Best Time to Buy and Sell Stock III

We are required to find the maximum profit that can be achieved with at most two transactions. We will use dynamic programming to handle this problem.

Let’s initialize four variables:

  • buy1: Maximum amount of money we have left after the first purchase.
  • sell1: Maximum amount of money we have left after the first sale.
  • buy2: Maximum amount of money we have left after the second purchase.
  • sell2: Maximum amount of money we have left after the second sale.

Here’s the code that does exactly this:

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class Solution:
    def maxProfit(self, prices: List[int]) -> int:
        # Initialize the four variables with suitable values
        buy1, sell1 = -prices[0], 0
        buy2, sell2 = float('-inf'), 0
        
        # Iterate through the prices
        for price in prices:
            # If we buy the stock on this day, how much money will we have left?
            buy1 = max(buy1, -price)
            
            # If we sell the stock on this day, how much money will we have left?
            sell1 = max(sell1, buy1 + price)
            
            # If we buy the stock for the second time on this day, how much money will we have left?
            buy2 = max(buy2, sell1 - price)
            
            # If we sell the stock for the second time on this day, how much money will we have left?
            sell2 = max(sell2, buy2 + price)
            
        # The maximum profit will be in sell2
        return sell2

The code iteratively updates the variables as it considers each price, thus capturing all possible transactions. The result, which is the maximum profit that can be achieved with at most two transactions, is found in sell2.

Identifying Problem Isomorphism

“Best Time to Buy and Sell Stock III” can be mapped to “Maximum Subarray II”.

In “Best Time to Buy and Sell Stock III”, you’re given an array where the ith element is the price of a given stock on day i and the objective is to find the maximum profit. The task is that you must complete at most two transactions. However, you may not engage in multiple transactions simultaneously (i.e., you must sell the stock before you buy again).

In “Maximum Subarray II”, the task is to find two non-overlapping subarrays A and B, which lead to the maximum value of the sum of the subarray A and the sum of the subarray B. The connection here is that you are dividing an array into two parts to maximize some value, similar to the Stock problem where you’re trying to maximize profit with two transactions.

“Maximum Subarray II” is simpler than “Best Time to Buy and Sell Stock III” as the latter has an additional constraint that you cannot engage in multiple transactions simultaneously.

This is an extension of the array manipulation and dynamic programming concepts introduced in previous problems of the same series. Here are 10 problems to prepare for it:

  1. 121. Best Time to Buy and Sell Stock: It’s the first problem in the series which introduces the fundamental concept of the problem.

  2. 122. Best Time to Buy and Sell Stock II: It’s the second problem in the series where you’re allowed to complete as many transactions as you like.

  3. 53. Maximum Subarray: This problem introduces the concept of maximum subarray sum, which is used in the stock buying problem.

  4. 152. Maximum Product Subarray: This problem expands on the maximum subarray concept, asking for the maximum product instead of sum.

  5. 198. House Robber: This problem introduces a similar dynamic programming concept where you have to make a decision at each step.

  6. 70. Climbing Stairs: A simpler dynamic programming problem, but the logic is similar to stock problems where the current state depends on previous ones.

  7. 300. Longest Increasing Subsequence: Though not directly related, the dynamic programming concept used here is useful.

  8. 139. Word Break: This problem offers good practice for dynamic programming problems where you need to make optimal choices at each step.

  9. 188. Best Time to Buy and Sell Stock IV: This problem is a more general form of the problem where you are allowed to complete at most k transactions.

  10. 714. Best Time to Buy and Sell Stock with Transaction Fee: This problem adds a transaction fee to each sale, adding another layer of complexity.

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class Solution:
    def maxProfit(self, prices: List[int]) -> int:
				dp_2_hold, dp_2_not_hold = -float('inf'), 0
        dp_1_hold, dp_1_not_hold = -float('inf'), 0

        for stock_price in prices:
            dp_2_not_hold = max( dp_2_not_hold, dp_2_hold + stock_price )
            dp_2_hold = max( dp_2_hold, dp_1_not_hold - stock_price )
            dp_1_not_hold = max( dp_1_not_hold, dp_1_hold + stock_price )
            dp_1_hold = max( dp_1_hold, 0 - stock_price )

        return dp_2_not_hold

This problem requires identifying the maximum profit that can be obtained by performing at most two non-overlapping transactions. This problem requires understanding and analyzing price trends, hence Price Trend Analysis / Optimization.

Clarification Questions

What are the clarification questions we can ask about this problem?

Language Agnostic Coding Drills

  1. Understanding Basic Data Structures:

    • Creating and updating arrays/list.
    • Create an array with 5 elements and then update the 3rd element.

  2. Understanding Infinity:

    • Infinity can be used as an initial value when we want to get a minimum value or when we’re dealing with edges cases.
    • Find the minimum number from a list of positive numbers. Use infinity as an initial value for comparison.

  3. Working with Sequences:

    • Looping over a sequence is often a fundamental part of solving problems.
    • Loop over a sequence of numbers and calculate the square of each number.

  4. Dynamic Programming with Multiple States:

    • Solving dynamic programming problems often involve maintaining and updating multiple states as you progress through the problem.
    • Write a dynamic programming problem where you have to maintain two states. The classic problem “Rod Cutting” would be an example. You have to decide for each length of rod whether to cut it or not, and then find the maximum profit.

  5. Problem Specific Drill:

    • Sometimes, a problem might require a very specific technique or concept.
    • In this case, understanding the stock trading problem where you can perform two transactions is key. For each day, you have to make four decisions: Buy the stock for the first time, Sell the stock for the first time, Buy the stock for the second time, Sell the stock for the second time. Write a problem where you need to make multiple decisions and track multiple states. An example would be the Knapsack problem, where for each item you decide to take or not take it.

Targeted Drills in Python

This problem involves understanding how dynamic programming (DP) works, particularly how to track multiple states within the DP process. The states here involve whether to hold or not to hold the stock and whether we are on the first or second transaction. The problem also involves understanding how to deal with sequences (in this case, stock prices over time) and the concept of infinity (used as an initial value).

Here are the coding drills:

  1. Understanding Basic Data Structures:
    • Create a list, update the list.
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# Python
prices = [7,1,5,3,6,4]
prices[0] = 10
print(prices) # Output: [10, 1, 5, 3, 6, 4]
  1. Understanding Infinity:
    • Use of positive and negative infinity in computations.
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# Python
print(max(-float('inf'), 0, 1)) # Output: 1
print(min(float('inf'), 0, 1)) # Output: 0
  1. Working with Sequences:
    • Loop over a sequence and perform operations.
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# Python
for price in prices:
    print(price*2) # Output: double of each price in the list
  1. Dynamic Programming with Multiple States:
    • Maintain several states in a dynamic programming problem.
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# Python
dp_a, dp_b = 0, -float('inf')

for price in prices:
    dp_b = max(dp_b, dp_a - price)
    dp_a = max(dp_a, dp_b + price)

print(dp_a) # Output: max profit from one transaction

Problem Specific Drill:

The specific problem requires understanding of the stock trading problem where you can perform two transactions. For this, you need to manage four states - dp_1_hold, dp_1_not_hold, dp_2_hold, dp_2_not_hold. Here is a drill that deals with similar multi-state dynamic programming:

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# Python
dp_1_hold, dp_1_not_hold = -float('inf'), 0
dp_2_hold, dp_2_not_hold = -float('inf'), 0

for stock_price in prices:
    dp_2_not_hold = max( dp_2_not_hold, dp_2_hold + stock_price )
    dp_2_hold = max( dp_2_hold, dp_1_not_hold - stock_price )
    dp_1_not_hold = max( dp_1_not_hold, dp_1_hold + stock_price )
    dp_1_hold = max( dp_1_hold, 0 - stock_price )

print(dp_2_not_hold) # Output: max profit from two transactions

This drill demonstrates the approach to solving this kind of problem. You track the maximum profit at each step, deciding whether to buy (hold) or sell (not hold) at each price point for each transaction. You then return the maximum profit after two transactions.