Solve the Best Time to Buy and Sell Stock Problem

You are given an integer array of prices where prices[i] is the price of a given stock on an ith day. Design an algorithm to find the maximum profit.

Some constraints are as follows:

Example 1:

Example 2:

The problem is to find out the maximum profit you can get for a stock during a certain period of time.

We have 2 states for a stock on a Day(i):

Hence, there are 3 possible actions you can take on Day(i):

Let’s rearrange the above into actions and states.
The goal is to calculate the max profit while making sure that the state (buy or sell) stays on that day and there is no transaction overlap.

Now let’s add the transaction count constraint.
The goal is to calculate the max profit while making sure that the state (buy or sell) stays on that day, there is no transaction overlap and the transaction count is within limits.

If we calculate the above states for all days and transaction counts, we will eventually find out the maximum profit on the last day with the Sell state. Sell should be the last state since we want a transaction that’s closed towards the end since a Buy at the end would be a wasteful loss of money.

The above O(n * k) solution works very well if k less than the maximum number of transactions possible.

But what if the transaction count was unlimited or in other words greater than the maximum number of mutually-exclusive transactions possible during those days? Can we do better?

A simpler solution for this case is to iterate through the days and start a transaction each time we see a local valley and sell on the local peak.

The following code implements the logic explained above in Java.
Buy state is represented with a 1 and Sell state with 0.

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