Problem Description

In this problem, you are given an array nums containing n integers, indexed from 0 to n-1. You are also given an integer k, which defines a radius. The task is to calculate the k-radius average for each element in the array. The k-radius average for an element at index i includes all the elements from index i - k to i + k, inclusive. To qualify for calculating the average, there must be at least k elements before and after index i. If an element doesn't have enough neighbors to satisfy the radius k, the average for that element will be -1.

It is important to note that the average is computed using integer division, meaning you add all the elements within the range and then divide by the total number of elements, truncating the result to remove the decimal part.

For example, for the array [1,3,5,7,9] and k = 1, the result would be [-1, 3, 5, 7, -1] because:

• elements at indices 0 and 4 don't have enough neighbors for radius 1;
• element at index 1 has an average of (1+3+5)/3, which truncates to 3;
• and so on for the elements at indices 2 and 3.

Intuition

The straightforward approach would be to calculate the average for each element individually, which would involve summing elements within the radius for every index, resulting in a time-consuming process with a time complexity of O(n*k).

However, a more efficient method is to use a moving sum (or sliding window). By maintaining a running sum of the last 2k+1 elements, we can compute the k-radius average for the current index by simply dividing this sum by 2k+1. After computing the average, move the window by increasing the index i, add the next number in the array to the sum, and subtract the number that just left the window (which would be at index i - 2k). This way, we only need one pass through the array, resulting in a time complexity of O(n).

The provided Python function getAverages implements this efficient sliding window approach. An array ans of the same length as nums is initially filled with -1. This will store our results. The variable s is the sliding sum, which is updated as we iterate over nums with index i. If i is large enough (i >= k * 2), it means we can compute an average for the element at index i - k. We add the current value v to s, and if the window is valid, we compute the average and update ans[i - k]. One important detail is the use of integer division s // (k * 2 + 1) as required by the problem statement. Finally, we return the completed array ans.

Learn more about Sliding Window patterns.

Solution Approach

The solution provided uses a sliding window technique to efficiently compute each k-radius average in a single pass through the input array nums.

Here's an in-depth explanation of the algorithm's steps:

1. Initialize a running sum, s, which will keep track of the sum of elements in the current window. We will also initialize an array ans with the same length as nums filled with -1s, to store our results.

2. Iterate through the array nums with both index and value (i and v). Increment the running sum s by the value v.

3. Check if the current index i allows us to have a complete window of 2k+1 elements. This is determined by the condition i >= k * 2. If i is less than k * 2, we cannot calculate the average for i - k since we do not have enough elements before index i.

4. If we do have enough elements, calculate the average for the element at index i - k as the running sum s divided by 2k+1. We use integer division // here as specified by the problem constraints. The result is stored in ans[i - k].

5. Then, to maintain the size of the window, subtract the element at index i - 2k from the running sum s. This is the element that's falling out of our window as we move forward.

6. Continue this process until all elements have been visited, and the ans array is fully populated with the k-radius averages where possible or -1 where the average cannot be calculated.

The algorithm makes use of simple data structures which are a running sum variable s and an array ans to hold the results. The sliding window pattern here avoids redundantly recalculating the sum for overlapping parts of the window, thus optimizing the process to a time complexity of O(n) where n is the length of the input array.

Here's the core code snippet that illustrates the algorithm:

s = 0
ans = [-1] * len(nums)
for i, v in enumerate(nums):
    s += v
    if i >= k * 2:
        ans[i - k] = s // (k * 2 + 1)
        s -= nums[i - k * 2]

By maintaining the sliding window and updating the running sum in this manner, the algorithm efficiently computes the required averages without redundant calculations.

Example Walkthrough

Let's walk through the solution approach with a small example. Assume we have an array nums = [2, 4, 6, 8, 10] and k = 2. We want to find the k-radius average for each element.

1. Initialize Variables: We start by initializing the running sum s = 0 and an array ans = [-1, -1, -1, -1, -1] to store the results.

2. Iterate Through nums: We begin iterating over nums. At the start, our running sum s will start accumulating values as we iterate:

   • At index 0, s = 0 + 2 = 2. We cannot compute an average yet, as we don't have enough elements before this index.
   • At index 1, s = 2 + 4 = 6. We still cannot compute an average for i - k = -1.
   • At index 2, s = 6 + 6 = 12. We reach a point where we could compute an average for the first element (i = 2, k = 2), but since it's the first element, we can't.

3. Check For Complete Window: Now, we reach the first index where a complete window 2k+1 is available:

   • At index 3, s = 12 + 8 = 20.
   • As i = 3 is not less than k * 2 (4), we can't calculate the average for i - k = 1 yet.

4. Calculate k-Radius Averages:

   • At index 4, s = 20 + 10 = 30. Since i = 4 >= k * 2, we can now calculate the average for i - k = 2. We compute it as s // (2 * 2 + 1) = 30 // 5 = 6. The running sum at this point includes nums[0] through nums[4] (values 2, 4, 6, 8, 10). We store 6 in ans[2].

5. Slide The Window: Before we move to the next iteration (which we don't have in this example, since we've reached the end of the array), we subtract the number that's falling out of the window. Here that's nums[4 - 2 * 2], which is nums[0] = 2. Thus, we update the running sum s = 30 - 2 = 28.

If we had more elements in the array, we would continue this process. However, since we've reached the end, we're done, and our result array ans looks like this: [-1, -1, 6, -1, -1]. The k-radius average is only calculated for the element at index 2 since those at indices 0, 1, 3 and 4 do not have enough neighbors.

This example illustrates the efficiency of the sliding window technique in avoiding redundant calculations. We avoid iterating over each window separately for each element, which significantly reduces the time complexity from O(nk) to O(n).

Solution Implementation

Time and Space Complexity

The time complexity of the given code is O(n), where n is the length of the input list nums. This is because there is a single loop that iterates over all elements of nums once. Within the loop, operations are done in constant time including calculating the sum for the average and updating the sum by subtracting the element that is falling out of the sliding window.

The space complexity of the code is O(n), where n is the length of the input list nums. This is due to the ans list which is initialized to the same length as nums. There are no other data structures that depend on the size of the input that would increase the space complexity.

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