Given n arrays of different length consisting of integers, the target is to pick atmost one subarray from an array such that the combined length of all picked subarrays does not become greater than m and also sum of their elements is maximum.(also given that value of n can not be more than 100)

Prerequisite : Knapsack Problem

Examples :

Input :
n = 5, m = 6
arr[][m] = {{3, 2, 3, 5},
            {2, 7, -1},
            {2, 8, 10},
            {4, 5, 2, 6, 1},
            {3, 2, 3, -2}};
Output : Maximum sum can be obtained is 39
Explanation : We are allowed to pick at most
one subarray from every array. 
We get total sum 39 as ((5) + (7) + (8 + 10) + 
(4 + 5))

Input :
n = 3, m = 4
arr[][m] = {{2, 3, 2},  
            {3, -1, 7, 10},
            {4, 8, 10, -5, 3}};
Output : Maximum sum can be obtained is 35

This problem is similar to Knapsack problem.where you have to either pick an element or leave it. We will have the same strategy here.
Given that the total number of elements in these n arrays is atmost 10^5. It is also known that m is atmost 10^3 and the input arrays can contain negative numbers. First, make a DP table (2D array) of size n * m and then, pre-compute the cumulative sum of an array so that maximum sum of every length from 1 to n of that array can be easily computed, so that for every given array there can be a maximum contiguous sum for length k, where k is from 1 to length of the array.
In detail, process input arrays one by one. First, compute the maximum sum subarrays of processed array for all sizes from 1 to length. Then, update our dynamic programming table with these values and we start processing the next array.

Algorithm :

1. Pick one array from n arrays and start processing it.
2. Calculate maximum contiguous sum for length k, k is from 1 to length of the array and save it in the array maxSum.
3. Now, fill the DP table by storing maximum sum possible for every length 0 to m.
4. In the last step we traverse last row(nth row) of DP table and pick maximum sum possible and return it.

   Below is the implementation of above approach :
   

   C++

   
   

   
   
   

   // A Dynamic Programming based C++ program to find  // maximum sum of array of size less than or  // equal to m from given n arrays  #include <bits/stdc++.h>  using namespace std;     /* N and M to define sizes of arr,  dp, current_arr and maxSum */ #define N 105  #define M 1001     // INF to define min value  #define INF -1111111111     // Function to find maximum sum  int maxSum(int arr[][N])  {      // dp array of size N x M      int dp[N][M];         // current_arr of size M      int current_arr[M];         // maxsum of size M      int maxsum[M];         memset(dp, -1, sizeof(dp[0][0]) * N * M);         current_arr[0] = 0;         // if we have 0 elements      // from 0th array      dp[0][0] = 0;         for (int i = 1; i <= 5; i++) {          int len = arr[i - 1][0];             // compute the cumulative sum array          for (int j = 1; j <= len; j++) {              current_arr[j] = arr[i - 1][j];              current_arr[j] += current_arr[j - 1];              maxsum[j] = INF;          }             // calculating the maximum contiguous          // array for every length j, j is from          // 1 to lengtn of the array          for (int j = 1; j <= len && j <= 6; j++)              for (int k = 1; k <= len; k++)                  if (j + k - 1 <= len)                      maxsum[j] = max(maxsum[j],                                  current_arr[j + k - 1] -                                  current_arr[k - 1]);             // every state is depending on          // its previous state          for (int j = 0; j <= 6; j++)              dp[i][j] = dp[i - 1][j];             // computation of dp table similar          // approach as knapsack problem          for (int j = 1; j <= 6; j++)              for (int cur = 1; cur <= j && cur <= len; cur++)                  dp[i][j] = max(dp[i][j],                              dp[i - 1][j - cur] +                                          maxsum[cur]);      }         // now we have done processing with      // the last array lets find out      // what is the maximum sum possible      int ans = 0;      for (int i = 0; i <= 6; i++)          ans = max(ans, dp[5][i]);             return ans;  }     // Driver program  int main()  {      // first element of each      // row is the size of that row      int arr[][N] = { { 3, 2, 3, 5 },                      { 2, 7, -1 },                      { 2, 8, 10 },                      { 4, 5, 2, 6, 1 },                      { 3, 2, 3, -2 } };         cout << "Maximum sum can be obtained "         << "is : " << maxSum(arr) << "\n";  }

   Java

   
   

   
   
   

   // Java program to find maximum sum  // of array of size less than or  // equal to m from given n arrays  import java.io.*;     public class GFG  {      /* N and M to define      sizes of arr,      dp, current_arr      and maxSum */     static int N = 105;      static int M = 1001;             // INF to define      // min value      static int INF = -1111111111;             // Function to find      // maximum sum      static int maxSum(int [][]arr)      {          // dp array of size N x M          int [][]dp = new int[N][M];                 // current_arr of size M          int []current_arr = new int[M];                 // maxsum of size M          int []maxsum = new int[M];                     for (int i = 0; i < N; i++)          {              for (int j = 0; j < M ; j++)                  dp[i][j] = -1;          }                 current_arr[0] = 0;                 // if we have 0 elements          // from 0th array          dp[0][0] = 0;                 for (int i = 1; i <= 5; i++)          {              int len = arr[i - 1][0];                     // compute the cumulative              // sum array              for (int j = 1; j <= len; j++)              {                  current_arr[j] = arr[i - 1][j];                  current_arr[j] += current_arr[j - 1];                  maxsum[j] = INF;              }                     // calculating the maximum              // contiguous array for every              // length j, j is from 1 to              // lengtn of the array              for (int j = 1; j <= len && j <= 6; j++)                  for (int k = 1; k <= len; k++)                      if (j + k - 1 <= len)                          maxsum[j] = Math.max(maxsum[j],                                      current_arr[j + k - 1] -                                      current_arr[k - 1]);                     // every state is depending              // on its previous state              for (int j = 0; j <= 6; j++)                  dp[i][j] = dp[i - 1][j];                     // computation of dp table              // similar approach as              // knapsack problem              for (int j = 1; j <= 6; j++)                  for (int cur = 1; cur <= j &&                                  cur <= len; cur++)                      dp[i][j] = Math.max(dp[i][j],                                          dp[i - 1][j - cur] +                                              maxsum[cur]);          }                 // now we have done processing          // with the last array lets          // find out what is the maximum          // sum possible          int ans = 0;          for (int i = 0; i <= 6; i++)              ans = Math.max(ans, dp[5][i]);                 return ans;      }             // Driver Code      public static void main(String args[])      {          // first element of each          // row is the size of that row          int[][] arr = {                  { 3, 2, 3, 5 },                  { 2, 7, -1 },                  { 2, 8, 10 },                  { 4, 5, 2, 6, 1 },                  { 3, 2, 3, -2 }          };          System.out.println("Maximum sum can be " +                              "obtained is : " +                      maxSum(arr));      }  }     // This code is contributed by  // Manish Shaw(manishshaw1)

   Python3

   
   

   
   
   

   # A Dynamic Programming based Python3  # program to find maximum sum of array # of size less than or equal to m from # given n arrays    # N and M to define sizes of arr, # dp, current_arr and maxSum  N = 105 M = 1001    # INF to define min value INF = -1111111111    # Function to find maximum sum def maxSum(arr):        # dp array of size N x M     dp = [[-1 for x in range(M)]               for y in range(N)]        # current_arr of size M     current_arr = [0] * M        # maxsum of size M     maxsum = [0] * M        current_arr[0] = 0        # If we have 0 elements     # from 0th array     dp[0][0] = 0        for i in range(1, 6):         len = arr[i - 1][0]            # Compute the cumulative sum array         for j in range(1, len + 1):             current_arr[j] = arr[i - 1][j]             current_arr[j] += current_arr[j - 1]             maxsum[j] = INF            # Calculating the maximum contiguous         # array for every length j, j is from         # 1 to lengtn of the array         j = 1         while j <= len and j <= 6:             for k in range(1, len + 1):                 if (j + k - 1 <= len):                     maxsum[j] = max(maxsum[j],                                current_arr[j + k - 1] -                                current_arr[k - 1])                                               j += 1            # Every state is depending on         # its previous state         for j in range(7):             dp[i][j] = dp[i - 1][j]            # computation of dp table similar         # approach as knapsack problem         for j in range(1, 7):             cur = 1             while cur <= j and cur <= len:                 dp[i][j] = max(dp[i][j],                                dp[i - 1][j - cur] +                                       maxsum[cur])                                                          cur += 1        # Now we have done processing with     # the last array lets find out     # what is the maximum sum possible     ans = 0     for i in range(7):         ans = max(ans, dp[5][i])         return ans    # Driver code if __name__ == "__main__":        # First element of each     # row is the size of that row     arr = [ [ 3, 2, 3, 5 ],             [ 2, 7, -1 ],             [ 2, 8, 10 ],             [ 4, 5, 2, 6, 1 ],             [ 3, 2, 3, -2 ] ]        print("Maximum sum can be obtained",            "is : ", maxSum(arr))    # This code is contributed by chitranayal

   C#

   
   

   
   
   

   // C# program to find maximum sum  // of array of size less than or  // equal to m from given n arrays  using System;     class GFG  {      /* N and M to define      sizes of arr,      dp, current_arr      and maxSum */     static int N = 105;      static int M = 1001;             // INF to define      // min value      static int INF = -1111111111;             // Function to find      // maximum sum      static int maxSum(int [][]arr)      {          // dp array of size N x M          int [,]dp = new int[N, M];                 // current_arr of size M          int []current_arr = new int[M];                 // maxsum of size M          int []maxsum = new int[M];                     for (int i = 0; i < N; i++)          {              for (int j = 0; j < M ; j++)                  dp[i, j] = -1;          }                 current_arr[0] = 0;                 // if we have 0 elements          // from 0th array          dp[0, 0] = 0;                 for (int i = 1; i <= 5; i++)          {              int len = arr[i - 1][0];                     // compute the cumulative              // sum array              for (int j = 1; j <= len; j++)              {                  current_arr[j] = arr[i - 1][j];                  current_arr[j] += current_arr[j - 1];                  maxsum[j] = INF;              }                     // calculating the maximum              // contiguous array for every              // length j, j is from 1 to              // lengtn of the array              for (int j = 1; j <= len && j <= 6; j++)                  for (int k = 1; k <= len; k++)                      if (j + k - 1 <= len)                          maxsum[j] = Math.Max(maxsum[j],                                      current_arr[j + k - 1] -                                      current_arr[k - 1]);                     // every state is depending              // on its previous state              for (int j = 0; j <= 6; j++)                  dp[i, j] = dp[i - 1, j];                     // computation of dp table              // similar approach as              // knapsack problem              for (int j = 1; j <= 6; j++)                  for (int cur = 1; cur <= j &&                                  cur <= len; cur++)                      dp[i, j] = Math.Max(dp[i, j],                                          dp[i - 1, j - cur] +                                              maxsum[cur]);          }                 // now we have done processing          // with the last array lets          // find out what is the maximum          // sum possible          int ans = 0;          for (int i = 0; i <= 6; i++)              ans = Math.Max(ans, dp[5, i]);                 return ans;      }             // Driver Code      static void Main()      {          // first element of each          // row is the size of that row          int[][] arr = new int[][]          {              new int[]{ 3, 2, 3, 5 },              new int[]{ 2, 7, -1 },              new int[]{ 2, 8, 10 },              new int[]{ 4, 5, 2, 6, 1 },              new int[]{ 3, 2, 3, -2 }          };          Console.Write ("Maximum sum can be " +                              "obtained is : " +                          maxSum(arr) + "\n");      }  }     // This code is contributed by  // Manish Shaw(manishshaw1)

   Output :

   Maximum sum can be obtained is : 39
   

   Time complexity : O(n)

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