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Touch up on appendix Lemma 3.9 proof.

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Aaron Huber 2022-06-08 09:22:02 -04:00
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app_hard_linsys.tex
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@ -38,7 +38,7 @@ We follow the same process in deriving an equality for $G^{(2)}$. Replacing occ
&-\left[\numocc{\graph{2}}{\threepath}\prob+3\numocc{\graph{2}}{\tri}\prob\right]-\left[\numocc{\graph{2}}{\twopathdis}\prob^2-3\numocc{\graph{2}}{\threedis}\prob^2\right]\nonumber\\
&+\left(4\numocc{G}{\oneint}+\left[6\numocc{G}{\twopathdis}+18\numocc{G}{\threedis}\right]+\left[4\numocc{G}{\threepath}+12\numocc{G}{\tri}\right]\right)(3\prob^2 - \prob^3)\label{eq:b2-final}
\end{align}
The steps to obtaining \cref{eq:b2-final} are analogous to the derivation immediately preceding. As in the previous derivation, note that the LHS of \Cref{eq:b2-final} is the same as $\vct{M}[2]\cdot \vct{x}[2]$. The RHS of \Cref{eq:b2-final} has terms all computable (by equations (\ref{eq:1e})-(\ref{eq:3p-3tri})) in $O(m)$ time. Setting $\vct{b}[2]$ to the RHS then completes the proof of step 1.
The steps to obtaining \cref{eq:b2-final} are analogous to the derivation of~\Cref{eq:b1-alg-2}. As in the previous derivation, note that the LHS of \Cref{eq:b2-final} is the same as $\vct{M}[2]\cdot \vct{x}[2]$. The RHS of \Cref{eq:b2-final} has terms all computable (by equations (\ref{eq:1e})-(\ref{eq:3p-3tri})) in $O(m)$ time. Setting $\vct{b}[2]$ to the RHS then completes the proof of step 1.
Note that if $\vct{M}$ has full rank then one can compute $\numocc{G}{\tri}$ and $\numocc{G}{\threedis}$ in $O(1)$ using Gaussian elimination.