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

This problem is an extension of Maximum sum such that no two elements are adjacent. Only thing to be changed is to take maximum element of both row of a particular column. We traverse column by column and maintain maximum sum considering two cases.

1) An element of current column is included. In this case we take maximum of two elements in current column.

2) An element of current column is excluded (or not included)

Below is the implementation of 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; ` `} ` |

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## 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. ` |

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## 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 ` |

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## 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 ` |

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## 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.. ` `?> ` |

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**Output:**

24

**Time Complexity:** O(n)

**Space Complexity:** O(2n) which is equal to O(n)

This article is contributed by **Sahil Chhabra**. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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