# Maximum profit after buying and selling the stocks with transaction fees | Set 2

Given an array arr[] of positive integers representing prices of stocks and an integer transactionFee, the task is to find the maximum profit possible after buying and selling stocks any number of times and giving the transaction fee for each transaction.Examples:Input: arr[] = {6, 1, 7, 2, 8, 4}, transactionFee = 2Output: 8Explanation: A maximum profit of 8 can be obtained by two transactions.Transaction 1: Buy at price 1 and sell at price 7. Profit = 7 – 1 – 2 = 4.Transaction 2: Buy at price 2 and sell at price 8. Profit = 8 – 2 – 2 = 4.Therefore, total profit = 4 + 4 = 8, which is the maximum possible.Input: arr[] = {2, 7, 5, 9, 6, 4}, transactionFee = 1Output: 7Naive Approach: Refer to the previous post for the simplest approach to solve the problem.Time Complexity: O(N2)Auxiliary Space: O(1)Efficient Approach: To optimize the above approach, the idea is to use Dynamic Programming. For each day, maintain the maximum profit, if stocks are bought on that day (buy) and the maximum profit if all stocks are sold on that day (sell). For each day, update buy and sell using the following relations:buy = max(sell – arr[i], buy)sell = max(buy +arr[i] – transactionFee, sell)Below is the implementation of the above approach:C++#include using namespace std;  int MaxProfit(int arr[], int n,              int transactionFee){      int buy = -arr;    int sell = 0;              for (int i = 1; i < n; i++) {        int temp = buy;                  buy = max(buy, sell - arr[i]);        sell = max(sell,                   temp + arr[i] - transactionFee);    }          return max(sell, buy);}  int main(){        int arr[] = { 6, 1, 7, 2, 8, 4 };    int n = sizeof(arr) / sizeof(arr);    int transactionFee = 2;          cout 