Maximum sum such that no two elements are adjacent

Software Development
#1

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)