ODIn/paper-BagRelationalPDBsAreHard
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Working on generalizing v_t

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Aaron Huber 2019-07-25 09:59:33 -04:00
parent 95be054863
commit 8270b01aab
1 changed files with 3 additions and 1 deletions
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hash_const.tex
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@ -65,7 +65,9 @@ Using the same argument as in $\gIJ$ yields
\end{equation*}
Setting $T_3 = \sum\limits_{\wVec \in \pw \st \sketchPolarParam{\wVec} = 0}\kMapParam{\wVec}$, $T_4 = \sum\limits_{\wVec \in \pw \st \sketchPolarParam{\wVec} = 1}\kMapParam{\wVec}$ gives an exact calculation for each term given a fixed $\buck$:
\begin{equation*}
T_3 = \gVt{(k \cdot)} | \{\wVec \st \matrixH \cdot \wVec = \buck^{(0)}, \kMapParam{\wVec} = \gVt{(k) }1\}\rightarrow T_3 \in [0, 2^{\numTup - rank(\matrixH')}]
T_3 = \gVt{\sum_{\substack{k \in \{\wVec \st \\
\matrixH \cdot \wVec = \buck^{(0)},\\
\kMapParam{\wVec} = k\}}}k} | \{\wVec \st \matrixH \cdot \wVec = \buck^{(0)}, \kMapParam{\wVec} = \gVt{(k) }1\}\rightarrow T_3 \in [0, 2^{\numTup - rank(\matrixH')}]
\end{equation*}
\begin{equation*}
T_4 = \gVt{(k \cdot)} | \{\wVec \st \matrixH \cdot \wVec = \buck^{(1)}, \kMapParam{\wVec} = \gVt{(k) 1}\}\rightarrow T_4 \in [0, 2^{\numTup - rank(\matrixH')}]