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

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
Explanation: transactions = [buy, sell, cooldown, buy, sell]
```

**Example 2:**

```
Input: prices = [1]
Output: 0
```

 

**Constraints:**

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

### Solution:

每次 `sell` 之后要等一天才能继续交易。只要把这个特点融入上一题的状态转移方程即可：

```python
dp[i][0] = max(dp[i-1][0], dp[i-1][1] + prices[i])
dp[i][1] = max(dp[i-1][1], dp[i-2][0] - prices[i])
解释：第 i 天选择 buy 的时候，要从 i-2 的状态转移，而不是 i-1 。
```

翻译成代码：

```java
// 原始版本
int maxProfit_with_cool(int[] prices) {
    int n = prices.length;
    int[][] dp = new int[n][2];
    for (int i = 0; i < n; i++) {
        if (i - 1 == -1) {
            // base case 1
            dp[i][0] = 0;
            dp[i][1] = -prices[i];
            continue;
        }
        if (i - 2 == -1) {
            // base case 2
            dp[i][0] = Math.max(dp[i-1][0], dp[i-1][1] + prices[i]);
            // i - 2 小于 0 时根据状态转移方程推出对应 base case
            dp[i][1] = Math.max(dp[i-1][1], -prices[i]);
            //   dp[i][1] 
            // = max(dp[i-1][1], dp[-1][0] - prices[i])
            // = max(dp[i-1][1], 0 - prices[i])
            // = max(dp[i-1][1], -prices[i])
            continue;
        }
        dp[i][0] = Math.max(dp[i-1][0], dp[i-1][1] + prices[i]);
        dp[i][1] = Math.max(dp[i-1][1], dp[i-2][0] - prices[i]);
    }
    return dp[n - 1][0];
}

// 空间复杂度优化版本
int maxProfit_with_cool(int[] prices) {
    int n = prices.length;
    int dp_i_0 = 0, dp_i_1 = Integer.MIN_VALUE;
    int dp_pre_0 = 0; // 代表 dp[i-2][0]
    for (int i = 0; i < n; i++) {
        int temp = dp_i_0;
        dp_i_0 = Math.max(dp_i_0, dp_i_1 + prices[i]);
        dp_i_1 = Math.max(dp_i_1, dp_pre_0 - prices[i]);
        dp_pre_0 = temp;
    }
    return dp_i_0;
}
```


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