Minimize Range of the Array

Given an array A of size N. In one operation, you can select any number from the array and reduce it to a divisor greater than 1. For example, you can reduce 16 to 2, 4, 8 or 16. You need to find the minimum range of the array by doing any number of operations on the array. You can do multiple operations on a single element as well. The range of the array is defined as the difference between the maximum element and minimum element of the array.

Examples:

Input: N=3 , A=[2 4 8]
Output: 0
Explanation: We can convert the array into [2, 2, 2]. So the difference between the maximum element and minimum element is zero.

Input: N=3 ,A=[3 8 9]
Output:1
Explanation: We can convert the array into [3, 2, 3]. So the difference between the maximum element and minimum element is one.

Approach: We can solve this problem using the below idea

The first observation is that the duplicate elements does not add any thing to the result. for Example [2, 4, 8, 8] in this example if the first 8 is reduce to 2 likewise the other 8 can also be reduced to 2. The main aim of the problem is to minimize the difference between maximum and minimum element of the array. So in order to minimize the difference, we have two options one is to maximize the minimum element of the array or minimize the maximum element of the array. Lets concentrate on maximize the minimum element of the array. Consider an example of [2, 4, 8]. Listing out the factors ( > 1) of these numbers:
Indexes = [0, 1, 2]
2 = [2]
4 = [2, 4]
8 = [2, 4, 8]

We will keep pick up the minimum element from all of the factors of a particular number and push them into a heap. And will keep track of maximum element in the maximum variable. We will pop the smallest element from the heap now difference the smallest element and maximum element that we have kept track of can be the answer. But if there is another factor for the element we have popped we push the factor into the heap again by doing so we can increase the minimum element of the array. We will continue doing the same procedure until the heap becomes empty. And will return minimum of the all those.

Steps to Solve this problem:

1. Find all the factors in the range [2, 10⁴] using sieve.
2. Remove all duplicates elements in the array A. By inserting them into a set.
3. Initialize a variable maximum to zero. To keep track of the maximum element.
4. Initialize a vector startIndexes which will hold the current index of the factor for a particular element in the array.
5. Now iterate over the unique sorted elements. Push all the minimum factor for all the elements in the array into a min heap and store maximum of them in variable maximum.
6. Initialize a variable ans which will store the final result to difference of maximum and top element in the min heap.
7. If the minimum element having some more factors, we move ahead to the next factor in the list. We will pop the top element from the priority queue and add the next factor and according the increment the startIndexes array.
8. Continue this procedure until the priority queue becomes empty or there are no more factors exists for top most element.
9. Return the final result.

Below is the Implementation of the above approach:

1

Time Complexity: O(NlogN). Where N is the number of elements of the array
Auxiliary space: O(N * sqrt(N)), Where N is the number of elements of the array.