# Maximum sum such that no two elements are adjacent

Hello Everyone,

Given an array of positive numbers, find the maximum sum of a subsequence with the constraint that no 2 numbers in the sequence should be adjacent in the array. So 3 2 7 10 should return 13 (sum of 3 and 10) or 3 2 5 10 7 should return 15 (sum of 3, 5 and 7).Answer the question in most efficient way.

Examples :

Input : arr[] = {5, 5, 10, 100, 10, 5}
Output : 110

Input : arr[] = {1, 2, 3}
Output : 4

Input : arr[] = {1, 20, 3}
Output : 20

Algorithm:
Loop for all elements in arr[] and maintain two sums incl and excl where incl = Max sum including the previous element and excl = Max sum excluding the previous element.
Max sum excluding the current element will be max(incl, excl) and max sum including the current element will be excl + current element (Note that only excl is considered because elements cannot be adjacent).
At the end of the loop return max of incl and excl.

Example:

arr[] = {5, 5, 10, 40, 50, 35}

incl = 5
excl = 0

For i = 1 (current element is 5)
incl = (excl + arr[i]) = 5
excl = max(5, 0) = 5

For i = 2 (current element is 10)
incl = (excl + arr[i]) = 15
excl = max(5, 5) = 5

For i = 3 (current element is 40)
incl = (excl + arr[i]) = 45
excl = max(5, 15) = 15

For i = 4 (current element is 50)
incl = (excl + arr[i]) = 65
excl = max(45, 15) = 45

For i = 5 (current element is 35)
incl = (excl + arr[i]) = 80
excl = max(65, 45) = 65

And 35 is the last element. So, answer is max(incl, excl) = 80

Implementation:

`//c++ program for the above approach`

`#include <bits/stdc++.h>`

`using` `namespace` `std;`

`/*Function to return max sum such that no two elements`

` ` `are adjacent */`

`int` `FindMaxSum(vector<` `int` `> arr, ` `int` `n)`

`{`

` ` `int` `incl = arr[0];`

` ` `int` `excl = 0;`

` ` `int` `excl_new;`

` ` `int` `i;`

` ` `for` `(i = 1; i < n; i++)`

` ` `{`

` ` `/* current max excluding i */`

` ` `excl_new = (incl > excl) ? incl : excl;`

` ` `/* current max including i */`

` ` `incl = excl + arr[i];`

` ` `excl = excl_new;`

` ` `}`

` ` `/* return max of incl and excl */`

` ` `return` `((incl > excl) ? incl : excl);`

`}`

`// Driver program to test above functions`

`int` `main()`

`{`

` ` `vector<` `int` `> arr = {5, 5, 10, 100, 10, 5};`

` ` `cout<<FindMaxSum(arr, arr.size());`

`}`

Output:

110

Time Complexity: O(n)