What is the Maximum Average Subarray Problem?

Finding the maximum average subarray is a common problem encountered in technical interviews. This problem requires identifying a contiguous subarray within a given array of numbers that has the highest possible average. The challenge is not only to find this subarray but also to do so efficiently.

Problem Definition

Given an array of integers and a number k, the task is to find the contiguous subarray of length k that has the maximum average. The average is calculated by summing the elements of the subarray and dividing by k.

For example, consider the array:

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If k = 4, the contiguous subarrays of length 4 are:

1. [1, 12, -5, -6] → average = (1 + 12 - 5 - 6) / 4 = 0.5
2. [12, -5, -6, 50] → average = (12 - 5 - 6 + 50) / 4 = 12.75
3. [-5, -6, 50, 3] → average = (-5 - 6 + 50 + 3) / 4 = 10.5

The maximum average in this case is 12.75.

Approach

One efficient way to solve this problem is by using the sliding window technique. This method avoids the need to calculate the sum from scratch for every subarray by maintaining a running sum. Here’s a general outline for the solution:

1. First, calculate the sum of the first k elements. This is your initial sum.
2. Then, slide the window across the array. For each step, subtract the element that is leaving the window and add the new element that is entering the window.
3. Track the maximum average seen so far.

Example Implementation

Here’s how you might implement this in Python:

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Example Interview Questions

Below are a few sample interview questions related to the maximum average subarray problem:

Question 1

Given an array and an integer k, write a function to find the maximum average of a subarray of length k.

Answer: Use the sliding window technique as explained. Utilize a running sum while iterating through the array. Ensure you handle edge cases like when k is greater than the array size.

Question 2

How would you modify your solution to handle floating-point numbers in the array?

Answer: The approach remains largely unchanged, as the sum and average calculations in Python handle floating-point arithmetic automatically. For language that does not handle floating points well, be cautious about precision issues.

Question 3

What is the time complexity of your solution?

Answer: The time complexity is O(n), where n is the length of the array. This is because we iterate through the array once to calculate the maximum sum for the subarrays of length k.

Question 4

Can you solve this problem without using extra space?

Answer: Yes, the sliding window approach uses a fixed amount of extra space (which is O(1)). The sums are calculated in place and no additional data structures are required.

Question 5

How would the approach change if you need to find the maximum average subarray of a variable length k?

Answer: The algorithm would need to be modified to check subarrays of all possible lengths. A nested loop can be used to iterate through all possible lengths, recalculating sums for each, leading to a higher time complexity of O(n^2).