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Minimum Replacements to Sort the Array

Number: 2450

Difficulty: Hard

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Problem Description

Given a 0-indexed integer array nums, you are allowed to replace any element with any two elements that add up to it. The task is to compute the minimum number of such operations required to transform the array into one that is sorted in non-decreasing order.

Key Insights

  • Process the array from right to left, using the rightmost element as the initial upper bound.
  • For each element moving left, if it exceeds the current bound, split it into several pieces so that each piece does not exceed the bound.
  • The number of pieces (k) needed is determined by the ceiling of (current element / bound), and the number of operations is (k - 1).
  • Update the new bound to be the maximum allowable value for the parts (which is the floor division of the current element by k).
  • Use greedy splitting to minimize the operations while ensuring the non-decreasing sequence requirement is met in the final array.

Space and Time Complexity

Time Complexity: O(n), where n is the number of elements in nums, since we perform a single pass from right to left. Space Complexity: O(1), as we use a constant amount of extra space.

Solution

The solution uses a greedy approach by iterating from the end of the array towards the beginning. The rightmost element sets the allowable maximum for any preceding segments. For every element to its left:

  1. If the current element is greater than the allowable maximum (current bound), calculate the minimum number of pieces it should be split into using ceiling division.
  2. The number of operations needed is one less than the number of pieces.
  3. Update the allowable maximum based on the new pieces.
  4. Continue until the array is processed. This ensures that after all operations, the array can be rearranged (implicitly, by splitting) into a non-decreasing order while using the minimum number of operations.

Code Solutions

# Python solution with detailed comments

from math import ceil
from typing import List

def minimumReplacement(nums: List[int]) -> int:
    operations = 0
    # current is the allowed maximum value for the previous piece.
    current = nums[-1]
    
    # Traverse the array backwards (ignoring the last element)
    for num in reversed(nums[:-1]):
        # If the current number is greater than the allowed maximum,
        # it needs to be split into smaller pieces.
        if num > current:
            # Calculate the minimum number of pieces to split 'num'
            # such that each piece is <= current.
            pieces = (num + current - 1) // current  # Equivalent to ceil(num/current)
            # The number of operations is pieces - 1.
            operations += pieces - 1
            # Update current to be the maximum allowed for the new pieces.
            current = num // pieces
        else:
            # Otherwise, update current to be the current number.
            current = num
    return operations

# Example usage:
if __name__ == "__main__":
    print(minimumReplacement([3, 9, 3]))  # Expected output: 2
    print(minimumReplacement([1, 2, 3, 4, 5]))  # Expected output: 0
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