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

Difficulty Level : Medium
Last Updated : 20 Jan, 2021

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. 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;
}

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..
?>

Output:

Time Complexity: O(n)
Space Complexity: O(1)
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