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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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This model corresponds to the classical set-relational approach to \abbrTIDB{}s, where we can handle the case of each input tuple having its own multiplicity by replacing each input tuple with as many copes as its multiplicity. To make each duplicate tuple unique in a set-\abbrTIDB we can assign unique keys across all duplicates. This increases the size of the input but this overhead is negligible when each input tuple has constant multiplicity. %$\tup$ in $\pdb$.
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This model corresponds to the classical set-relational approach to \abbrTIDB{}s, where we can handle the case of each input tuple having its own multiplicity by replacing each input tuple with as many copes as its multiplicity. To make each duplicate tuple unique in a set-\abbrTIDB we can assign unique keys across all duplicates. This increases the size of the input but this overhead is negligible when each input tuple has constant multiplicity. %$\tup$ in $\pdb$.
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%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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We further generalize this model in \cref{sec:background} and beyond.
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}.
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We will denote the $n$ input tuples by $t_1,\dots,t_\numvar$ and each of the $2^n$ possible database instance in $\Omega$ can be encoded as a string in $\{0,1\}^\numvar$. In particular, any vector $\vct{W}=\inparen{W_1,\dots,W_n}\in \{0,1\}^\numvar$ represents a database instance that has $t_i$ in it if and only if $w_i=1$. Further $\pd$ is compactly described by tuple $\vct{p}=\inparen{p_1,\dots,p_n}$, which induces the Bernoulli distrbution over vectors $\vct{W}\in\{0,1\}^\numvar$ where each $i\in [n]$, $\probOf(W_i=1)=p_i$. Finally for each $\vct{W}\in\{0,1\}^\numvar$, define $\pdb_{\vct{W}}$ the deterministic database represented by $\vct{W}$.
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%Atri: Stuff below was confusing, so am re-writing it.
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%A \abbrTIDB encodes a compatible $\pdb$ as a deterministic database $\encodedDB$ with $\numvar$ tuples, each annotated with a probability $\prob_\tup$, and with $\pd$
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%A \abbrTIDB encodes a compatible $\pdb$ as a deterministic database $\encodedDB$ with $\numvar$ tuples, each annotated with a probability $\prob_\tup$, and with $\pd$
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%with a deterministic table $\encodedDB$ which is a set of $\numvar$ tuples, encoding the set of possible worlds $\idb$. The probability distribution $\pd$ over the set of database instances (possible worlds) is the one
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%being the distribution induced from the requirement that each tuple $\tup \in \encodedDB$ be treated as an independent Bernoulli distributed random variable with probability $\prob_\tup$.
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%being the distribution induced from the requirement that each tuple $\tup \in \encodedDB$ be treated as an independent Bernoulli distributed random variable with probability $\prob_\tup$.
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%The possible worlds of a \abbrTIDB can be encoded by the vector $\vct{W}$, such that each of the $\numvar$ tuples in $\vct{W}$ has its own unique Bernoulli-distributed random variable, i.e. $\vct{W} = \inparen{W_{\tup_1},\ldots, W_{\tup_\numvar}}$, and for each tuple $\tup$, $\probOf(W_\tup) = \prob_\tup$.
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%Given a vector $\vct{X}$ such that each $\tup \in \encodedDB$ has a unique formal variable annotation $X_\tup \in \vct{X}$, for a boolean domain $\{0,1\}^\numvar$, denote by $\pdb_{\vct{X}}$ the deterministic database consisting of exactly those tuples $\tup$ where $X_\tup = 1$.
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When $\pdb$ is a \abbrTIDB, for every output tuple $\tup$, $\query\inparen{\pdb}\inparen{\tup}$ can be encoded by a polynomial, with variables in $\vct{X}$.
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Green, Karvounarakis, and Tannen established (\cite{DBLP:conf/pods/GreenKT07}; see \cref{fig:nxDBSemantics}) that for any $\raPlus$ query $\query$ and \abbrTIDB $\pdb$, there exists a polynomial $\poly_\tup\inparen{\vct{X}}$ following the standard addition and multiplication operators over Natural numbers (i.e., $\semN$-semiring semantics), such that $\query\inparen{\pdb_{\vct{W}}}\inparen{\tup} = \poly_\tup\inparen{\vct{W}}$.
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This in turn implies that $\expct\pbox{\query\inparen{\pdb}\inparen{\tup}} = \expct_{\vct{W}\sim\pd}\pbox{\poly_\tup\inparen{\vct{\randWorld}}}$.
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This in turn implies that $\expct\pbox{\query\inparen{\pdb}\inparen{\tup}} = \expct_{\vct{W}\sim\pd}\pbox{\poly_\tup\inparen{\vct{\randWorld}}}$.
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Thanks to linearity of expectation, simple polynomial-time algorithms exist
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Thanks to linearity of expectation, simple polynomial-time algorithms exist
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% The algo is trivial so I think putting in a 2010 cite seems like bit too much
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%\cite{kennedy:2010:icde:pip})
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%\cite{kennedy:2010:icde:pip})
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for computing exact results for bag-probabilistic count queries $Q$ over \abbrTIDB{}s. On the other hand, it is also known that since we are considering data complexity, the {\em deterministic} query processing for the same query $Q$ can also be done in polynomial time. If our notion of efficiency was polynomial time algorithms, then we would be done. However, in practice (and in theory), we care about the {\em fine-grained} complexity of deterministic query processing (i.e. we care about the exact exponent in our polynomial runtime). Given that there is a huge literature on fine grained complexity of determinitic query complexity, here is a natural (informal) specialization of~\cref{prob:bag-pdb-query-eval}:
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\begin{Problem}[Informal problem statement]
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For any query $Q$, is it the case the {\em fine-grained complexity} of bag-PDB processing of $Q$ can be asymptotically as fast as the `best' deterministic query processing of $Q$?
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@ -42,7 +46,7 @@ For any query $Q$, is it the case the {\em fine-grained complexity} of bag-PDB p
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%In this paper, we explore the \emph{fine-grained complexity} of bag-probabilistic database query evaluation.
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%Atri: I'm not sure if this comment makes much sense here-- it sort of breaks the flow I think. I'll refer to this when talking about our results.
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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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%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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%In this paper, we begin to explore whether the problem of bag-probabilistic query evaluation (which we relate to deterministic query processing more precisely below) falls into this same complexity class.
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We note that an answer in the affirmative for~\cref{prob:informal} indicates that bag-probabilistic databases can be competitive with classical deterministic databases, opening the door for deployment in practice.
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@ -51,11 +55,11 @@ We note that an answer in the affirmative for~\cref{prob:informal} indicates tha
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\mypar{Relationship to Set-Probabilistic Query Evaluation}
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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
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%Atri: If we get a reviewer who does not know what a propositional formula is then we are in trouble-- I did move some of the footnote text to the main part though
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%\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 ($\semB$-semiring semantics).}
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%\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 ($\semB$-semiring semantics).}
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whose evaluation follows the standard Boolean semi-ring semantics (i.e. addition is logical OR and multiplication is logical AND), 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
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%Atri: Again if we have a reviewer who does not know what \sharpp is then we are in trouble
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%\footnote{\sharpp is the counting version for problems residing in the NP complexity class.}
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in general, and proved that a dichotomy exists for this problem, where the runtime of $\query(\pdb)$ is either polynomial or \sharpphard $Q$ in data complexity. %for any polynomial-time deterministic query.
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%\footnote{\sharpp is the counting version for problems residing in the NP complexity class.}
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in general, and proved that a dichotomy exists for this problem, where the runtime of $\query(\pdb)$ is either polynomial or \sharpphard $Q$ in data complexity. %for any polynomial-time deterministic query.
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Thus, for the hard queries the answer to~\cref{prob:informal} is {\em no} for set-PDBs (under the standard complexity assumption that $\sharpp\ne \polytime$.
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Concretely, easy queries in this dichotomy can be answered through so-called \emph{extensional} query evaluation, where probability computation is inlined into normal deterministic query processing.
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@ -76,19 +80,25 @@ However, there exist some queries for which \emph{bag}-\abbrPDB\xplural are a mo
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%BEGIN Needs to be noted.
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%As noted, bag-\abbrPDB query output is a probability distribution over the possible multiplicities of $\poly_\tup\inparen{\vct{X}}$, a stark contrast to the marginal probability %($\expct\pbox{\poly\inparen{\vct{X}}}$)
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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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% 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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We show in \cref{sec:circuit-runtime} that, assuming a standard $\raPlus$ query evaluation algorithm, the cost of constructing the lineage polynomial for all tuples in a query result is upper-bounded by runtime of generating those tuples through deterministic query evaluation.
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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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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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>>>>>>> 3f771b6cf71fd4f2693fd44b1d8f097a5e324f1c
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\begin{figure}
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\begin{align*}
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@ -136,8 +160,9 @@ as the representation system of $\poly_\tup(\vct{X})$, which are a natural fit t
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\end{align*}\\[-10mm]
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\caption{Evaluation semantics $\evald{\cdot}{\db}$ for $\semNX$-DBs~\cite{DBLP:conf/pods/GreenKT07}.}
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\label{fig:nxDBSemantics}
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\end{figure}
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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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>>>>>>> 3f771b6cf71fd4f2693fd44b1d8f097a5e324f1c
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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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@ -223,18 +267,24 @@ Let $\pdb$ be a \abbrTIDB over $n$ input tuples
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For any \abbrTIDB-lineage polynomial $\poly\inparen{\vct{X}}$ based on $\query\inparen{\pdb}$ the following holds:
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\begin{equation*}
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\expct_{\vct{W} \sim \pd}\pbox{\poly\inparen{\vct{W}}} = \rpoly\inparen{\probAllTup}.
|
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\end{equation*}
|
||||
\expct_{\vct{W} \sim \pd}\pbox{\poly\inparen{\vct{W}}} = \rpoly\inparen{\probAllTup}.
|
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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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=======
|
||||
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 $Q$).
|
||||
|
||||
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\\
|
||||
&\leq\prob_A\prob_X\prob_B + \prob_B\prob_Y\prob_D + \prob_B\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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2\prob_A\prob_X\prob_B\prob_Y\prob_D + 2\prob_A\prob_X\prob_B\prob_Z\prob_C + 2\prob_B\prob_Y\prob_D\prob_Z\prob_C \\
|
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&= \rpoly\inparen{\vct{p}}
|
||||
&= \rpoly\inparen{\vct{p}}
|
||||
%\inparen{0.9\cdot 1.0\cdot 1.0 + 0.5\cdot 1.0\cdot 1.0 + 0.5\cdot 1.0\cdot 0.5}^2 = 2.7225 < 3.45 = \rpoly^2\inparen{\probAllTup}
|
||||
\end{align*}
|
||||
If we assume that all of the seven probability values are at least $p_0>0$,
|
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|
|
@ -245,3 +295,9 @@ To get an $(1\pm \epsilon)$-multiplicative approximation we uniformly sample mon
|
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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