LeetCode #188: Best Time to Buy and Sell Stock IV

Problem Statement

(Source) 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).

Analysis

dp[i][j]: the maximum profit can be achieved in the previous j days with at most i transactions.

• Time Complexity: O(kn2)
• Space Complexity: O(kn)

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Tags: Dynamic Programming.

Solution

class Solution(object):    def maxProfit(self, k, prices):        """        :type k: int        :type prices: List[int]        :rtype: int        """        if k <= 0 or (not prices) or len(prices) < 2:            return 0        n = len(prices)        res = 0        if k >= (n >> 1): # Arbitray transactions allowed in this case.            dp = [0 for i in xrange(n + 1)]            for i in xrange(2, n + 1):                dp[i] = dp[i - 1]                j = i - 1                while j > 0 and prices[i-1] > prices[j-1]:                    dp[i] = max(dp[i], dp[j-1] + prices[i-1] - prices[j-1])                    j -= 1            return dp[-1]        dp = [[0 for j in xrange(n)] for i in xrange(k + 1)]        for i in xrange(1, k + 1):            for j in xrange(1, n):                dp[i][j] = dp[i][j-1]                # Buy at day t and sell at day j.                t = j - 1                while t >= 0 and prices[t] < prices[j]:                    profit = max(prices[j] - prices[t], 0)                    if t - 1 > 0:                        profit += dp[i-1][t-1]                    dp[i][j] = max(dp[i][j], profit)                    t -= 1                res = max(res, dp[i][j])        return res