# 309. Best Time to Buy and Sell Stock with Cooldown

## 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 as many transactions as you like (i.e., buy one and sell one share of the stock multiple times) with the following restrictions:

• After you sell your stock, you cannot buy stock on the next day (i.e., cooldown one day).

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

Example 1:

Input: prices = [1,2,3,0,2]
Output: 3


Example 2:

Input: prices = [1]
Output: 0


Constraints:

• 1 <= prices.length <= 5000
• 0 <= prices[i] <= 1000

## Solutions

Solution 1: Memoization Search

We design a function $dfs(i, j)$, which represents the maximum profit that can be obtained starting from the $i$th day with state $j$. The values of $j$ are $0$ and $1$, respectively representing currently not holding a stock and holding a stock. The answer is $dfs(0, 0)$.

The execution logic of the function $dfs(i, j)$ is as follows:

If $i \geq n$, it means that there are no more stocks to trade, so return $0$;

Otherwise, we can choose not to trade, then $dfs(i, j) = dfs(i + 1, j)$. We can also trade stocks. If $j > 0$, it means that we currently hold a stock and can sell it, then $dfs(i, j) = prices[i] + dfs(i + 2, 0)$. If $j = 0$, it means that we currently do not hold a stock and can buy, then $dfs(i, j) = -prices[i] + dfs(i + 1, 1)$. Take the maximum value as the return value of the function $dfs(i, j)$.

The answer is $dfs(0, 0)$.

To avoid repeated calculations, we use the method of memoization search, and use an array $f$ to record the return value of $dfs(i, j)$. If $f[i][j]$ is not $-1$, it means that it has been calculated, and we can directly return $f[i][j]$.

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

Solution 2: Dynamic Programming

We can also use dynamic programming to solve this problem.

We define $f[i][j]$ to represent the maximum profit that can be obtained on the $i$th day with state $j$. The values of $j$ are $0$ and $1$, respectively representing currently not holding a stock and holding a stock. Initially, $f[0][0] = 0$, $f[0][1] = -prices[0]$.

When $i \geq 1$, if we currently do not hold a stock, then $f[i][0]$ can be obtained by transitioning from $f[i - 1][0]$ and $f[i - 1][1] + prices[i]$, i.e., $f[i][0] = \max(f[i - 1][0], f[i - 1][1] + prices[i])$. If we currently hold a stock, then $f[i][1]$ can be obtained by transitioning from $f[i - 1][1]$ and $f[i - 2][0] - prices[i]$, i.e., $f[i][1] = \max(f[i - 1][1], f[i - 2][0] - prices[i])$. The final answer is $f[n - 1][0]$.

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

We notice that the transition of state $f[i][]$ is only related to $f[i - 1][]$ and $f[i - 2][0]$, so we can use three variables $f$, $f_0$, $f_1$ to replace the array $f$, optimizing the space complexity to $O(1)$.

Solution 3: Greedy

• The solution uses dynamic programming to keep track of the profit at each step. It defines three states for each day:
1. f1 represents the maximum profit achievable on day i if you own a stock. This means you’ve bought a stock and haven’t sold it yet.
2. f2 represents the maximum profit achievable on day i if you’re in a cooldown period or have sold the stock and not bought another yet.
3. f3 represents the maximum profit achievable on day i if you’re free to buy a stock, meaning you’ve neither bought a stock nor are in a cooldown period.
• The initial conditions are set as follows:
• f1 is initialized to -prices[0] because buying a stock costs money, which is considered a negative profit on day 0.
• f2 and f3 are initialized to 0 because you haven’t made any transactions yet.
• The for loop iterates over the prices starting from the second day (prices[1:]), and for each day, it calculates:
• The new f1 as the maximum of staying in the current f1 state or moving from the f3 state to the f1 state by buying a stock (pf3 - price). This decision reflects either holding onto the stock bought earlier or buying a new stock after a cooldown.
• The new f2 as the maximum of staying in the current f2 state or moving from the f1 state to the f2 state by selling a stock (pf1 + price). This captures the profit from selling a stock you own.
• The new f3 as the maximum of staying in the current f3 state or moving from the f2 state to the f3 state (pf2). This transition reflects the cooldown period after selling a stock before you can buy again.
• Finally, the function returns f2, which represents the maximum profit at the end of the trading period under the condition that you’ve sold your stock and possibly are in a cooldown period or not looking to buy immediately. Since you want to maximize profit, ending with an owned stock (f1) is not considered for the final answer because you haven’t realized the profit from selling it. Ending in the f3 state doesn’t necessarily reflect the maximum profit, as it’s possible to have sold a stock on the last day, entering the cooldown with the profit included in f2.
• class Solution {
public int maxProfit(int[] prices) {
int f = 0, f0 = 0, f1 = -prices[0];
for (int i = 1; i < prices.length; ++i) {
int g0 = Math.max(f0, f1 + prices[i]);
f1 = Math.max(f1, f - prices[i]);
f = f0;
f0 = g0;
}
return f0;
}
}

• class Solution {
public:
int maxProfit(vector<int>& prices) {
int f = 0, f0 = 0, f1 = -prices[0];
for (int i = 1; i < prices.size(); ++i) {
int g0 = max(f0, f1 + prices[i]);
f1 = max(f1, f - prices[i]);
f = f0;
f0 = g0;
}
return f0;
}
};

• class Solution:
def maxProfit(self, prices: List[int]) -> int:
f1, f2, f3 = -prices[0], 0, 0
for price in prices[1:]:
pf1, pf2, pf3 = f1, f2, f3
f1 = max(pf1, pf3 - price)
f2 = max(pf2, pf1 + price)
f3 = max(pf3, pf2) # cooldown
return f2

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

class Solution:
def maxProfit(self, prices: List[int]) -> int:
f, f0, f1 = 0, 0, -prices[0]
for x in prices[1:]:
f, f0, f1 = f0, max(f0, f1 + x), max(f1, f - x)
return f0

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

class Solution(object):
def maxProfit(self, prices):
"""
:type prices: List[int]
:rtype: int
"""
if len(prices) < 2:
return 0
sell = [0] * len(prices)
sell[0] = 0
sell[1] = max(prices[1] - prices[0], 0)
for i in range(2, len(prices)):
sell[i] = max(prices[i] + buy[i - 1], sell[i - 1])
return max(sell)


• func maxProfit(prices []int) int {
f, f0, f1 := 0, 0, -prices[0]
for _, x := range prices[1:] {
f, f0, f1 = f0, max(f0, f1+x), max(f1, f-x)
}
return f0
}

• function maxProfit(prices: number[]): number {
let [f, f0, f1] = [0, 0, -prices[0]];
for (const x of prices.slice(1)) {
[f, f0, f1] = [f0, Math.max(f0, f1 + x), Math.max(f1, f - x)];
}
return f0;
}