Best Time To Buy And Sell Stock With Cooldown Leetcode 309 [Python]

If you're on a journey to become a proficient problem-solver in the world of coding, you've come to the right place.

Today, we're going to tackle the Best Time To Buy And Sell Stock With Cooldown problem.

This LeetCode problem, known as Problem 309, falls under the category of 2-D Dynamic Programming, and it's a medium-level challenge.

So, whether you're preparing for coding interviews or just looking to enhance your problem-solving skills, this guide will walk you through the solution step by step.

Problem Overview

Let's begin by understanding the problem.

You are given an array prices where prices[i] represents the price of a given stock on the ith day.

Your goal is to maximize the profit you can achieve.

Sounds simple, right?

Well, there's a twist.

You can complete as many transactions as you like, which means you can buy and sell shares multiple times.

However, there are some restrictions:

1. After selling a stock, you cannot buy a stock on the next day (cooldown period).
2. You cannot engage in multiple transactions simultaneously, which means you must sell the stock before you buy again.

To clarify, let's look at an example:

Example 1:

Input: prices = [1,2,3,0,2]

Output: 3

Explanation: The sequence of transactions is as follows: buy, sell, cooldown, buy, sell.

So, how can we approach this problem efficiently?

Let's dive into the details.

Understanding Constraints

Before we jump into the solution, it's crucial to understand the constraints provided in the problem:

• 1 <= prices.length <= 5000: The length of the prices array can be up to 5000.
• 0 <= prices[i] <= 1000: The stock prices can range from 0 to 1000. Keeping these constraints in mind, we need to devise an efficient algorithm to find the maximum profit.

Best Time to Buy And Sell Stock With Cooldown LeetCode Problem Solution

Bruteforce Approach

A brute force approach for solving this problem would be to consider all possible combinations of buying and selling on each day and calculate the maximum profit.

However, this approach has an exponential time complexity of O(2^n), making it impractical for larger input sizes.

An Efficient Approach with Dynamic Programming

To achieve an efficient solution, we can use dynamic programming.

The key idea is to maintain the state of whether we are buying or selling the stock.

We can create a dynamic programming table dp, where dp[(i, buying)] stores the maximum profit we can obtain at day i in either a buying or selling state.

Let's break down the recursive algorithm:

def maxProfit(prices):
    # Create a dynamic programming table dp
    dp = {}

    def dfs(i, buying):
        # Base case: If we go beyond the array's bounds, return 0. if i >= len(prices):
            return 0

        # Check if the result for this state is already cached.

if (i, buying) in dp:
            return dp[(i, buying)]

        # Initialize the profit in the cooldown state.

cooldown = dfs(i + 1, buying)

        if buying:
            # If we are in a buying state, we have two options: buy or cooldown.

buy = dfs(i + 1, not buying) - prices[i]
            dp[(i, buying)] = max(buy, cooldown)
        else:
            # If we are in a selling state, we can either sell or cooldown.

sell = dfs(i + 2, not buying) + prices[i]
            dp[(i, buying)] = max(sell, cooldown)

        return dp[(i, buying)]

    # Start the recursive function from day 0 in the buying state.

return dfs(0, True)

This dynamic programming solution eliminates redundant calculations by caching results using the dp table, which reduces the time complexity to O(n).

Python Code Solution

Here's the Python code that implements the efficient dynamic programming solution for the Best Time To Buy And Sell Stock With Cooldown problem:

def maxProfit(prices):
    dp = {}

    def dfs(i, buying):
        if i >= len(prices):
            return 0

        if (i, buying) in dp:
            return dp[(i, buying)]

        cooldown = dfs(i + 1, buying)

        if buying:
            buy = dfs(i + 1, not buying) - prices[i]
            dp[(i, buying)] = max(buy, cooldown)
        else:
            sell = dfs(i + 2, not buying) + prices[i]
            dp[(i, buying)] = max(sell, cooldown)

        return dp[(i, buying)]

    return dfs(0, True)

Time and Space Complexity

Now, let's analyze the time and space complexity of our solution:

Time Complexity: The dynamic programming approach has a time complexity of O(n), where n is the length of the prices array.

This is because we calculate the maximum profit for each day, and the caching mechanism avoids redundant calculations.

Space Complexity: The space complexity is O(n) as well.

This is due to the space used by the dp table to store the results for different states of each day.

Reasoning Behind Our Approach

The key to understanding the dynamic programming approach lies in recognizing the buying and selling states and their impact on the overall profit.

By efficiently calculating the maximum profit for each day while considering the cooldown period, we can optimize our solution.

Our approach ensures that we don't repeat calculations for the same state, which is crucial for solving problems with larger inputs efficiently.

It's a powerful technique to have in your problem-solving toolkit.

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Conclusion

In this blog post, we've explored the Best Time To Buy And Sell Stock With Cooldown problem on LeetCode.

We've discussed the problem statement, outlined the constraints, and presented an efficient dynamic programming solution in Python.

Understanding dynamic programming and how to cache results for different states is an essential skill for tackling complex coding challenges.

With the efficient approach presented here, you can confidently solve this problem and similar ones in coding interviews or competitive programming.

If you found this guide helpful, please feel free to comment, ask questions, make suggestions, and share the content.

Your engagement and feedback are highly appreciated.

Question Link: Best Time to Buy And Sell Stock With Cooldown

Now, go ahead and apply your newfound knowledge to other coding problems and keep refining your problem-solving skills.

Happy coding!