Best Time to Buy and Sell Stock II - Python Solution and Time Complexity Analysis

Say you have an array for which the i-th element is the price of a given stock on day i.

Design an algorithm to find the maximum profit. You may complete at most k transactions.

Note:
You may not engage in multiple transactions at the same time (ie, you must sell the stock before you buy again).

Example 1:

Input: [2,4,1], k = 2
Output: 2
Explanation: Buy on day 1 (price = 2) and sell on day 2 (price = 4), profit = 4-2 = 2.

Example 2:

Input: [3,2,6,5,0,3], k = 2
Output: 7
Explanation: Buy on day 2 (price = 2) and sell on day 3 (price = 6), profit = 6-2 = 4.
             Then buy on day 5 (price = 0) and sell on day 6 (price = 3), profit = 3-0 = 3.

Understanding the Problem

The core challenge of this problem is to maximize the profit from stock transactions given the constraint that you can only hold one stock at a time and you can complete at most k transactions. This problem is significant in financial markets where traders aim to maximize their returns by buying and selling stocks at optimal times.

Common pitfalls include misunderstanding the constraint of not holding multiple stocks simultaneously and not considering the optimal number of transactions.

Approach

To solve this problem, we can use dynamic programming. The idea is to maintain a table where the entry at index (i, j) represents the maximum profit achievable using at most i transactions up to day j.

We can start with a naive solution that tries all possible pairs of buy and sell days, but this would be inefficient with a time complexity of O(n^2). Instead, we can optimize this using dynamic programming.

Optimized Solution

We use a 2D DP array where dp[i][j] represents the maximum profit using at most i transactions up to day j. The state transition can be derived as follows:

For each day j, we can either not trade, in which case dp[i][j] = dp[i][j-1], or we can sell the stock on day j. To find the best day to buy the stock, we need to consider all previous days m and calculate the profit as prices[j] - prices[m] + dp[i-1][m].

To optimize this, we can keep track of the maximum value of dp[i-1][m] - prices[m] as we iterate through the days.

Algorithm

Here is a step-by-step breakdown of the algorithm:

Initialize a 2D DP array dp with dimensions (k+1) x n, where n is the number of days.
Iterate over the number of transactions from 1 to k.
For each transaction, iterate over the days from 1 to n-1.
Keep track of the maximum value of dp[i-1][m] - prices[m] as max_diff.
Update dp[i][j] as the maximum of dp[i][j-1] and prices[j] + max_diff.

Code Implementation

def maxProfit(k, prices):
    # If there are no prices or k is 0, return 0 profit
    if not prices or k == 0:
        return 0
    
    n = len(prices)
    
    # If k is greater than n//2, it's equivalent to unlimited transactions
    if k >= n // 2:
        return sum(max(prices[i] - prices[i - 1], 0) for i in range(1, n))
    
    # Initialize the DP table
    dp = [[0] * n for _ in range(k + 1)]
    
    # Fill the DP table
    for i in range(1, k + 1):
        max_diff = -prices[0]
        for j in range(1, n):
            dp[i][j] = max(dp[i][j - 1], prices[j] + max_diff)
            max_diff = max(max_diff, dp[i - 1][j] - prices[j])
    
    return dp[k][-1]

# Example usage
print(maxProfit(2, [2, 4, 1]))  # Output: 2
print(maxProfit(2, [3, 2, 6, 5, 0, 3]))  # Output: 7

Complexity Analysis

The time complexity of this solution is O(k * n), where k is the number of transactions and n is the number of days. The space complexity is also O(k * n) due to the DP table.

This is a significant improvement over the naive O(n^2) approach.

Edge Cases

Consider the following edge cases:

Empty prices array: The output should be 0.
k = 0: No transactions allowed, so the output should be 0.
k >= n // 2: This is equivalent to unlimited transactions, and we can use a simpler greedy approach.

Testing

To test the solution comprehensively, consider the following test cases:

Simple cases with small arrays.
Cases with large arrays to test performance.
Edge cases as mentioned above.

Using a testing framework like unittest in Python can help automate and manage these tests.

Thinking and Problem-Solving Tips

When approaching such problems, it's essential to:

Understand the constraints and edge cases.
Start with a naive solution and identify its limitations.
Think about how to optimize the solution using techniques like dynamic programming.
Practice similar problems to improve problem-solving skills.

Conclusion

In this blog post, we discussed the problem of finding the maximum profit from stock transactions with at most k transactions. We explored a dynamic programming approach to solve the problem efficiently and analyzed its complexity. Understanding and solving such problems is crucial for developing strong algorithmic skills.

Additional Resources

For further reading and practice, consider the following resources: