Largest Sum of Averages

Problem Description

You are given an integer array nums and an integer k. You can partition the array into at most k non-empty adjacent subarrays. The score of a partition is the sum of the averages of each subarray. Note that the partition must use every integer in nums, and that the score is not necessarily an integer. Return the maximum score you can achieve of all the possible partitions. Answers within 10^{-6} of the actual answer will be accepted.

Key Insights

• The problem involves partitioning the array into adjacent subarrays while maximizing the sum of their averages.
• Dynamic programming can be used to explore all possible partitions and calculate their scores efficiently.
• Prefix sums can help in calculating the average of subarrays quickly without repeated summation.

Space and Time Complexity

Time Complexity: O(n^2 * k)
Space Complexity: O(n * k)

Solution

The solution employs dynamic programming to keep track of the maximum score obtainable with a given number of partitions. The core idea is to use a 2D DP array where dp[i][j] represents the maximum score achievable with the first i elements of the array partitioned into j groups. We also use a prefix sum array to efficiently compute the sum of any subarray, which allows us to calculate averages quickly.

1. Initialize a prefix sum array to store cumulative sums of nums.
2. Use a nested loop where the outer loop iterates through the number of elements and the inner loop iterates through the number of partitions.
3. For each element and partition count, calculate the possible partitions and update the DP table.
4. The final result will be found in the DP table at the position representing all elements and k partitions.

Code Solutions