**π‘ Problem Formulation:** In the stock market, the goal is to maximize profit by choosing the best time to buy and sell a stock within a given period. The challenge lies in determining the optimal buying day when prices are low, and selling day when prices peak. For example, if we’re given a list of daily stock prices like [7, 1, 5, 3, 6, 4], the maximum profit we can achieve by buying and selling once would be 5 (buying at 1 and selling at 6).

## Method 1: Brute Force

The Brute Force method involves comparing the price of each day with every subsequent day to find the maximum possible profit. This is computationally intensive and has a complexity of O(n^2).

Here’s an example:

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

Output: 5

This code snippet defines a function `maxProfit`

which takes a list of stock prices. It uses two nested loops to consider every possible pair of buying and selling days, keeping track of the highest profit found.

## Method 2: Dynamic Programming

The Dynamic Programming approach involves keeping track of the minimum price encountered so far and the maximum profit that can be obtained by selling on the current day. It reduces the time complexity to O(n).

Here’s an example:

def maxProfit(prices): min_price = float('inf') max_profit = 0 for price in prices: min_price = min(min_price, price) profit = price - min_price max_profit = max(max_profit, profit) return max_profit stock_prices = [7, 1, 5, 3, 6, 4] print(maxProfit(stock_prices))

Output: 5

This Python function keeps track of the minimum stock price so far and calculates the profit for each day by subtracting the current price from the minimum price, updating the maximum profit when a better opportunity is found.

## Method 3: Divide and Conquer

Divide and Conquer involves dividing the dataset into two halves, recursively finding the maximum profit in each half, and then finding the maximum profit crossing the divide. It has a complexity of O(n log n).

Here’s an example:

# This method is more of a conceptual approach and is less practical for this problem # compared to other methods given that it is more complex to implement and has higher time complexity # than the dynamic programming solution.

For brevity and practicality, the code for this method is omitted as it is complex and not as efficient as Method 2.

## Method 4: Using Libraries

Python’s powerful libraries like NumPy and Pandas can be utilized for efficient calculations, taking advantage of vectorized operations and convenient data structures respectively.

Here’s an example:

import numpy as np def maxProfit(prices): prices = np.array(prices) min_price = np.inf max_profit = 0 for price in prices: min_price = min(min_price, price) profit = price - min_price max_profit = max(max_profit, profit) return max_profit stock_prices = [7, 1, 5, 3, 6, 4] print(maxProfit(stock_prices))

Output: 5

This snippet uses NumPy for array manipulation, which can significantly speed up operations for large datasets due to NumPy’s optimized C backend.

## Bonus One-Liner Method 5: List Comprehension with min()

A Pythonic one-liner solution is possible by building a list of profits and selecting the maximum. It’s an elegant but less efficient variation of the Brute Force method.

Here’s an example:

stock_prices = [7, 1, 5, 3, 6, 4] print(max([j - i for i in stock_prices for j in stock_prices[stock_prices.index(i)+1:]]))

Output: 5

The code snippet calculates the profit for each pair of days using list comprehension and then finds the maximum profit in the produced list of profits, but it still retains a time complexity of O(n^2).

## Summary/Discussion

**Method 1:**Brute Force. Simple to understand and implement. Not scalable due to O(n^2) complexity.**Method 2:**Dynamic Programming. Efficient and scalable with O(n) complexity. Best for large datasets.**Method 3:**Divide and Conquer. More theoretical and complex. O(n log n) complexity makes it less practical than Method 2.**Method 4:**Using Libraries. Simplifies implementation using Python libraries. Efficiency gains on large datasets.**Method 5:**List Comprehension with min(). Pythonic and concise. Inefficient for massive datasets.

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.