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@ -115,9 +115,9 @@ In this work we consider bag semantics, where each tuple is associated with a mu
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Example}\label{ex:intro-tbls}
Consider the bag-\ti relations shown in \Cref{fig:ex-shipping-simp}. We define a \ti under bag semantics analog to the set case: each tuple is associated with a probability of having a multiplicity of one (and otherwise has multiplicity zero) and tuples are independent random events. Ignore column $\Phi$ for now. In this example, we have shipping routes that are certain (probability 1.0) and information about whether shipping at locations is on time (with a certain probability). Query $\query_1$ shown below returns starting points of shipping routes where processing of shipping is on time.
Consider the bag-\ti relations shown in \Cref{fig:ex-shipping-simp}. We define a \ti under bag semantics analog to the set case: each tuple is associated with a probability of having a multiplicity of one (and otherwise multiplicity zero), and tuples are independent random events. Ignore column $\Phi$ for now. In this example, we have shipping routes that are certain (probability 1.0) and information about whether shipping at locations is on time (with a certain probability). Query $\query_1$ shown below returns starting points of shipping routes where processing of shipping is on time.
$$Q_1 := \pi_{\text{City}_1}(Loc \bowtie_{\text{City}_\ell = \text{City}_1} Route)$$
$$Q_1(\text{City}) :- Loc(\text{City}), Route(\text{City}, \_)$$
\Cref{subfig:ex-shipping-simp-queries} shows the possible results of this query.
For example, there is a 90\% probability there is a single route starting in Buffalo that is on time, and the expected multiplicity of this result tuple is $0.9$.
@ -126,25 +126,25 @@ Since the Chicago location has a 50\% probability of being on schedule (we assum
\end{Example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
A well-known result in probabilistic databases is that under set semantics the marginal probability of a query result $\tup$ can be computed based on the tuple's lineage. The lineage of a tuple is a Boolean formula (an element of the semiring $\text{PosBool}[\vct{X}]$ of positive Boolean expressions over variables $\vct{X}=(X_1,\dots,X_n)$~\cite{DBLP:conf/pods/GreenKT07}) over random variables that encode the existence of input tuples. Each possible world $\db$ corresponds to an assignment $\{0,1\}^\numvar$ of the variables in $\vct{X}$ to either true (the tuple exists in this world) or false (the tuple does not exist in this world). Importantly, the following holds: if the lineage formula for $t$ evaluates to true over the assignment for a world $\db$, then $\tup \in \query(\db)$.
A well-known result in probabilistic databases is that under set semantics the marginal probability of a query result $\tup$ can be computed based on the tuple's lineage. The lineage of a tuple is a Boolean formula (an element of the semiring $\text{PosBool}[\vct{X}]$ of positive Boolean expressions over variables $\vct{X}=(X_1,\dots,X_n)$~\cite{DBLP:conf/pods/GreenKT07}) over random variables that encode the existence of input tuples. Each possible world $\db$ corresponds to an assignment $\{0,1\}^\numvar$ of the variables in $\vct{X}$ to either true (the tuple exists in this world) or false (the tuple does not exist in this world). Importantly, the following holds: if the lineage formula for $t$ evaluates to true under the assignment for a world $\db$, then $\tup \in \query(\db)$.
Thus, the marginal probability of tuple $\tup$ is equal to the probability that its lineage evaluates to true (with respect to the obvious analog of probability distribution $\probDist$ defined over $\vct{X}$).
For bag semantics, the lineage of a tuple is a polynomial over variables $\vct{X}=(X_1,\dots,X_n)$ with % \in \mathbb{N}^\numvar$ with
coefficients in the set of natural numbers $\mathbb{N}$ (an element of semiring $\mathbb{N}[\vct{X}]$).
Analogously to the set case, evaluating the lineage for $t$ over an assignment corresponding to a possible world (mapping variables to natural numbers representing input tuple multiplicities in this world) yields the multiplicity of the result tuple $\tup$ in this world. Thus, instead of using \cref{eq:intro-bag-expectation} to compute the expected result multiplicity of a tuple $\tup$, we can equivalently compute the expectation of the lineage polynomial of $\tup$ which we will denote as $\linsett{\query}{\pdb}{\tup}$ or $\Phi$ if the parameters are clear from the context. In this work, we study the complexity of computing the expectation of such polynomials encoded as arithmetic circuits.
Analogously to sets, evaluating the lineage for $t$ over an assignment corresponding to a possible world yields the multiplicity of the result tuple $\tup$ in this world. Thus, instead of using \cref{eq:intro-bag-expectation} to compute the expected result multiplicity of a tuple $\tup$, we can equivalently compute the expectation of the lineage polynomial of $\tup$ which we will denote as $\linsett{\query}{\pdb}{\tup}$ or $\Phi$ if the parameters are clear from the context. In this work, we study the complexity of computing the expectation of such polynomials encoded as arithmetic circuits.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Example}\label{ex:intro-lineage}
Associating a lineage variable with every input tuple as shown in \cref{fig:ex-shipping-simp}, we can compute the lineage of every result tuple as shown in \cref{subfig:ex-shipping-simp-route}. For example, the tuple Chicago is in the result, because $L_b$ joins with both $R_b$ and $R_c$. Its lineage is $\Phi = L_b \cdot R_b + L_b \cdot R_c$. The expected multiplicity of this result tuple is calculated by summing the multiplicity of the result tuple, weighted by its probability, over all possible worlds.
In this example, $\Phi$ is a sum of products (SOP), and so observe that we can use linearity of expectation to solve the problem in linear time (in the size of $\linsett{\query}{\pdb}{\tup}$)
The expectation of the sum is the sum of the expectations of each monomial.
In this example, $\Phi$ is a sum of products (SOP), and so we can use linearity of expectation to solve the problem in linear time (in the size of $\linsett{\query}{\pdb}{\tup}$).
The expectation of the sum is the sum of the expectations of monomials.
The expectation of each monomial is then computed by multiplying the probabilities of the variables (tuples) occurring in the monomial.
The expected multiplicity of Chicago is $1.0$.
\end{Example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The expected multiplicity of a query result can be computed in linear time (in the size of the result's lineage) if the lineage is in SOP form.
However, this need not be true for compressed representations of polynomials such as factorized polynomials and arithmetic circuits.
However, this need not be true for compressed representations of polynomials, including factorized polynomials or arithmetic circuits.
For instance, \Cref{subfig:ex-proj-push-circ-q4} shows two circuits encoding the lineage of the result tuple $(Chicago)$ from \Cref{ex:intro-lineage}.
The left circuit encodes the lineage as a SOP while the right circuit uses distributivity to push the addition gate below the multiplication, resulting in a smaller circuit.
Given that there is a large body of work that can output such compressed representations~\cite{DBLP:conf/pods/KhamisNR16,factorized-db}, %\BG{cite FDBs and FAQ},
@ -153,14 +153,14 @@ If the answer is in the affirmative, and if lineage formulas can also be compute
Unfortunately, we prove that this is not the case: computing the expected count of a query result tuple is super-linear under standard complexity assumptions (\sharpwonehard) in the size of a lineage circuit.
Concretely, we make the following contributions:
(i) We show that computing the expected result multiplicity problem for conjunctive queries for bag-$\ti$ is \sharpwonehard in the size of a lineage circuit by reduction from counting the number of $k$-matchings over an arbitrary graph;
(i) We show that the expected result multiplicity problem for conjunctive queries for bag-$\ti$s is \sharpwonehard in the size of a lineage circuit by reduction from counting the number of $k$-matchings over an arbitrary graph;
(ii) We present an $(1\pm\epsilon)$-\emph{multiplicative} approximation algorithm for bag-$\ti$s and show that for typical database usage patterns (e.g. when the circuit is a tree or is generated by recent worst-case optimal join algorithms or its FAQ followups~\cite{DBLP:conf/pods/KhamisNR16}) its complexity is linear in the size of the compressed lineage encoding; %;\BG{Fix not linear in all cases, restate after 4 is done}
(iii) We generalize the approximation algorithm to bag-$\bi$s, a more general model of probabilistic data;
(iv) We further prove that for \raPlus queries\AR{Some places we use \raPlus and UCQ in others: we should use one consistently (assuming they are both the same)}, we can approximate the expected output tuple multiplicities with only $O(\log{Z})$ 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).
%\mypar{Implications of our Results} As mentioned above
\mypar{Overview of our Techniques} All of our results rely on working with a {\em reduced} form of the lineage polynomial $\Phi$. In fact, it turns out that for TIDB (and BIDB) case, computing the expected multiplicity is {\em exactly} the same as evaluating this reduced polynomial over the probabilities that define the TIDB/BIDB. Next, we motivate this reduced polynomial by continuing~\Cref{ex:intro-tbls}.
\mypar{Overview of our Techniques} All of our results rely on working with a {\em reduced} form of the lineage polynomial $\Phi$. In fact, it turns out that for the TIDB (and BIDB) case, computing the expected multiplicity is {\em exactly} the same as evaluating this reduced polynomial over the probabilities that define the TIDB/BIDB. Next, we motivate this reduced polynomial by continuing~\Cref{ex:intro-tbls}.
%Moving forward, we focus exclusively on bags.
Consider the query $Q():-$$OnTime(\text{City}), Route(\text{City}_1, \text{City}_2),$ $OnTime(\text{City}')$\OK{Should we be using RA- or Datalog-style query notation?} over the bag relations of \cref{fig:ex-shipping-simp}. It can be verified that $\Phi$ for $Q$ is $L_aL_b + L_bL_d + L_bL_c$. Now consider the product query $\poly^2():- Q \times Q$.