Problem $4($Shared$-$Bus Take II$).(20$ points$)$ Remember how you helped the famous ride$-$sharing company Ubyft to created a new "shared$-$bus" service during the first midterm? Well that didn't work very well. Turns out that the customers don't like to specify two points, they just want to give a single destination. Ubyft is redesigning their algorithm and they need your help.

The bus runs on a path, each point on the path is denoted by its distance from the starting point. In this new problem, there are $n$ passengers. They all board the bus at the starting point $0.$ Passenger i specifies a destination $x_i$ where the passenger wants to go$.$ You can assume that the $x_i\text{'}$s are already sorted in increasing order.

If passenger i leaves the bus at a position $y,$ then the "unhappiness" of the passenger is $(x_i-y{)}^2.$

Similar to the previous problem, there are no fixed bus stops. Instead, the stops will be decided after gathering the user information. The bus can make at most $k$ stops, and your goal is to design an algorithm that minimizes the sum of unhappiness of all the passengers.

As an example, if there are $4$ passengers with with $x_1=1,x_2=18,x_3=19,x_4=23,$ and $k=2,$ then the optimal solution will make two stops at points $1$ and $20.$ The first stop serves the first passenger; the second stop serves the remaining three passengers. The unhappiness for passengers $1,2,3,4$ are $0,4,1,9$ respectively and the total unhappiness is $14.$

$($a$)(12$ points$)$ Define the state $($sub$-$problems$),$ write the transition function and specify the base cases.

$($b$)(8$ points$)$ Design an algorithm for finding the locations where the bus needs to stop. $($Your algorithm only needs to output the optimal total unhappiness of the passengers$).$ Analyze the running time of your algorithm. $($Your algorithm should run in time $O(k{n}^2).)$