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\section{Introduction}

In practice, modern production databases, e.g., Postgres, Oracle, etc. use bag semantics.  In contrast, most implementations of modern probabilistic databases (PDBs) are built in the setting of set semantics, and this contributes to slow computation time. When we consider PDBs in the bag setting, it is then the case that each tuple is annotated with a polynomial, which describes the tuples contributing to the given tuple's presence in the output.  The polynomial is composed of $+$ and $\times$ operators, with constants from the set $\mathbb{N}$ and variables from the set of variables $\vct{X}$.  The polynomial is an encoding of the multiplicity of the tuple in the output, while in general, as we allude later one, the polynomial can also represent set semantics, access levels, and other encodings.  Should we attempt to make computations over the output polynomial, the naive algorithm cannot hope to do better than linear time in the size of the polynomial.  However, in the set semantics setting, when e.g., computing the expectation of the output polynomial given values for each variable in the polynomial's set of variables $\vct{X}$, this problem is \#P-hard. %of the output polynomial of a result tuple for an arbitrary query.  In contrast, in the bag setting, one cannot generate a result better than linear time in the size of the polynomial.

There is limited work and results in the area of bag semantic PDBs.  This work seeks to leverage prior work in factorized databases (e.g. Olteanu et. al.)~\cite{DBLP:conf/tapp/Zavodny11} with PDB implementations to improve efficient computation over output polynomials.  When considering PDBs in the bag setting a subtelty arises that is easily overlooked due to the \textit{oversimplification} of PDBs in the set setting, i.e., in set semantics expectation doesn't have linearity over disjunction, and a consequence of this it is not true in the general case that a compressed polynomial has an equivalent expectation to its DNF form.  In the bag PDB setting, however, expectation does enjoy linearity over addition, and the expectation of a compressed polynomial and its equivalent SOP are indeed the same.

For almost all modern PDB implementations, an output polynomial is only ever considered in its expanded SOP form.
\BG{I don't think this statement holds up to scrutiny. For instance, ProvSQL uses circuits.}
Any computation over the polynomial then cannot hope to be in less than linear in the number of monomials, which is known to be exponential in the size of the input in the general case.

The landscape of bags changes when we think of annotation polynomials in a compressed form rather than as traditionally implemented in 
disjunctive normal form (DNF).  The implementation of PDBs has followed a course of producing output tuple annotation polynomials as a 
giant pile of monomials, aka DNF.  This is seen in many implementations, including MayBMS, MystiQ, GProM, Orion, etc.),  ~\cite{DBLP:conf/icde/AntovaKO07a}, ~\cite{DBLP:conf/sigmod/BoulosDMMRS05}, ~\cite{AF18}, ~\cite{DBLP:conf/sigmod/SinghMMPHS08}, all of which use an 
encoding that is essentially an enumeration through all the monomials in the DNF.  The reason for this is because of the customary fixed data 
size rule for attributes in the classical approach to building DBs.  Such an approach allows practitioners to know how big a tuple will be,
and thus pave the way for several optimizations.  The goal is to avoid the situation where a field might get too big.  However, annotations
break this convention, since, e.g., a projection can produce an annotation that is arbitrarily greater in the size of the data.  Other RA operators, such as join
grow the annotations unboundedly as well, albeit at a lesser rate.  As a result, the aforementioned PDBs want to avoid creating arbitrarily sized
fields, and the strategy has been to take the provenance polynomial, flatten it into individual monomials, storing each individual monomial in 
a table.  This restriction carries with it $O(n^2)$ in the size of the input tables to materialize the monomials.  Obviously, such an approach 
disallows doing anything clever.  For those PDBs that do allow factorized polynomials, e.g., Sprout ~\cite{DBLP:conf/icde/OlteanuHK10}, they assume such an encoding in the set semantics.  With compressed encodings, the problem in bag semantics is actually hard in a non-obvious way.


It turns out, that should an implementation allow for a compressed form of the polynomial, that computations over the polynomial can be done in better than linear runtime, again, linear in the number of monomials of the expanded Sum of Products (SOP) form.  While it is true that the runtime in the number of monomials versus runtime in the size of the polynomial is the same when the polynomial is given in SOP form, the runtimes are not the same when we allow for compressed versions of the polynomial as input to the desired computation.  While the naive algorithm to compute the expectation of a polynomial with respective probability values for all variables in $\vct{X}$ is to generate all the monomials and compute each of their probabilities, factorized polynomials in the bag setting allow us to compute the probability in the number of terms (with their corresponding probability values substituted in for their respective variables) that make up the compressed representation.  For clarity, the probability we are considering is whether or not the tuple exists in the output.


As implied above, we define hard to be anything greater than linear in the number of monomials when the polynomial is in SOP form.  In this work, we show, that computing the expectation over the output polynomial for even the query class of $CQ$, which allow for only projections and joins. over a bag $\ti$ where all tuples have probability $\prob$ is hard in the general case.  However, allowing for compressed versions of the polynomial paves the way for an approximation algorithm that performs in linear time with $\epsilon/\delta$ guarantees.  Also, while implied in the preceeding, in this work, the input size to the approximation algorithm is considered to be the query polynomial as opposed to the input database.


The richness of the problem we explore gives us a lower and upper bound in the compressed form of the polynomial, and its size in SOP form, specifically the range [compressed, SOP].  In approximating the expectation, an expression tree to model the query output polynomial, which indeed facilitates polyomials in compressed form.

\paragraph{Problem Definition/Known Results/Our Results/Our Techniques}
This work addresses the problem of performing computations over the output query polynomial efficiently.  We specifically focus on computing the
expectation over the polynomial that is the result of a query over a bag PDB.  This is a problem where, to the best of our knowledge, there has not
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
in the field of graph theory.  Further we introduce a linear approximation time algorithm with guaranteed confidence bounds.  We then prove the
claimed runtime and confidence bounds.  The algorithm accepts an expression tree which models the output polynomial, samples uniformly from the
expression tree, and then outputs an approximation within the claimed bounds in the claimed runtime. 

\paragraph{Interesting Mathematical Contributions}
This work shows an equivalence between the polynomial $\poly$ and $\rpoly$, where $\rpoly$ is the polynomial $\poly$ such that all
exponents $e > 1$ are set to $1$ across all variables over all monomials.  The equivalence is realized when $\vct{X}$ is in $\{0, 1\}^\numvar$.  
This setting then allows for yet another equivalence, where we prove that $\rpoly(\prob,\ldots, \prob)$ is indeed $\expct\pbox{\poly(\vct{X})}$.
This realization facilitates the building of an algorithm which approximates $\rpoly(\prob,\ldots, \prob)$ and in turn the expectation of 
$\poly(\vct{X})$.

Another interesting result in this work is the reduction of the computation of $\rpoly(\prob,\ldots, \prob)$ to finding the number of 
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
hard.  We show in Thm 2.1 that the exact computation of $\rpoly(\prob, \ldots, \prob)$ is indeed hard.   We finally propose and prove
an approximation algorithm of $\rpoly(\prob,\ldots, \prob)$, a linear time algorithm with guaranteed $\epsilon/\delta$ bounds.  The algorithm
leverages the efficiency of compressed polynomial input by taking in an expression tree of the output polynomial, which allows for factorized
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
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
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
on the characterization of the query.



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