**Question:** 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.

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**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} inc = 5 exc = 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(5, 15) = 65 And 35 is the last element. So, answer is max(incl, excl) = 80

Thanks to Debanjan for providing code.

**Implementation:**

## C/C++

#include<stdio.h> /*Function to return max sum such that no two elements are adjacent */ int FindMaxSum(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 function */ int main() { int arr[] = {5, 5, 10, 100, 10, 5}; int n = sizeof(arr) / sizeof(arr[0]); printf("%d \n", FindMaxSum(arr, 6)); return 0; }

## Java

class MaximumSum { /*Function to return max sum such that no two elements are adjacent */ int FindMaxSum(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 public static void main(String[] args) { MaximumSum sum = new MaximumSum(); int arr[] = new int[]{5, 5, 10, 100, 10, 5}; System.out.println(sum.FindMaxSum(arr, arr.length)); } } // This code has been contributed by Mayank Jaiswal

## Python

# Function to return max sum such that # no two elements are adjacent def find_max_sum(arr): incl = 0 excl = 0 for i in arr: # Current max excluding i (No ternary in # Python) new_excl = excl if excl>incl else incl # Current max including i incl = excl + i excl = new_excl # return max of incl and excl return (excl if excl>incl else incl) # Driver program to test above function arr = [5, 5, 10, 100, 10, 5] print find_max_sum(arr) # This code is contributed by Kalai Selvan

Output:

110

**Time Complexity: **O(n)

Now try the same problem for array with negative numbers also.

Please write comments if you find any bug in the above program/algorithm or other ways to solve the same problem.