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paper-BagRelationalPDBsAreHard
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Finished S2 pass.

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Aaron Huber 2021-09-08 12:17:19 -04:00
parent 513b345ccb
commit fa241123d5
5 changed files with 48 additions and 46 deletions
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2
macros.tex
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@ -206,7 +206,7 @@
\newcommand{\apolyqdt}{\polyqdt{\query}{\pdb}{\tup}}
\newcommand{\tupvar}[2]{X_{#1,#2}}
\newcommand{\atupvar}{\tupvar{\rel}{\tup}}
\newcommand{\polyX}{\poly\inparen{\pVar}}%<---let's see if this proves handy
\newcommand{\polyX}{\poly\inparen{\vct{\pVar}}}%<---let's see if this proves handy
\newcommand{\rpoly}{\widetilde{\poly}}%r for reduced as in reduced 'Q'
\newcommand{\rpolyX}{\rpoly\inparen{\pVar}}%<---if this isn't something we use much, we can get rid of it
\newcommand{\biDisProd}{\mathcal{B}}%bidb disjoint tuple products (def 2.5)

2
main.tex
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@ -47,7 +47,7 @@ sensitive=true
}
\lstset{style=psql}
%%%%%%%%%%%%%%%%%%BORROWED FROM UADB paper^-----
\usepackage{wrapfig}
\usepackage{fancyvrb}
\usepackage{caption}
\usepackage{subcaption}

8
poly-form.tex
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@ -149,10 +149,10 @@ Let $\pdb$ be a \abbrBIDB over $\numvar$ input tuples such that the probability
\end{equation*}
\end{Lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Note that in the preceding lemma, we have assigned $\vct{p}$
%(introduced in \Cref{subsec:def-data})
to the variables $\vct{X}$. Intuitively, \Cref{lem:exp-poly-rpoly} states that when we replace each variable $X_i$ with its probability $\prob_i$ in the reduced form of a \bi-lineage polynomial and evaluate the resulting expression in $\mathbb{R}$, then the result is the expectation of the polynomial.
By \Cref{lem:exp-poly-rpoly} and linearity of expectation, the following corollary results.
%Note that in the preceding lemma, we have assigned $\vct{p}$
%%(introduced in \Cref{subsec:def-data})
%to the variables $\vct{X}$. Intuitively, \Cref{lem:exp-poly-rpoly} states that when we replace each variable $X_i$ with its probability $\prob_i$ in the reduced form of a \bi-lineage polynomial and evaluate the resulting expression in $\mathbb{R}$, then the result is the expectation of the polynomial.

80
prob-def.tex
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@ -11,11 +11,11 @@ We represent lineage polynomials via {\em arithmetic circuits}~\cite{arith-compl
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Definition}[Circuit]\label{def:circuit}
A circuit $\circuit$ is a Directed Acyclic Graph (DAG) whose source gates (in degree of $0$) consist of elements in either $\domN$ or $\vct{X}$. The internal gates have binary input and are either sum ($\circplus$) or product ($\circmult$) gates.
A circuit $\circuit$ is a Directed Acyclic Graph (DAG) whose source gates (in degree of $0$) consist of elements in either $\domN$ or $\vct{X}$. For each output tuple there exists one source gate. The internal gates have binary input and are either sum ($\circplus$) or product ($\circmult$) gates.
%
Each gate has the following members: \type, \vpartial, \vari{input}, \degval, \vari{Lweight}, and \vari{Rweight}, where \type is the value type $\{\circplus, \circmult, \var, \tnum\}$ and \vari{input} the list of inputs. Source gates have an additional member \val storing the value. $\circuit_\linput$ ($\circuit_\rinput$) denotes the left (right) input.
Each gate has the following members: \type, \vpartial, \vari{input}, \degval, \vari{Lweight}, and \vari{Rweight}, where \type is the value type $\{\circplus, \circmult, \var, \tnum\}$ and \vari{input} the list of inputs. Source gates have an additional member \val storing the value. $\circuit_\linput$ ($\circuit_\rinput$) denotes the left (right) input of \circuit.
\end{Definition}
When the underlying DAG is a tree (with edges pointing towards the root), the structure is an expression tree \etree. In such a case, the root of \etree is analogous to the sink of \circuit. We ignore the fields \vari{partial}, \degval, \vari{Lweight}, and \vari{Rweight} until \Cref{sec:algo}.\AH{\emph{I think} \degval is used only in appendix proofs.}
When the underlying DAG is a tree (with edges pointing towards the root), the structure is an expression tree \etree. In such a case, the root of \etree is analogous to the sink of \circuit. The fields \vari{partial}, \degval, \vari{Lweight}, and \vari{Rweight} are used in the proofs of \Cref{sec:proofs-approx-alg}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -25,36 +25,38 @@ When the underlying DAG is a tree (with edges pointing towards the root), the st
\begin{Example}
The circuit \circuit in \Cref{fig:circuit-express-tree} encodes the polynomial $XY + WZ$. Note that circuit \circuit encodes a tree, with edges pointing towards the root.
The circuits in \Cref{fig:two-step} encode their respective polynomials in column $\poly$.
%\circuit in \Cref{fig:circuit-express-tree} encodes the polynomial $XY + WZ$.
Note that each circuit \circuit encodes a tree, with edges pointing towards the root.
\end{Example}
\begin{figure}[t]
\begin{subfigure}[b]{0.45\linewidth}
\centering
\begin{tikzpicture}[thick]
\node[tree_node] (a1) at (0, 0){$\boldsymbol{X}$};
\node[tree_node] (b1) at (1, 0){$\boldsymbol{Y}$};
\node[tree_node] (c1) at (2, 0){$\boldsymbol{W}$};
\node[tree_node] (d1) at (3, 0){$\boldsymbol{Z}$};
\node[tree_node] (a2) at (0.5, 1){$\boldsymbol{\circmult}$};
\node[tree_node] (b2) at (2.5, 1){$\boldsymbol{\circmult}$};
\node[tree_node] (a3) at (1.5, 2){$\boldsymbol{\circplus}$};
\draw[->] (a1) -- (a2);
\draw[->] (b1) -- (a2);
\draw[->] (c1) -- (b2);
\draw[->] (d1) -- (b2);
\draw[->] (a2) -- (a3);
\draw[->] (b2) -- (a3);
\draw[->] (a3) -- (1.5, 2.5);
\end{tikzpicture}
\caption{Circuit encoding $XY + WZ$, a special case of an expression tree}
\label{fig:circuit-express-tree}
\end{subfigure}
\hspace{5mm}
\begin{subfigure}[b]{0.45\linewidth}
%\begin{figure}[t]
% \begin{subfigure}[b]{0.45\linewidth}
% \centering
% \begin{tikzpicture}[thick]
% \node[tree_node] (a1) at (0, 0){$\boldsymbol{X}$};
% \node[tree_node] (b1) at (1, 0){$\boldsymbol{Y}$};
% \node[tree_node] (c1) at (2, 0){$\boldsymbol{W}$};
% \node[tree_node] (d1) at (3, 0){$\boldsymbol{Z}$};
%
% \node[tree_node] (a2) at (0.5, 1){$\boldsymbol{\circmult}$};
% \node[tree_node] (b2) at (2.5, 1){$\boldsymbol{\circmult}$};
%
% \node[tree_node] (a3) at (1.5, 2){$\boldsymbol{\circplus}$};
%
% \draw[->] (a1) -- (a2);
% \draw[->] (b1) -- (a2);
% \draw[->] (c1) -- (b2);
% \draw[->] (d1) -- (b2);
% \draw[->] (a2) -- (a3);
% \draw[->] (b2) -- (a3);
% \draw[->] (a3) -- (1.5, 2.5);
% \end{tikzpicture}
% \caption{Circuit encoding $XY + WZ$, a special case of an expression tree}
% \label{fig:circuit-express-tree}
% \end{subfigure}
% \hspace{5mm}
\begin{wrapfigure}{l}{0.45\linewidth}
\centering
\begin{tikzpicture}[thick]
\node[tree_node] (a1) at (0, 0) {$\boldsymbol{X}$};
@ -87,11 +89,10 @@ The circuit \circuit in \Cref{fig:circuit-express-tree} encodes the polynomial $
\end{tikzpicture}
\caption{Circuit encoding of $(X + 2Y)(2X - Y)$}
\label{fig:circuit}
\end{subfigure}
\caption{Example circuit encodings}
\end{figure}
We next formally define the relationship of circuits with polynomials.
\end{wrapfigure}
% \caption{Example circuit encodings}
%\end{figure}
We next formally define the relationship of circuits with polynomials. While the definition assumes one sink for notational convenience, it easily generalizes to the multiple sinks case.
\begin{Definition}[$\polyf(\cdot)$]\label{def:poly-func}
Denote $\polyf(\circuit)$ to be the function from circuit $\circuit$ to its corresponding polynomial (in \abbrSMB).\footnote{Recall our assumption that unless otherwise mentioned, all polynomials are considered in $\abbrSMB$.} $\polyf(\cdot)$ is recursively defined on $\circuit$ as follows, with addition and multiplication following the standard interpretation for polynomials:
\begin{equation*}
@ -119,11 +120,12 @@ The circuit of \Cref{fig:circuit} is an element of $\circuitset{2X^2+3XY-2Y^2}$.
\noindent We are now ready to formally state our \textbf{main problem}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Definition}[The Expected Result Multiplicity Problem]\label{def:the-expected-multipl}
Let $\vct{X} = (X_1, \ldots, X_n)$ and $\pdb$ be an arbitrary \abbrBIDB-PDB over $\vct{X}$ with probability distribution $\pd$ over assignments $\vct{X} \to \{0,1\}^\numvar$. Fix a query $\query$ and an output tuple $\tup$.
Let $\pdb$ be an arbitrary \abbrBIDB-PDB and $\vct{X}$ be the set of variables annotating tuples in $\dbbase$. Fix a query $\query$ and an output tuple $\tup$.
The \expectProblem is defined as follows:\\[-7mm]
\begin{center}
\textbf{Input}: A circuit $\circuit \in \circuitset{\polyX}$ for $\polyX = \query(\pxdb)(t)$
\hspace*{5mm}\textbf{Output}: $\expct_{\vct{W} \sim \pd}[\poly(\vct{W})]$
\textbf{Input}: $\circuit \in \circuitset{\polyX}$ for $\polyX = \apolyqdt$
\hspace*{2mm}
\textbf{Output}: $\expct_{\vct{W} \sim \pdassign}[\apolyqdt(\vct{W})]$
\end{center}
\end{Definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

2
two-step-model.tex
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@ -106,7 +106,7 @@
\end{tabular}
};
%label below rectangle
\node[below=0.2cm of rect]{{\LARGE $\query(\pdb)\inparen{\tup}\equiv \poly_\tup\inparen{\vct{X}}$}};
\node[below=0.2cm of rect]{{\LARGE $\query(\pdb)\inparen{\tup}\equiv \poly\inparen{\vct{X}}$}};
%Second arrow
\node[single arrow, right=0.25 of rect, draw=black, fill=black!65, text=white, minimum height=0.75cm, minimum width=0.25cm](arrow2) {\textbf{\abbrStepTwo}};
%Expectation computation; (output of step 2)