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Reduction Operations to Make the Array Elements Equal

Difficulty: Medium

Problem Description

Given an integer array nums, your goal is to make all elements in nums equal. To complete one operation, follow these steps:

  1. Find the largest value in nums. Let its index be i (0-indexed) and its value be largest. If there are multiple elements with the largest value, pick the smallest i.
  2. Find the next largest value in nums strictly smaller than largest. Let its value be nextLargest.
  3. Reduce nums[i] to nextLargest.

Return the number of operations to make all elements in nums equal.

Key Insights

  • The problem requires reducing the largest element to the next largest, and this needs to be done iteratively until all elements are equal.
  • We can take advantage of sorting to identify the unique values in the array and count how many operations it takes to reduce each element to the smallest value.
  • The number of operations needed to reduce all elements can be derived from the count of occurrences of each unique value.

Space and Time Complexity

Time Complexity: O(n log n) - due to sorting the array. Space Complexity: O(n) - for storing the count of elements.

Solution

To solve the problem, we can follow these steps:

  1. Sort the array to easily identify the unique elements.
  2. Count how many times each unique element occurs.
  3. Calculate the number of operations needed to reduce each unique element to the next smaller unique element.
  4. The total number of operations is the sum of the counts of all elements above the smallest unique element.

Code Solutions

def reductionOperations(nums):
    # Step 1: Sort the array
    nums.sort()
    # Step 2: Initialize variables
    operations = 0
    current_count = 0
    
    # Step 3: Count the number of operations needed
    for i in range(1, len(nums)):
        if nums[i] != nums[i - 1]:  # Unique value found
            current_count += 1  # Increment operation count
            operations += current_count  # Add current count to operations
    
    return operations
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