**π‘ Problem Formulation:** Trading stocks often involves a transaction fee, which affects the profit from buying and selling stocks. Given a list of stock prices and a fixed transaction fee, the goal is to determine the maximum profit that can be obtained from an unlimited number of transactions. To solve this, we explore five Python methods, addressing the challenge of optimizing transactions to maximize profit minus fees. Assume prices are given as `[1, 3, 2, 8, 4, 9]`

and the fee is `2`

, the desired output is the maximum profit which would be `8`

.

## Method 1: Dynamic Programming

This method involves using a dynamic programming approach where two arrays, `cash`

and `hold`

, are maintained. `cash`

represents the maximum profit with no stock in hand, while `hold`

represents the maximum profit with a stock in hand at the end of each day. With each price update, these arrays are updated accordingly to maximize the profit considering the transaction fee.

Here’s an example:

def maxProfit(prices, fee): cash, hold = 0, -prices[0] for i in range(1, len(prices)): cash = max(cash, hold + prices[i] - fee) hold = max(hold, cash - prices[i]) return cash prices = [1, 3, 2, 8, 4, 9] fee = 2 print(maxProfit(prices, fee))

Output: `8`

The `maxProfit`

function calculates the maximum profit by iterating over each price and updating the `cash`

and `hold`

values. The `cash`

value represents the best profit after selling the stocks and deducting the fee, while `hold`

represents the best profit after buying the stock. The solution dynamically adapts to price changes to maintain optimal profits.

## Method 2: Greedy Approach

This method uses a greedy approach that goes through the stock prices and decides whether to buy, sell, or skip an action based on the current price, previous buying price, and the fee. The algorithm ensures that a transaction is made only when it is profitable after deducting the fee.

Here’s an example:

def maxProfit(prices, fee): max_profit = 0 hold = -prices[0] for i in range(1, len(prices)): max_profit = max(max_profit, hold + prices[i] - fee) hold = max(hold, max_profit - prices[i]) return max_profit prices = [1, 3, 2, 8, 4, 9] fee = 2 print(maxProfit(prices, fee))

Output: `8`

By comparing potential profits at each step with the `maxProfit`

function, the greedy algorithm decides the most beneficial point to buy or sell the stock. This strategy ensures that each transaction contributes to increasing the maximum profit, accounting for the transaction fee.

## Method 3: State Machine

The state machine method models the problem as a finite state machine with states representing whether the stock is held or not. Transitions between these states are based on the current price and transaction fee. This method visualizes the problem in terms of state changes and associated profits.

Here’s an example:

def maxProfit(prices, fee): state_sell, state_buy = 0, -prices[0] for price in prices[1:]: state_sell = max(state_sell, state_buy + price - fee) state_buy = max(state_buy, state_sell - price) return state_sell prices = [1, 3, 2, 8, 4, 9] fee = 2 print(maxProfit(prices, fee))

Output: `8`

In this approach, `state_sell`

represents the profit when we don’t hold a stock, while `state_buy`

represents the profit when we hold a stock. The state machine transitions are executed for each price to track profits.

## Method 4: Optimized Space Dynamic Programming

This method is an optimized version of dynamic programming that uses only two variables instead of arrays to keep track of the cash and hold states. This reduces space complexity from O(n) to O(1).

Here’s an example:

def maxProfit(prices, fee): cash, hold = 0, -prices[0] for price in prices[1:]: cash = max(cash, hold + price - fee) hold = max(hold, cash - price) return cash prices = [1, 3, 2, 8, 4, 9] fee = 2 print(maxProfit(prices, fee))

Output: `8`

Here, we have refined the space usage in dynamic programming by maintaining only the required variables to track the maximum profit instead of using whole arrays, thus saving memory while still rendering the same optimal solution.

## Bonus One-Liner Method 5: Functional Approach

This one-liner uses Python’s functional programming capabilities, such as `reduce()`

, to calculate the maximum profit in a concise way. It’s not as readable but still an interesting and non-trivial use of Python’s functional tools.

Here’s an example:

from functools import reduce maxProfit = lambda prices, fee: reduce(lambda r, p: (max(r[0], r[1]+p-fee), max(r[1], r[0]-p)), prices, (0, -prices[0]))[0] prices = [1, 3, 2, 8, 4, 9] fee = 2 print(maxProfit(prices, fee))

Output: `8`

This one-liner uses the `reduce()`

function to accumulate the results while iteratively calculating the cash and hold states concisely. It demonstrates the power of lambda functions and tuple unpacking in Python.

## Summary/Discussion

**Method 1: Dynamic Programming.**Effective for various inputs. Accurate. Potentially high space complexity for large input arrays.**Method 2: Greedy Approach.**Simple logic. Efficient. May be less intuitive to understand the transaction decision process.**Method 3: State Machine.**Good for visualizing transaction states. Clear state transitions. Can be overkill for simpler problems.**Method 4: Optimized Space Dynamic Programming.**Space-efficient. Retains accuracy of dynamic programming. May be harder to extend for more complex variations.**Bonus Method 5: Functional Approach.**Compact code. Harder to read and debug. An elegant demonstration of functional programming.