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

28 May, 2022
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
Java
Python3
C#
PHP
Javascript
// C++ program to find maximum sum in a grid such that// no two elements are adjacent.#include<bits/stdc++.h>#define MAX 1000using namespace std; // Function to find max sum without adjacentint 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 codeint 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 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 adjacentint 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 codeint 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 Code for Maximum sum in a 2 x n grid// such that no two elements are adjacentimport 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 program to find maximum sum in a grid such that# no two elements are adjacent. # Function to find max sum without adjacentdef 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 codeif __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# program Code for Maximum sum// in a 2 x n grid such that no two// elements are adjacentusing System;    class GFG{ // Function to find max sum// without adjacentpublic 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 Codepublic 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 program to find maximum sum// in a grid such that no two elements// are adjacent. // Function to find max sum// without adjacentfunction 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..?>
<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 adjacentfunction 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.This article is contributed by Sahil Chhabra. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
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