment $\{\;W_a\mapsto\top, W_b \mapsto\top, W_c \mapsto\bot\;\}$ and the analogous bag assignment:
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{FH13} the use of Shannon decomposition, which is at worst exponential in the size of the input.
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.
Recall the nice property of $\query$ that its expected count could be computed by evaluating its lineage on the probability vector (i.e., \Cref{eqn:can-inline-probabilities-into-polynomial}).
This property does not hold for $\poly^2$ (i.e., $\expct\pbox{\poly^2}\neq\poly^2(\probOf\pbox{W_a}, \probOf\pbox{W_b}, \probOf\pbox{W_c})$), but does suggest a related closed form formula.
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 $1$.
Note that the reduced polynomial is a closed form of the expected count (i.e., $\expct\pbox{\poly^2}=\rpoly(\probOf\pbox{W_a=1}, \probOf\pbox{W_b=1}, \probOf\pbox{W_c=1})$).
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.
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 $2\cdot3^k$.
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?}
% 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.
(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;
(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;
(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.
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 $\rpoly_G^k(\prob,\dots,\prob)$ can be written as $\sum_{i=0}^{2k} c_i\cdot\prob^i$ 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 $\rpoly_G^k(\prob_i,\dots,\prob_i)$ for distinct values of $\prob_i$ for $0\le i\le2k$, 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 $\rpoly_G^k(\prob,\dots, \prob)$ for a {\em single specific} value of $\prob$ might be easy: indeed it is easy for $\prob=0$ or $\prob=1$. However, we are able to show that for any other value of $\prob$, computing $\rpoly_G^k(\prob,\dots, \prob)$ 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.
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 $\prob^k$ approximation to the value $\rpoly(\prob,\dots, \prob)$ that we are after. If $\prob$ 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).
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 higher moments of the tuple multiplicity (instead of just the expectation).
\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}.