Problem Description

The problem presents an array of integers, nums, with n elements, and another integer, k. The objective is to find a contiguous subarray within nums that has a length exactly equal to k and which has the maximum average value compared to all other possible subarrays of the same length.

Constraints that might apply to this problem, such as the range of values for n, k, and the individual elements in nums, are not stated here. The answer needs to be sufficiently accurate such that a calculation error less than 10^-5 is acceptable.

Intuition

To solve this problem effectively, we can use the "sliding window" technique. The key idea behind this technique is to maintain a "window" of size k that moves through the array, one element at a time, updating the sum of the window as it goes. By doing this, we avoid recalculating the sum for each subarray from scratch, which leads to significant time savings.

Here's the intuition step-by-step:

• First, calculate the sum of the initial window, which is the sum of the first k elements.
• This sum is our current maximum sum (and, as it stands, the maximum average since we would divide this by k).
• Now, slide the window by one element to the right. This means adding the element that comes after the current window and subtracting the element that is leaving the window.
• Update the maximum sum if the new window's sum is greater.
• Repeat this process of sliding the window and updating the maximum sum until we have visited all possible windows of size k.
• Since the maximum sum will have been found by the time we've checked all windows, we can simply take the maximum sum and divide it by k to find the maximum average.

This approach is efficient because it calculates the necessary sum in constant time for each step of the sliding window instead of linear time if we were to add up k elements again and again. Therefore, the overall time complexity is O(n), where n is the number of elements in the array.

Learn more about Sliding Window patterns.

Solution Approach

The solution is implemented using the sliding window pattern as mentioned:

1. Initialization: The first step is to calculate the sum of the first k elements. This is done via the expression sum(nums[:k]). We initialize a variable s with this initial sum, and also we initialize an ans variable with the same value because, at this point, this is the maximum sum we have found, and hence the maximum average as well.

2. Sliding the Window:

   • The for loop with for i in range(k, len(nums)): begins the process of sliding the window across the array. The variable i represents the right boundary of the current window.
   • As the window slides, we add the new rightmost element to our sum with s += nums[i] and remove the leftmost element of the previous window with - nums[i - k]. This enables us to maintain the sum of exactly k elements.
   • After adjusting the sum, we then check if the new sum (s) is greater than the previously recorded maximum sum (ans). If it is, we update ans with ans = max(ans, s).

3. Calculating the Maximum Average:

   • Once the window has slid from the start to the end of the array, the maximum sum of any subarray with length k will be stored in ans.
   • The final step is to divide ans by k to find the maximum average value, which is returned as the result of the function: return ans / k.

By following this approach, the algorithm ensures that each element is added and subtracted exactly once from the sum, leading to an efficient O(n) time complexity. No extra space is needed apart from a few variables, so the space complexity is O(1).

Example Walkthrough

Let's use a small example to illustrate the solution approach. Consider the array nums = [1, 12, -5, -6, 50, 3] and k = 4. We want to find the contiguous subarray of length k with the maximum average value.

Here's how the sliding window approach can be applied:

1. Initialization: First, we calculate the sum of the first k elements, which are 1 + 12 - 5 - 6 = 2. We set s to 2, and also ans is initialized to 2.

2. Sliding the Window:

   • Begin the process with i at index k, or 4, which corresponds to the first number outside of our initial window, 50.
   • The for loop starts, i.e., for i in range(4, 6).
   • At i = 4, we add nums[4] (which is 50) to s, and subtract nums[4 - 4] (which is 1, the leftmost element of the previous window). Now, s = s + 50 - 1 = 51.
   • We now check if s is greater than ans. Since 51 is greater than 2, we update ans to 51.
   • The window is now [12, -5, -6, 50].

3. Continue Sliding the Window:

   • Increment i to 5, add nums[5] to s, and subtract nums[5 - 4]. So s = 51 + 3 - 12 = 42.
   • We check if s is greater than the current ans. It is not (42 is less than 51), so ans remains 51.
   • The window is now [-5, -6, 50, 3].

4. Calculating the Maximum Average:

   • We have now checked all windows of size k. ans is holding the maximum sum, which is 51.
   • The final step is to divide ans (which is 51) by k (which is 4) to find the maximum average. maximum average = ans / k = 51 / 4 = 12.75

Therefore, the subarray [12, -5, -6, 50] returns the maximum average of 12.75 for the given array when the subarray length is 4. This example demonstrates that we only pass through the array once, updating our sum as we go and only carrying out constant work for each element, thereby giving us an O(n) time complexity.

Solution Implementation

Time and Space Complexity

The time complexity of the provided code is O(N) where N is the number of elements in the input list nums. This is because the code iterates through the list once after calculating the initial sum of the first k elements. Each iteration performs a constant number of operations: one addition, one subtraction, and one comparison.

The space complexity of the code is O(1) because it uses a fixed amount of extra space: variables s and ans to keep the rolling sum and the maximum sum, respectively. The space used does not depend on the size of the input list nums.

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