Problem Description

You are provided with an array nums which consists of unique integers. The goal is to rearrange the elements in the array in such a way that no element is equal to the average of its immediate neighbors. To clarify, for every index i of the rearranged array (1 <= i < nums.length - 1), the value at nums[i] should not be equal to (nums[i-1] + nums[i+1]) / 2. The task is to return any version of the nums array that satisfies this condition.

Intuition

The key observation to arrive at the solution is that if we arrange the numbers in a sorted manner and then alternate between the smaller and larger halves of the sorted array, we can ensure that no element will be the average of its neighbors. This is because the numbers are distinct and sorted, so a number from the smaller half will always be less than the average of its neighbors, and a number from the larger half will always be greater than the average of its neighbors.

Therefore, first, we sort nums. Then, we divide the sorted array into two halves - the first half containing the smaller elements and the second half containing the larger elements. We then interleave these two halves such that we first take an element from the first half, followed by an element from the second half, and so on until we run out of elements.

Here's how the thought process breaks down:

1. Sort the array to easily identify the smaller and larger halves.
2. Identify the middle index to divide the sorted array into two halves.
3. Create an empty array ans to store the final rearranged sequence.
4. Iterate through each of the elements in the first half and alternate by adding an element from the second half.
5. Return the rearranged array which should now meet the requirements of the problem.

Learn more about Greedy and Sorting patterns.

Solution Approach

The solution is pretty straightforward once we understand the intuition behind the problem. The approach makes use of the sorting algorithm, which is a very common and powerful technique in many different problems to create order from disorder.

Here's a step-by-step explanation of the implementation with reference to the provided solution code:

1. We start by sorting the array nums. Sorting is done in-place and can use the TimSort algorithm (Python's default sorting algorithm), which has a time complexity of O(n log n), where n is the number of elements in the array.

   1nums.sort()

2. Calculate the middle index m of the array. It represents the starting index of the second half of the sorted array. If the array is of odd length, the middle index will include one more element in the first half.

   1m = (n + 1) >> 1

   The >> operator is a right bitwise shift, which is equivalent to dividing by two, but since we are dealing with arrays that are zero-indexed, (n + 1) ensures that the first half includes the middle element when n is odd. For even n, this does not change the result.

3. An empty list ans is initialized to store the final rearranged sequence of nums.

   1ans = []

4. Use a for-loop to iterate over the indices of the first half of nums, which runs from 0 to m - 1 inclusive. For each i, add nums[i] to ans. Then, check if the corresponding index in the second half i + m is within bounds (i + m < n), and if so, add nums[i + m] to ans. This ensures that elements are alternated from both halves.

   1for i in range(m):
   2    ans.append(nums[i])
   3    if i + m < n:
   4        ans.append(nums[i + m])

5. The completed ans list is now a rearranged version of nums that satisfies the problem conditions and is returned as the result.

   1return ans

By implementing this solution, we ensure that elements from the lower half and the upper half of the sorted array alternate in the final arrangement, which prevents any element from being the average of its neighbors due to the distinct and sorted nature of nums.

Example Walkthrough

Let's walk through a small example to illustrate the solution approach described above using an array nums. Suppose our input array is:

1nums = [1, 3, 2, 5, 4, 6]

Following the steps in the approach:

1. We start by sorting the array nums, which gives us:

   1nums.sort() => [1, 2, 3, 4, 5, 6]

2. Calculate the middle index m. There are 6 elements (n = 6), so the middle index will be:

   1m = (6 + 1) >> 1 => 3

   This means that the first half is [1, 2, 3] and the second half is [4, 5, 6].

3. An empty list ans is initialized to store the final rearranged sequence of nums.

   1ans = []

4. Use a for-loop to iterate over the indices of the first half of nums. We take an element from the first half, then an element from the second half, and repeat:

   1for i in range(3):  # range(m)
   2    ans.append(nums[i])     # Add from the first half
   3    if i + 3 < 6:           # Check if the second half index is within bounds
   4        ans.append(nums[i + 3])  # Add from the second half

   When we iterate through, the ans array gets filled as follows:

   • i = 0: Append nums[0] which is 1 and then nums[0 + 3] which is 4, so ans = [1, 4].
   • i = 1: Append nums[1] which is 2 and then nums[1 + 3] which is 5, so ans = [1, 4, 2, 5].
   • i = 2: Append nums[2] which is 3 and then nums[2 + 3] which is 6, so ans = [1, 4, 2, 5, 3, 6].

5. The reordered list ans is now [1, 4, 2, 5, 3, 6] and satisfies the problem condition where no element is equal to the average of its immediate neighbors, and this is returned as the result.

The final output for the initial nums array is:

1return ans => [1, 4, 2, 5, 3, 6]

The process effectively rearranges the original array preventing any value from being the average of its neighbors by interleaving the first and second halves of the sorted list.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the code consists of the sort operation and the loop that rearranges the elements.

1. Sorting the nums array takes O(n log n) time, where n is the number of elements in nums.
2. The for loop runs for m = (n + 1) >> 1 iterations, which is essentially n/2 iterations for large n. Each iteration executes constant time operations, making this part O(n/2) which simplifies to O(n).

Hence, the overall time complexity is dominated by the sort operation: O(n log n).

Space Complexity

The space complexity is determined by the additional space used besides the input nums array.

1. The ans list, which contains n elements, requires O(n) space.
2. There are also constant-size extra variables like n, m, and i which use O(1) space.

Thus, the total space complexity is O(n) for the ans list.

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