Problem Description

In this challenge, you are given an array prices where prices[i] represents the price of a given stock on the i-th day. Your task is to calculate the maximum profit that can be achieved through making as many buy-and-sell transactions as you wish, with a specific restriction: after selling a stock, you must wait one day before buying again.

To be more explicit, you can buy one share and then sell it. However, after you sell, you need to skip a day before you can initiate another purchase. This is called a cooldown period. Additionally, it is important to note that you cannot hold more than one share at a time; before you can buy another share, you must have already sold the previous one. Your goal is to find out the best strategy to maximize your profit under these terms.

Intuition

The intuition behind the solution involves dynamic programming. Since we cannot engage in a new transaction on the day following a sale, we can make a decision for each day - to buy, sell, or rest (do nothing). The key is to track the profits of these decisions while considering the cooldown.

We can maintain two states for each day that represent the decisions and conditions:

1. f0: The maximum profit we can have if we rest (do not buy or sell) on the current day or if we sell on the current day.
2. f1: The maximum profit we can have if we buy on the current day.

For day 0, our states would be initialized as f0 = 0 (we start with no stock and no cooldown) and f1 = -prices[0] (since buying the stock would cost us the price of the stock).

We need a third temporary variable f to store the previous value of f0 (from the day before), which will be used to update f1 since we cannot buy the stock on the same day we sell.

The dynamic programming transitions can be described as follows:

• f0 will be the maximum of itself (choose to rest, and profit doesn't change) or f1 + x (choosing to sell the stock bought at a cost of f1, which means adding the current price x).
• f1 will be the maximum of itself (choosing to do nothing with the previously bought stock) or f - x (buying a new stock after cooldown, which means subtracting the current price x from the profit made before cooldown indicated by f).

In each iteration, we go through the prices starting with the second day, because the first day's decisions are predetermined as f0 = 0 and f1 = -prices[0]. The result returned will be f0, which indicates the maximum profit that could be made with the last action being a rest or a sale.

The implementation of this solution successfully combines the conditions of the problem into a dynamic iterative process that captures the essence of buy-and-sell strategies while complying with the cooldown constraint.

Learn more about Dynamic Programming patterns.

Solution Approach

The solution is implemented using a dynamic programming approach, which utilizes iteration and optimal substructure properties common in such problems.

• Initialization: Before the loop begins, we prepare our initial state variables:

  • f0 is initialized to 0, representing the profit with no stock on hand at the start.
  • f1 is set to -prices[0], accounting for the cost of buying stock on the first day.
  • A temporary variable f will hold the previous value of f0 during iteration, serving as a memory for the state of profits two days ago.

• Iteration: We begin iterating through the prices array from the second day onward (since the first day's decisions have already been set up). During each iteration, we perform two key updates to represent possible actions and their outcomes:

  • f0 gets updated to max(f0, f1 + x), where x is the current day's stock price. This represents the maximum profit between not selling/buying anything on the current day (f0) and selling the stock bought at f1 price (therefore, f1 + x).
  • f1 gets updated to max(f1, f - x), which represents the maximum profit between keeping the stock obtained before (f1) or buying today (which requires the previous day's profit f minus the current price x; note this action is only permitted after a cooldown).

• Data Structures: We use three integer variables f0, f1, and f, which change with each iteration to keep track of our decision impacts. No additional data structures are used, making the space complexity linear.

• Pattern: The pattern applied here is reminiscent of state machine logic used in dynamic programming. Multiple states (f0 and f1) are considered, with transitions based on conditions specified by the problem statement. The decision on each day depends on the state of the previous days (either one or two days back).

• Algorithm Completion: The loop continues to make decisions and update states based on the stock prices for each day. The algorithm completes after the last day in the input array. The return value is f0 because it represents the maximum profit obtainable with the last action being rest or sale, aligned with the problem requirements.

The algorithm's overall complexity is O(n), as it involves a single pass through the prices array, and the operations within each iteration are constant time. This approach effectively balances the problem constraints and the need to make optimized decisions at each step, leading to an efficient and intuitive solution.

Example Walkthrough

Let's consider a small example using the solution approach outlined above. Suppose we have the following stock prices over a series of days: prices = [1, 2, 3, 0, 2]. Using the provided algorithm and dynamic programming approach, we'll calculate the maximum profit possible.

• Day 0: Before the loop starts, let's initialize our variables.

  • f0 is set to 0, because we haven't made any transaction yet.
  • f1 is set to -prices[0] which is -1, because we bought one share of stock at the price of 1.

• Day 1:

  • Update f to the previous value of f0 which is 0.
  • Calculate f0 as max(f0, f1 + prices[1]) = max(0, -1 + 2) = 1.
  • Update f1 as max(f1, f - prices[1]) = max(-1, 0 - 2) = -1.
  • At the end of Day 1, f0 is 1 and f1 is -1.

• Day 2:

  • Update f to the previous value of f0 which is 1.
  • Calculate f0 as max(f0, f1 + prices[2]) = max(1, -1 + 3) = 2.
  • Update f1 as max(f1, f - prices[2]) = max(-1, 1 - 3) = -1.
  • At the end of Day 2, f0 is 2 and f1 is -1.

• Day 3:

  • Update f to the previous value of f0 which is 2.
  • Calculate f0 as max(f0, f1 + prices[3]) = max(2, -1 + 0) = 2.
  • Update f1 as max(f1, f - prices[3]) = max(-1, 2 - 0) = 2.
  • At the end of Day 3, f0 is 2 and f1 is 2.

• Day 4:

  • Update f to the previous value of f0 which is 2.
  • Calculate f0 as max(f0, f1 + prices[4]) = max(2, 2 + 2) = 4.
  • Update f1 as max(f1, f - prices[4]) = max(2, 2 - 2) = 2.
  • At the end of Day 4, f0 is 4 and f1 is 2.

After iterating through all the days, our algorithm concludes and the maximum profit that we can yield from the given prices is stored in f0, which is 4. This means the best strategy would have resulted in a total profit of 4, considering all the buy-and-sell transactions and the compulsory cooldown period after each sale.

Solution Implementation

Time and Space Complexity

The given Python code defines a method maxProfit designed to find the maximum profit that can be achieved from a sequence of stock prices, where prices is a list of stock prices.

Time Complexity

The time complexity of the code is driven by a single loop that iterates through the list of stock prices once (excluding the first price which is used for initial setup). Inside the loop, the code performs a fixed number of comparisons and arithmetic operations for each price. Because these operations occur within the loop and their number does not depend on the size of prices, they constitute constant-time operations.

Therefore, the time complexity is determined solely by the number of iterations, which is n - 1, where n is the length of prices. Considering big O notation, which focuses on the upper limit of performance as the input size grows, the time complexity of this algorithm is O(n).

Space Complexity

The space complexity is assessed based on the additional memory used by the algorithm as a function of the input size. In this code, only a fixed number of variables f, f0, and f1 are used, regardless of the size of the input list prices. There is no use of any additional data structures that would grow with the input size.

Thus, the space complexity of the algorithm is O(1), denoting constant space usage.

Learn more about how to find time and space complexity quickly using problem constraints.