algoadvance

Given the array nums, obtain a subsequence of the array such that the sum of its elements is strict greater than the sum of the non-included elements. If there are multiple solutions, return the subsequence with the minimum length and if there still are multiple solutions, return the subsequence with the maximum total sum. The result should be in non-increasing order.

Clarifying Questions:

  1. What is the valid range of the elements in the array nums?
    • The elements can be both negative and positive integers, and the length of the array can vary with typical constraints seen in competitive programming (e.g., up to (10^4) elements).
  2. Can nums contain duplicate values?
    • Yes, the array nums can contain duplicate values.
  3. What should be done if the array is empty?
    • Given the constraints and the problem context, it can be assumed that the array is non-empty.

Strategy:

  1. Sort the Array:
    • First, sort the array nums in descending order.
  2. Select Elements:
    • Iterate through the sorted array and keep adding elements to the subsequence until the sum of the selected elements is greater than the remaining elements.
  3. Check Sum:
    • Maintain two pointers or counters for the sums of the selected and remaining elements. Keep track of the total sum of the array and subtract from it as you add elements to the subsequence.
  4. Output the Result:
    • Return the list of selected elements.

Code:

def minSubsequence(nums):
    nums.sort(reverse=True)  # Sort in non-increasing order
    total_sum = sum(nums)
    subsequence_sum = 0
    subsequence = []
    
    for num in nums:
        subsequence_sum += num
        subsequence.append(num)
        if subsequence_sum > total_sum - subsequence_sum:
            return subsequence

# Example usage
nums = [4, 3, 10, 9, 8]
print(minSubsequence(nums))  # Output should be: [10, 9]

Time Complexity:

Thus, the overall time complexity is O(n log n) due to the sorting step.

This solution ensures that we find the smallest subsequence in non-increasing order such that it has a sum greater than the remaining elements. The sorting guarantees that we always consider larger elements first, which helps in finding the minimal length subsequence effectively.

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