Finished @oliver 081221 comments for intro.
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%root: main.tex
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\section{Introduction (Rewrite - 070921)}
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\input{two-step-model}
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A tuple independent probabilistic database\footnote{In \cref{sec:background} and beyond, we generalize the data model.} (\abbrTIDB) $\pdb$ is a tuple $\inparen{\db, \pd}$ where $\db$ is a set of $\numvar$ tuples. The probability distribution $\pd$ over $\db$ is the one induced from the requirement that each tuple be treated as an independent Bernoulli distributed random variable. In bag query semantics the random variable $\query\inparen{\pdb}\inparen{\tup}$ computes the multiplicity of its corresponding tuple $\tup$. In addition to traditional deterministic query evaluation requirements (for a given query class), the query evaluation problem in bag-\abbrPDB semantics further requires the following condition:
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A tuple independent probabilistic database\footnote{In \cref{sec:background} and beyond, we generalize the data model.} (\abbrTIDB) $\pdb$ is a tuple $\inparen{\db, \pd}$ where $\db$ is a set of $\numvar$ tuples. The probability distribution $\pd$ over the set of database instances (possible worlds) encoded in $\db$ is the one induced from the requirement that each tuple be treated as an independent Bernoulli distributed random variable. In bag query semantics the random variable $\query\inparen{\pdb}\inparen{\tup}$ computes the multiplicity of its corresponding tuple $\tup$. In addition to traditional deterministic query evaluation requirements (for a given query class), the query evaluation problem in bag-\abbrPDB semantics further requires the following condition:
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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 ($\raPlus$),\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).} compute the expected multiplicity ($\expct\pbox{\query\inparen{\pdb}\inparen{\tup}}$) of output tuple $\tup$.
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\end{Problem}
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There exists a polynomial $\poly_\tup\inparen{\vct{X}}$ such that $\expct\pbox{\query\inparen{\pdb}\inparen{\tup}} = \expct\pbox{\poly_\tup\inparen{\vct{\randWorld}}}$, where $\vct{X} = \inparen{X_1,\ldots, X_\numvar}$ is the set of variables annotating the tuples in $\pdb$ and $\vct{W}$ the set of random variables corresponding to $\vct{X}$ drawn from $\pd$. The expectation is any Bernoulli distribution $\pd$ over $\{0, 1\}^\numvar$, whose evaluation semantics follow the standard interpretation of addition and multiplication operators over the natural numbers, i.e. $\semN$-semiring semantics. While the aforementioned assumes set \abbrTIDB inputs, this is not limiting, since one can reduce a bag-\abbrTIDB input to a set-\abbrTIDB by assigning unique keys across all $\tup$ in $\pdb$. Such a generalization has an $\bigO{c}$ increase in size, for $c = \max_{\tup \in \db}\db\inparen{\tup}$, where $\db\inparen{\tup}$ denotes $\tup$'s multiplicity.
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%BEGIN MOVE TO S. 2
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%Our work focuses on the following setting for query computation. Inputs of $\query$ are set-\abbrPDB\xplural, while the output of $\query$ is a bag-\abbrPDB. This setting, however, is not limiting as a simple generalization exists, reducing a bag \abbrPDB to a set \abbrPDB with typically only an $O(c)$ increase in size, for $c = \max_{\tup \in \db}\db\inparen{t}$.
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%END MOVE
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$\query\inparen{\pdb}\inparen{\tup}$ can be encoded by a polynomial, with variables in the vector $\vct{X}$, such that each of the $\numvar$ tuples in $\vct{X}$ has its own unique variable, i.e. $\vct{X} = \inparen{X_1,\ldots, X_\numvar}$. Since $\raPlus$ operators have one to one correspondence to polynomial operators (\cref{fig:nxDBSemantics}), then there exists a polynomial $\poly_\tup\inparen{\vct{X}}$ such that $\expct\pbox{\query\inparen{\pdb}\inparen{\tup}} = \expct\pbox{\poly_\tup\inparen{\vct{\randWorld}}}$, where $\vct{W}$ the set of random variables corresponding to $\vct{X}$ drawn from $\pd$. The expectation is any Bernoulli distribution $\pd$ over $\{0, 1\}^\numvar$ (the set of possible worlds), whose evaluation semantics follow the standard interpretation of addition and multiplication operators over the natural numbers, i.e. $\semN$-semiring semantics. While the aforementioned assumes set \abbrTIDB inputs, this is not limiting, as a simple generalization from bag-\abbrPDB\xplural to set-\abbrPDB\xplural exists.\footnote{A bag-\abbrTIDB can be reduced to a set-\abbrTIDB by assigning unique keys across all $\tup$ in $\pdb$. This typically has an $\bigO{c}$ increase in size, for $c = \max_{\tup \in \db}\db\inparen{\tup}$, where $\db\inparen{\tup}$ denotes $\tup$'s multiplicity in the encoding.}
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The problem of deterministic query evaluation is known to be \sharpwonehard\footnote{A problem is in \sharpwone if the runtime of the most efficient known algorithm to solve it is lower bounded by some function $f$ of a parameter $k$, where the growth in runtime is polynomially dependent on $f(k)$, i.e. $\Omega\inparen{\numvar^{f(k)}}$.} in data complexity for general $\query$. For example, the counting $k$-cliques query problem (where the parameter $k$ is the size of the clique) is \sharpwonehard since (under standard complexity assumptions) it cannot run in time faster than $n^{f(k)}$ for some strictly increasing $f(k)$.
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This result is unsatisfying when considering complexity of evaluating $\query$ over \abbrPDB\xplural, since it does not account for computing $\expct\pbox{\poly_\tup\inparen{\vct{\randWorld}}}$, entirely ignoring the `P' in \abbrPDB.
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Simply considering deterministic query evaluation, however, is unsatisfying when considering the complexity of evaluating $\query$ over \abbrPDB\xplural, since it does not account for computing $\expct\pbox{\poly_\tup\inparen{\vct{\randWorld}}}$, entirely ignoring the `P' in \abbrPDB.
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\Cref{prob:bag-pdb-query-eval} has been extensively studied in the context of \emph{set}-\abbrPDB\xplural, where each output tuple appears at most once. Here, $\poly_\tup\inparen{\vct{X}}$ is a propositional formula\footnote{To be precise, $\poly_\tup\inparen{\vct{X}}$ is a propositional formula composed of boolean variables and the logical disjunction and conjunction connectives. Evaluating such a formula follows the standard semantics of the said operators on boolean variables.} whose evaluation follows the standard semantics ($\semB$-semiring semantics), denoting the presence or absence of $\tup$. Computing $\expct\pbox{\poly_\tup\inparen{\vct{\randWorld}}}$ determines the marginal probability of $\tup$ appearing in the output. Dalvi and Suicu \cite{10.1145/1265530.1265571} showed that the complexity of the query computation problem over set-\abbrPDB\xplural is \sharpphard\footnote{\sharpp is the counting version for problems residing in the NP complexity class.} in general, and proved that a dichotomy exists for this problem, where the runtime of $\query(\pdb)$ is either polynomial or \sharpphard for any polynomial-time \abbrStepOne.
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\Cref{prob:bag-pdb-query-eval} has been extensively studied in the context of \emph{set}-\abbrPDB\xplural, where each output tuple appears at most once. Here, $\poly_\tup\inparen{\vct{X}}$ is a propositional formula\footnote{To be precise, $\poly_\tup\inparen{\vct{X}}$ is a propositional formula composed of boolean variables and the logical disjunction and conjunction connectives. Evaluating such a formula follows the standard semantics of the said operators on boolean variables.} whose evaluation follows the standard semantics ($\semB$-semiring semantics), denoting the presence or absence of $\tup$. Computing $\expct\pbox{\poly_\tup\inparen{\vct{\randWorld}}}$ determines the marginal probability of $\tup$ appearing in the output. Dalvi and Suicu \cite{10.1145/1265530.1265571} showed that the complexity of the query computation problem over set-\abbrPDB\xplural is \sharpphard\footnote{\sharpp is the counting version for problems residing in the NP complexity class.} in general, and proved that a dichotomy exists for this problem, where the runtime of $\query(\pdb)$ is either polynomial or \sharpphard for any polynomial-time deterministic query.
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%BEGIN Needs to be said
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%Since the hardness is in data complexity (the size of the input, $\Theta(\numvar$)), techniques such as parameterized complexity (bounding complexity by another parameter other than $\numvar$) and fine grained analysis (complexity analysis that asks what precisely is the value of this other parameter, for example, what is the value of $f(k)$ given a \sharpwone algorithm) of \abbrStepTwo will not refine the hardness results from \sharpphard.
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%END NEeds to be said
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@ -27,22 +22,22 @@ There exist some queries for which \emph{bag}-\abbrPDB\xplural are a more natura
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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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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. 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 neatly followed by semiring provenance, where $\semNX$-DB\footnote{An $\semNX$-DB is a database whose tuples are annotated with standard polynomials, i.e. elements from $\semNX$ connected by addition operators.} query processing \cite{DBLP:conf/pods/GreenKT07} first computes the query and polynomial, and the $\semNX$-polynomial is 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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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 standard polynomials, i.e. elements from $\semNX$ connected by multiplication and addition operators.} 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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Let $\timeOf{\abbrStepOne}$ denote the runtime of \abbrStepOne and similarly for $\timeOf{\abbrStepTwo}$.
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In bag-\abbrPDB\xplural, by linearity of expectation and independence of \abbrTIDB, $\timeOf{\abbrStepTwo}$ is $\bigO{\abs{\poly_\tup}}$\footnote{$\abs{\poly_\tup}$ denotes the size of $\poly_\tup$, i.e., the number of arithmetic operations.} when $\poly_\tup$ is in \abbrSMB. Given bag-\abbrPDB query $\query$ and \abbrTIDB $\pdb$ with $\numvar$ tuples, let us go a step further and assume that\footnote{This assumption is not necessary for the following results, but more clearly illustrates the point.} computing $\poly_\tup$ is lower bounded by the runtime of determistic query computation of $\query$ for the following situations. Such an assumption can occur when $\query$ is a sequence of query algorithms (e.g. $\project\inparen{\join}$) whose evaluation is precisely mirrored in the \abbrSMB representation of $\poly_\tup$. When $\poly_\tup$ is in \abbrSMB, 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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Let $\timeOf{\abbrStepOne}$ denote the runtime of \abbrStepOne and similarly for $\timeOf{\abbrStepTwo}$.
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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 $\query$ for the following situations, i.e. 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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%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 $p_i^h x_i^h = p_i \cdot 1^{h-1}x_i^h$.} Here is another special case where $\timeOf{\abbrStepTwo}$ is $\bigO{\timeOf{\abbrStepOne}}$ and we again achieve deterministic query runtime for $\query\inparen{\pdb}$ (up to a constant factor).
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% When solving \cref{prob:bag-pdb-query-eval}, $\timeOf{\abbrStepTwo}$ lies somewhere between $\bigO{\timeOf{\abbrStepOne}}$ and $\bigO{\timeOf{\abbrStepOne}^k}$, since when $\poly_\tup$ is in \abbrSMB\footnote{\abbrSMB is akin to the sum of products expansion but with the added requirement that all monomials are unique.} computing $\expct\pbox{\poly_\tup\inparen{\vct{X}}}$ is linear (due to linearity of expectation and the independence assumption of \abbrTIDB), while the case of a factorized $\poly_\tup$ has a worst case (since in general product terms must be expanded) of $\timeOf{\abbrStepTwo}$ being $\timeOf{\abbrStepOne}^k$ for a $k$-wise factorization.
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These observations introduce our next problem statement:
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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 $p_i^h x_i^h = p_i \cdot 1^{h-1}x_i^h$.} Here is another special case where $\timeOf{\abbrStepTwo}$ is $\bigO{\timeOf{\abbrStepOne}}$ and we again achieve deterministic query runtime for $\query\inparen{\pdb}$ (up to a constant factor). These observations introduce our next problem statement:
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\begin{Problem}\label{prob:big-o-step-one}
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Given a \abbrPDB $\pdb$ and $\raPlus$ query $\query$, is it \emph{always} the case that $\timeOf{\abbrStepTwo}$ is always $\bigO{\timeOf{\abbrStepOne}}$?
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\end{Problem}
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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.
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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 of $\poly_\tup(\vct{X})$. To capture such factorizations, this work uses (arithmetic) circuits
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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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@ -67,7 +62,7 @@ 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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Above we have seen, given a circuit \circuit, if \circuit is in \abbrSMB, then we have that $\timeOf{\abbrStepTwo}$ is indeed $\bigO{\timeOf{\abbrStepOne}}$. 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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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 \abbrPDB $\pdb$ and $\raPlus$ query $\query$, is it always the case that $\timeOf{\abbrStepTwo}$ is $\bigO{\abbrStepOne}$?
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\end{Problem}
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@ -78,9 +73,9 @@ Given \abbrPDB $\pdb$ and $\raPlus$ query $\query$, is it always the case that $
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Concretely, we make the following contributions:
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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 for a specific %cubic
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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 approximation algorithm to a class of bag-Block Independent Disjoint Databases (see \cref{subsec:tidbs-and-bidbs}) (\abbrBIDB\xplural), a more general model of probabilistic data; (iv) We further prove that for \raPlus queries
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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 considerred 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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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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6
main.bib
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main.bib
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year = {2018}
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}
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@inproceedings{Imielinski1989IncompleteII,
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title={Incomplete Information in Relational Databases},
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author={T. Imielinski and W. Lipski},
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year={1989}
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}
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@inproceedings{10.1145/1265530.1265571,
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author = {Dalvi, Nilesh and Suciu, Dan},
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booktitle = {PODS},
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