Given n arrays of different lengths consisting of integers, the target is to pick atmost one subarray from an array such that the combined length of all picked sub arrays 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 at most 10^5. It is also known that m is at most 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 the 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 continuous 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 the 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 :
- Pick one array from n arrays and start processing it.
- Calculate the maximum contiguous sum for length k, k is from 1 to the length of the array, and save it in the array maxSum.
- Now, fill the DP table by storing the maximum sum possible for every length 0 to m.
- In the last step, we traverse the last row(nth row) of DP table and pick the maximum sum possible and return it.
Below is the implementation of the 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 = 105M = 1001# INF to define min valueINF = -1111111111# Function to find maximum sumdef 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 codeif __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) |
Javascript
// Javascript 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 */ let N = 105; let M = 1001; // INF to define // min value let INF = -1111111111; // Function to find // maximum sum function maxSum(arr) { // dp array of size N x M let dp = new Array(N); for(let i = 0;i<N;i++){ dp[i] = new Array(M); } // current_arr of size M let current_arr = new Array(M); // maxsum of size M let maxsum = new Array(M); for (let i = 0; i < N; i++) { for (let 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 (let i = 1; i <= 5; i++) { let len = arr[i - 1][0]; // compute the cumulative // sum array for (let 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 (let j = 1; j <= len && j <= 6; j++) for (let 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 (let j = 0; j <= 6; j++) dp[i][j] = dp[i - 1][j]; // computation of dp table // similar approach as // knapsack problem for (let j = 1; j <= 6; j++) for (let 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 let ans = 0; for (let i = 0; i <= 6; i++) ans = Math.max(ans, dp[5][i]); return ans; } // first element of each // row is the size of that row let arr = [ [ 3, 2, 3, 5 ], [ 2, 7, -1 ], [ 2, 8, 10 ], [ 4, 5, 2, 6, 1 ], [ 3, 2, 3, -2 ] ]; console.log("Maximum sum can be " + "obtained is : " + maxSum(arr)); // This code is contributed by aadityaburujwale. |
Output
Maximum sum can be obtained is : 39
Time complexity : O(
n)
Auxiliary Space: O(n*m)
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