Best Time to Buy and Sell Stock with Cooldown

Problem Statement

In the given problem, you are provided with an array called prices, where each element prices[i] represents the price of a stock on the ith day. Your objective is to determine the maximum profit possible from buying and selling stocks over the course of these days. You are permitted to conduct as many buy-and-sell transactions as you deem fit. However, there are specific rules that must be followed:

• You cannot hold multiple stocks simultaneously; sell before buying another.
• After a sale, there is a mandatory cooldown period of one day before you can purchase again.

This need to manage the timing of buys and sells, along with the cooldown constraint, adds a layer of complexity to the strategic planning of transactions.

Examples

Example 1

Input:

prices = [1,2,3,0,2]

Output:

3

Explanation:

transactions = [buy, sell, cooldown, buy, sell]

Example 2

Input:

prices = [1]

Output:

0

Constraints

• 1 <= prices.length <= 5000
• 0 <= prices[i] <= 1000

Approach and Intuition

To approach this problem, understanding the behavior of stock prices and the impact of transaction timings is crucial. The primary goal is to buy low and sell high repetitively, harvesting profits from each transaction while obeying the cooldown period after each sale.

1. Review the sequence of stock prices to identify potential profit opportunities—a rise in stock prices between days suggests a beneficial buy-sell cycle.
2. Implement a dynamic programming based approach that maintains a record of profits in different states:
   • state_0: The maximum profit achieved so far without holding any stock and not being in a cooldown.
   • state_1: The maximum profit achieved so far holding one stock.
   • state_2: The maximum profit achieved having sold the stock and being in a cooldown.
3. The state transitions for subsequent days are defined by:
   • Transition from state_0 to state_1 by buying stock, involving a profit decrement by the stock price of the current day.
   • Transition from state_1 to state_0 by selling stock, augmenting profit by the stock price of the current day.
   • Transition from state_1 to state_2 directly after a sale.
   • From state_2, you can move to state_0 post cooldown to either buy on the current day or continue without buying.
4. Iterate over the array of prices while updating these state values according to the possible actions (buy, sell or cooldown).

The solution to this problem is highly reliant on correct state management with dynamic transitions based on historical data points and constraints, ensuring maximum profit extraction from the available series of prices.

Solutions

class Solution {
  public int calculateMaxProfit(int[] stockPrices) {
    int[] maxProfits = new int[stockPrices.length + 2];

    for (int i = stockPrices.length - 1; i >= 0; i--) {
      int maxProfitCase1 = 0;
      for (int sellIndex = i + 1; sellIndex < stockPrices.length; sellIndex++) {
        int potentialProfit = (stockPrices[sellIndex] - stockPrices[i]) + maxProfits[sellIndex + 2];
        maxProfitCase1 = Math.max(potentialProfit, maxProfitCase1);
      }

      int maxProfitCase2 = maxProfits[i + 1];

      maxProfits[i] = Math.max(maxProfitCase1, maxProfitCase2);
    }
    return maxProfits[0];
  }
}