💡 Problem Formulation: The “Best Time to Buy and Sell Stock III” problem requires one to identify the optimum times to buy and sell a stock within a given list of daily prices, where a maximum of two transactions are allowed. For input [3,3,5,0,0,3,1,4], the ideal output would be 6, which represents maximum profit from buying on day 4 and selling on day 6, then buying on day 7 and selling on day 8.
Method 1: Dynamic Programming
This method leverages dynamic programming to tackle the problem by breaking it into subproblems. Each day’s transaction decisions are based on the state of previous decisions, considering only local decisions to create a maximum profit globally. Dynamic programming stores interim results to avoid redundant calculations.
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Here’s an example:
def maxProfit(prices):
if not prices:
return 0
n = len(prices)
pre_profit, post_profit = [0]*n, [0]*n
min_price, max_profit = prices[0], 0
for i in range(n):
min_price = min(min_price, prices[i])
max_profit = max(max_profit, prices[i] - min_price)
pre_profit[i] = max_profit
max_price = prices[-1]
for i in range(n-1, -1, -1):
max_price = max(max_price, prices[i])
post_profit[i] = max(post_profit[i + 1], max_price - prices[i])
max_total_profit = 0
for i in range(n):
max_total_profit = max(max_total_profit, pre_profit[i] + post_profit[i])
return max_total_profit
prices = [3,3,5,0,0,3,1,4]
print(maxProfit(prices))
Output: 6
The provided code defines a function maxProfit that calculates the maximum potential profit one could earn by making at most two transactions. It does so by iterating twice over the list of prices – first, it computes the maximum possible profit at each day as if it were the selling point. It keeps track of the lowest price so far and computes the profit relative to that. Next, it traverses from the end to calculate the profit after the assumed selling point. Finally, it determines the maximum profit by summing the two profits at every possible selling point.
Method 2: State Machine Approach
The state machine method models the purchasing and selling of stocks as a finite state machine, with each day representing a possible state transition. The states represent the number of transactions and whether you’re holding a stock or not. This approach is effective in avoiding redundant calculations typical in more naive implementations.
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:
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
The function maxProfit uses four variables to represent the different states: holding the first stock (s1), sold the first stock (s2), holding the second stock (s3), and sold the second stock (s4). It iterates through each price update, transitioning between states to find the maximum potential profit, which is then returned as the maximum between 0 and s4.
Method 3: Simplified Dynamic Programming
This approach simplifies dynamic programming by reducing the number of states required and not explicitly storing the profits for each day, thus saving memory. The method keeps track of the minimum purchase prices and potential profits for two transactions.
Here’s an example:
def maxProfit(prices):
buy1 = buy2 = float('inf')
profit1 = profit2 = 0
for price in prices:
buy1 = min(buy1, price)
profit1 = max(profit1, price - buy1)
buy2 = min(buy2, price - profit1)
profit2 = max(profit2, price - buy2)
return profit2
prices = [3,3,5,0,0,3,1,4]
print(maxProfit(prices))
Output: 6
The code snippet executes a single pass through the list of prices while maintaining four variables to track the lowest purchase prices (buy1, buy2) and the maximum profits (profit1, profit2) after the first and second transactions. This approach greatly reduces space complexity and maintains linear time complexity.
Method 4: Using Less Space Complexity
This method attempts to solve the problem in a space-optimized manner. Instead of maintaining separate lists to store the profits of transactions, it retains only the relevant variables needed to calculate the maximum profits at each step thereby saving on space without sacrificing time complexity.
Here’s an example:
def maxProfit(prices):
t1_cost, t2_cost = float('inf'), float('inf')
t1_profit, t2_profit = 0, 0
for price in prices:
t1_cost = min(t1_cost, price)
t1_profit = max(t1_profit, price - t1_cost)
t2_cost = min(t2_cost, price - t1_profit)
t2_profit = max(t2_profit, price - t2_cost)
return t2_profit
prices = [3,3,5,0,0,3,1,4]
print(maxProfit(prices))
Output: 6
This method sets up four variables, t1_cost and t2_cost to record the lowest costs of buying the first and second stocks, and t1_profit and t2_profit to capture the profits from selling these stocks. For each price, it updates these variables to reflect the best achievable profit after each transaction. Ultimately, t2_profit will hold the maximum profit after the second sale.
Bonus One-Liner Method 5: Using List Comprehensions
Python’s list comprehensions provide a concise way to create lists and can be used to implement a clever, albeit less readable, solution for this problem. While it may not be the most efficient method, it’s an interesting use of Python’s syntactic sugar.
Here’s an example:
def maxProfit(prices):
return max([min(prices[:i]) - prices[i] + max(prices[i:]) - price for i, price in enumerate(prices)] + [0])
prices = [3,3,5,0,0,3,1,4]
print(maxProfit(prices))
Output: 6
The one-liner comprehension iterates through the prices while calculating the maximum profit that can be made with one transaction before and after the current day, thereby simulating two transactions. This solution, however, has a high time complexity due to repeated calculations for every index.
Summary/Discussion
- Method 1: Dynamic Programming. It’s thorough and guarantees an optimal solution. However, it may be less space-efficient due to the storage of interim results for every day.
- Method 2: State Machine Approach. Efficiently avoids redundant work and provides a clear conceptual model. The number of state variables is fixed, which is space-efficient. However, the readability might suffer due to the abstract nature of state transitions.
- Method 3: Simplified Dynamic Programming. This offers a good balance between space and time complexity. It’s also more intuitive than the state machine approach but requires a clear understanding of profit transitions between transactions.
- Method 4: Using Less Space Complexity. Very similar to Method 3, it provides a minimalistic but effective solution. The readability of this approach is decent and it’s space-optimized, making it practical for scenarios with large input data.
- Bonus Method 5: Using List Comprehensions. While succinct, this method’s brute force nature makes it inefficient for large datasets. It is an interesting demonstration of Python’s capabilities but may not be suitable for production code.