The mathematical derivation is easy.

Lemma: We have to equalize the salary, this means that difference between every workerâ€™s salary must be 0.

What operation is available to us? To increase salary of every worker except 1.

Think it like this, what effect does it have on the relative difference?

The difference, of course, decreases by 1, but theres another observation to this.

Observation: Increasing salary of all workers except 1 has same result as decreasing that guyâ€™s salary by 1 and keeping everyoneâ€™s salary as it is. We can mathematically prove that effect on difference of salaries is same (and hence on amount of operations to equalize), even though final salaries are different. [The answer is independent of final salary/absolute values. It depends on relative values, i.e. how is this array value in comparison with other array values]

So we transform operation to this-

â€śChose a guy and decrease his salary by 1â€ť , since in terms of operation needed to equalize salary, it will give same result.

Now it becomes simple. We just pick up the minimum element, and find operations needed to convert each element to minimum element.

Challenge: If you understood the concept, give a hand at this problem- https://www.hackerrank.com/challenges/equal/problem

Steps-

Arr=[1,2,3]

Minimum element=1.

1st element - arr[0]=1. Already minimum. Operations needed=0;

2nd element- arr[1]=2. To convert to minimum element, need 1 operation. Operations=1

3rd element- arr[2]=3, to convert to minimum element, need 2 operations. Operations=3

Q- Why the minimum element?

Because we showed that the transformation giving in question is equivalent to picking up a worker and reducing his salary by 1 (in terms of difference of salaries).

So, obviously this operation cannot increase any workerâ€™s salary. So to equalize salaries, we need to reduce salaries of all workers till they are equal. The only possible option is to convert each to minimum element.