algoadvance

You are given an integer n. A 0-indexed integer array nums of length n + 1 is generated in the following way:

Return the maximum integer in the array nums.

Clarifying Questions

  1. What should be returned if n is 0?
    • Return 0 as nums[0] is the only element in the array.
  2. What is the range of n?
    • According to the problem statement constraints, n is in the range [0, 100].
  3. Are there any constraints on the runtime of the solution?
    • Since n is relatively small (up to 100), an O(n) solution is acceptable.

Strategy

To solve the problem, follow these steps:

  1. Initialization: Create an array nums of size n + 1 and initialize the first two elements nums[0] and nums[1] as given.
  2. Array Generation: Iterate from 2 to n, and populate elements of nums based on whether the current index i is even or odd.
  3. Maximum Value: Track the maximum value encountered while generating the array.
  4. Return Result: Return the tracked maximum value.

Code

def getMaximumGenerated(n: int) -> int:
    if n == 0:
        return 0
    if n == 1:
        return 1
    
    nums = [0] * (n + 1)
    nums[0], nums[1] = 0, 1
    max_val = 1
    
    for i in range(2, n + 1):
        if i % 2 == 0:
            nums[i] = nums[i // 2]
        else:
            nums[i] = nums[i // 2] + nums[(i // 2) + 1]
        
        max_val = max(max_val, nums[i])
    
    return max_val

Time Complexity

  1. Initialization: This takes O(1) time and space.
  2. Array Generation: The loop runs n times, and each iteration takes O(1) time, so the total complexity for this part is O(n).
  3. Tracking Maximum: Updating the maximum value during the loop also takes O(1) time per iteration, leading to O(n) cumulatively.

Hence, the overall time complexity is O(n), and the space complexity is O(n) due to the array nums.

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