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You are given an integer array arr. You need to find the permutation of the array arr that has the minimum sum of absolute differences between consecutive elements. More specifically, you need to reorder the elements in arr such that the total sum of absolute differences between each consecutive pair of elements is minimized. Return the permutation that achieves this smallest sum.

Clarifying Questions

  1. Input Constraints:
    • What are the constraints on the length of the array?
    • What are the constraints on the elements of the array?
  2. Output:
    • Should we return the array itself or the sum of the absolute differences?
    • Are there any specific requirements for the format of the output?
  3. Edge Cases:
    • How to handle arrays with a single element?
    • How to handle arrays with all elements the same?

Given the problem requires a permutation with a minimal sum of absolute differences, a natural strategy is to order the elements in some optimal configuration. Here is a structured approach to tackle this problem:

Strategy

  1. Sorting the Array:
    • If we sort the array in ascending order, the sum of absolute differences between consecutive elements will naturally be minimized. This is because, for any two elements ( a_i ) and ( a_{i+1} ) in a sorted array, ( a_{i+1} - a_i ) is minimized. Consecutive differences in sorted arrays tend to be the smallest possible within the given array.
  2. Greedy Approach:
    • Since sorting gives the minimal adjacent differences, iterating through a sorted array from start to end will give us the minimal possible sum of absolute differences.

Time Complexity

Thus, the overall time complexity is ( O(n \log n) ).

Let’s implement this approach in code:

Code

def min_cost_permutation(arr):
    # Sorting the input array
    arr_sorted = sorted(arr)
    
    # Returning the sorted array as it is the required permutation
    return arr_sorted

# Example usage
arr = [4, 2, 1, 3]
print(min_cost_permutation(arr))  # Output: [1, 2, 3, 4]

Explanation

  1. Sorting:
    • We first sort the array, which ensures that the difference between consecutive elements is minimized.
  2. Result:
    • The sorted array itself is returned since it inherently represents the permutation with the minimal sum of absolute differences between consecutive elements.

Feel free to ask more specific questions or for further elaboration on any part of the solution!

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