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This approach utilizes dynamic programming combined with prefix sums to efficiently calculate the sum of subarray averages when partitioned optimally. The prefix sum array allows for quick computation of subarray sums, and dynamic programming is used to build the optimal solution up to k partitions.
Time Complexity: O(n^2 * k), where n is the length of the array, and k is the number of partitions.
Space Complexity: O(n * k) due to the dp table.
1using System;
2
3public class Solution {
4 public double LargestSumOfAverages(int[] nums, int k) {
5 int n = nums.Length;
6 double[] prefixSum = new double[n + 1];
7 double[,] dp = new double[n, k];
8
9 for (int i = 0; i < n; ++i) {
10 prefixSum[i + 1] = prefixSum[i] + nums[i];
11 }
12
13 for (int i = 0; i < n; ++i) {
14 dp[i, 0] = (prefixSum[n] - prefixSum[i]) / (n - i);
15 }
16
17 for (int m = 1; m < k; ++m) {
18 for (int i = 0; i < n; ++i) {
19 for (int j = i + 1; j < n; ++j) {
20 dp[i, m] = Math.Max(dp[i, m], (prefixSum[j] - prefixSum[i]) / (j - i) + dp[j, m - 1]);
21 }
22 }
23 }
24 return dp[0, k - 1];
25 }
26
27 public static void Main() {
28 Solution s = new Solution();
29 int[] nums = {9, 1, 2, 3, 9};
30 int k = 3;
31 Console.WriteLine(s.LargestSumOfAverages(nums, k).ToString("F5"));
32 }
33}C# uses similar structures as C++ and Java implementations. The code carefully checks multiple partitioning strategies to find the one that provides the maximum sum of averages.
This approach uses recursion with memoization to solve for the maximum sum of averages by exploring all possible partition strategies and storing computed results for reuse. This avoids redundant computations and optimizes performance compared to plain recursion.
Time Complexity: O(n^2 * k) as each subproblem is solved once.
Space Complexity: O(n * k) for memoization table.
1
In Python, this implementation employs recursive function calls with memoization. It stores prefix sums for quick access while computing potential maximum average scores through recursive partition exploration.