Given a rectangular grid of dimension 2 x n. We need to find out the maximum sum such that no two chosen numbers are adjacent, vertically, diagonally, or horizontally.
Examples:
Input: 1 4 5
2 0 0
Output: 7
If we start from 1 then we can add only 5 or 0.
So max_sum = 6 in this case.
If we select 2 then also we can add only 5 or 0.
So max_sum = 7 in this case.
If we select from 4 or 0 then there is no further
elements can be added.
So, Max sum is 7.
Input: 1 2 3 4 5
6 7 8 9 10
Output: 24
Approach:
This problem is an extension of Maximum sum such that no two elements are adjacent. The only thing to be changed is to take a maximum element of both rows of a particular column. We traverse column by column and maintain the maximum sum considering two cases.
1) An element of the current column is included. In this case, we take a maximum of two elements in the current column.
2) An element of the current column is excluded (or not included)
Below is the implementation of the above steps.
C++
// C++ program to find maximum sum in a grid such that // no two elements are adjacent. #include<bits/stdc++.h> #define MAX 1000 using namespace std; // Function to find max sum without adjacent int maxSum(int grid[2][MAX], int n) { // Sum including maximum element of first column int incl = max(grid[0][0], grid[1][0]); // Not including first column's element int excl = 0, excl_new; // Traverse for further elements for (int i = 1; i<n; i++ ) { // Update max_sum on including or excluding // of previous column excl_new = max(excl, incl); // Include current column. Add maximum element // from both row of current column incl = excl + max(grid[0][i], grid[1][i]); // If current column doesn't to be included excl = excl_new; } // Return maximum of excl and incl // As that will be the maximum sum return max(excl, incl); } // Driver code int main() { int grid[2][MAX] = {{ 1, 2, 3, 4, 5}, { 6, 7, 8, 9, 10}}; int n = 5; cout << maxSum(grid, n); return 0; } |
C
// C program to find maximum sum in a grid such that // no two elements are adjacent. #include <stdio.h> #define MAX 1000 // Function to find max sum without adjacent int maxSum(int grid[2][MAX], int n) { // Sum including maximum element of first column int max = grid[0][0]; if(max < grid[1][0]) max = grid[1][0]; int incl = max; // Not including first column's element int excl = 0, excl_new; // Traverse for further elements for (int i = 1; i<n; i++ ) { // Update max_sum on including or excluding // of previous column max = excl; if(max < incl) max = incl; excl_new = max; // Include current column. Add maximum element // from both row of current column max = grid[0][i]; if(max < grid[1][i]) max = grid[1][i]; incl = excl + max; // If current column doesn't to be included excl = excl_new; } // Return maximum of excl and incl // As that will be the maximum sum max = excl; if(max < incl) max = incl; return max; } // Driver code int main() { int grid[2][MAX] = {{ 1, 2, 3, 4, 5}, { 6, 7, 8, 9, 10}}; int n = 5; printf("%d",maxSum(grid, n)); return 0; } // This code is contributed by kothavvsaakash. |
Java
// Java Code for Maximum sum in a 2 x n grid // such that no two elements are adjacent import java.util.*; class GFG { // Function to find max sum without adjacent public static int maxSum(int grid[][], int n) { // Sum including maximum element of first // column int incl = Math.max(grid[0][0], grid[1][0]); // Not including first column's element int excl = 0, excl_new; // Traverse for further elements for (int i = 1; i < n; i++ ) { // Update max_sum on including or // excluding of previous column excl_new = Math.max(excl, incl); // Include current column. Add maximum element // from both row of current column incl = excl + Math.max(grid[0][i], grid[1][i]); // If current column doesn't to be included excl = excl_new; } // Return maximum of excl and incl // As that will be the maximum sum return Math.max(excl, incl); } /* Driver program to test above function */ public static void main(String[] args) { int grid[][] = {{ 1, 2, 3, 4, 5}, { 6, 7, 8, 9, 10}}; int n = 5; System.out.println(maxSum(grid, n)); } } // This code is contributed by Arnav Kr. Mandal. |
Python3
# Python3 program to find maximum sum in a grid such that # no two elements are adjacent. # Function to find max sum without adjacent def maxSum(grid, n) : # Sum including maximum element of first column incl = max(grid[0][0], grid[1][0]) # Not including first column's element excl = 0 # Traverse for further elements for i in range(1, n) : # Update max_sum on including or excluding # of previous column excl_new = max(excl, incl) # Include current column. Add maximum element # from both row of current column incl = excl + max(grid[0][i], grid[1][i]) # If current column doesn't to be included excl = excl_new # Return maximum of excl and incl # As that will be the maximum sum return max(excl, incl) # Driver code if __name__ == "__main__" : grid = [ [ 1, 2, 3, 4, 5], [ 6, 7, 8, 9, 10] ] n = 5 print(maxSum(grid, n)) // This code is contributed by Ryuga |
C#
// C# program Code for Maximum sum // in a 2 x n grid such that no two // elements are adjacent using System; class GFG { // Function to find max sum // without adjacent public static int maxSum(int [,]grid, int n) { // Sum including maximum element // of first column int incl = Math.Max(grid[0, 0], grid[1, 0]); // Not including first column's // element int excl = 0, excl_new; // Traverse for further elements for (int i = 1; i < n; i++ ) { // Update max_sum on including or // excluding of previous column excl_new = Math.Max(excl, incl); // Include current column. Add // maximum element from both // row of current column incl = excl + Math.Max(grid[0, i], grid[1, i]); // If current column doesn't // to be included excl = excl_new; } // Return maximum of excl and incl // As that will be the maximum sum return Math.Max(excl, incl); } // Driver Code public static void Main(String[] args) { int [,]grid = {{ 1, 2, 3, 4, 5}, { 6, 7, 8, 9, 10}}; int n = 5; Console.Write(maxSum(grid, n)); } } // This code is contributed // by PrinciRaj1992 |
PHP
<?php // PHP program to find maximum sum // in a grid such that no two elements // are adjacent. // Function to find max sum // without adjacent function maxSum($grid, $n) { // Sum including maximum element // of first column $incl = max($grid[0][0], $grid[1][0]); // Not including first column's element $excl = 0; $excl_new; // Traverse for further elements for ($i = 1; $i < $n; $i++ ) { // Update max_sum on including or // excluding of previous column $excl_new = max($excl, $incl); // Include current column. Add maximum // element from both row of current column $incl = $excl + max($grid[0][$i], $grid[1][$i]); // If current column doesn't // to be included $excl = $excl_new; } // Return maximum of excl and incl // As that will be the maximum sum return max($excl, $incl); } // Driver code $grid = array(array(1, 2, 3, 4, 5), array(6, 7, 8, 9, 10)); $n = 5; echo maxSum($grid, $n); // This code is contributed by Sachin.. ?> |
Javascript
<script> // JavaScript program Code for Maximum sum // in a 2 x n grid such that no two // elements are adjacent // Function to find max sum // without adjacent function maxSum(grid,n) { // Sum including maximum element // of first column let incl = Math.max(grid[0][0], grid[1][0]); // Not including first column's // element let excl = 0, excl_new; // Traverse for further elements for (let i = 1; i < n; i++ ) { // Update max_sum on including or // excluding of previous column excl_new = Math.max(excl, incl); // Include current column. Add // maximum element from both // row of current column incl = excl + Math.max(grid[0][i], grid[1][i]); // If current column doesn't // to be included excl = excl_new; } // Return maximum of excl and incl // As that will be the maximum sum return Math.max(excl, incl); } // Driver Code let grid =[[ 1, 2, 3, 4, 5], [6, 7, 8, 9, 10]]; let n = 5; document.write(maxSum(grid, n)); // This code is contributed // by PrinciRaj1992 </script> |
Output:
24
Time Complexity: O(n) where n is number of elements in given array. As, we are using a loop to traverse N times so it will cost us O(N) time
Auxiliary Space: O(1), as we are not using any extra space.
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