algoadvance

You are given an array nums of length n consisting of positive integers. The goal is to find the smallest missing integer greater than the sequential prefix sum of the array.

The “sequential prefix sum” of the array is a new array where each element at index i is the sum of all previous elements in nums plus the current element nums[i].

For example, the sequential prefix sum of [1,2,3] is [1, 1+2, 1+2+3] which is [1, 3, 6].

Given this, find the smallest integer greater than the maximum element in this sequential prefix sum array that does not appear in the array nums.

Clarifying Questions

  1. Are the given integers always positive?
    • Yes, the problem statement specifies that the array nums consists of positive integers.
  2. Should we consider only integers greater than the maximum sequential prefix sum, or all integers?
    • We are looking for the smallest integer greater than the sequential prefix sum array and which is not present in nums.
  3. Is the array sorted?
    • No, the given array nums is not necessarily sorted.

Strategy

  1. Compute Sequential Prefix Sum:
    • Iterate through the nums array and compute the sequential prefix sum.
  2. Find the Maximum of Prefix Sum:
    • Identify the maximum value from the computed prefix sums.
  3. Identify the Missing Integer:
    • Initialize a candidate integer and check if it’s present in nums. Increment the candidate until we find a missing integer.

Code

def smallest_missing_integer(nums):
    # Step 1: Compute Sequential Prefix Sum
    seq_prefix_sums = []
    current_sum = 0
    
    for num in nums:
        current_sum += num
        seq_prefix_sums.append(current_sum)
    
    # Step 2: Find the maximum value in the prefix sums
    max_prefix_sum = max(seq_prefix_sums)
    
    # Step 3: Find the smallest missing integer greater than max_prefix_sum
    candidate = max_prefix_sum + 1
    
    while candidate in nums:
        candidate += 1
    
    return candidate

# Example usage:
nums = [1, 2, 3]
print(smallest_missing_integer(nums))  # Output should be the smallest missing integer > 6 not in [1, 2, 3]

Time Complexity

The time complexity can be broken down as follows:

  1. Computing Prefix Sums: This takes O(n) time because we iterate through the nums array once.
  2. Finding the Maximum Prefix Sum: This also takes O(n) time because we iterate through the prefix sums.
  3. Finding the Missing Integer: In the worst case, this could take O(n) time if the missing integer is a bit greater than the maximum prefix sum but close to any known large gaps in nums.

Therefore, the overall time complexity is O(n).

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