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Maximum Split of Positive Even Integers

Number: 2279

Difficulty: Medium

Paid? No

Companies: Google

Problem Description

Given an integer finalSum, split it into a sum of a maximum number of unique positive even integers. Return a list containing one valid split with the maximum number of integers. If no valid split exists, return an empty list.

Key Insights

  • Since only even integers are used, if finalSum is odd, the answer is immediately an empty list.
  • A greedy approach is effective: start from the smallest even integer (2) and keep adding the next even number until the next addition would exceed the remaining sum.
  • After adding numbers in increasing order, if a remainder exists, add it to the last number to ensure the final sum equals finalSum while maintaining uniqueness.

Space and Time Complexity

Time Complexity: O(k), where k is the number of split integers, which is approximately O(√finalSum) since the sum of the first k even integers is k(k+1). Space Complexity: O(k) due to storage of the list of even integers.

Solution

First, check if finalSum is even; if not, return an empty list. Then, using a greedy algorithm, iteratively add the smallest available even integer (starting from 2) until the next even integer exceeds the remaining finalSum. If there’s a remainder at the end, adjust the last number by adding the remainder to it. This ensures that the sum equals finalSum and that the maximum number of unique even integers are used.

Code Solutions

# Python solution for Maximum Split of Positive Even Integers

def maximumEvenSplit(finalSum):
    # If finalSum is odd, return an empty list since a valid split is impossible.
    if finalSum % 2 != 0:
        return []
    
    # List to store the unique positive even integers
    even_integers = []
    current_even = 2
    
    # Keep adding the next smallest even integer until finalSum is too small to add it.
    while finalSum >= current_even:
        even_integers.append(current_even)
        finalSum -= current_even
        current_even += 2  # Move to the next even integer.
    
    # If there's any remaining finalSum, add it to the last element.
    if finalSum > 0:
        even_integers[-1] += finalSum
        
    return even_integers

# Example usage:
print(maximumEvenSplit(12))  # Expected output: [2, 4, 6]
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