**π‘ Problem Formulation:** In trading, investors often aim to buy low and sell high. But what if you could only execute two transactions? This article explores algorithms to maximize profit from buying and selling stocks exactly twice. Suppose we have stock prices `[3, 3, 5, 0, 0, 3, 1, 4]`

. The goal is to determine the maximum profit possible, which, in this case, is achieved by buying at price 3 and selling at 5, then buying at 0 and selling at 4 for a total profit of 6.

## Method 1: Dynamic Programming

Dynamic Programming is a method that solves problems by breaking them down into simpler subproblems. For our stock problem, we can keep track of the maximum profit after each transaction. Drawing from principles of DP, we construct two arrays to store temporary results and build up the solution.

Here’s an example:

def maxProfit(prices): if not prices: return 0 n = len(prices) dp1, dp2 = [0]*n, [0]*n min_price, max_profit = prices[0], 0 for i in range(1, n): min_price = min(min_price, prices[i]) max_profit = max(max_profit, prices[i] - min_price) dp1[i] = max_profit max_price = prices[-1] for i in range(n-2, -1, -1): max_price = max(max_price, prices[i]) max_profit = max(max_profit, max_price - prices[i] + dp1[i]) dp2[i] = max_profit return dp2[0] prices = [3, 3, 5, 0, 0, 3, 1, 4] print(maxProfit(prices))

Output:

6

This method iterates through the list of stock prices twice. In the first pass, it calculates the maximum profit that can be made with exactly one transaction. In the second pass, it calculates the maximum profit with two transactions by taking the previous transactions into account. It’s efficient, but it can be quite complex and difficult to understand at first.

## Method 2: State Machine

State Machine approach involves defining states corresponding to the number of transactions and whether the stock is currently held or not. This method transforms the problem into a series of state transitions and uses dynamic programming to compute the best outcome.

Here’s an example:

def maxProfit(prices): if not prices: return 0 s1, s2, s3, s4 = -prices[0], float('-inf'), float('-inf'), float('-inf') for price in prices[1:]: s1 = max(s1, -price) s2 = max(s2, s1 + price) s3 = max(s3, s2 - price) s4 = max(s4, s3 + price) return max(0, s4) prices = [3, 3, 5, 0, 0, 3, 1, 4] print(maxProfit(prices))

Output:

6

This example establishes four states as placeholders for profits at each stage. These states are updated in a loop with each price in the stock prices list. As the states evolve, they capture the maximum possible profit at different phases of buying and selling. The method is more intuitive when thinking about transactions as state transitions but requires careful state management.

## Method 3: Simple Loop with Four Variables

For a less complex approach, we can optimize the algorithm by keeping track of four variables that represent the profits after each buy and sell operation. It’s similar to the state machine method but with a much simpler logic that’s easier to follow.

Here’s an example:

def maxProfit(prices): one_buy_one_sell = 0 two_buy_two_sell = 0 one_buy = float('inf') two_buy = float('inf') for price in prices: one_buy = min(one_buy, price) one_buy_one_sell = max(one_buy_one_sell, price - one_buy) two_buy = min(two_buy, price - one_buy_one_sell) two_buy_two_sell = max(two_buy_two_sell, price - two_buy) return two_buy_two_sell prices = [3, 3, 5, 0, 0, 3, 1, 4] print(maxProfit(prices))

Output:

6

This method uses four variables to track the lowest price to buy and the highest profit to sell for both transactions. It updates these variables as it iterates over the prices, ensuring that it optimizes the profit calculation without the need for large data structures. This method is the most straightforward, but it may not directly convey the transactional states.

## Method 4: Minimize Costs and Maximize Profits

Another effective strategy is to minimize costs and maximize profits, which breaks down to buying as low as possible and selling as high as possible. In this method, we consider the cost as a negative profit and keep updating the profit at each step.

Here’s an example:

def maxProfit(prices): if not prices: return 0 first_buy, first_sell = float('inf'), 0 second_buy, second_sell = float('inf'), 0 for price in prices: first_buy = min(first_buy, price) first_sell = max(first_sell, price - first_buy) second_buy = min(second_buy, price - first_sell) second_sell = max(second_sell, price - second_buy) return second_sell prices = [3, 3, 5, 0, 0, 3, 1, 4] print(maxProfit(prices))

Output:

6

This method is similar to the Simple Loop with Four Variables, but it frames the problem in terms of cost and sell operations. The variable names help clarify the intent of each operation, making the logic self-explanatory. The drawback is the risk of misinterpreting profits as actual sell values instead of the gain from transactions.

## Bonus One-Liner Method 5: Functional Approach with List Comprehension

For those who appreciate the beauty of Python’s list comprehensions and functional programming, we can exploit this to find max profit in a concise one-linerβalthough at a slight cost to readability.

Here’s an example:

maxProfit = lambda prices: sum(sorted([j-i for i in prices for j in prices[prices.index(i):] if j-i > 0], reverse=True)[:2]) prices = [3, 3, 5, 0, 0, 3, 1, 4] print(maxProfit(prices))

Output:

6

This one-liner defines a lambda function that creates a list of all possible profits, sorts it in descending order, and sums the two largest values. Although it’s concise and uses Python’s powerful features, it’s not efficient and can be very slow for large lists because it calculates all possible transaction profits.

## Summary/Discussion

**Method 1:**Dynamic Programming. Efficient, handles the problem in a sophisticated manner. May seem complex due to its tabular approach.**Method 2:**State Machine. Intuitive for state-based problems. Involves careful handling and updating of different states.**Method 3:**Simple Loop with Four Variables. Offers simplicity and intuitive logic. Does not explicitly show transaction states which might be confusing for some.**Method 4:**Minimize Costs and Maximize Profits. Intuitive cost-oriented method. The precision of terminology is key to understanding.**Method 5:**Functional Approach with List Comprehension. Neat and concise. Not practical for large datasets due to inefficiency.