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
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; } |
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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. |
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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 |
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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 |
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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.. ?> |
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Output:
24
Time Complexity: O(n)
Space Complexity: O(2n) which is equal to O(n)
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