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 
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.

Try It Yourself

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 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# 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:

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.

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

Explore