# 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 0Output: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 10Output: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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