Problem Description

You are given an integer array prices, where prices[i] represents the stock price on day i. Your task is to find the maximum profit you can achieve with at most two transactions.

Constraints:

• You must sell the stock before buying again.
• If no profit can be achieved, you may choose not to make any transactions.

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Examples

1. Input: prices = [3,3,5,0,0,3,1,4]
   Output: 6
   Explanation:
   • First transaction: Buy on day 4 (price = 0), sell on day 6 (price = 3), profit = 3.
   • Second transaction: Buy on day 7 (price = 1), sell on day 8 (price = 4), profit = 3.
   • Total profit = 3 + 3 = 6.
2. Input: prices = [1,2,3,4,5]
   Output: 4
   Explanation:
   • Buy on day 1 (price = 1), sell on day 5 (price = 5), profit = 4.
3. Input: prices = [7,6,4,3,1]
   Output: 0
   Explanation: No transactions are possible for a profit.

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Approach

This problem can be solved using Dynamic Programming (DP) combined with Recursion and Memoization. Here's a step-by-step explanation:

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Key Insights

1. Divide the Problem:
   • Find the maximum profit for one transaction before or up to day i.
   • Find the maximum profit for one transaction after or from day i.
   • Combine the two profits to find the global maximum.
2. State Representation:
   • i: Current day (0 to n-1).
   • t: Number of completed transactions (0, 1, or 2).
   • can_buy: A flag indicating if you can buy (1 = buy allowed, 0 = sell allowed).
3. Base Case:
   • If i == n (past the last day) or t == 2 (completed two transactions), the profit is 0.
4. State Transition:
   • If can_buy == 1: Choose to buy or skip.
     profit = max(-prices[i] + dfs(i+1, t, 0), dfs(i+1, t, 1))
   • If can_buy == 0: Choose to sell or skip.
     profit = max(prices[i] + dfs(i+1, t+1, 1), dfs(i+1, t, 0))

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Code Implementation

        
def maxProfit(prices):
    n = len(prices)
    memo = {}

    # Recursive helper function
    def dfs(day, transactions, can_buy):
        # Base cases
        if day == n or transactions == 2:
            return 0
        if (day, transactions, can_buy) in memo:
            return memo[(day, transactions, can_buy)]
        
        # Skip current day
        profit = dfs(day + 1, transactions, can_buy)
        
        if can_buy:
            # Buy or skip
            profit = max(profit, -prices[day] + dfs(day + 1, transactions, 0))
        else:
            # Sell or skip
            profit = max(profit, prices[day] + dfs(day + 1, transactions + 1, 1))
        
        # Memoize and return result
        memo[(day, transactions, can_buy)] = profit
        return profit

    return dfs(0, 0, 1)
        
    

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Interview Simulation

Interviewer Question 1: How did you divide the problem?
Candidate Answer:
"The problem is split into two phases: finding the maximum profit before a given day and finding the maximum profit after that day. Combining these two gives the total profit. The DP approach efficiently handles overlapping subproblems using memoization."

Interviewer Question 2: How would this algorithm change if we allowed unlimited transactions?
Candidate Answer:
"For unlimited transactions, we wouldn’t limit transactions to 2. Instead, we’d iterate through the days while tracking the profit. The state would then depend only on whether we can buy or sell, reducing the complexity."

Interviewer Question 3: Can you optimize space usage?
Candidate Answer:
"Yes, instead of storing results for all days, we can use a rolling array to store only the results of the current and previous days, thereby reducing space complexity from O(n*2*3) to O(2*3)."

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Example Walkthrough

1. Input: prices = [3,3,5,0,0,3,1,4]
   • Day 4: Buy (price = 0).
   • Day 6: Sell (price = 3), profit = 3.
   • Day 7: Buy (price = 1).
   • Day 8: Sell (price = 4), profit = 3.
   • Total profit = 6.
2. Input: prices = [7,6,4,3,1]
   • No transactions are made; profit = 0.

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Challenges and CSOAhelp’s Role

• Dynamic Problem-Solving: Candidates often struggle with recursion and memoization. CSOAhelp simplifies these concepts with step-by-step examples and detailed explanations.
• Mock Interviews: By simulating the actual interview process and providing detailed prompts, CSOAhelp prepares candidates to handle tough questions confidently.
• Time Complexity Optimization: Our hints help candidates avoid redundant calculations and explain their approach clearly to interviewers.

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