Largest Sum of Averages

Largest Sum of Averages

1. What is this problem about?

The Largest Sum of Averages coding problem asks you to partition an array into at most k non-empty adjacent subarrays. The goal is to maximize the sum of the averages of these subarrays. Each element must belong to exactly one subarray, and the order of elements must be preserved.

2. Why is this asked in interviews?

This "Medium" difficulty problem is a favorite at Amazon and Google because it tests Dynamic Programming skills and the ability to optimize range queries using prefix sums. It requires you to make a sequence of decisions (where to split the array) where each decision affects the remaining possibilities, a hallmark of DP. It also evaluates your ability to handle floating-point arithmetic and maximize a specific objective function.

3. Algorithmic pattern used

The problem utilizes the Dynamic Programming and Prefix Sum interview pattern. Let dp[i][k] be the maximum sum of averages for the first i elements partitioned into k groups. To calculate dp[i][k], we iterate through all possible split points j < i, calculating the average of the subarray from j to i and adding it to dp[j][k-1]. Prefix sums are used to calculate the sum of any subarray in O(1) time.

4. Example explanation

Array: [9, 1, 2, 3], k = 3. Possible partition: [9], [1], [2, 3]

• Average 1: 9/1 = 9
• Average 2: 1/1 = 1
• Average 3: (2+3)/2 = 2.5
• Total sum: 9 + 1 + 2.5 = 12.5. Another partition: [9, 1], [2], [3]
• Total sum: (10/2) + 2 + 3 = 10. The DP will explore all such valid partitions to find the maximum sum.

5. Common mistakes candidates make

One common mistake is using a greedy approach (e.g., trying to put large numbers in their own subarrays), which doesn't work for this problem. Another is forgetting that we can use at most k subarrays, though the optimal solution usually uses exactly k (unless the average would decrease). Failing to use prefix sums leads to an O(N³) or O(N⁴) solution, which may be too slow.

6. Interview preparation tip

When you see "partition an array into k groups" and "maximize some sum," think Dynamic Programming. Practice identifying the "state" of your DP—usually the number of elements processed and the number of partitions used.