# 123. Best Time to Buy and Sell Stock III

## Description

You are given an array prices where prices[i] is the price of a given stock on the ith day.

Find the maximum profit you can achieve. You may complete at most two transactions.

Note: You may not engage in multiple transactions simultaneously (i.e., you must sell the stock before you buy again).

Example 1:

Input: prices = [3,3,5,0,0,3,1,4]
Output: 6
Explanation: Buy on day 4 (price = 0) and sell on day 6 (price = 3), profit = 3-0 = 3.
Then buy on day 7 (price = 1) and sell on day 8 (price = 4), profit = 4-1 = 3.

Example 2:

Input: prices = [1,2,3,4,5]
Output: 4
Explanation: Buy on day 1 (price = 1) and sell on day 5 (price = 5), profit = 5-1 = 4.
Note that you cannot buy on day 1, buy on day 2 and sell them later, as you are engaging multiple transactions at the same time. You must sell before buying again.


Example 3:

Input: prices = [7,6,4,3,1]
Output: 0
Explanation: In this case, no transaction is done, i.e. max profit = 0.


Constraints:

• 1 <= prices.length <= 105
• 0 <= prices[i] <= 105

## Solutions

Solution 1: Dynamic Programming

We define the following variables:

• f1 represents the maximum profit after the first purchase of the stock;
• f2 represents the maximum profit after the first sale of the stock;
• f3 represents the maximum profit after the second purchase of the stock;
• f4 represents the maximum profit after the second sale of the stock.

During the traversal, we directly calculate f1, f2, f3, f4. We consider that buying and selling on the same day will result in a profit of $0$, which will not affect the answer.

Finally, return f4.

The time complexity is $O(n)$, where $n$ is the length of the prices array. The space complexity is $O(1)$.

• public class Solution {
public int MaxProfit(int[] prices) {
int f1 = -prices[0], f2 = 0, f3 = -prices[0], f4 = 0;
for (int i = 1; i < prices.Length; ++i) {
f1 = Math.Max(f1, -prices[i]);
f2 = Math.Max(f2, f1 + prices[i]);
f3 = Math.Max(f3, f2 - prices[i]);
f4 = Math.Max(f4, f3 + prices[i]);
}
return f4;
}
}

• class Solution {
public:
int maxProfit(vector<int>& prices) {
int f1 = -prices[0], f2 = 0, f3 = -prices[0], f4 = 0;
for (int i = 1; i < prices.size(); ++i) {
f1 = max(f1, -prices[i]);
f2 = max(f2, f1 + prices[i]);
f3 = max(f3, f2 - prices[i]);
f4 = max(f4, f3 + prices[i]);
}
return f4;
}
};

• '''
-   f1 表示第一次买入股票后的最大利润；
-   f2 表示第一次卖出股票后的最大利润；
-   f3 表示第二次买入股票后的最大利润；
-   f4 表示第二次卖出股票后的最大利润。
'''
class Solution:
def maxProfit(self, prices: List[int]) -> int:
# 第一次买入，第一次卖出，第二次买入，第二次卖出
f1, f2, f3, f4 = -prices[0], 0, -prices[0], 0
for price in prices[1:]:
f1 = max(f1, -price)
f2 = max(f2, f1 + price)
f3 = max(f3, f2 - price)
f4 = max(f4, f3 + price)
return f4

class Solution:
def maxProfit(self, prices: List[int]) -> int:
if not prices:
return 0

# 0 to i, max profit
left_max = [0] * len(prices)

tmp_min = prices[0]
for i in range(len(prices)):
tmp_min = min(tmp_min, prices[i])
left_max[i] = max(left_max[i-1], prices[i]-tmp_min) if i > 0 else 0

# i to end, max profit
right_max = [0] * len(prices)

tmp_max = prices[-1]
for i in range(len(prices)-1, -1, -1):
tmp_max = max(tmp_max, prices[i])
right_max[i] = max(right_max[i+1], tmp_max-prices[i]) if i < len(prices)-1 else 0

return max(left_max[i] + right_max[i] for i in range(len(prices)))

############

class Solution(object):
def maxProfit(self, prices):
"""
:type prices: List[int]
:rtype: int
"""
sell1 = sell2 = 0

for i in range(len(prices)):
sell1 = max(prices[i] + buy1, sell1)
sell2 = max(sell2, prices[i] + buy2)
return max(sell1, sell2)


• func maxProfit(prices []int) int {
f1, f2, f3, f4 := -prices[0], 0, -prices[0], 0
for i := 1; i < len(prices); i++ {
f1 = max(f1, -prices[i])
f2 = max(f2, f1+prices[i])
f3 = max(f3, f2-prices[i])
f4 = max(f4, f3+prices[i])
}
return f4
}

• function maxProfit(prices: number[]): number {
let [f1, f2, f3, f4] = [-prices[0], 0, -prices[0], 0];
for (let i = 1; i < prices.length; ++i) {
f1 = Math.max(f1, -prices[i]);
f2 = Math.max(f2, f1 + prices[i]);
f3 = Math.max(f3, f2 - prices[i]);
f4 = Math.max(f4, f3 + prices[i]);
}
return f4;
}


• public class Solution {
public int MaxProfit(int[] prices) {
int f1 = -prices[0], f2 = 0, f3 = -prices[0], f4 = 0;
for (int i = 1; i < prices.Length; ++i) {
f1 = Math.Max(f1, -prices[i]);
f2 = Math.Max(f2, f1 + prices[i]);
f3 = Math.Max(f3, f2 - prices[i]);
f4 = Math.Max(f4, f3 + prices[i]);
}
return f4;
}
}

• impl Solution {
pub fn max_profit(prices: Vec<i32>) -> i32 {
let mut f1 = -prices[0];
let mut f2 = 0;
let mut f3 = -prices[0];
let mut f4 = 0;
let n = prices.len();

for i in 1..n {
f1 = std::cmp::max(f1, -prices[i]);
f2 = std::cmp::max(f2, f1 + prices[i]);
f3 = std::cmp::max(f3, f2 - prices[i]);
f4 = std::cmp::max(f4, f3 + prices[i]);
}

f4
}
}