Maximum average sum partition of an array

Given an array, we partition a row of numbers A into at most K adjacent (non-empty) groups, then the score is the sum of the average of each group. What is the maximum score than can be scored ?

Examples:

Input : A = { 9, 1, 2, 3, 9 }
K = 3
Output : 20
Explanation : We can partition A into [9], [1, 2, 3], [9]. The answer is 9 + (1 + 2 + 3) / 3 + 9 = 20.
We could have also partitioned A into [9, 1], [2], [3, 9]. That partition would lead to a score of 5 + 2 + 6 = 13, which is worse.

Input : A[] = { 1, 2, 3, 4, 5, 6, 7 }
K = 4
Output : 20.5
Explanation : We can partition A into [1, 2, 3, 4], [5], [6], [7]. The answer is 2.5 + 5 + 6 + 7 = 20.5.



A simple solution is to use recursion. An efficient solution is memorization where we keep the largest score upto k i.e. for 1, 2, 3… upto k;

Let memo[i][k] be the best score portioning A[i..n-1] into at most K parts. In the first group, we partition A[i..n-1] into A[i..j-1] and A[j..n-1], then our candidate partition has score average(i, j) + score(j, k-1)), where average(i, j) = (A[i] + A[i+1] + … + A[j-1]) / (j – i). We take the highest score of these.

In total, our recursion in the general case is :
memo[n][k] = max(memo[n][k], score(memo, i, A, k-1) + average(i, j))
for all i from n-1 to 1 .

// CPP program for maximum average sum partition
#include <bits/stdc++.h>
using namespace std;
  
#define MAX 1000
  
double memo[MAX][MAX];
  
// bottom up approach to calculate score
double score(int n, vector<int>& A, int k)
{
    if (memo[n][k] > 0)
        return memo[n][k];
    double sum = 0;
    for (int i = n - 1; i > 0; i--) {
        sum += A[i];
        memo[n][k] = max(memo[n][k], score(i, A, k - 1) +
                                          sum / (n - i));
    }
    return memo[n][k];
}
  
double largestSumOfAverages(vector<int>& A, int K)
{
    int n = A.size();
    double sum = 0;
    memset(memo, 0.0, sizeof(memo));
    for (int i = 0; i < n; i++) {
        sum += A[i];
  
        // storing averages from starting to each i ; 
        memo[i + 1][1] = sum / (i + 1);
    }
    return score(n, A, K);
}
  
int main()
{
    vector<int> A = { 9, 1, 2, 3, 9 };
    int K = 3; // atmost partioning size
    cout << largestSumOfAverages(A, K) << endl;
    return 0;
}

Output:

20

Above problem can now be easily understood as dynamic programming.
Let dp(i, k) be the best score partioning A[i:j] into at most K parts. If the first group we partition A[i:j] into ends before j, then our candidate partition has score average(i, j) + dp(j, k-1)). Recursion in the general case is dp(i, k) = max(average(i, N), (average(i, j) + dp(j, k-1))). We can precompute the prefix sums for fast execution of out code.

// CPP program for maximum average sum partition
#include <bits/stdc++.h>
using namespace std;
  
double largestSumOfAverages(vector<int>& A, int K)
{
    int n = A.size();
  
    // storing prefix sums
    double pre_sum[n+1]; 
    pre_sum[0] = 0;
    for (int i = 0; i < n; i++)
        pre_sum[i + 1] = pre_sum[i] + A[i];
  
    // for each i to n storing averages 
    double dp[n] = {0};
    double sum = 0;
    for (int i = 0; i < n; i++) 
        dp[i] = (pre_sum[n] - pre_sum[i]) / (n - i);
      
    for (int k = 0; k < K - 1; k++) 
        for (int i = 0; i < n; i++) 
            for (int j = i + 1; j < n; j++) 
                dp[i] = max(dp[i], (pre_sum[j] -
                         pre_sum[i]) / (j - i) + dp[j]);
      
    return dp[0];
}
  
// Driver code
int main()
{
    vector<int> A = { 9, 1, 2, 3, 9 };
    int K = 3; // atmost partioning size
    cout << largestSumOfAverages(A, K) << endl;
    return 0;
}

Output:

20


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