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Aaron Huber 2020-07-14 11:45:57 -04:00
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22
poly-form.tex
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%root: main.tex %root: main.tex
%!TEX root = ./main.tex %!TEX root = ./main.tex
%\onecolumn
\section{Polynomial Formulation} \section{Polynomial Formulation}
We can think of $\poly(\vct{w})$ as a function whose input are the variables $X_1,\ldots, X_M$ as in $\poly(X_1,\ldots, X_M)$. Denote the sum of products expansion of $\poly(X_1,\ldots, X_\numTup)$ as $\poly(X_1,\ldots, X_\numTup)_{\Sigma}$ We can think of $\poly(\vct{w})$ as a function whose input are the variables $X_1,\ldots, X_M$ as in $\poly(X_1,\ldots, X_M)$. Denote the sum of products expansion of $\poly(X_1,\ldots, X_\numTup)$ as $\poly(X_1,\ldots, X_\numTup)_{\Sigma}$
@ -206,10 +206,10 @@ The following is an option.
\item Build $G_2$ from $G_1$, where each edge in $G_1$ gets replaced by a 2 path. \item Build $G_2$ from $G_1$, where each edge in $G_1$ gets replaced by a 2 path.
\end{enumerate} \end{enumerate}
Then $\numocc{G}{\tri}_2 = 0$, and if we can prove that Then $\numocc{G_2}{\tri} = 0$, and if we can prove that
\begin{itemize} \begin{itemize}
\item $\numocc{G}{\threepath}_2 = 2 \cdot \numocc{G}{\twopath}_1$ \item $\numocc{G_2}{\threepath} = 2 \cdot \numocc{G_1}{\twopath}$
\item $\numocc{G}{\threedis}_2 = 8 \cdot \numocc{G}{\threedis}_1 + 6 \cdot \numocc{G}{\twopathdis}_1 + 4 \cdot \numocc{G}{\oneint}_1 + 4 \cdot \numocc{G}{\threepath}_1 + 2 \cdot \numocc{G}{\tri}_1$ \item $\numocc{G_2}{\threedis} = 8 \cdot \numocc{G_1}{\threedis} + 6 \cdot \numocc{G_1}{\twopathdis} + 4 \cdot \numocc{G_1}{\oneint} + 4 \cdot \numocc{G_1}{\threepath} + 2 \cdot \numocc{G_1}{\tri}$
\end{itemize} \end{itemize}
we solve our problem for $q_E^3$ based on $G_2$ and we can compute $\numocc{G}{\threedis}$, a hard problem. we solve our problem for $q_E^3$ based on $G_2$ and we can compute $\numocc{G}{\threedis}$, a hard problem.
\end{proof} \end{proof}
@ -248,7 +248,7 @@ The fact that there is a \textit{fixed} number of possible subgraphs that can be
Earlier, we claimed the following. Earlier, we claimed the following.
\[\numocc{G}{\threedis}_2 = 8 \cdot \numocc{G}{\threedis}_1 + 6 \cdot \numocc{G}{\twopathdis}_1 + 4 \cdot \numocc{G}{\oneint}_1 + 4 \cdot \numocc{G}{\threepath}_1 + 2 \cdot \numocc{G}{\tri}_1\] \[\numocc{G_2}{\threedis} = 8 \cdot \numocc{G_1}{\threedis} + 6 \cdot \numocc{G_1}{\twopathdis} + 4 \cdot \numocc{G_1}{\oneint} + 4 \cdot \numocc{G_1}{\threepath} + 2 \cdot \numocc{G_1}{\tri}\]
Beginning with the leftmost of RHS terms and proceeding to the consecutive rightmost terms, let us show this to be the case. Beginning with the leftmost of RHS terms and proceeding to the consecutive rightmost terms, let us show this to be the case.
@ -268,19 +268,19 @@ For Tri, note that it is the case that the graph $G_2$ is a 'triangle of two pat
\end{proof} \end{proof}
\qed \qed
\AH{Linear Equation computing 3-matchings in $G_3$ using all 3-edge subgraphs in $G_1$.}
In a similar way we can count the number of 3-matchings in graph $G_3$, where each edge in a given $G_1$ gets replaced with a disjoint 3-path, disjoint meaining that no other 3-path intersects another 3-path, except at its endpoints as in the original graph. Because of $G_3$ construction, we now need to also account for two paths in $G_1$. In a similar way we can count the number of 3-matchings in graph $G_3$, where each edge in a given $G_1$ gets replaced with a disjoint 3-path, disjoint meaining that no other 3-path intersects another 3-path, except at its endpoints as in the original graph. Because of $G_3$ construction, we now need to also account for two paths in $G_1$.
The linear combination for 3-matchings in $G_3$ follows. The linear combination of 3-edge $G_1$ subgraphs to compute the number of 3-matchings in $G_3$ follows is
\begin{align*} \begin{align*}
\numocc{G}{\threedis}_3 = &4\pbrace{\numocc{G_1}{\twopath}} + 6\pbrace{\numocc{G_1}{\twodis}} + 30\pbrace{\numocc{G_1}{\tri}} + 35\pbrace{\numocc{G_1}{\threepath}}\\ \numocc{G_3}{\threedis} = &4\pbrace{\numocc{G_1}{\twopath}} + 6\pbrace{\numocc{G_1}{\twodis}} + 30\pbrace{\numocc{G_1}{\tri}} + 35\pbrace{\numocc{G_1}{\threepath}}\\
&+ 40\pbrace{\numocc{G_1}{\twopathdis}} + 32\pbrace{\numocc{G_1}{\oneint}} + 45\pbrace{\numocc{G_1}{\threedis}} &+ 40\pbrace{\numocc{G_1}{\twopathdis}} + 32\pbrace{\numocc{G_1}{\oneint}} + 45\pbrace{\numocc{G_1}{\threedis}}.
\end{align*} \end{align*}
\AH{Justification next.} \AH{Justification.}
Enumerate through the RHS in a similar fashion. Beginning with a two path $\twopath$ in $G_1$, it is the case that in $G_3$ this becomes a six-path. As discussed previously, this yields four three matching subgraphs. For subgragh of two disjoint edges, $\twodis$, this becomes two disjoint 3-paths. It is the case in one 3-path, that we have one subgraph of two disjoint edges, where a third disjoint edge can be picked from any of the three edges in the remaining disjoint 3-path. The process can be repeated starting with the alternative 3-path, giving $2 * 3$ unique 3-matchings. Enumerate through the RHS in a similar fashion. Beginning with a two path $\twopath$ in $G_1$, it is the case that in $G_3$ this becomes a six-path. As discussed previously, this yields four three matching subgraphs. For subgragh of two disjoint edges, $\twodis$, this becomes two disjoint 3-paths. It is the case in one 3-path, that we have one subgraph of two disjoint edges, where a third disjoint edge can be picked from any of the three edges in the remaining disjoint 3-path. The process can be repeated starting with the alternative 3-path, giving $2 * 3$ unique 3-matchings.
Now for the 3-edge subgraphs, starting with a triangle. When a triangle in $G_1$ is transformed into $G_3$, it becomes a 'triangle' where each leg is a three-path. This is very similar to a 9-path, with the caveat that the first and last edge cannot be in the same 3-matching set together. Iterating through all possible combinations producing 3-matchings, i.e. $(e_1, e_3, e_5),\ldots, (e_1, e_3, e_8), (e_1, e_4, e_6),\ldots, (e_1, e_4, e_8),$\newline$\ldots, (e_1, e_6, e_8),\ldots, (e_5, e_7, e_9)$ gives a total of Now for the 3-edge subgraphs, starting with a triangle. When a triangle in $G_1$ is transformed into $G_3$, it becomes a 'triangle' where each leg is a three-path. This is very similar to a 9-path, with the caveat that the first and last edge cannot be in the same 3-matching set together. Iterating through all possible combinations producing 3-matchings, i.e. $(e_1, e_3, e_5),\ldots, (e_1, e_3, e_8), (e_1, e_4, e_6),\ldots, (e_1, e_4, e_8),$\newline$\ldots, (e_1, e_6, e_8),\ldots, (e_5, e_7, e_9)$ gives a total of
@ -293,4 +293,4 @@ matchings. Consider next a 3-path in $G_1$, where the resulting subgraph in $G_
Given the $Fan$ subgraph, where 3 distinct edges are connected at one common endpoint, occurring in $G_1$. In $G_3$, this becomes 3 distinct 9-paths, with each 9-path intersecting the others at one common and shared endpoint. If we consider the outermost non-intersecting edges along with the middle non-intersecting edges, we have $2 * 2 * 2 = 8$ possible 3-matchings. Considering the inner, intersecting edges, we have the condition that only one can appear at a time in a 3-matching set. When we pick an arbitrary inner edge, we have one of two possibilities, we can pick the outer edge of the same 3-path the inner edge is located on, while picking any of the other 4 remaining edges in the middle and outer edges of the other two 3-paths. This gives $4 * 3$ more unique 3-matchings. The remaining possibility exists in combining the arbitrary inner edge with any of the 4 combinations of the middle and outer edges of the other 3-paths. This yields again $3 * 4 = 12$ unique three-matchings, together which make $8 + 12 + 12 = 32$ three-matchings. Given the $Fan$ subgraph, where 3 distinct edges are connected at one common endpoint, occurring in $G_1$. In $G_3$, this becomes 3 distinct 9-paths, with each 9-path intersecting the others at one common and shared endpoint. If we consider the outermost non-intersecting edges along with the middle non-intersecting edges, we have $2 * 2 * 2 = 8$ possible 3-matchings. Considering the inner, intersecting edges, we have the condition that only one can appear at a time in a 3-matching set. When we pick an arbitrary inner edge, we have one of two possibilities, we can pick the outer edge of the same 3-path the inner edge is located on, while picking any of the other 4 remaining edges in the middle and outer edges of the other two 3-paths. This gives $4 * 3$ more unique 3-matchings. The remaining possibility exists in combining the arbitrary inner edge with any of the 4 combinations of the middle and outer edges of the other 3-paths. This yields again $3 * 4 = 12$ unique three-matchings, together which make $8 + 12 + 12 = 32$ three-matchings.
Given the $\threedis$ subgraph occurring in $G_1$, the resulting graph consists of three disjoint 3-paths in $G_3$. There are two considerations. First, if we pull one edge from each disjoint 3-path, we have three choices from each path, which is $3^3 = 27$ three-matchings. The second consideration is that we can pull a two matching from any of the given disjoint 3-paths, matching it with a third disjoint edge from any of the other edges in the other 2 three-paths, giving $3 * 6 = 18$ more unique 3-matchings for a total of $27 + 18 = 45$ 3-matchings. Given the $\threedis$ subgraph occurring in $G_1$, the resulting graph consists of three disjoint 3-paths in $G_3$. There are two considerations. First, if we pull one edge from each disjoint 3-path, we have three choices from each path, which is $3^3 = 27$ three-matchings. The second consideration is that we can pull a two matching from any of the given disjoint 3-paths, matching it with a third disjoint edge from any of the other edges in the other 2 three-paths, giving $3 * 6 = 18$ more unique 3-matchings for a total of $27 + 18 = 45$ three-matchings.

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ra-to-poly.tex
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%root: main.tex %root: main.tex
%!TEX root=./main.tex %!TEX root=./main.tex
\onecolumn
\section{Query translation into polynomials} \section{Query translation into polynomials}
%\AH{This section will involve the set of queries (RA+) that we are interested in, the probabilistic/incomplete models we address, and the outer aggregate functions we perform over the output \textit{annotation} %\AH{This section will involve the set of queries (RA+) that we are interested in, the probabilistic/incomplete models we address, and the outer aggregate functions we perform over the output \textit{annotation}
%1) RA notation %1) RA notation