# Introduction

Welcome to Day 2 of the Blind 75 Challenge! Today I will be tackling the problem of finding the maximum profit by buying and selling stock once, a common problem in algorithm interviews and coding competitions. In this blogpost, I will explore a simple and efficient algorithm to solve this problem in Python, using only one pass/iteration through the array of stock prices.

# Problem

You are given an array `prices` where `prices[i]` is the price of a given stock on the `ith` day. You want to maximize your profit by choosing a single day to buy one stock and choosing a different day in the future to sell that stock. Return the maximum profit you can achieve from this transaction. If you cannot achieve any profit, return 0.

# Problem Definition and Explanation

In this question, we are given an array of stock prices, where each element in the array represents the price in a particular day, and the `ith` index location of a day of the stock price corresponds the the `ith` day. We are expected to find a solution that will provide the maximum profit. The maximum profit here is defined as the largest difference between the largest positive number between the selling price and the buying price. As it is we would want to maximize the profit by buying the stock on one day and selling it on a different day in the future.

For example given the array below

``````[1, 2, 3, 7, 4, 3]
``````

The maximum profit would be `6`. Since the minimum price in the array is `1` and the maximum price is `7` (which comes at a later day). Again, the task is to find the solution that gets the maximum profit from the array.

# Intuition behind the solution

Naive solution One possible approach for finding the maximum profit by buying and selling stock is to first find the minimum and maximum values in the array and then calculate the difference between them. This can be implemented as follows:

``````minimum_price = min(input_list)
maximum_[price = max(input_list)
``````

Then get the maximum profit by finding the difference between the maximum price and minimum price.

``````maximum_profit = maximum price -minimum price
``````

While finding the minimum and maximum values in the array and subtracting them to get the maximum profit might work in some cases, it is not a correct solution in all cases. This approach is not always correct as it fails to consider cases where buying the stock on a day preceding the selling day would result in a greater profit.

Consider the following example:

``````[3, 2, 6, 5, 0, 3]
``````

If we simply find the minimum value (0) and the maximum value (6), we would get a profit of 6 - 0 = 6, which is incorrect. The correct maximum profit that can be made in this case is 6 - 2 = 4, by buying the stock on day 2 (price 2) and selling it on day 3 (price 6). Since you can only buy on a day preceding the selling day. Therefore, finding the minimum and maximum values in the array and subtracting them is not a correct solution for this problem. Instead, we need to use an algorithm that finds the maximum profit that can be made by buying and selling the stock once.

Using One-Pass Algorithm

To overcome the limitations of the naive approach, a one-pass algorithm can be used. This algorithm processes each element of the data structure only once and keeps track of the minimum price seen so far and the maximum profit that can be made from selling the stock at the current price.

Here are the steps for implementing the one-pass algorithm:

1. First check if the list is empty. If empty return 0 as maximum profit.
``````if not prices:
return 0
``````
2. Initialize the minimum price to the first element in the array.

``````maximum_profit = 0
``````
3. Traverse through the array.

``````for price in input_list:
``````
4. Check if the current price is lower than the minimum price.

`````` if price < minimum_price:
``````
5. If it is, update the minimum price (since no profit can be made from a lower price.

``````   minimum_price = price
``````
6. Else calculate the profit that can be made by selling the stock at the current price. This is the difference between the current price and the minimum price so far.

``````        else:
profit = price - minimum_price
``````
7. Finally, compare the current profit with the maximum profit seen so far and update the profit if the current profit is greater.

``````         if profit > maximum_profit_seen:
maximum_profit_seen = profit
``````
8. Return the maximum profit obtained. ```python return maximum_profit_seen

Using this algorithm, we can find the maximum profit that can be made by buying and selling the stock once, taking into account the constraint that the buying day must precede the selling day.

# Putting it altogether - Code

``````def maximum_profit_buy(input_list: list):
# Check if the input list is empty
if len (input_list) == 0:
return 0

# Initialize the minimum price and maximum profit seen so far
minimum_price = input_list
maximum_profit_seen = 0

# Traverse through the input list
for price in input_list:
# Update the minimum price seen so far
if price < minimum_price:
minimum_price = price
else:
# Calculate the profit that can be made by selling at the current price
profit = price - minimum_price
# Update the maximum profit seen so far if the current profit is greater
if profit > maximum_profit_seen:
maximum_profit_seen = profit

# Return the maximum profit seen so far
return maximum_profit_seen
``````

# Testing

Let's test the `maximum_profit_buy` function:

``````print(maximum_profit_buy([7,6,4,3,1])) # Expected output: 0
print (maximum_profit_buy ([1, 2, 3, 7, 4, 3])) # Expected Output 6
``````

The First test case represents the array `[7,6,4,3,1]`, where the stock price decreases every day. In this case, no profit can be made, so the expected output is `0`. For the second test case, we have an array `[1, 2, 3, 7, 4, 3]` and the maximum profit that can be made by buying stock on `day 1` `price 1` is and selling it on `day 4` `price 7` is 6 which is the expected output.

# Time and Space Complexity

The function has a time complexity of O(n), where n is the length if the input array. This is so since we need to iterate through the array only once. The space complexity is O(1) since we only use a constant amount of extra space to store the minimum price seen so far and the maximum profit.

# Use cases

The problem of finding the maximum profit by buying and selling a stock once is a common problem in coding interviews and competitions. It can also be used in finance and economics to analyze the performance of stocks and investments.

# Conclusion

In this blog post, we explored a simple and efficient algorithm to solve the problem of finding the maximum profit that can be made by buying and selling a stock once. By using the one-pass approach and keeping track of the minimum price seen so far and the maximum profit that can be made by selling the stock at the current price, we can solve this problem in O(n) time complexity, where n is the length of the input array.