主要讲解的是以动态规划的方式来解决算法问题，虽然部分题目也可以使用其他更加快速方法解决，但本篇关注的是动态规划的思想

Best Time to Buy and Sell Stock

只能买卖一次，求最大利润

从前往后遍历数组，求第i天卖出可以获得的最大利润，即第i天的价格（卖出） - 第1~i-1天最低的价格（买入）。

状态转移方程：

1. max_profit[i] = max(price[i] - min_buy_price[i-1], max_profit[i-1])
2. min_buy_price[i] = min(price[i], min_buy_price[i-1])

import unittest

class Solution:
    def maxProfit(self, prices):
        """
        :type prices: List[int]
        :rtype: int
        """
        if len(prices) < 2:
            return 0

        max_profit = 0 
        min_buy_price = prices[0]

        for price in prices:
            if price > min_buy_price:
                max_profit = max(max_profit, price - min_buy_price)
            else:
                min_buy_price = min(min_buy_price, price)
        
        return max_profit

class TestSolution(unittest.TestCase):
    def setUp(self):
        self.s = Solution()

    def test_one(self):
        input = [7, 1, 5, 3, 6, 4]
        self.assertEqual(self.s.maxProfit(input), 5)
    
    def test_two(self):
        input = [7, 6, 4, 3, 1]
        self.assertEqual(self.s.maxProfit(input), 0)

if __name__ == "__main__":
    unittest.main()

Best Time to Buy and Sell Stock II

允许多次买卖，求最大利润

第i天有2种状态：拥有股票own[i]，未拥有股票no_own[i]，那么其状态方程可以表示为：

1. 在当天过后手头拥有股票的条件下，有两种情况：①当天买入的；②当天没有操作，之前买入的；own[i] = max(own[i-1], no_own[i-1] - price[i])
2. 在当天过后手头未拥有股票的条件下，有两种情况：①当天卖出的；②当天没有操作，之前卖出的；no_own[i] = max(own[i-1] + price[i], no_own[i-1])

状态转移方程：

1. own[i] = max(own[i-1], no_own[i-1] - price[i])
2. no_own[i] = max(own[i-1] + price[i], no_own[i-1])

import unittest

class Solution(object):
    def maxProfit(self, prices):
        """
        :type prices: List[int]
        :rtype: int
        """

        if len(prices) < 2:
            return 0

        own = [0] * len(prices)
        no_own = [0] * len(prices)
        own[0] = -prices[0]

        for i in range(1, len(prices)):
            own[i] = max(own[i-1], no_own[i-1] - prices[i])
            no_own[i] = max(no_own[i-1], own[i] + prices[i])

        return no_own[-1]

class TestSolution(unittest.TestCase):
    def test_one(self):
        prices = [7,1,5,3,6,4]
        s = Solution()
        print(s.maxProfit(prices))

    def test_two(self):
        prices = [1, 2, 3, 4, 5]
        s = Solution()
        print(s.maxProfit(prices))


if __name__ == "__main__":
    unittest.main()

Best Time to Buy and Sell Stock III

最多两次的任意买卖，求最大利润

第i天有2种状态，k表示这是第k次来回交易：拥有股票own[k][i]，未拥有股票no_own[k][i]，那么其状态方程可以表示为：

1. 在当天过后手头拥有股票的条件下，有两种情况：①当天是第k次轮回买入的；②当天没有操作，之前第k次买入的；own[k][i] = max(own[k][i-1], no_own[k-1][i-1] - price[i])
2. 在当天过后手头未拥有股票的条件下，有两种情况：①当天是第k次轮回卖出的，即在之前第k次轮回买入的基础上计算；②当天没有操作，之前第k次轮回卖出的；no_own[k][i] = max(own[k][i-1] + price[i], no_own[k][i-1])

状态转移方程：

1. own[k][i] = max(own[k][i-1], no_own[k-1][i-1] - price[i])
2. no_own[k][i] = max(own[k][i-1] + price[i], no_own[k][i-1])

import unittest
from pprint import pprint

class Solution:
    def maxProfit(self, prices):
        if len(prices) < 2:
            return 0

        length = len(prices)

        own = [[0]*(length) for i in range(3)]
        no_own = [[0]*(length) for i in range(3)]

        own[1][0] = own[2][0] = -prices[0]

        for k in range(1,3):
            for j in range(1, length):
                own[k][j] = max(own[k][j-1], no_own[k-1][j-1] - prices[j])
                no_own[k][j] = max(no_own[k][j-1], own[k][j-1] + prices[j])
        return no_own[-1][-1]

class TestSolution(unittest.TestCase):
    def setUp(self):
        self.s = Solution()

    def test_one(self):
        input = [3, 3, 5, 0, 0, 3, 1, 4]
        self.assertEqual(self.s.maxProfit(input), 6)
    
    def test_two(self):
        input = [1, 2, 3, 4, 5]
        self.assertEqual(self.s.maxProfit(input), 4)

if __name__ == "__main__":
    unittest.main()

Best Time to Buy and Sell Stock IV

最多k次的任意买卖，求最大利润

同上，不过由于k可能较大，为了节省空间，我们利用k%2来节省空间，因为每第k次交易只与上一次交易有关。
仅仅就行上面的优化是不够的，因为k可能大，单当k>=len(prices)//2时，就转化为Best Time to Buy and Sell Stock II的问题了，这样减少了许多计算量。

import unittest

class Solution:
    def maxProfit(self, k, prices):
        if len(prices) < 2:
            return 0
        
        length = len(prices)

        if k >= length//2:
            own = [0] * length
            no_own = [0] * length
            own[0] = -prices[0]

            for i in range(1, length):
                own[i] = max(own[i-1], no_own[i-1] - prices[i])
                no_own[i] = max(no_own[i-1], own[i] + prices[i])

            return no_own[-1]

        own = [[0]*(length) for i in range(2)]
        no_own = [[0]*(length) for i in range(2)]

        own[0][0] = own[1][0] = -prices[0]

        for k in range(0, k):
            k = k % 2
            own[k][0] = -prices[0]
            for j in range(1, length):
                own[k][j] = max(own[k][j-1], no_own[k-1][j-1] - prices[j])
                no_own[k][j] = max(no_own[k][j-1], own[k][j-1] + prices[j])
        return max(no_own[0][-1], no_own[1][-1])

class TestSolution(unittest.TestCase):
    def setUp(self):
        self.s = Solution()

    def test_one(self):
        input = [2, 4, 1]
        self.assertEqual(self.s.maxProfit(2,input), 2)
    
    def test_two(self):
        input = [3, 2, 6, 5, 0, 3]
        self.assertEqual(self.s.maxProfit(2, input), 7)

    def test_three(self):
        input = [1, 2]
        self.assertEqual(self.s.maxProfit(1, input), 1)
if __name__ == "__main__":
    unittest.main()

Best Time to Buy and Sell Stock with Cooldown

可以进行任意次交易，但卖出股票的第二天不允许进行交易

第i天有2种状态：拥有股票own[i]，未拥有股票no_own[i]，那么其状态方程可以表示为：

1. 在当天过后手头拥有股票的条件下，有两种情况：①当天买入的，由于条件限制，因此只能在第i-1之前卖出；②当天没有操作，之前买入的；own[i] = max(own[i-1], no_own[i-2] - price[i])
2. 在当天过后手头未拥有股票的条件下，有两种情况：①当天卖出的；②当天没有操作，之前卖出的；no_own[i] = max(own[i-1] + price[i], no_own[i-1])

状态转移方程：

1. own[i] = max(own[i-1], no_own[i-2] - price[i])
2. no_own[i] = max(own[i-1] + price[i], no_own[i-1])

import unittest

class Solution:
    def search(self, nums, target):
        if len(nums) == 0:
            return -1

        l = 0
        h = len(nums) - 1

        while l < h:
            m = l + (h-l)//2
            if nums[m] <= target:
                l = m + 1
            elif nums[m] >= target:
                h = m
        
        pos = -1 
        if nums[l] > target:
            pos = l

        return pos

class TestSolution(unittest.TestCase):
    def setUp(self):
        self.s = Solution()

    def test_one(self):
        input = [-1, 0, 3, 3, 5, 9, 12]
        self.assertEqual(self.s.search(input, 3), 4)
    
    def test_two(self):
        input = [-1, 0, 3, 3, 5, 9, 12]
        self.assertEqual(self.s.search(input, 2), 2)

    def test_three(self):
        input = [0]
        self.assertEqual(self.s.search(input, 0), -1)

if __name__ == "__main__":
    unittest.main()

Best Time to Buy and Sell Stock with Transaction Fee

可以交易任意多笔，但每次买卖需要支付一定的费用fee

第i天有2种状态：拥有股票own[i]，未拥有股票no_own[i]，那么其状态方程可以表示为：

1. 在当天过后手头拥有股票的条件下，有两种情况：①当天买入的；②当天没有操作，之前买入的；own[i] = max(own[i-1], no_own[i-1] - price[i])
2. 在当天过后手头未拥有股票的条件下，有两种情况：①当天卖出的，此时需要支付交易费；②当天没有操作，之前卖出的；no_own[i] = max(own[i-1] + price[i] - fee, no_own[i-1])

状态转移方程：

1. own[i] = max(own[i-1], no_own[i-1] - price[i])
2. no_own[i] = max(own[i-1] + price[i] - fee, no_own[i-1])

import unittest

class Solution:
    def maxProfit(self, prices, fee):
        """
        :type prices: List[int]
        :type fee: int
        :rtype: int
        """

        length = len(prices)

        if len(prices) < 2:
            return 0

        last_own = -prices[0]
        last_no_own = 0

        for i in range(1, length):
            own = max(last_own, last_no_own - prices[i])
            last_no_own = max(last_no_own, last_own + prices[i] - fee)
            last_own = own
        
        return last_no_own

class TestSolution(unittest.TestCase):
    def setUp(self):
        self.s = Solution()

    def test_one(self):
        input = [1, 3, 2, 8, 4, 9]
        self.assertEqual(self.s.maxProfit(input, 2), 8)

    def test_two(self):
        pass

if __name__ == "__main__":
    unittest.main()

总结

一开始做这些题的时候，不知道从何下手，但经过这几道题的训练之后，对于大部分的动态规划还是能分析出来，只要状态方程推导出来以后，确定好初始值，就可以得到最终的结果。