Exercise $3(10$ points$).$ Consider a CSP over variables X$1,...,$ X$8,$ each over the domain $\{0,1,2,3,4,5\}$ and with the following binary constraints:

X$1<$ X$2$; X$5!=$ X$6$; X$2<$ X$6$; X$6=$ X$71$; X$7=$ X$4$; X$7$ X$83$; X$7+$ X$33$

$1.(1$ points$)$ Draw the constraint graph. Is it a tree? $($PLEASE DRAW IT OUT$)$

$2.(9$ points$)$ Solve the CSP $($i$.$e$.,$ find a solution if one exists$)$ by using the tree CSP algorithm $($fig$.6\mathrm{.}11).$ Be explicit about every step in the algorithm $($indicate how you construct the topological sort, how the domain of each variable evolves, how you pick a value for each variable, etc.$).$ If needed to break ties within any step, use the order of the variables.

For Question $2,$ use the following psuedo$-$code and rewrite it using the answer from Question $1.$

function TREE$-$CSP$-$SOLVER$($csp$)$ returns a solution, or failure

inputs: csp$,$ a CSP with components X$,$ D$,$ C

n$$number of variables in X

assignment$$an empty assignment

root$$any variable in X

X$$TOPOLOGICALSORT$($X$,$root$)$

for j $=$ n down to $2$ do

MAKE$-$ARC$-$CONSISTENT$($PARENT$($Xj$),$Xj$)$

if it cannot be made consistent then return failure

for i $=1$ to n do

assignment$[$Xi$]$any consistent value from Di

if there is no consistent value then return failure

return assignment