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
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, n)); 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 adjacentdef 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 functionarr = [5, 5, 10, 100, 10, 5]print find_max_sum(arr) # This code is contributed by Kalai Selvan |
C#
/* Program to return max sum such that no two elements are adjacent */using System; class GFG { /* Function to return max sum such that no two elements are adjacent */ static 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() { int []arr = new int[]{5, 5, 10, 100, 10, 5}; Console.Write( FindMaxSum(arr, arr.Length)); }} // This code has been contributed by// nitin mittal |
PHP
<?php// PHP code to find Maximum sum // such that no two elements // are adjacent /* Function to return max sum such that no two elements are adjacent */function FindMaxSum($arr, $n){ $incl = $arr[0]; $excl = 0; $excl_new; $i; for ($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 : $excl);} // Driver Code$arr = array(5, 5, 10, 100, 10, 5);$n = sizeof($arr);echo FindMaxSum($arr, $n); // This code is contributed by Ajit?> |
Output:
110
Time Complexity: O(n)
Refer Find maximum possible stolen value from houses for more explanation.
Now try the same problem for an 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.
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