714. Best Time to Buy and Sell Stock with Transaction Fee

1. Description

You are given an array prices where prices[i] is the price of a given stock on the $i^{th}$ day, and an integer fee representing a transaction fee.
Find the maximum profit you can achieve. You may complete as many transactions as you like, but you need to pay the transaction fee for each transaction.

Note:

• You may not engage in multiple transactions simultaneously (i.e., you must sell the stock before you buy again).
• The transaction fee is only charged once for each stock purchase and sale.

2. Example

Example 1

Input: prices = [1,3,2,8,4,9], fee = 2
Output: 8
Explanation: The maximum profit can be achieved by:

• Buying at prices[0] = 1
• Selling at prices[3] = 8
• Buying at prices[4] = 4
• Selling at prices[5] = 9

The total profit is ((8 - 1) - 2) + ((9 - 4) - 2) = 8.

Example 2

Input: prices = [1,3,7,5,10,3], fee = 3
Output: 6

3. Constraints

• 1 <= prices.length <= 5 * $10^4$
• 1 <= prices[i] < 5 * $10^4$
• 0 <= fee < 5 * $10^4$

4. Solutions

Dynamic Programming

n = prices.size()
Time complexity: O(n)
Space complexity: O(1)

// Let dp[i][1] be the maximum profit on day i when holding one stock
// Let dp[i][0] be the maximum profit on day i when holding no stock
//
// State transitions:
//   dp[i][1] = max(dp[i-1][1], dp[i-1][0] - prices[i] - fee)
//              (keep holding, or buy today with transaction fee)
//   dp[i][0] = max(dp[i-1][0], dp[i-1][1] + prices[i])
//              (keep not holding, or sell today)
//
// Since dp[i] only depends on dp[i-1], we compress the DP states
// into constant space using two variables for each state.
class Solution {
public:
    int maxProfit(const vector<int> &prices, const int fee) {
        int a1 = -prices[0] - fee, b1 = 0, a2 = 0, b2 = 0;
        const int n = prices.size();
        for (int i = 1; i < n; ++i) {
            a2 = max(a1, b1 - prices[i] - fee);
            b2 = max(b1, a1 + prices[i]);

            a1 = a2;
            b1 = b2;
        }

        return max(b2, 0);
    }
};