Consider a binary variable Markov Random Field $\backslash ($p$(\backslash$mathbf$\{$x$\})=$ Z$^\{-1\}\backslash$prod$\_\{$i$>$j$\}\backslash$phi$($x$\_$i$,$ x$\_$j$)\backslash ),$ defined on the $\backslash ($n $\backslash$times n$\backslash )$ lattice with $\backslash (\backslash$phi$($x$\_$i$,$ x$\_$j$)=$ e$^\{\backslash$left$(\backslash$prod $[[$x$\_$i $=$ x$\_$j$]]\backslash$right$)\}\backslash )$ for $\backslash ($i$\backslash )$ a neighbor of $\backslash ($j$\backslash )$ on the lattice and $\backslash ($i$>$j$\backslash ).$ A naive way to perform inference is to first stack all the variables in the $\backslash ($t$^\{$th$\}\backslash )$ column and call this cluster variable $\backslash ($X$\_$t$\backslash ),$ as shown below for a $\backslash (3\backslash$times $3\backslash )$ lattice. The resulting graph is then singly connected. What is the complexity of computing the normalization constant based on this cluster representation? Compute $\backslash (\backslash$log Z$\backslash )$ for $\backslash ($n $=10\backslash )?$

You need to provide your software code for answering $\backslash (\backslash$log$($Z$)\backslash )$ for $\backslash ($n$=10\backslash ).$ Give also your answer for $\backslash (\backslash$log$($Z$)\backslash )$ so that we do not need to run code.