Maximize Profit with at Most K Transactions

Question

The question asks you to determine the maximum profit that can be achieved from stock transactions, given that you can make at most K transactions.

Codemia · Accepted Answer

To solve the problem of maximizing profit with at most K transactions, we need to understand that each transaction consists of a buy and a sell operation. The goal is to find the best days to buy and sell stocks to maximize our profits while adhering to the limit of K transactions.

We can use a dynamic programming approach to tackle this problem. We will create a 2D array `dp` where `dp[i][j]` represents the maximum profit achievable with at most `i` transactions by the end of day `j`. The idea is to fill this table based on previous computations.

The algorithm can be summarized in the following steps:

1. Initialize a 2D array `dp` of size `(K+1) x (n)` where `n` is the number of days.
2. Iterate over the number of transactions from 1 to K.
3. For each transaction, iterate through the days and calculate the maximum profit by considering the best day to sell.
4. Update the `dp` table accordingly.

Here is the Python implementation:

```python
def maxProfit(K, prices):
    n = len(prices)
    if n == 0 or K == 0:
        return 0
    
    if K >= n // 2:
        return sum(max(prices[i + 1] - prices[i], 0) for i in range(n - 1))
    
    dp = [[0] * n for _ in range(K + 1)]
    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][n - 1]
```

The time complexity of this algorithm is O(K * n), where K is the number of transactions and n is the number of days. The space complexity is O(K * n) as well due to the `dp` table. However, we can optimize the space complexity to O(n) by using a rolling array approach.

Interviewers are looking for a clear understanding of dynamic programming concepts, the ability to articulate the thought process behind the algorithm, and proficiency in coding. They also appreciate candidates who can recognize and discuss potential optimizations and edge cases.

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