Problem Description

The problem presents an integer array nums and defines a specific type of array called base[n]. The base[n] is an array of length n + 1 that contains each integer from 1 to n - 1 exactly once and includes the integer n exactly twice. For example, base[3] would be [1, 2, 3, 3]. The primary task is to determine if the given array nums is a permutation of any base[n] array.

A permutation in this context means any rearrangement of the elements. So, if nums is a rearrangement of all the numbers from 1 to n - 1 and the number n appears exactly twice, the array is considered "good," and the function should return true, otherwise false. The problem simplifies to checking whether the array contains the correct count of each number to match the base definition.

Intuition

The intuition behind the solution involves counting the occurrences of each number in the array nums.

• Identify n as the length of the input array nums minus one, since a base[n] array has a length of n + 1. This identification is based on the definition mentioned in the problem description.

• Utilize a Counter (a collection type that conveniently counts occurrences of elements) to tally how often each number appears in nums.

• Subtract 2 from the count of n because n should appear exactly twice in a good base array.

• Then subtract 1 from the counts of all other numbers from 1 to n - 1 because each of these numbers should appear exactly once in a good base array.

• Finally, check if all counts are zero using all(v == 0 for v in cnt.values()). If they are, it means that the input array has the exact count for each number as required for it to be a permutation of a base[n] array, and the function should return true. If even one count is not zero, it indicates that there's a discrepancy in the required number frequencies, and the function should return false.

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Solution Approach

The implementation of the solution utilizes a standard Python library called collections.Counter, which is a subclass of dictionary specifically designed to count hashable objects.

Here's the step-by-step breakdown:

1. Compute n as the length of the array nums minus one. This is because we expect the array to be a permutation of a base[n], which has length n + 1.

   n = len(nums) - 1

2. Initialize a Counter with nums which automatically counts the frequency of each element in the array.

   cnt = Counter(nums)

3. Adjust the counted frequencies to match the expectations of a base[n] array. According to base[n], the number n should appear twice, and all the numbers from 1 to n - 1 should appear once. To reflect this in our counter, we subtract 2 from the count of n and subtract 1 from the counts of all other numbers within the range 1 to n.

   cnt[n] -= 2
   for i in range(1, n):
       cnt[i] -= 1

4. After the adjustments, a "good" array would leave all counts in the Counter at zero. Verify that this is true by applying the all function on a generator expression that checks if all values in the Counter are zero.

   return all(v == 0 for v in cnt.values())

5. If any value in the Counter is not zero, then the array cannot be a permutation of "base" because it does not contain the correct frequency of numbers. In such a case, the function will return false.

This solution approach utilizes the Counter data structure to perform frequency counting efficiently. The adjustment steps ensure that the counts match the unique base[n] array's requirements. The final all check succinctly determines the validity of nums being a good array, keeping the implementation both effective and elegant.

Example Walkthrough

Let's use an example to illustrate the solution approach.

Example Input

Consider the array nums = [3, 1, 2, 3].

Steps

1. First, calculate n by taking the length of nums and subtracting one to account for the fact that there should be n + 1 elements in a good base array. In this case, len(nums) - 1 equals 4 - 1 which is 3. Hence, n = 3.

   n = len(nums) - 1  # n = 3

2. Initialize a Counter with the array nums. This will count how many times each number appears in nums.

   cnt = Counter(nums)  # cnt = {3: 2, 1: 1, 2: 1}

3. Alter the counted frequencies to mimic a base[n] array. The number n should appear twice, so we subtract 2 from its count. All other numbers from 1 to n - 1 should appear once, so we subtract 1 from their counts.

   cnt[n] -= 2  # cnt[3] -= 2 gives cnt = {3: 0, 1: 1, 2: 1}
   for i in range(1, n):
       cnt[i] -= 1  # iterating and subtracting 1, cnt = {3: 0, 1: 0, 2: 0}

4. All values in the Counter should now be zero for a good base array. Using the all function we check each value:

   return all(v == 0 for v in cnt.values())  # This evaluates to `True`

5. Since each number from 1 to n - 1 is included exactly once and n is included exactly twice, and all adjusted counts are zero, the function will return true. This means nums is indeed a permutation of a base[3] array.

Following this approach, the given array nums = [3, 1, 2, 3] is confirmed to be a good array and thus the expected output for this example would be True as it fits the criteria of a base[n] array permutation.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the provided function is determined by a few major steps:

1. n = len(nums) - 1: This is a constant time operation, O(1).
2. cnt = Counter(nums): Building the counter object from the nums list is O(N), where N is the number of elements in nums.
3. cnt[n] -= 2: Another constant time operation, O(1).
4. The loop for i in range(1, n): cnt[i] -= 1: This will execute N-1 times (since n = len(nums) - 1), and each operation inside the loop is constant time, resulting in O(N) complexity.
5. The all function combined with the generator expression all(v == 0 for v in cnt.values()): Since counting the values in a Counter object and then iterating through them is O(N), this step is O(N) as well.

Adding up all the parts, the overall time complexity is O(N) + O(N) + O(N) which simplifies to O(N), because in Big O notation we keep the highest order term and drop the constants.

Space Complexity

The space complexity is also determined by a few factors:

1. cnt = Counter(nums): Storing the count of each number in nums requires additional space which is proportional to the number of unique elements in nums. In the worst case, if all elements are unique, this will be O(N).
2. The for loop and the all function does not use extra space that scales with the size of the input, as they only modify the existing Counter object.

Therefore, the space complexity is O(N), where N is the number of elements in nums and assuming all elements are unique.

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