algoadvance

You are given an array nums of non-negative integers. nums is considered special if there exists a number x such that there are exactly x numbers in nums that are greater than or equal to x.

Return x if the array is special, otherwise return -1. Note that x does not have to be an element in nums.

Clarifying Questions

  1. Q: Can nums contain duplicate values?
    • A: Yes, nums can contain duplicate values.
  2. Q: What is the range of the length of nums?
    • A: The length can range from 1 to 100.
  3. Q: What is the range of the values in nums?
    • A: The values are non-negative integers and can range from 0 to 1000.
  4. Q: What should be the return value if no such x exists?
    • A: Return -1 if no such x exists.

Strategy

  1. Sort the Array: We will start by sorting the array in non-decreasing order.
  2. Check for Each x: Iterate over possible values of x from 0 to n (length of array):
    • For each x, count how many numbers in the array are greater than or equal to x.
    • If the count matches x, then return x.
  3. Return -1: If no such x is found during the iteration, return -1.

Code

Here’s the implementation in Python:

def specialArray(nums):
    nums.sort()
    n = len(nums)
    
    for x in range(n + 1):
        count = sum(1 for num in nums if num >= x)
        if count == x:
            return x
    
    return -1

Explanation

  1. Sorting: We first sort nums to ensure that we can easily count numbers greater than or equal to x.
  2. Iteration and Counting:
    • We use a list comprehension within a sum() function to count the number of elements in nums that are greater than or equal to the current x.
    • If this count matches x, we return x as it satisfies the condition.
  3. Returning -1: If the loop completes without finding such an x, we return -1.

Time Complexity

Therefore, the overall time complexity is O(n log n + n^2), which is dominated by O(n^2) due to the nested counting operation.

This approach should be efficient given the constraints (length up to 100).

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