Best Time To Buy And Sell Stock With Cooldown

John Blinks

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Aug 02, 2024

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Best Time to Buy and Sell Stock with Cooldown: A Comprehensive Guide

The "Best Time to Buy and Sell Stock with Cooldown" problem is a classic dynamic programming challenge in computer science. This problem explores the optimal strategy for maximizing profit when buying and selling stocks, with a cooldown period required between transactions.

Understanding the Problem

Imagine you're a stock trader with a unique rule: you can buy a stock, sell it for profit, and then must wait for a certain period (the cooldown) before buying again. This problem asks you to devise a plan that maximizes your profit over a given timeline of stock prices.

Approach: Dynamic Programming

Dynamic programming is a powerful technique for solving optimization problems. Here's how it works for the "Best Time to Buy and Sell Stock with Cooldown" problem:

1. Define Subproblems: We break the problem into smaller, overlapping subproblems.

   • dp[i][0]: The maximum profit achievable after day i with no stock in hand.
   • dp[i][1]: The maximum profit achievable after day i with a stock in hand.
   • dp[i][2]: The maximum profit achievable after day i in cooldown.

2. Base Cases:

   • dp[0][0] = 0: Starting with no stock, the profit is 0.
   • dp[0][1] = -prices[0]: Buying on day 0, the profit is the negative of the price.
   • dp[0][2] = 0: Initially in cooldown, the profit is 0.

3. Recursive Relation:

   • dp[i][0] = max(dp[i-1][0], dp[i-1][2]): On day i, either we had no stock before (and still don't) or we were in cooldown and now have no stock.
   • dp[i][1] = max(dp[i-1][1], dp[i-1][0] - prices[i]): On day i, either we had a stock before or we bought a stock on day i.
   • dp[i][2] = dp[i-1][1] + prices[i]: On day i, we are in cooldown, meaning we sold the stock on the previous day.

4. Final Result: The maximum profit is stored in dp[n-1][0] (where n is the total number of days).

Example:

Let's say the stock prices are: [1, 2, 3, 0, 2].

The maximum profit achieved is 8.

Implementation (Python):

def maxProfit(prices):
    n = len(prices)
    if n <= 1:
        return 0
    dp = [[0] * 3 for _ in range(n)]
    dp[0][0] = 0
    dp[0][1] = -prices[0]
    dp[0][2] = 0
    for i in range(1, n):
        dp[i][0] = max(dp[i-1][0], dp[i-1][2])
        dp[i][1] = max(dp[i-1][1], dp[i-1][0] - prices[i])
        dp[i][2] = dp[i-1][1] + prices[i]
    return dp[n-1][0]

Time and Space Complexity:

• Time Complexity: O(n), where n is the number of days.
• Space Complexity: O(n) for the dynamic programming table.

Conclusion

The "Best Time to Buy and Sell Stock with Cooldown" problem is a great example of how dynamic programming can be used to solve complex optimization problems. By breaking the problem down into smaller, overlapping subproblems, we can efficiently calculate the optimal solution.