Problem Description

The problem presents us with an integer array called nums, which contains distinct positive integers. This means that all the elements in the array are unique and greater than zero. The task is to identify a number in this array that is neither the smallest (minimum) nor the largest (maximum) value present. If such a number exists, it should be returned as the output. If no such number can be found, which would likely be the case if the array has too few elements to have a middle value, then the function should return -1.

The problem specifies that the function should return any such "middle" value. This means that if there are multiple numbers satisfying the condition of being neither the minimum nor the maximum, any one of them can be chosen as the correct answer.

Intuition

To solve this problem, we can follow a straightforward approach. Since we are given that all the numbers in the array are distinct, we know that there will be one unique minimum value and one unique maximum value.

The first step is to find these two values, the smallest and the largest number in the array, which we can do efficiently by using the min and max functions respectively.

Once we have the smallest and largest values, we can then iterate through the array to find an element that is neither of these. As soon as we find such an element, we can return it, since the problem does not require us to do anything other than find one such 'middle' number.

• We use a loop to check each number.
• For each number, if it is not equal to the minimum and not equal to the maximum, we have found a valid "middle" number and we return it immediately.
• If the loop completes without finding such an element, this means that all elements are either the minimum or the maximum, and we return -1 as per the problem's instructions.

This approach ensures that we look at each element of the array at most once, making the solution efficient and straightforward.

Learn more about Sorting patterns.

Solution Approach

The implementation of the solution uses a very basic algorithmic approach with simple control structures. No advanced data structures or design patterns are required, just the built-in Python list and the straightforward use of the min and max functions.

The core parts of the solution can be broken down as follows:

1. Finding the Minimum and Maximum: We first find the smallest (mi) and largest (mx) numbers in the array using the min(nums) and max(nums) functions. These functions traverse the array, and each one completes in O(n) time where n is the number of elements in the array.

1mi, mx = min(nums), max(nums)

2. Iterating Through the Array: Next, we iterate through each element x in the array nums. This is done using a for loop.

1for x in nums:

3. Checking the Condition: In each iteration, we check if the current element x is not equal to the minimum value mi and not equal to the maximum value mx.

1if x != mi and x != mx:

4. Returning the Middle Value: If x is neither the minimum nor maximum, we have found a valid number that fits the problem's criteria, and we can return it immediately.

1return x

5. Returning -1 If No Number Found: If we go through the entire array without finding a number that satisfies the condition, it implies that no such number exists (for example, the array might only contain the minimum and maximum values). In that case, after the loop concludes, we return -1.

1return -1

This algorithm is linear in nature since it involves a single pass through the array to find min and max and another pass to find a non-minimum and non-maximum element. Hence, the overall time complexity is O(n) because the array is traversed at most twice.

Note: The assumption is that the input array nums provided to the solution function is valid as per the problem description and contains distinct positive integers only.

Example Walkthrough

Let's illustrate the solution approach with an example:

Assume our input array nums is [3, 1, 4, 2].

1. Finding the Minimum and Maximum: We find the smallest number (mi) is 1 and the largest number (mx) is 4.

   1mi, mx = min(nums), max(nums) # mi = 1, mx = 4

2. Iterating Through the Array: We iterate through each element x in the array [3, 1, 4, 2], using a for loop.

3. Checking the Condition:

   • First iteration: x = 3, x != mi is True, and x != mx is also True. Since both conditions are satisfied, we have found a "middle" number.

   1for x in nums:
   2    if x != mi and x != mx:
   3        return x

4. Returning the Middle Value: We return 3 immediately because it satisfies the condition of being neither the minimum nor the maximum value in the array.

In this case, the function would have returned 3 on the first iteration, illustrating that the algorithm efficiently finds a number that is neither the smallest nor the largest in the array without needing to check the rest of the elements. If the array had been [2, 1], the loop would have completed without finding a suitable middle number, so the function would return -1.

Solution Implementation

Time and Space Complexity

// The time complexity of the findNonMinOrMax method is O(n), where n is the number of elements in the input list nums. This is because min(nums) and max(nums) both iterate over the entire list individually, each taking O(n) time. The subsequent for-loop also iterates over the list once, leading to a linear time complexity overall.

// The space complexity of the method is O(1) since it only uses a fixed amount of additional space for the variables mi, mx, and x regardless of the size of the input list.

Learn more about how to find time and space complexity quickly using problem constraints.