Minimum Replacements to Sort the Array

Problem Description

You are given a 0-indexed integer array nums. In one operation you can replace any element of the array with any two elements that sum to it. Return the minimum number of operations to make the array sorted in non-decreasing order.

Key Insights

• The operation allows replacing a single number with two numbers that sum to it, effectively breaking one number into two smaller parts.
• To ensure the array is sorted, each number must not exceed the previous number in the sorted order.
• The minimum number of operations can be calculated by comparing each number with its predecessor and determining how many replacements are needed to maintain the non-decreasing order.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1)

Solution

To solve this problem, we iterate through the array from the second element to the last. For each element, we compare it with the previous element. If the current element is less than the previous one, we calculate how many operations are needed to break down the previous element into smaller parts that can accommodate the current element.

We can use the following approach:

1. Start from the second element and iterate through the list.
2. If the current element is less than the previous element:
   • Calculate the number of operations required to break down the previous element so that all resulting elements are less than or equal to the current element.
   • Count the operations needed.
3. Return the total count of operations.

Code Solutions