Saving my intro in another file
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%!TEX root=./main.tex
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%root: main.tex
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\section{Introduction (Rewrite - 070921)}\label{sec:intro-rewrite-070921}
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\section{Introduction (Atri's pass - 08/26/21)}\label{sec:intro-rewrite-070921}
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\input{two-step-model}
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A probabilistic database (or PDB) $\pdb$ is a pair $\inparen{\idb, \pd}$ such that $\idb$ is a set of deterministic database instances (possible worlds) and $\pd$ is a probability distribution over $\idb$.
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In bag count-query\AR{Why are we restricting ourselves to a couont query here? Why not just say `bag query'?} semantics the random variable $\query\inparen{\pdb}\inparen{\tup}$ is the multiplicity of its corresponding output tuple $\tup$ (in a random database instance in $\idb$ chosen according to $\pd$).
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A probabilistic database (PDB) $\pdb$ is a tuple $\inparen{\idb, \pd}$ such that $\idb$ is a set of deterministic database instances called possible worlds and $\pd$ is a probability distribution over $\idb$.
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A commonly studied problem in probabilistic databases is given a query $\query$, PDB $\pdb$, and possible query result tuple $\tup$, to compute the tuple's \textit{marginal probability} to be in the query's result, i.e., computing the expectation of a Boolean random variable over $\pd$ that is $1$ for every $\db \in \idb$ for which $\tup \in \query(\db)$ and $0$ otherwise. In this work, we are interested in bag semantics where each tuple $\tup$ is associated with a multiplicity $\db(\tup)$ from $\semN$ in each possible world.\footnote{We find it convenient to use the notation from~\cite{DBLP:conf/pods/GreenKT07} which models bag relations as function that map tuples to their multiplicity.} The natural generalization of the problem of computing marginal probabilities of query result tuples to bag semantics is to compute the expectation of a random variable over $\pd$ that is $m$ for world $\db$ iff $\query(\db)(\tup) = m$.
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In bag count-query semantics the random variable $\query\inparen{\pdb}\inparen{\tup}$ computes the multiplicity of its corresponding tuple $\tup$.
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In addition to traditional deterministic query evaluation requirements (for a given query class), the count-query evaluation problem in bag-\abbrPDB semantics can be formally stated as:
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\begin{Problem}\label{prob:bag-pdb-query-eval}
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Given a query $\query$ from the set of positive relational algebra queries\footnote{The class of $\raPlus$ queries consists of all queries that can be composed of the positive (monotonic) relational algebra operators: selection, projection, join, and union (SPJU).} ($\raPlus$), compute the expected\footnote{Unless stated otherwise, we assume the implicity probability distribution $\pd$, and for notational convenience use $\expct\pbox{\cdot}$ instead of $\expct_\pd\pbox{\cdot}$.}
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multiplicity ($\expct\pbox{\query\inparen{\pdb}\inparen{\tup}}$)
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of output tuple $\tup$. We will be interested in the data complexity of this problem (i.e. we think of $Q$ as being of constant size).
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\end{Problem}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Solving~\cref{prob:bag-pdb-query-eval} for arbitrary $\pd$ is hopeless since we need exponential space to repreent an arbitrary $\pd$.
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We initially focus on tuple-independent probabilistic bag-databases (\abbrTIDB), a compressed encoding of probabilistic databases where the presence of each individual tuple (out of a total of $\numvar$ input tuples) in a possible world can be modeled as an independent probabilistic event\footnote{
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@ -79,9 +83,14 @@ However, there exist some queries for which \emph{bag}-\abbrPDB\xplural are a mo
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% paradigm of set-\abbrPDB\xplural. To address the question of whether or not bag-\abbrPDB\xplural are easy,
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%END Needs to be noted.
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<<<<<<< HEAD
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% Atri: Removing stuff below as per conversation with Oliver on matrix on Aug 26
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%A natural question is whether or not we can quantify the complexity of computing $\expct\pbox{\poly_\tup\inparen{\vct{\randWorld}}}$ separately from the complexity of deterministic query evaluation, effectively dividing \abbrPDB query evaluation into two steps: deterministic query evaluation\footnote{Given input $\pdb$, this step includes outputting every tuple $\tup$ that satisfies $\query$, annotated with its lineage polynomial ($\poly_\tup$) which is computed inline across the query operators of $\query$.\cite{Imielinski1989IncompleteII}\cite{DBLP:conf/pods/GreenKT07}} and computing expectation. Viewing \abbrPDB query evaluation as these two seperate steps is also known as intensional evaluation \cite{DBLP:series/synthesis/2011Suciu}, illustrated in \cref{fig:two-step}.
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%The first step, which we will refer to as \termStepOne (\abbrStepOne), consists of computing both $\query\inparen{\db}$ and $\poly_\tup(\vct{X})$.\footnote{Assuming standard $\raPlus$ query processing algorithms, computing the lineage polynomial of $\tup$ is upperbounded by the runtime of deterministic query evaluation of $\tup$, as we show in \cref{sec:circuit-runtime}.} The second step is \termStepTwo (\abbrStepTwo), which consists of computing $\expct\pbox{\poly_\tup(\vct{\randWorld})}$. Such a model of computation is nicely followed in set-\abbrPDB semantics \cite{DBLP:series/synthesis/2011Suciu}, where $\poly_\tup\inparen{\vct{X}}$ must be computed separate from deterministic query evaluation to obtain exact output when $\query(\pdb)$ is hard since evaluating the probability inline with query operators (extensional evaluation) will only approximate the actual probability in such a case. The paradigm of \cref{fig:two-step} is also analogous to semiring provenance, where $\semNX$-DB\footnote{An $\semNX$-DB is a database whose tuples are annotated with elements from the set of polynomials with variables in $\vct{X}$ and natural number coeficients and exponents.} query processing \cite{DBLP:conf/pods/GreenKT07} first computes the query and polynomial, and the $\semNX$-polynomial can then subsequently evaluated over a semantically appropriate semiring, e.g. $\semN$ to model bag semantics. Further, in this work, the intensional model lends itself nicely in separating the concerns of deterministic computation and the probability computation.
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=======
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A natural question is whether or not we can quantify the complexity of computing $\expct\pbox{\poly_\tup\inparen{\vct{\randWorld}}}$ separately from the complexity of deterministic query evaluation, effectively dividing \abbrPDB query evaluation into two steps: deterministic query evaluation\footnote{Given input $\pdb$, this step includes outputting every tuple $\tup$ that satisfies $\query$, annotated with its lineage polynomial ($\poly_\tup$) which is computed inline across the query operators of $\query$.\cite{Imielinski1989IncompleteII}\cite{DBLP:conf/pods/GreenKT07}} and computing expectation. Viewing \abbrPDB query evaluation as these two seperate steps is also known as intensional evaluation \cite{DBLP:series/synthesis/2011Suciu}, illustrated in \cref{fig:two-step}.
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The first step, which we will refer to as \termStepOne (\abbrStepOne), consists of computing both $\query\inparen{\db}$ and $\poly_\tup(\vct{X})$.\footnote{Assuming standard $\raPlus$ query processing algorithms, computing the lineage polynomial of $\tup$ is upperbounded by the runtime of deterministic query evaluation of $\tup$, as we show in \cref{sec:circuit-runtime}.} The second step is \termStepTwo (\abbrStepTwo), which consists of computing $\expct\pbox{\poly_\tup(\vct{\randWorld})}$. Such a model of computation is nicely followed in set-\abbrPDB semantics \cite{DBLP:series/synthesis/2011Suciu}, where $\poly_\tup\inparen{\vct{X}}$ must be computed separate from deterministic query evaluation to obtain exact output when $\query(\pdb)$ is hard since evaluating the probability inline with query operators (extensional evaluation) will only approximate the actual probability in such a case. The paradigm of \cref{fig:two-step} is also analogous to semiring provenance, where $\semNX$-DB\footnote{An $\semNX$-DB is a database whose tuples are annotated with elements from the set of polynomials with variables in $\vct{X}$ and natural number coeficients and exponents.} query processing \cite{DBLP:conf/pods/GreenKT07} first computes the query and polynomial, and the $\semNX$-polynomial can then subsequently evaluated over a semantically appropriate semiring, e.g. $\semN$ to model bag semantics. Further, in this work, the intensional model lends itself nicely in separating the concerns of deterministic computation and the probability computation.
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>>>>>>> 3f771b6cf71fd4f2693fd44b1d8f097a5e324f1c
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Analogous to set-probabilistic databases, we focus on the intensional model of query evaluation, as illustrated in \cref{fig:two-step}.
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Given input $\pdb$ and $\query$, the first step, which we will refer to as \termStepOne (\abbrStepOne), outputs every tuple $\tup$ that satisfies $\query$, annotated with its lineage polynomial ($\poly_\tup$) which is computed inline across the query operators of $\query$~\cite{Imielinski1989IncompleteII}\cite{DBLP:conf/pods/GreenKT07}.
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@ -89,6 +98,7 @@ We show in \cref{sec:circuit-runtime} that, assuming a standard $\raPlus$ query
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In other words, the first step is in \sharpwonehard, allowing us to focus on the complexity of the second step.
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The second step is \termStepTwo (\abbrStepTwo), which consists of computing $\expct\pbox{\poly_\tup(\vct{\randWorld})}$.
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<<<<<<< HEAD
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We observe that the paradigm of \cref{fig:two-step} is also analogous to semiring provenance~\cite{DBLP:conf/pods/GreenKT07}, where $\semNX$-DB\footnote{An $\semNX$-DB is a database whose tuples are annotated with standard polynomials, i.e. elements from $\semNX$ connected by multiplication and addition operators.} query processing first computes the query and polynomial, and the $\semNX$-polynomial can then subsequently evaluated over a semantically appropriate semiring, e.g. $\semN$ to model bag semantics. Further, in this work, the intensional model lends itself nicely in separating the concerns of deterministic computation and the probability computation. \AR{Need to state/justify that intensional model is the "norm" in existing PDB systems.}
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For PDB $\pdb$ and query $Q$, let $\timeOf{\abbrStepOne}(Q,\pdb)$ denote the runtime of \abbrStepOne (Lineage Computation) and similarly for $\timeOf{\abbrStepTwo}(Q,\pdb)$ (Expectation Computation).
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@ -98,6 +108,14 @@ When $\poly_\tup(\vct{X})$ is in standard monomial basis (\abbrSMB)\footnote{A p
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% Replaced the stuff below with something more auccint
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%For output tuple $\tup'$ of $\query\inparen{\pdb}$, computing $\expct\pbox{\poly_{\tup'}\inparen{\vct{\randWorld}}}$ is linear in
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%$\abs{\poly_\tup}$
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=======
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We observe that the paradigm of \cref{fig:two-step} is also analogous to semiring provenance~\cite{DBLP:conf/pods/GreenKT07}, where $\semNX$-DB\footnote{An $\semNX$-DB is a database whose tuples are annotated with standard polynomials, i.e. elements from $\semNX$ connected by multiplication and addition operators.} query processing first computes the query and polynomial, and the $\semNX$-polynomial can then subsequently evaluated over a semantically appropriate semiring, e.g. $\semN$ to model bag semantics. Further, in this work, the intensional model lends itself nicely in separating the concerns of deterministic computation and the probability computation.
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\subsection{Intensional Bag-Probabilistic Query Evaluation}
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Let $\timeOf{\abbrStepOne}$ denote the runtime of \abbrStepOne (Lineage Computation) and similarly for $\timeOf{\abbrStepTwo}$ (Expectation Computation).
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Given bag-\abbrPDB query $\query$ and \abbrTIDB $\pdb$ with $\numvar$ tuples, let us go a step further and assume that computing $\poly_\tup$ is lower bounded by the runtime of determistic query computation of $\query$ (e.g. when $\abs{\textnormal{input}} \leq \abs{\textnormal{output}}$). When $\poly_\tup$ is in standard monomial basis (\abbrSMB)\footnote{A polynomial is in \abbrSMB when it consists of a sum of unique products.}, by linearity of expectation and independence of \abbrTIDB, it follows that $\timeOf{\abbrStepTwo}$ is indeed $\bigO{\timeOf{\abbrStepOne}}$. Let $\prob_i$ denote the probability of tuple $\tup_i$ ($\probOf\pbox{X_i = 1}$) for $i \in [\numvar]$. Consider another special case when for all $i$ in $[\numvar]$, $\prob_i = 1$. For output tuple $\tup'$ of $\query\inparen{\pdb}$, computing $\expct\pbox{\poly_{\tup'}\inparen{\vct{\randWorld}}}$ is linear in
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$\abs{\poly_\tup}$
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>>>>>>> 3f771b6cf71fd4f2693fd44b1d8f097a5e324f1c
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%the size of the arithemetic circuit
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%, since we can essentially push expectation through multiplication of variables dependent on one another.\footnote{For example in this special case, computing $\expct\pbox{(X_iX_j + X_\ell X_k)^2}$ does not require product expansion, since we have that $p_i^h x_i^h = p_i \cdot 1^{h-1}x_i^h$.}
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In this case, we have for any output tuple $\tup$ $\expct\pbox{\Phi_\tup(\vct{W})}=\Phi(1,\dots,1)$.
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@ -109,6 +127,7 @@ Given a \abbrPDB $\pdb$, $\raPlus$ query $\query$ and output tuple $\tup$, is it
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If the answer to \cref{prob:big-o-step-one} is yes, then the query evaluation problem over bag \abbrPDB\xplural is of the same complexity as deterministic query evaluation, and probabilistic databases can offer performance competitive with deterministic databases.
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<<<<<<< HEAD
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The main insight of the paper is that to answer~\Cref{prob:big-o-step-one}, the representation of $\Phi_\tup(\vct{X})$ matters. One can have compact representations of $\poly_\tup(\vct{X})$ resulting from, for example, optimizations like projection push-down which produce factorized representations
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%Atri: footnote below was not informative: used an example instead
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%\footnote{A factorized representation is a representation of a polynomial that is not in \abbrSMB form.}
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@ -116,6 +135,11 @@ of $\poly_\tup(\vct{X})$ (e.g. in~\Cref{fig:two-step}, $B(Y+Z)$ is a factorized
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\footnote{An arithmetic circuit has variable and/or numeric inputs, with internal nodes representing either an addition or multiplication operator.}
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as the representation system of $\poly_\tup(\vct{X})$, which are a natural fit to $\raPlus$ queries as each operator maps to either a $\circplus$ or $\circmult$ operation \cite{DBLP:conf/pods/GreenKT07}. The standard query evaluation semantics depicted in \cref{fig:nxDBSemantics} illustrate this.
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=======
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The main insight of the paper is that we should not stop here. One can have compact representations of $\poly_\tup(\vct{X})$ resulting from, for example, optimizations like projection push-down which produce factorized representations\footnote{A factorized representation is a representation of a polynomial that is not in \abbrSMB form.} of $\poly_\tup(\vct{X})$. To capture such factorizations, this work uses (arithmetic) circuits
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\footnote{An arithmetic circuit has variable and/or numeric inputs, with internal nodes each of which can take on a value of either an addition or multiplication operator.}
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as the representation system of $\poly_\tup(\vct{X})$, which are a natural fit to $\raPlus$ queries as each operator maps to either a $\circplus$ or $\circmult$ operation \cite{DBLP:conf/pods/GreenKT07}. The standard query evaluation semantics depicted in \cref{fig:nxDBSemantics} nicely illustrate this.
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\begin{figure}
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\begin{align*}
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@ -138,6 +162,7 @@ as the representation system of $\poly_\tup(\vct{X})$, which are a natural fit t
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\label{fig:nxDBSemantics}
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\end{figure}
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<<<<<<< HEAD
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In other words, we can capture the size of a factorized lineage polynomial by the size of its correspoding arithmetic circuit $\circuit$ (which we denote by $|\circuit|$).
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More importantly, our result in \cref{sec:circuit-runtime} shows that, assuming a standard $\raPlus$ query evaluation algorithm for \termStepOne, given the arithmetic circuit $\circuit$ corresponding to lineage polynomial output at the end of \termStepOne, we always have $|\circuit|\le \bigO{\timeOf{\abbrStepOne}(Q,\pdb)}$. Given this, we study the following stronger version of~\Cref{prob:big-o-step-one}:
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@ -145,6 +170,9 @@ More importantly, our result in \cref{sec:circuit-runtime} shows that, assuming
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%Re-stating our earlier observation, given a circuit \circuit, if \circuit is in \abbrSMB (i.e. every sink to source path has a prefix of addition nodes and the rest of the internal nodes are multiplication nodes), then we have that $\timeOf{\abbrStepTwo}(Q,\pdb)$ is indeed $\bigO{\timeOf{\abbrStepOne}(Q,\pdb)}$. We note that \abbrSMB representations are produced by queries with a projection operation on top of a join operation.
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% the form $\project, \project\inparen{\join},$ etc.
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%Suppose, on the contrary, that \circuit is not in \abbrSMB and rather in some factorized form. Then to naively compute \abbrStepTwo, one needs to convert \circuit into \circuit' such that \circuit' is in \abbrSMB, and then compute $\expct\pbox{\poly_\tup\inparen{\vct{\randWorld}}}$, which takes $\bigO{|\circuit|^k}$ time for the case that $k$ is the degree of the polynimial $\Phi_\tup(\vct{X})$. Since $|\circuit'|$ lies between $\bigO{|\circuit|}$ and $\bigO{|\circuit|^k}$, it behooves us to determine which of these extremes is true for the general \circuit. This leads us to the main problem statement of our paper:
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=======
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Above we have seen, given a circuit \circuit, if \circuit is in \abbrSMB, then we have that $\timeOf{\abbrStepTwo}$ is indeed $\bigO{\timeOf{\abbrStepOne}}$. Such representations are produced by queries with the form $\project, \project\inparen{\join},$ etc. Suppose, on the contrary, that \circuit is not in \abbrSMB and rather in some factorized form. Then to naively compute \abbrStepTwo, one needs to convert \circuit into \circuit' such that \circuit' is in \abbrSMB, and then compute $\expct\pbox{\poly_\tup\inparen{\vct{\randWorld}}}$, which takes $\bigO{\abbrStepOne^k}$ time for the general $k$-wise factorization. Since \abbrStepTwo lies between $\bigO{\abbrStepOne}$ and $\bigO{\abbrStepOne^k}$, it behooves us to determine which of these extremes is true for the general \circuit. This leads us to the main problem statement of our paper:
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\begin{Problem}\label{prob:intro-stmt}
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Given a circuit $\circuit$ for \termStepOne for \abbrPDB $\pdb$ and $\raPlus$ query $\query$, is it always the case that $\timeOf{\abbrStepTwo}(Q,\pdb)$ is $\bigO{|\circuit|}$?
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%\OK{This doesn't parse. What is $\bigO{\abbrStepOne}$? Should this be $\bigO{\poly}$?}
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@ -156,6 +184,7 @@ Note that an answer in the affirmative to the above question, implies an affirma
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%%%%%%%%%%%%%%%%%%%%%%%%%
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\mypar{Our Results} In this paper we tackle~\Cref{prob:big-o-step-one} to~\Cref{prob:intro-stmt}.
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Concretely, we make the following contributions:
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<<<<<<< HEAD
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(i) %Under fine grained hardness assumption,
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We show that the answer to~\Cref{prob:big-o-step-one} is no in general for exact computation. %\cref{prob:intro-stmt} for bag-\abbrTIDB\xplural is not true in general
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% \sharpwonehard in the size of the lineage circuit
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@ -180,6 +209,21 @@ we can approximate the expected output tuple multiplicities (for all output tupl
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Consider the query $\query(\pdb) \coloneqq \project_\emptyset(OnTime \join_{City = City_1} Route \join_{{City}_2 = City'}\rename_{City' \leftarrow City}(OnTime)$\AR{$\rename$ is not defined. Any reason why we do not just associate the attribute names with the relation. The datalog notation was much cleaner to me.}
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%$Q()\dlImp$$OnTime(\text{City}), Route(\text{City}, \text{City}'),$ $OnTime(\text{City}')$
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=======
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(i) Under fine grained hardness assumption, we show that \cref{prob:intro-stmt} for bag-\abbrTIDB\xplural is not true in general
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% \sharpwonehard in the size of the lineage circuit
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by reduction from counting the number of $k$-matchings over an arbitrary graph; we further show superlinear hardness in the size of \circuit for a specific %cubic
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graph query for the special case of all $\prob_i = \prob$ for some $\prob$ in $(0, 1)$;
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(ii) We present an $(1\pm\epsilon)$-\emph{multiplicative} approximation algorithm for bag-\abbrTIDB\xplural and $\raPlus$ queries that makes \cref{prob:intro-stmt} true again; we further 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 their Functional Aggregate Query (FAQ) followups~\cite{DBLP:conf/pods/KhamisNR16}), the approximation algorithm has runtime linear in the size of the compressed lineage encoding (in contrast, known approximation techniques in set-\abbrPDB\xplural are at most quadratic\footnote{Note that this doesn't rule out queries for which approximation is linear}); (iii) We generalize the \abbrPDB data model considered by the approximation algorithm to a class of bag-Block Independent Disjoint Databases (see \cref{subsec:tidbs-and-bidbs}) (\abbrBIDB\xplural); (iv) We further prove that for \raPlus queries
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\AH{This point \emph{\Large seems} weird to me. I thought we just said that the approximation complexity is linear in step one, but now it's as if we're saying that it's $\log{\text{step one}} + $ the runtime of step one. Where am I missing it?}
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\OK{Atri's (and most theoretician's) statements about complexity always need to be suffixed with ``to within a log factor''}
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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).
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\mypar{Overview of our Techniques} All of our results rely on working with a {\em reduced} form of the lineage polynomial $\poly_\tup$. 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 in what follows.
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Consider the query $\query(\pdb) \coloneqq \project_\emptyset(OnTime \join_{City = City_1} Route \join_{{City}_2 = City'}\rename_{City' \leftarrow City}(OnTime)$
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%$Q()\dlImp$$OnTime(\text{City}), Route(\text{City}, \text{City}'),$ $OnTime(\text{City}')$
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>>>>>>> 3f771b6cf71fd4f2693fd44b1d8f097a5e324f1c
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over the bag relations of \cref{fig:two-step}. It can be verified that $\poly_\tup\inparen{A, B, C, D, X, Y, Z}$ for $Q$ is $AXB + BYD + BZC$. Now consider the product query $\query^2(\pdb) = \query(\pdb) \times \query(\pdb)$.
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The lineage polynomial for $Q^2$ is given by $\poly^2\inparen{A, B, C, D, X, Y, Z}$:
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\end{equation*}
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\end{Lemma}
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<<<<<<< HEAD
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To prove our hardness result we show that for the same $Q$ considered in the example above, for an arbitrary product width $k$, the query $Q^k$ is able to encode various hard graph-counting problems\footnote{While $\query$ is the same, our results assume $\bigO{\numvar}$ tuples rather than the constant number of tuples appearing in \cref{fig:two-step}}. We do so by analyzing how the coefficients in the (univariate) polynomial $\widetilde{\Phi}\left(p,\dots,p\right)$ relate to counts of various sub-graphs on $k$ edges in an arbitrary graph $G$ (which is used to define the $Route$ relation in $Q$).
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For an upper bound on approximating the expected count, it is easy to check that if all the probabilties are constant then ${\Phi}\left(\prob_1,\dots, \prob_n\right)$ (i.e. evaluating the original lineage polynomial over the probability values) is a constant factor approximation. For example, using $\query^2$ from above, using $\prob_A$ to denote $\probOf\pbox{A = 1}$ (and similarly for the other six variables), we can see that
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=======
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To prove our hardness result we show that for the same $Q$ considered in the query above, for an arbitrary product width $k$, the query $Q^k$ is able to encode various hard graph-counting problems\footnote{While $\query$ is the same, our results assume $\bigO{\numvar}$ tuples rather than the constant number of tuples appearing in \cref{fig:two-step}}. We do so by analyzing how the coefficients in the (univariate) polynomial $\widetilde{\Phi}\left(p,\dots,p\right)$ relate to counts of various sub-graphs on $k$ edges in an arbitrary graph $G$ (which is used to define the relations in $Q$).
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For an upper bound on approximating the expected count, it is easy to check that if all the probabilties are constant then ${\Phi}\left(\probOf\pbox{X_1=1},\dots, \probOf\pbox{X_n=1}\right)$ (i.e. evaluating the original lineage polynomial over the probability values) is a constant factor approximation. For example, using $\query^2$ from above, using $\prob_A$ to denote $\probOf\pbox{A = 1}$, we can see that
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>>>>>>> 3f771b6cf71fd4f2693fd44b1d8f097a5e324f1c
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\begin{align*}
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\poly^2\inparen{\probAllTup} &= \prob_A^2\prob_X^2\prob_B^2 + \prob_B^2\prob_Y^2\prob_D^2 + \prob_B^2\prob_Z^2\prob_C^2 + 2\prob_A\prob_X\prob_B^2\prob_Y\prob_D + 2\prob_A\prob_X\prob_B^2\prob_Z\prob_C + 2\prob_B^2\prob_Y\prob_D\prob_Z\prob_C\\
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&\leq\prob_A\prob_X\prob_B + \prob_B\prob_Y\prob_D + \prob_B\prob_Z\prob_C +
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@ -245,3 +295,9 @@ To get an $(1\pm \epsilon)$-multiplicative approximation we uniformly sample mon
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\mypar{Paper Organization} We present relevant background and notation in \Cref{sec:background}. We then prove our main hardness results in \Cref{sec:hard} and present our approximation algorithm in \Cref{sec:algo}. We present some (easy) generalizations of our results in \Cref{sec:gen} and also discuss extensions from computing expectations of polynomials to the expected result multiplicity problem (\Cref{def:the-expected-multipl})\AH{Aren't they the same?}. Finally, we discuss related work in \Cref{sec:related-work} and conclude in \Cref{sec:concl-future-work}.
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "main"
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%%% End:
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Reference in a new issue