**π‘ Problem Formulation:** Investors are often faced with the challenge of maximizing profit through a limited number of buy and sell transactions. Given an array representing the price of a stock on different days and an integer `k`

, representing the maximum number of transactions allowed (buy and sell constituting one transaction), this article explores Python methods to calculate the maximum profit one can obtain. Example input: prices = [3,2,6,5,0,3], k = 2. Desired output: 7 (Buy on day 2 at 2, sell on day 3 at 6, buy on day 5 at 0, and sell on day 6 at 3).

## Method 1: Dynamic Programming Approach

This method involves using dynamic programming to store the maximum profit that can be made up to the i-th day with at most j transactions. The main idea is to build a 2D array where each element (i, j) represents the maximum profit from at most j transactions up to day i. This method is robust but can be memory-intensive for large datasets.

Here’s an example:

def maxProfit(prices, k): n = len(prices) if not prices or k == 0: return 0 dp = [[0] * (k + 1) for _ in range(n)] for i in range(1, k + 1): max_diff = -prices[0] for j in range(1, n): dp[j][i] = max(dp[j-1][i], prices[j] + max_diff) max_diff = max(max_diff, dp[j][i-1] - prices[j]) return dp[-1][-1] print(maxProfit([3, 2, 6, 5, 0, 3], 2))

The output of this code snippet is: `7`

.

This code defines a function `maxProfit`

that calculates the maximum profit given stock prices and the number of transactions `k`

. It uses a dynamic table to keep track of profits and iterates through prices to find the maximum profit with an optimally efficient solution.

## Method 2: State Machine Approach

The state machine approach models the buying and selling of stocks as a series of states. Each state represents a specific number of transactions and whether the stock is currently held or not. This method can be easier to reason about and sometimes more intuitive than dynamic programming.

Here’s an example:

def maxProfit(prices, k): if not prices: return 0 n = len(prices) buy = [-float('inf')]*(k+1) sell = [0]*(k+1) for price in prices: for i in range(1, k+1): buy[i] = max(buy[i], sell[i-1] - price) sell[i] = max(sell[i], buy[i] + price) return sell[k] print(maxProfit([3,2,6,5,0,3], 2))

The output of this code snippet is: `7`

.

This snippet uses the state machine approach, where `buy`

and `sell`

arrays represent the state of having bought or sold stocks after a certain transaction. It updates these states with the new price information to calculate the maximum profit efficiently.

## Method 3: Space Optimized Dynamic Programming

This method is a variation of the dynamic programming approach where space complexity is optimized. Instead of maintaining a full 2D array, two 1D arrays are used, to store the current and previous profits, thus reducing space requirements significantly.

Here’s an example:

def maxProfit(prices, k): if not prices: return 0 n = len(prices) dp = [0] * (k + 1) for j in range(1, n): prev_dp = dp.copy() for i in range(1, k + 1): max_diff = -prices[0] for l in range(1, j+1): max_diff = max(max_diff, prev_dp[i-1] - prices[l]) dp[i] = max(dp[i], max_diff + prices[j]) return dp[k] print(maxProfit([3,2,6,5,0,3], 2))

The output of this code snippet is: `7`

.

In this method, each iteration updates an array of maximum profits for up to the current transaction. It optimizes space by reusing two arrays to track the changing state of profits throughout the price changes.

## Method 4: Quick Solution for Unlimited Transactions

If the number of transactions k is not a constraint (i.e., it is very large compared to the number of days), the problem simplifies to finding all ascending sequences of prices. This method adds up the profit from each of these increasing segments.

Here’s an example:

def maxProfit(prices): return sum(max(prices[i+1] - prices[i], 0) for i in range(len(prices)-1)) print(maxProfit([3,2,6,5,0,3]))

The output of this code snippet is: `7`

.

This method assumes an unlimited number of transactions and simply calculates the positive price differences through the entire array. It is a quick method for profit calculation, but only applicable when transaction limits are not a factor.

## Bonus One-Liner Method 5: Recursion with Memoization

This method employs recursion with memoization, storing the results of subproblems to avoid redundant calculations. It’s elegant but could lead to a stack overflow for very large inputs or exceed time limits due to its recursive nature.

Here’s an example:

from functools import lru_cache @lru_cache(maxsize=None) def maxProfit(i, transactions_remaining, holding, prices): if i == len(prices) or transactions_remaining == 0: return 0 do_nothing = maxProfit(i+1, transactions_remaining, holding, prices) if holding: return max(do_nothing, prices[i] + maxProfit(i+1, transactions_remaining-1, not holding, prices)) return max(do_nothing, -prices[i] + maxProfit(i+1, transactions_remaining, not holding, prices)) print(maxProfit(0, 2, False, (3,2,6,5,0,3)))

The output of this code snippet is: `7`

.

This snippet uses Python’s `@lru_cache`

decorator to memoize the function calls, thereby optimizing the recursive solution that explores all possible transactions to find the maximum profit.

## Summary/Discussion

**Method 1:**Dynamic Programming Approach. Strengths include a clear understanding and robustness for smaller datasets. Weaknesses: potentially high memory usage for large datasets.

**Method 2:**State Machine Approach. Strengths: intuitive understanding of transaction states, good scalability. Weaknesses: may be less intuitive for those unfamiliar with state machines.

**Method 3:**Space Optimized Dynamic Programming. Strengths: reduced space complexity while maintaining DP’s benefits. Weaknesses: still requires careful implementation to avoid errors.

**Method 4:**Quick Solution for Unlimited Transactions. Strengths: extremely fast and simple for the right conditions. Weaknesses: not applicable for limited k transactions.

**Method 5:**Recursion with Memoization. Strengths: elegant and clear logic. Weaknesses: risk of stack overflow and possibly slower due to recursive calls.

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.