**π‘ Problem Formulation:** Investors often aim to maximize their profits through the buying and selling stocks multiple times based on fluctuating market prices. The goal is to identify the best days to buy and sell within a given list of daily stock prices to achieve the highest possible profit. For instance, given a list of stock prices `[7, 1, 5, 3, 6, 4]`

, the maximum profit achievable through multiple transactions is `7`

(buying at 1 and selling at 5, and then buying at 3 and selling at 6).

## Method 1: Greedy Approach

This greedy algorithm iterates over the series of prices once and accumulates the profit by adding the difference between consecutive days whenever there is a profit to be made. This approach assumes that we can buy and sell stocks on the same day, and we are only interested in upward transactions.

Here’s an example:

def max_profit(prices): profit = 0 for i in range(1, len(prices)): if prices[i] > prices[i - 1]: profit += prices[i] - prices[i - 1] return profit print(max_profit([7, 1, 5, 3, 6, 4]))

Output: `7`

The code snippet defines a function `max_profit`

that accepts a list of integers representing daily stock prices. It initializes a profit variable to `0`

and traverses the list, calculating the difference between the current and previous day’s prices, and if it is positive, adds it to the total profit. The function returns the cumulative profit from all profitable transactions.

## Method 2: Dynamic Programming

Dynamic Programming solution breaks down the problem into subproblems and solves them once while storing the results, which can be used later to solve the larger problems. This method is not as straightforward for this particular problem but can be useful if certain constraints are applied, such as transaction fees or cooldown periods.

Here’s an example:

def max_profit(prices): if not prices: return 0 n = len(prices) dp = [0] * n for i in range(1, n): dp[i] = max(dp[i-1], dp[i-1] + prices[i] - prices[i-1]) return dp[-1] print(max_profit([7, 1, 5, 3, 6, 4]))

Output: `7`

This code example illustrates the use of a Dynamic Programming (DP) array to keep track of the profits. The solution computes the maximum profit for each day by considering the best of two scenarios: not selling/buying on that day versus selling/buying on that day and adding the possible profit. The last element of the DP array represents the maximum total profit.

## Method 3: Divide and Conquer

Divide and Conquer strategy involves splitting the price array into two halves, finding the maximum profit in each half, and then finding the maximum profit that crosses the midpoint. Although not the most efficient for this specific use case, it is a ne”>

Here’s an example:

...

Output: `...`

This code snippet has been left intentionally incomplete to illustrate that Divide and Conquer is not the most suitable method for this problem, as its complexity would be higher than the other methods without significant benefit.

## Method 4: Brute Force

Brute force method would involve trying every possible set of transactions to determine which combination yields the highest profit. This approach is computationally intensive and not suggested for large datasets due to its exponential time complexity. However, it can be useful for educational purposes to understand the problem space.

Here’s an example:

...

Output: `...`

Similar to the Divide and Conquer method, we leave this example incomplete, emphasizing that brute force is unoptimized and impractical for real-world applications when dealing with stock prices.

## Bonus One-Liner Method 5: Functional Approach using Python’s Built-in Functions

A succinct Pythonic one-liner can achieve the same result using comprehensions and the `zip`

function, demonstrating Pythonβs capacity to implement complex logic in a single line. Here the sum is taken of all positive differences between consecutive days, representing profitable trades.

Here’s an example:

max_profit = lambda prices: sum(max(b - a, 0) for a, b in zip(prices[:-1], prices[1:])) print(max_profit([7, 1, 5, 3, 6, 4]))

Output: `7`

This lambda function `max_profit` uses list comprehension and `zip` to iterate over pairs of consecutive values from the list `prices`. The `max` function ensures that only positive differences (profits) are summed, mirroring the logic of the Greedy Approach, but in a more concise manner.

## Summary/Discussion

**Method 1:**Greedy Approach. This method is practical and efficient for its simplicity and speed. It may not be suitable for advanced stock trading algorithms that consider transaction costs or limits on transactions.**Method 2:**Dynamic Programming. This method is more versatile and can handle additional constraints. However, it is more complex and can be overkill for this problem where simpler methods suffice.**Method 3:**Divide and Conquer (Incomplete). Not recommended for this problem due to greater complexity with no added benefit.**Method 4:**Brute Force (Incomplete). It ensures a thorough search but is inefficient and impractical for large inputs.**Method 5:**Functional Approach. Demonstrates Python’s ability for compact code but may be less readable for those unfamiliar with functional programming paradigms.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.