Minimum Total Cost

Given an array of integers, the goal is to make all the elements in the array have equal values by applying

some number of operations.

The rules of the operations are:

To apply an operation, one needs to choose a prefix of the array and an integer $x(x$ can be negative$).$

In this operation, add $x$ to each element inside this prefix.

The cost of this operation is $\frac{?}{x}/($Absolute value of $x).$

For example, if the array is $1,4,2,1$ and the prefix of length $2$ and $x=-3$ are chosen, the array would

now become $-2,1,2,1$ and the cost of this operation would be $|x|=|-3|=3.$

The total cost is the sum of costs of each operation applied. Find the minimum total cost of making all

the elements in the array have equal value.

Note: It is guaranteed that there always exists a series of operations by which all the elements in any

array can be equal. These operations can be applied any number of times.

Example

Consider $n=4$ and arr $=[1,2,1,5].$

The array can be made equal using the following three operations:

Choose the prefix of length $2$ and $x=-1,$ Hence the array now becomes $0,1,1,5.$ The cost of this operation

is $|x|=|-1|=1$

Choose the prefix of length $3$ and $x=4,$ Hence the array now becomes $4,5,5,5.$ The cost of this operation is

$|x|=|4|=4$

Choose the prefix of length $1$ and $x=1,$ Hence the array now becomes $5,5,5,5.$ The cost of this operation is

$|x|=|1|=1$

Thus, the total cost $=1+4+1=6,$ which is the minimum possible.

Any hint for this? when arr is $[1,1,1,1]$ cost should be zero when arr is $[1,2,3,1,2]$ cost is $5$