Best Time to Buy and Sell Stock II - C++ 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 by buying and selling stocks given the constraint of at most k transactions. This problem is significant in financial markets where traders aim to maximize their returns. A common pitfall is to misunderstand the constraint of not engaging in multiple transactions simultaneously.

Approach

To solve this problem, we need to consider both a naive and an optimized approach:

Naive Solution

The naive solution involves trying all possible pairs of buy and sell days, which is not optimal due to its high time complexity of O(n^2). This approach is impractical for large inputs.

Optimized Solution

We can use dynamic programming to optimize the solution. The idea is to maintain two arrays: one for the maximum profit up to day i with at most j transactions, and another for the maximum profit up to day i with exactly j transactions.

Algorithm

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

Initialize a 2D array dp where dp[i][j] represents the maximum profit up to day i with at most j transactions.
Iterate over the number of transactions from 1 to k.
For each transaction, iterate over the days and calculate the maximum profit by considering whether to sell on that day or not.
Update the dp array accordingly.

Code Implementation

#include <vector>
#include <algorithm>
using namespace std;

// Function to calculate the maximum profit
int maxProfit(int k, vector<int>& prices) {
    int n = prices.size();
    if (n == 0) return 0;

    // If k is greater than n/2, it's equivalent to unlimited transactions
    if (k >= n / 2) {
        int maxProfit = 0;
        for (int i = 1; i < n; ++i) {
            if (prices[i] > prices[i - 1]) {
                maxProfit += prices[i] - prices[i - 1];
            }
        }
        return maxProfit;
    }

    // DP array to store the maximum profit
    vector<vector<int>> dp(k + 1, vector<int>(n, 0));

    for (int i = 1; i <= k; ++i) {
        int maxDiff = -prices[0];
        for (int j = 1; j < n; ++j) {
            dp[i][j] = max(dp[i][j - 1], prices[j] + maxDiff);
            maxDiff = max(maxDiff, dp[i - 1][j] - prices[j]);
        }
    }

    return dp[k][n - 1];
}

Complexity Analysis

The time complexity of the optimized 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 2D DP array.

Edge Cases

Consider the following edge cases:

Empty price array: The output should be 0.
Single day price: The output should be 0 as no transactions can be made.
k greater than n/2: The algorithm should handle this by treating it as unlimited transactions.

Testing

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

Simple cases with small arrays.
Cases with increasing and decreasing prices.
Cases with random prices and varying k values.

Thinking and Problem-Solving Tips

When approaching such problems, consider breaking down the problem into smaller subproblems and using dynamic programming to optimize the solution. Practice similar problems to improve your problem-solving skills.

Conclusion

In this blog post, we discussed the problem of finding the best time to buy and sell stock with at most k transactions. We explored both naive and optimized solutions, provided a detailed algorithm, and analyzed the complexity. Understanding and solving such problems is crucial for improving algorithmic thinking and problem-solving skills.

Additional Resources

For further reading and practice, consider the following resources: