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Maximize Subarray Sum After Removing All Occurrences of One Element

Difficulty: Hard

Problem Description

You are given an integer array nums. You can do the following operation on the array at most once: choose any integer x such that nums remains non-empty on removing all occurrences of x, and then remove all occurrences of x from the array. Return the maximum subarray sum across all possible resulting arrays.

Key Insights

  • The problem requires evaluating the maximum subarray sum after potentially removing all occurrences of a specific integer.
  • The Kadane's algorithm can be used to find the maximum subarray sum efficiently.
  • We need to consider the case where we do not remove any elements, as this might yield the highest sum.
  • A frequency count of elements can help in determining which elements to consider for removal.

Space and Time Complexity

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

Solution

To solve the problem, we will utilize Kadane's algorithm to calculate the maximum subarray sum efficiently. We will also maintain a frequency count of the elements in the array to determine which elements can be removed. For each unique element, we will simulate the removal of that element, and calculate the maximum subarray sum of the resulting array. Finally, we will compare the sums obtained from all simulations, including the case where no elements are removed, to determine the overall maximum.

Code Solutions

from collections import Counter

def max_subarray_sum_after_removal(nums):
    def kadane(arr):
        max_sum = float('-inf')
        current_sum = 0
        for num in arr:
            current_sum += num
            max_sum = max(max_sum, current_sum)
            if current_sum < 0:
                current_sum = 0
        return max_sum

    total_sum = kadane(nums)
    freq = Counter(nums)
    max_sum = total_sum
    
    for x in freq:
        modified_nums = [num for num in nums if num != x]
        if modified_nums:  # ensure array is non-empty
            max_sum = max(max_sum, kadane(modified_nums))
    
    return max_sum
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