Given an array arr[] of size N and an integer K, the task is to find the maximum subarray sum by removing at most K elements from the array.
Examples:
Input: arr[] = { -2, 1, 3, -2, 4, -7, 20 }, K = 1
Output: 26
Explanation:
Removing arr[5] from the array modifies arr[] to { -2, 1, 3, -2, 4, 20 }
Subarray with maximum sum is { 1, 3, -2, 4, 20 }.
Therefore, the required output is 26.Input:arr[] = { -1, 1, -1, -1, 1, 1 }, K=2
Output: 3
Explanation:
Removing arr[2] and arr[3] from the array modifies arr[] to { – 1, 1, 1, 1}
Subarray with maximum sum is { 1, 1, 1 }.
Therefore, the required output is 3.
Approach: The problem can be solved using Dynamic Programming. The idea is to use the concept of Kadane’s algorithm. Follow the steps below to solve the problem:
- Traverse the array arr[] and for every array element following two operations needs to be performed:
- Remove the current array element from the subarray.
- Include the current array element in the subarray.
- Therefore, the recurrence relation to solve this problem is as follows:
mxSubSum(i, j) = max(max(0, arr[i] + mxSubSum(i – 1, j)), mxSubSum(i – 1, j – 1))
i: Stores index of array element
j: Maximum count of elements that can be removed from the subarray
mxSubSum(i, j): Return maximum subarray sum from the subarray { arr[i], arr[N – 1] } by removing K – j array elements.
- Initialize a 2D array, say dp[][], to store the overlapping subproblems of the above recurrence relation.
- Fill the dp[][] array using memoization.
- Find the maximum element from the dp[][] array say, res.
- Initialize a variable, say Max, to store the largest element present in the array arr[].
- If all the array elements are negative, then update res = max(res, Max).
- Finally, print res as the required answer.
Below is the implementation of the above approach:
C++
// C++ program to implement// the above approach#include <bits/stdc++.h>using namespace std;#define M 100// Function to find the maximum subarray// sum greater than or equal to 0 by// removing K array elementsint mxSubSum(int i, int* arr, int j, int dp[][M]){ // Base case if (i == 0) { return dp[i][j] = max(0, arr[i]); } // If overlapping subproblems // already occurred if (dp[i][j] != -1) { return dp[i][j]; } // Include current element in the subarray int X = max(0, arr[i] + mxSubSum(i - 1, arr, j, dp)); // If K elements already removed // from the subarray if (j == 0) { return dp[i][j] = X; } // Remove current element from the subarray int Y = mxSubSum(i - 1, arr, j - 1, dp); return dp[i][j] = max(X, Y);}// Utility function to find the maximum subarray// sum by removing at most K array elementsint MaximumSubarraySum(int n, int* arr, int k){ // Stores overlapping subproblems // of the recurrence relation int dp[M][M]; // Initialize dp[][] to -1 memset(dp, -1, sizeof(dp)); mxSubSum(n - 1, arr, k, dp); // Stores maximum subarray sum by // removing at most K elements int res = 0; // Calculate maximum element // in dp[][] for (int i = 0; i < n; i++) { for (int j = 0; j <= k; j++) { // Update res res = max(res, dp[i][j]); } } // If all array elements are negative if (*max_element(arr, arr + n) < 0) { // Update res res = *max_element(arr, arr + n); } return res;}// Driver Codeint main(){ int arr[] = { -2, 1, 3, -2, 4, -7, 20 }; int K = 1; int N = sizeof(arr) / sizeof(arr[0]); cout << MaximumSubarraySum(N, arr, K) << endl; return 0;} |
Java
// Java program to implement// the above approachimport java.util.*;class GFG{static final int M = 100;// Function to find the maximum subarray// sum greater than or equal to 0 by// removing K array elementsstatic int mxSubSum(int i, int []arr, int j, int dp[][]){ // Base case if (i == 0) { return dp[i][j] = Math.max(0, arr[i]); } // If overlapping subproblems // already occurred if (dp[i][j] != -1) { return dp[i][j]; } // Include current element in the subarray int X = Math.max(0, arr[i] + mxSubSum(i - 1, arr, j, dp)); // If K elements already removed // from the subarray if (j == 0) { return dp[i][j] = X; } // Remove current element from the subarray int Y = mxSubSum(i - 1, arr, j - 1, dp); return dp[i][j] = Math.max(X, Y);}// Utility function to find the maximum subarray// sum by removing at most K array elementsstatic int MaximumSubarraySum(int n, int []arr, int k){ // Stores overlapping subproblems // of the recurrence relation int [][]dp = new int[M][M]; // Initialize dp[][] to -1 for (int i = 0; i < M; i++) for (int j = 0; j < M; j++) dp[i][j] = -1; mxSubSum(n - 1, arr, k, dp); // Stores maximum subarray sum by // removing at most K elements int res = 0; // Calculate maximum element // in dp[][] for (int i = 0; i < n; i++) { for (int j = 0; j <= k; j++) { // Update res res = Math.max(res, dp[i][j]); } } // If all array elements are negative if (Arrays.stream(arr).max().getAsInt() < 0) { // Update res res = Arrays.stream(arr).max().getAsInt(); } return res;}// Driver Codepublic static void main(String[] args){ int arr[] = { -2, 1, 3, -2, 4, -7, 20 }; int K = 1; int N = arr.length; System.out.print(MaximumSubarraySum(N, arr, K) +"\n");}}// This code is contributed by 29AjayKumar |
Python3
# Python3 program to implement# the above approachM = 100# Function to find the maximum subarray# sum greater than or equal to 0 by# removing K array elementsdef mxSubSum(i, arr, j): global dp # Base case if (i == 0): dp[i][j] = max(0, arr[i]) return dp[i][j] # If overlapping subproblems # already occurred if (dp[i][j] != -1): return dp[i][j] # Include current element in the subarray X = max(0, arr[i] + mxSubSum(i - 1, arr, j)) # If K elements already removed # from the subarray if (j == 0): dp[i][j] = X return X # Remove current element from the subarray Y = mxSubSum(i - 1, arr, j - 1) dp[i][j] = max(X, Y) return dp[i][j]# Utility function to find the maximum subarray# sum by removing at most K array elements# Utility function to find the maximum subarray# sum by removing at most K array elementsdef MaximumSubarraySum(n, arr, k): mxSubSum(n - 1, arr, k) # Stores maximum subarray sum by # removing at most K elements res = 0 # Calculate maximum element # in dp[][] for i in range(n): for j in range(k + 1): # Update res res = max(res, dp[i][j]) # If all array elements are negative if (max(arr) < 0): # Update res res = max(arr) return res# Driver Codeif __name__ == '__main__': dp = [[-1 for i in range(100)] for i in range(100)] arr = [-2, 1, 3, -2, 4, -7, 20] K = 1 N = len(arr) print(MaximumSubarraySum(N, arr, K))# This code is contributed by mohit kumar 29 |
C#
// C# program to implement// the above approachusing System;using System.Collections;class GFG{ static int M = 100;// Function to find the maximum subarray// sum greater than or equal to 0 by// removing K array elementsstatic int mxSubSum(int i, int []arr, int j, int [,]dp){ // Base case if (i == 0) { return dp[i, j] = Math.Max(0, arr[i]); } // If overlapping subproblems // already occurred if (dp[i, j] != -1) { return dp[i, j]; } // Include current element in the subarray int X = Math.Max(0, arr[i] + mxSubSum(i - 1, arr, j, dp)); // If K elements already removed // from the subarray if (j == 0) { return dp[i, j] = X; } // Remove current element from the subarray int Y = mxSubSum(i - 1, arr, j - 1, dp); return dp[i, j] = Math.Max(X, Y);}// Utility function to find the maximum subarray// sum by removing at most K array elementsstatic int MaximumSubarraySum(int n, int []arr, int k){ // Stores overlapping subproblems // of the recurrence relation int [,]dp = new int[M, M]; // Initialize dp[,] to -1 for(int i = 0; i < M; i++) for(int j = 0; j < M; j++) dp[i, j] = -1; mxSubSum(n - 1, arr, k, dp); // Stores maximum subarray sum by // removing at most K elements int res = 0; // Calculate maximum element // in dp[,] for(int i = 0; i < n; i++) { for(int j = 0; j <= k; j++) { // Update res res = Math.Max(res, dp[i, j]); } } Array.Sort(arr); // If all array elements are negative if (arr[n - 1] < 0) { // Update res res = arr[n - 1]; } return res;}// Driver Codepublic static void Main(String[] args){ int []arr = { -2, 1, 3, -2, 4, -7, 20 }; int K = 1; int N = arr.Length; Console.WriteLine(MaximumSubarraySum(N, arr, K));}}// This code is contributed by AnkThon |
Javascript
<script>// Javascript program to implement// the above approachvar M = 100;// Function to find the maximum subarray// sum greater than or equal to 0 by// removing K array elementsfunction mxSubSum(i, arr, j, dp){ // Base case if (i == 0) { dp[i][j] = Math.max(0, arr[i]); return dp[i][j]; } // If overlapping subproblems // already occurred if (dp[i][j] != -1) { return dp[i][j]; } // Include current element in the subarray var X = Math.max( 0, arr[i] + mxSubSum(i - 1, arr, j, dp)); // If K elements already removed // from the subarray if (j == 0) { dp[i][j] = X; return dp[i][j] } // Remove current element from the subarray var Y = mxSubSum(i - 1, arr, j - 1, dp); dp[i][j] = Math.max(X, Y); return dp[i][j]}// Utility function to find the maximum subarray// sum by removing at most K array elementsfunction MaximumSubarraySum(n, arr, k){ // Stores overlapping subproblems // of the recurrence relation var dp = Array.from(Array(M), () => Array(M).fill(-1)); mxSubSum(n - 1, arr, k, dp); // Stores maximum subarray sum by // removing at most K elements var res = 0; // Calculate maximum element // in dp[][] for(var i = 0; i < n; i++) { for(var j = 0; j <= k; j++) { // Update res res = Math.max(res, dp[i][j]); } } // If all array elements are negative if (arr.reduce((a, b) => Math.max(a, b)) < 0) { // Update res res = arr.reduce((a, b) => Math.max(a, b)); } return res;}// Driver Codevar arr = [ -2, 1, 3, -2, 4, -7, 20 ];var K = 1;var N = arr.length;document.write( MaximumSubarraySum(N, arr, K));// This code is contributed by rrrtnx</script> |
Output:
26
Time Complexity: O(N * K)
Auxiliary Space: O(N * K)
Efficient approach : Using DP Tabulation method ( Iterative approach )
The approach to solve this problem is same but DP tabulation(bottom-up) method is better then Dp + memorization(top-down) because memorization method needs extra stack space of recursion calls.
Steps to solve this problem :
- Create a vector to store the solution of the subproblems.
- Initialize the table with base cases
- Fill up the table iteratively
- Return the final solution
Implementation :
C++
// C++ program to implement// the above approach#include <bits/stdc++.h>using namespace std;#define M 100// Utility function to find the maximum subarray// sum by removing at most K array elementsint MaximumSubarraySum(int n, int* arr, int k){ // dp[i][j]: Stores maximum subarray sum // by removing at most j elements from // subarray ending at i-th index int dp[n][k + 1]; // Initialize dp[][] to 0 memset(dp, 0, sizeof(dp)); // Initialization for i = 0 dp[0][0] = max(0, arr[0]); // Calculate maximum subarray sum by // removing at most j elements for // subarrays ending at i-th index for (int i = 1; i < n; i++) { for (int j = 0; j <= k; j++) { // Include current element in the subarray int X = max(0, arr[i] + dp[i - 1][j]); // Remove current element from the subarray int Y = (j > 0) ? dp[i - 1][j - 1] : 0; dp[i][j] = max(X, Y); } } // Stores maximum subarray sum by // removing at most K elements int res = 0; // Calculate maximum element // in dp[][] for (int i = 0; i < n; i++) { for (int j = 0; j <= k; j++) { // Update res res = max(res, dp[i][j]); } } // If all array elements are negative if (*max_element(arr, arr + n) < 0) { // Update res res = *max_element(arr, arr + n); } return res;}// Driver Codeint main(){ int arr[] = { -2, 1, 3, -2, 4, -7, 20 }; int K = 1; int N = sizeof(arr) / sizeof(arr[0]); cout << MaximumSubarraySum(N, arr, K) << endl; return 0;}// this code is contributed by bhardwajji |
Java
import java.util.Arrays;class Main { // Utility function to find the maximum subarray // sum by removing at most K array elements static int maximumSubarraySum(int n, int[] arr, int k) { // dp[i][j]: Stores maximum subarray sum // by removing at most j elements from // subarray ending at i-th index int[][] dp = new int[n][k + 1]; // Initialize dp[][] to 0 for (int i = 0; i < n; i++) { Arrays.fill(dp[i], 0); } // Initialization for i = 0 dp[0][0] = Math.max(0, arr[0]); // Calculate maximum subarray sum by // removing at most j elements for // subarrays ending at i-th index for (int i = 1; i < n; i++) { for (int j = 0; j <= k; j++) { // Include current element in the subarray int X = Math.max(0, arr[i] + dp[i - 1][j]); // Remove current element from the subarray int Y = (j > 0) ? dp[i - 1][j - 1] : 0; dp[i][j] = Math.max(X, Y); } } // Stores maximum subarray sum by // removing at most K elements int res = 0; // Calculate maximum element // in dp[][] for (int i = 0; i < n; i++) { for (int j = 0; j <= k; j++) { // Update res res = Math.max(res, dp[i][j]); } } // If all array elements are negative if (Arrays.stream(arr).max().getAsInt() < 0) { // Update res res = Arrays.stream(arr).max().getAsInt(); } return res; } // Driver Code public static void main(String[] args) { int[] arr = { -2, 1, 3, -2, 4, -7, 20 }; int K = 1; int N = arr.length; System.out.println(maximumSubarraySum(N, arr, K)); }} |
Python
# Python program to implement# the above approach# Utility function to find the maximum subarray# sum by removing at most K array elementsdef MaximumSubarraySum(n, arr, k): # dp[i][j]: Stores maximum subarray sum # by removing at most j elements from # subarray ending at i-th index dp = [[0 for j in range(k+1)] for i in range(n)] # Initialization for i = 0 dp[0][0] = max(0, arr[0]) # Calculate maximum subarray sum by # removing at most j elements for # subarrays ending at i-th index for i in range(1, n): for j in range(k+1): # Include current element in the subarray X = max(0, arr[i] + dp[i - 1][j]) # Remove current element from the subarray Y = dp[i - 1][j - 1] if j > 0 else 0 dp[i][j] = max(X, Y) # Stores maximum subarray sum by # removing at most K elements res = 0 # Calculate maximum element in dp[][] for i in range(n): for j in range(k+1): # Update res res = max(res, dp[i][j]) # If all array elements are negative if max(arr) < 0: # Update res res = max(arr) return res# Driver Codeif __name__ == '__main__': arr = [-2, 1, 3, -2, 4, -7, 20] K = 1 N = len(arr) print(MaximumSubarraySum(N, arr, K)) |
C#
using System;using System.Linq;class MainClass { // Utility function to find the maximum subarray // sum by removing at most K array elements static int maximumSubarraySum(int n, int[] arr, int k) { // dp[i][j]: Stores maximum subarray sum // by removing at most j elements from // subarray ending at i-th index int[][] dp = new int[n][]; // Initialize dp[][] to 0 for (int i = 0; i < n; i++) { dp[i] = new int[k + 1]; for (int j = 0; j <= k; j++) { dp[i][j] = 0; } } // Initialization for i = 0 dp[0][0] = Math.Max(0, arr[0]); // Calculate maximum subarray sum by // removing at most j elements for // subarrays ending at i-th index for (int i = 1; i < n; i++) { for (int j = 0; j <= k; j++) { // Include current element in the subarray int X = Math.Max(0, arr[i] + dp[i - 1][j]); // Remove current element from the subarray int Y = (j > 0) ? dp[i - 1][j - 1] : 0; dp[i][j] = Math.Max(X, Y); } } // Stores maximum subarray sum by // removing at most K elements int res = 0; // Calculate maximum element // in dp[][] for (int i = 0; i < n; i++) { for (int j = 0; j <= k; j++) { // Update res res = Math.Max(res, dp[i][j]); } } // If all array elements are negative if (arr.Max() < 0) { // Update res res = arr.Max(); } return res; } // Driver Code public static void Main() { int[] arr = { -2, 1, 3, -2, 4, -7, 20 }; int K = 1; int N = arr.Length; Console.WriteLine(maximumSubarraySum(N, arr, K)); }} |
Javascript
// JavaScript program to implement// the above approach// Utility function to find the maximum subarray// sum by removing at most K array elementsfunction MaximumSubarraySum(n, arr, k) { // dp[i][j]: Stores maximum subarray sum // by removing at most j elements from // subarray ending at i-th index let dp = new Array(n); for (let i = 0; i < n; i++) { dp[i] = new Array(k + 1).fill(0); } // Initialization for i = 0 dp[0][0] = Math.max(0, arr[0]); // Calculate maximum subarray sum by // removing at most j elements for // subarrays ending at i-th index for (let i = 1; i < n; i++) { for (let j = 0; j <= k; j++) { // Include current element in the subarray let X = Math.max(0, arr[i] + dp[i - 1][j]); // Remove current element from the subarray let Y = (j > 0) ? dp[i - 1][j - 1] : 0; dp[i][j] = Math.max(X, Y); } } // Stores maximum subarray sum by // removing at most K elements let res = 0; // Calculate maximum element // in dp[][] for (let i = 0; i < n; i++) { for (let j = 0; j <= k; j++) { // Update res res = Math.max(res, dp[i][j]); } } // If all array elements are negative if (Math.max(...arr) < 0) { // Update res res = Math.max(...arr); } return res;}// Driver Codelet arr = [-2, 1, 3, -2, 4, -7, 20];let K = 1;let N = arr.length;console.log(MaximumSubarraySum(N, arr, K));// This code is contributed by Prajwal Kandekar |
Output:
26
Time Complexity: O(N * K)
Auxiliary Space: O(N * K)
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