Best Time to Buy and Sell Stock IV (H)
題目
You are given an integer array prices where prices[i] is the price of a given stock on the ith day.
Design an algorithm to find the maximum profit. You may complete at most k transactions.
Notice that you may not engage in multiple transactions simultaneously (i.e., you must sell the stock before you buy again).
Example 1:
Input: k = 2, prices = [2,4,1]
Output: 2
Explanation: Buy on day 1 (price = 2) and sell on day 2 (price = 4), profit = 4-2 = 2.
Example 2:
Input: k = 2, prices = [3,2,6,5,0,3]
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.
Constraints:
0 <= k <= 10^90 <= prices.length <= 10^40 <= prices[i] <= 1000
題意
股票買賣問題之四，允許最多k次交易(k次買入k次賣出)，
思路
解法在 0309. Best Time to Buy and Sell Stock with Cooldown 的基礎上增加一個k次交易的限制，
\(hold[i][j]\)表示在第i天仍持有股票，且最多已進行了j次交易，
\(sold[i][j]\)表示在第i天未持有任何股票，且最多已進行了j次交易，
可以得到如下遞推關系：
\[\begin{cases} hold[i][j]=max(sold[i-1][j-1]-prices[i],\ hold[i-1][j])\\\\ sold[i][j]=max(hold[i-1][j]+prices[i],\ sold[i-1][j]) \end{cases} \]邊界條件是：\(\forall{j\ge1},\ hold[0][j]=-prices[0]\)
需要注意的是，case可能會使壞給一個巨大的k值，導致超時，這里的一個技巧是，當k大于陣列長度一半時時，問題就等同于可以進行不限次數的交易，可以直接用 0122. Best Time to Buy and Sell Stock II 中的一次遍歷方法解決，
代碼實作
Java
class Solution {
public int maxProfit(int k, int[] prices) {
if (prices.length == 0) {
return 0;
}
if (k > prices.length / 2) {
return maxProfit(prices);
}
int[][] hold = new int[prices.length][k + 1];
int[][] sold = new int[prices.length][k + 1];
for (int i = 0; i < prices.length; i++) {
for (int j = 1; j <= k; j++) {
if (i == 0) {
hold[i][j] = -prices[i];
} else {
hold[i][j] = Math.max(hold[i - 1][j], sold[i - 1][j - 1] - prices[i]);
sold[i][j] = Math.max(hold[i - 1][j] + prices[i], sold[i - 1][j]);
}
}
}
return sold[prices.length - 1][k];
}
private int maxProfit(int[] prices) {
int profit = 0;
for (int i = 1; i < prices.length; i++) {
if (prices[i] > prices[i - 1]) {
profit += prices[i] - prices[i - 1];
}
}
return profit;
}
}
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