> For the complete documentation index, see [llms.txt](https://jimmylin1991.gitbook.io/practice-of-algorithm-problems/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://jimmylin1991.gitbook.io/practice-of-algorithm-problems/array/188.-best-time-to-buy-and-sell-stock-iv.md).

# 188. Best Time to Buy and Sell Stock IV

Say you have an array for which the ith element is the price of a given stock on day i.

Design an algorithm to find the maximum profit. You may complete at most **k** transactions.

**Note:**\
You may not engage in multiple transactions at the same time (ie, you must sell the stock before you buy again).

**Example 1:**

```
Input: [2,4,1], k = 2
Output: 2
Explanation: Buy on day 1 (price = 2) and sell on day 2 (price = 4), profit = 4-2 = 2.
```

**Example 2:**

```
Input: [3,2,6,5,0,3], k = 2
Output: 7
Explanation: Buy on day 2 (price = 2) and sell on day 3 (price = 6), profit = 6-2 = 4.
             Then buy on day 5 (price = 0) and sell on day 6 (price = 3), profit = 3-0 = 3.
```

```cpp
// Same as 122. Best Time to Buy and Sell Stock II
int maxProfit2(vector<int>& prices) {
    int res = 0;
    for (int i = 1; i < prices.size(); ++i) {
        res += max(0, prices[i] - prices[i - 1]);
    }
    return res;
}
// DP + Greedy
int maxProfit(int k, vector<int>& prices) { // time: O(n * k); space: O(k)
    int n = prices.size();
    if (n <= 1) return 0;
    else if (k >= n / 2) return maxProfit2(prices);
    else {
        vector<int> sell(k + 1, 0), buy(k + 1, INT_MIN);
        for (int i = 0; i < n; ++i) {
            for (int j = k; j >= 1; --j) {
                sell[j] = max(sell[j], buy[j] + prices[i]);
                buy[j] = max(buy[j], sell[j - 1] - prices[i]);
            }
        }
        return sell[k];
    }
}
```

{% content-ref url="/pages/-LcZ54IvQXd3g3F2gw\_k" %}
[121. Best Time to Buy and Sell Stock](/practice-of-algorithm-problems/array/121.-best-time-to-buy-and-sell-stock.md)
{% endcontent-ref %}

{% content-ref url="/pages/-LcZ6n2xqhPQ2Q7ok\_6v" %}
[122. Best Time to Buy and Sell Stock II](/practice-of-algorithm-problems/array/122.-best-time-to-buy-and-sell-stock-ii.md)
{% endcontent-ref %}

{% content-ref url="/pages/-LcaryW9cRMBkzf-Imuq" %}
[123. Best Time to Buy and Sell Stock III](/practice-of-algorithm-problems/array/123.-best-time-to-buy-and-sell-stock-iii.md)
{% endcontent-ref %}

{% content-ref url="/pages/-LcdATaC9ECKuxgkzJ-8" %}
[309. Best Time to Buy and Sell Stock with Cooldown](/practice-of-algorithm-problems/array/309.-best-time-to-buy-and-sell-stock-with-cooldown.md)
{% endcontent-ref %}

{% content-ref url="/pages/-LcdyvMsbxkf5JxGNdhv" %}
[714. Best Time to Buy and Sell Stock with Transaction Fee](/practice-of-algorithm-problems/array/714.-best-time-to-buy-and-sell-stock-with-transaction-fee.md)
{% endcontent-ref %}