80 lines
8 KiB
TeX
80 lines
8 KiB
TeX
%root: main.tex
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%!TEX root=./main.tex
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%\onecolumn
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\section{Background and Notation}\label{sec:background}
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\subsection{Polynomial Definition and Terminology}
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%We now introduce some terminology
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%and develop a reduced form of lineage polynomials for a \abbrBIDB or \abbrTIDB.
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%Note that
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\secrev{A }
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polynomial over $\vct{X}=(X_1,\dots,X_n)$ with individual degree $B <\infty$
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is formally defined as (where $c_{\vct{d}}\in \semN$):
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\begin{equation}
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\label{eq:sop-form}
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\poly\inparen{X_1,\dots,X_n}=\secrev{\sum_{\vct{d}\in\{0,\ldots,B\}^\tupset} c_{\vct{d}}\cdot \prod_{\tup\in\tupset} X_\tup^{d_\tup}.}
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\end{equation}
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%where $c_{\vct{d}}\in \semN$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{Definition}[Standard Monomial Basis]\label{def:smb}
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The term $\prod_{\tup\in\tupset} X_\tup^{d_\tup}$ in \Cref{eq:sop-form} is a {\em monomial}. A polynomial $\poly\inparen{\vct{X}}$ is in standard monomial basis (\abbrSMB) when we keep only the terms with $c_{\vct{d}}\ne 0$ from \Cref{eq:sop-form}.
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\end{Definition}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Unless othewise noted, we consider all polynomials to be in \abbrSMB representation.
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When it is unclear, we use $\smbOf{\poly}$ to denote the \abbrSMB form of a polynomial $\poly$.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{Definition}[Degree]\label{def:degree-of-poly}
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The degree of polynomial $\poly(\vct{X})$ is the largest \secrev{$\norm{\vct{d}}_1$}% = \sum_{\tup\in\tupset} d_\tup$
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such that $c_{(d_1,\dots,d_n)}\ne 0$. % maximum sum of exponents, over all monomials in $\smbOf{\poly(\vct{X})}$.
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\end{Definition}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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As an example, the degree of the polynomial $X^2+2XY^2+Y^2$ is $3$.
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Product terms in lineage arise only from join operations (\Cref{fig:nxDBSemantics}), so intuitively, the degree of a lineage polynomial is analogous to the largest number of joins needed to produce a result tuple.
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%in any clause of the $\raPlus$ query that created it.
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\secrev{
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We call a polynomial $\poly\inparen{\vct{X}}$ a \emph{\abbrCTIDB-lineage polynomial} (%resp., \emph{\ti-lineage polynomial},
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or simply lineage polynomial), if there exists a $\raPlus$ query $\query$, \abbrCTIDB $\pdb$, and result tuple $\tup$ such that $\poly\inparen{\vct{X}} = \apolyqdt\inparen{\vct{X}}.$
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}
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%Following the typical representation of bags in production databases, for query inputs, we will use \abbrBPDB\xplural with multiplicities $\{0, 1\}$ (see \Cref{sec:gener-results-beyond} for more on this choice).
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\subsubsection{Reduction to $1$-\abbrBIDB}
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\label{subsec:tidbs-and-bidbs}
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An \textit{incomplete database} $\Omega$ is a set of deterministic databases $\omega$ called possible worlds.
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\noindent\secrev{
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A \abbrCTIDB $\pdb$ is a pair $\inparen{\worlds, \bpd}$ such that $\worlds$ is an incomplete database whose set of possible worlds is the $c+1^\numvar$ tuple/multiplicity combinations for each $\tup\in\tupset$, where $\abs{\tupset} = \numvar$, $\tupset = \bigcup_{\worldvec\in\worlds,~\worldvec_{\tup}\geq 1}\tup$ is the set of possible tuples across possible worlds, and $\bpd$ is a probability distribution over $\worlds$.
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A block independent database (\abbrBIDB) is a related probabilistic data model $\pdb=\inparen{\Omega, \pdb}$ such that the base set of tuples $\tupset = \bigcup_{\omega\in\Omega,~\tup\in\omega}\tup$ is partitioned into a set of $\numvar$ independent blocks $\inset{\inparen{\block_\tup}_{\tup\in\pbox{\numvar}}}$ such that the set of tuples $\inset{\inparen{\tup_j}_{j\in\pbox{\abs{\block}}}}$ in block $\block_\tup$ are disjoint from one another. This construction produces the set of possible worlds $\Omega$ that consists of all unique combinations of tuples in $\tupset$ with the constraint that for any $\omega\in\Omega$, no two tuples $\tup_j, \tup_{j'}, j\neq j'$ from the same block $\block_\tup$ exist together. A $\bound$-\abbrBIDB has the further requirement that each block has a multiplicity of at most $c$. We present a reduction that is useful in producing our results:
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\begin{Definition}[\abbrCTIDB reduction]\label{def:ctidb-reduct}
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Given \abbrCTIDB $\pdb = \inparen{\worlds, \bpd}$, let $\pdb' = \inparen{\Omega, \bpd'}$ be the \abbrOneBIDB obtained in the following manner: for each $\tup\in\tupset$, create block $\block_\tup = \inset{\intup{\tup, X_{\tup, j}}_{j\in\pbox{\bound}}}$ %with $\bound$ disjoint copies, such that $\tup_j$ is annotated with variable $X_{\tup, j}$ for $j\in\pbox{\bound}$.
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The probability distribution $\bpd'$ is the one induced by $\vct{p} = \inparen{\inparen{\prob_{\tup, j}}_{\tup\in\tupset, j\in\pbox{\bound}}}$ and the \abbrBIDB disjoint requirement.
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\end{Definition}
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As previously noted, unlike $X_{\tup}\in\inset{0,\ldots,\bound}$ for $X_{\tup}\in\vars{\pdb}$, $X_{\tup, j}\in\inset{0,1}$ for $X_{\tup, j}\in\vars{\pdb'}$. In the reduced \abbrOneBIDB setting, the base case of~\Cref{fig:nxDBSemantics} now becomes $\poly\pbox{\rel,\tupset, \tup} = \sum_{j\in\pbox{\bound}}X_{\tup, j}$. Then given the disjoint requirement and the semantics for constructing the lineage polynomial over a \abbrOneBIDB, $\poly\pbox{\rel,\tupset',\tup}$ is of the same form as the reformulated polynomial $\refpoly$ of step i) from~\Cref{def:reduced-poly}, which then implies that $\rpoly$ is the reduced polynomial that results from step ii) of~\Cref{def:reduced-poly}, and further that~\Cref{lem:tidb-reduce-poly} immediately follows for \abbrOneBIDB polynomials: $\expct_{\rvworld\sim\bpd'}\pbox{\poly\inparen{\rvworld}} = \rpoly\inparen{\vct{\prob}}$.
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}
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%In this paper, we focus on two popular forms of \abbrPDB\xplural: Block-Independent (\bi) and Tuple-Independent (\ti) \abbrPDB\xplural.
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%%
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%A \bi $\pdb$ is a \abbrPDB with the constraint that
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%%(i) every tuple $\tup_i$ is annotated with a unique random variable $\randWorld_i \in \{0, 1\}$ and (ii) that
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%the tuples in $\dbbase$ can be partitioned into a set of $\ell$ blocks such that tuples $\tup_{i, j}, \tup_{k, j'}$ from separate blocks $(i\neq k)$ are independent of each other while tuples $\tup_{i, j}, \tup_{i, k}$ from the same block are disjoint events.\footnote{
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% Although only a single independent, $[\abs{\block_i}+1]$-valued variable is customarily used per block~\cite{DBLP:series/synthesis/2011Suciu}, we decompose it into $\abs{\block_i}$ correlated $\{0,1\}$-valued variables per block that can be used directly in polynomials (without an indicator function). For $t_{i, j} \in b_i$, the event $(\randWorld_{i,j} = 1)$ corresponds to the event $(\randWorld_i = j)$ in the customary annotation scheme.
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%}
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%Each tuple $\tup_{i, j}$ is annotated with a random variable $\randWorld_{i, j} \in \{0, 1\}$ denoting its presence in a possible world $\db$. The probability distribution $\pd$ over $\dbbase$ is the one induced from individual tuple probabilities $\prob_{i, j}\in \vct{\prob}=\inparen{\prob_{1, 1},\ldots,\prob_{\abs{\block},\ldots,\abs{\block_{\abs{\block}}}}}$ (where $\forall i$, $\sum_j p_{i,j}\le 1$) and the conditions on the blocks. A \abbrTIDB is a \abbrBIDB where each block has size exactly $1$.
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Instead of looking only at the possible worlds of $\pdb;$, one can consider all worlds, including those that cannot exist due to disjointness. The all worlds set can be modeled by $\worldvec\in \{0, 1\}^{\bound\numvar}$,\footnote{Here and later, especially in \Cref{sec:algo}, we will rename the variables as $X_1,\dots,X_n$, where $n=\sum_{i=1}^\ell \abs{b_i}$.} such that $\worldvec_{\tup, j} \in \worldvec$ represents whether or not the multiplicity of $\tup$ is $j$.%(where $k = \sum_{\ell = 1}^{i - 1} \abs{b_\ell} + j$).
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We denote a probability distribution over all $\worldvec \in \{0, 1\}^\numvar$ as $\bpd''$. When $\bpd''$ is the one induced from each $\prob_{\tup, j}$ while assigning $\probOf\pbox{\worldvec} = 0$ for any $\worldvec$ with $\worldvec_{\tup, j} = \worldvec_{\tup, j'} = 1$ for $j\neq j'$, we end up with a bijective mapping from $\bpd'$ to $\bpd''$, such that each mapping is equivalent, implying the distributions are equivalent.
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\Cref{subsec:supp-mat-ti-bi-def} has more details.
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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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