Mastering the Maximum Average Subarray I Problem in Python

The Maximum Average Subarray I problem from LeetCode 75 is a common interview question that tests your understanding of sliding window techniques, array traversal, and optimization. The task is straightforward: given an integer array and a window size k, find the contiguous subarray of length k that has the maximum average.

In this blog post, we’ll break down the Maximum Average Subarray I problem, walk through a clean Python solution and explain why it’s optimal for interviews and real-world applications.

Problem Statement

Given an integer array nums and an integer k, find the contiguous subarray of length k that has the maximum average value and return that maximum average.

Understanding the Problem

A subarray of length k is a consecutive sequence of k elements from the array. Instead of calculating the sum of every possible subarray (which is inefficient), we can use the sliding window technique to optimize the calculation.

Examples

Example 1:

Input: nums = [1,12,-5,-6,50,3], k = 4  
Output: 12.75  
Explanation: Subarray [12,-5,-6,50] has the maximum average = (12-5-6+50)/4 = 12.75

Example 2:

Input: nums = [5], k = 1  
Output: 5.0  
Explanation: Only one element forms the subarray.

Python Solution – Step-by-Step Explanation

Here’s a clean Python solution using the sliding window technique:

class Solution:
    def findMaxAverage(self, nums: List[int], k: int) -> float:
        n = len(nums)
        # Calculate the sum of the first window of size k
        window_sum = sum(nums[:k])
        max_average = window_sum / k

        # Iterate through the array, moving the sliding window
        for i in range(k, n):
            window_sum = window_sum - nums[i - k] + nums[i]
            # Update the maximum average if the current window has a higher average
            max_average = max(max_average, window_sum / k)

        return max_average

Step 1: Calculate the Initial Window Sum

window_sum = sum(nums[:k])
max_average = window_sum / k

Compute the sum of the first k elements.
Initialize max_average with this value.

Step 2: Slide the Window Across the Array

for i in range(k, n):
    window_sum = window_sum - nums[i - k] + nums[i]

Remove the element that leaves the window (nums[i - k]).
Add the element that enters the window (nums[i]).
This avoids recalculating the sum of the entire subarray each time.

Step 3: Update Maximum Average

max_average = max(max_average, window_sum / k)

After updating the window sum, calculate the average.
Keep track of the highest average found so far.

Step 4: Return the Result

return max_average

After sliding the window through the array, max_average contains the result.

Why This Solution Works

Sliding window ensures we compute each subarray sum in O(1) after the first window.
Avoids nested loops, resulting in linear time complexity.
Efficient and scalable for large arrays.

Time and Space Complexity

Metric	Value
Time Complexity	O(n) – iterate through array once
Space Complexity	O(1) – only variables for sum and max average

Edge Cases to Consider

Case	Output	Explanation
nums = [1,1,1,1], k = 2	1.0	All subarrays have same average
nums = [5], k = 1	5.0	Single element array
nums = [-1,-2,-3], k = 2	-1.5	Handles negative numbers correctly
nums = [], k = 0	Error	Invalid input, array cannot be empty

Real-World Applications

Stock analysis: Find the period with maximum average stock price.
Signal processing: Compute maximum average values over a moving window.
Performance analytics: Determine peak average performance over a fixed interval.

Conclusion

The Maximum Average Subarray I problem is a fundamental example of how the sliding window technique optimizes array traversal. By maintaining a running sum and updating it efficiently, we achieve linear time complexity without extra space.

Mastering this problem improves your understanding of window-based algorithms which are widely applicable in data analysis, stream processing and coding interviews.

External Links

GeeksforGeeks – Maximum Average Subarray in Python

Programiz – Python List Slicing & Sum

HackerRank – Subarray Problems Practice

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Mastering the Maximum Average Subarray I Problem in Python – LeetCode 75 Explained

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