ODIn/paper-BagRelationalPDBsAreHard
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Aaron Huber 2019-06-19 14:08:03 -04:00
parent 59cbff067f
commit 24cc338697
2 changed files with 20 additions and 18 deletions
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30
analysis.tex
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@ -48,31 +48,31 @@ Note that four-wise independence is assumed across all four random variables of
\end{equation}
it can be seen that for $\wOne, \wOneP \in \pw$ and $\wTwo, \wTwoP \in \pw'$, all four random variables in \eqref{eq:polar-product} take their values from $\pw$, although we have iteration over two separate sets $\pw$. Thus, there are four possible sets of $\wVec$ variable combinations, namely:
\begin{align*}
&\distPattern{1}:&\cOne\\
&\distPattern{2}:&\cTwo \textit{*} \\
&\distPattern{3}:&\cThree \textit{*} \\
&\distPattern{4}:&\cFour \textit{*}\\
&\distPattern{5}:&\cFive
&\distPattern{1}:&\forElems{\cOne}&\\
&\distPattern{2}:&\forElems{\cTwo}& \textit{*} \\
&\distPattern{3}:&\forElems{\cThree}& \textit{*} \\
&\distPattern{4}:&\forElems{\cFour}& \textit{*}\\
&\distPattern{5}:&\forElems{\cFive}&
\end{align*}
$$\text{ }^*\textit{(and all variants of the respective pattern)}$$
We are interested in those particular cases whose expecation does not equal zero, since an expectation of zero will not add to the summation of \eqref{eq:var-sum-w}. In expectation we have that
\begin{align}
&\expect{%\sum_{\substack{\elems \\
\forAllW{\distPattern{1}}&\rightarrow\expect{%\sum_{\substack{\elems \\
%\st \cOne}}
\polarProdEq \st \cOne} = 1 \label{eq:polar-prod-all}\\
&\expect{%\sum_{\substack{\elems \\
\polarProdEq} = 1 \label{eq:polar-prod-all}\\
\forAllW{\distPattern{2}}&\rightarrow\expect{%\sum_{\substack{\elems \\
%\st \cTwo}}
\polarProdEq \st \cTwo} = 1 \label{eq:polar-prod-two-and-two}\\
&\expect{%\sum_{\substack{\elems \\
\polarProdEq} = 1 \label{eq:polar-prod-two-and-two}\\
\forAllW{\distPattern{3}}&\rightarrow\expect{%\sum_{\substack{\elems \\
%\st \cThree}}
\polarProdEq \st \cThree} = 0 \nonumber \\
&\expect{%\sum_{\substack{\elems \\
\polarProdEq} = 0 \nonumber \\
\forAllW{\distPattern{4}}&\rightarrow\expect{%\sum_{\substack{\elems \\
%\st \cFour}}
\polarProdEq \st \cFour} = 0 \nonumber \\
&\expect{%\sum_{\substack{\elems \\
\polarProdEq} = 0 \nonumber \\
\forAllW{\distPattern{5}}&\rightarrow\expect{%\sum_{\substack{\elems \\
%\st \cFive}}
\polarProdEq \st \cFive} = 0 \nonumber
\polarProdEq} = 0 \nonumber
\end{align}
Only equations \eqref{eq:polar-prod-all} and \eqref{eq:polar-prod-two-and-two} influence the $\var$ computation.

8
macros.tex
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@ -43,10 +43,12 @@
%4-way cases
%%%%%%%%%%%%%%%%
\newcommand{\polarProdEq}{\sketchPolarParam{\wVec_1}\cdot\sketchPolarParam{\wVec_2}\cdot\sketchPolarParam{\wVecPrime_1}\cdot\sketchPolarParam{\wVecPrime_2}}
\newcommand{\elems}{\wOne, \wOneP, \wTwo, \wTwoP \in \pw}
\newcommand{\lab}[1]{\textit{#1}}
\newcommand{\distPattern}[1]{\lab{Set}{\textit{ {#1}}}}
\newcommand{\elems}{\wOne, \wOneP, \wTwo, \wTwoP}
\newcommand{\forAllW}[1]{\forall \elems \in {#1}}
\newcommand{\lab}[1]{#1}
\newcommand{\distPattern}[1]{\lab{S_{#1}}}
\newcommand{\vCase}[1]{\lab{Variant }{#1}}
\newcommand{\forElems}[1]{\{\elems \st {#1}\}}
\newcommand{\cOne}{\wOne = \wOneP = \wTwo = \wTwoP}
\newcommand{\cTwo}{\wOne = \wOneP \neq \wTwo = \wTwoP}
\newcommand{\cThree}{\wOne = \wOneP = \wTwo \neq \wTwoP}