metric methods in computer sience question

does the following solution is a true solution to the problem in the picture

if yes please provide a full proof of approximation gurantee

the proof should be fully formal explaining each step and in a very clear way

the solution:

As in the hint I will use the $\backslash$alpha $-$approximation for Max$-$Cut as a black box to solve the Maximum Linear Arrangement problem.

Algorithm:

Take the input graph G and run the Max$-$Cut $\backslash$alpha $-$approximation algorithm $,$ To split the vertices into two disjoint sets, V$$ and V$,$ let C be the cut that results.

$2.$ Using the labels that were acquired from the Max$-$Cut algorithm, sort the vertices in V$$ and V$.($For example, let V$1=\{$v$,...,$vi$\}$ and V$2=\{$u$,...,$uj$\})$

$\mathrm{.}3.$ Beginning with V$,$ assign locations to each vertex in V$$ and V$$ in turn. For instance, give the vertices in V$$ and V$$ the locations $\{1,3,5,...,2$k$1\}$ and $\{2,4,6,...,2$m$\}$ respectively.

$4.$ Determine the total edge length using the linear configuration established by the points given in step $3$: Total$\_$length is equal to $\backslash$Sigma $($u$,$v$)$ in E$|\backslash$pi $($u$)\backslash$pi $($v$)|.$

thank you