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# Maximum sum in a 2 x n grid such that no two elements are adjacent

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
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``#define MAX 1000``using` `namespace` `std;` `// Function to find max sum without adjacent``int` `maxSum(``int` `grid[MAX], ``int` `n)``{``    ``// Sum including maximum element of first column``    ``int` `incl = max(grid, grid);` `    ``// Not including first column's element``    ``int` `excl = 0, excl_new;` `    ``// Traverse for further elements``    ``for` `(``int` `i = 1; i

## C

 `// C program to find maximum sum in a grid such that``// no two elements are adjacent.``#include ` `#define MAX 1000` `// Function to find max sum without adjacent``int` `maxSum(``int` `grid[MAX], ``int` `n)``{``  ``// Sum including maximum element of first column``  ``int` `max = grid;``  ``if``(max < grid)``    ``max = grid;``  ``int` `incl = max;` `  ``// Not including first column's element``  ``int` `excl = 0, excl_new;` `  ``// Traverse for further elements``  ``for` `(``int` `i = 1; i

## 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

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## Javascript

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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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