<p>You are given an array <code>prices</code> where <code>prices[i]</code> is the price of a given stock on the <code>i<sup>th</sup></code> day.</p>

<p>Find the maximum profit you can achieve. You may complete <strong>at most two transactions</strong>.</p>

<p><strong>Note:</strong> You may not engage in multiple transactions simultaneously (i.e., you must sell the stock before you buy again).</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> prices = [3,3,5,0,0,3,1,4]
<strong>Output:</strong> 6
<strong>Explanation:</strong> Buy on day 4 (price = 0) and sell on day 6 (price = 3), profit = 3-0 = 3.
Then buy on day 7 (price = 1) and sell on day 8 (price = 4), profit = 4-1 = 3.</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> prices = [1,2,3,4,5]
<strong>Output:</strong> 4
<strong>Explanation:</strong> Buy on day 1 (price = 1) and sell on day 5 (price = 5), profit = 5-1 = 4.
Note that you cannot buy on day 1, buy on day 2 and sell them later, as you are engaging multiple transactions at the same time. You must sell before buying again.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> prices = [7,6,4,3,1]
<strong>Output:</strong> 0
<strong>Explanation:</strong> In this case, no transaction is done, i.e. max profit = 0.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= prices.length &lt;= 10<sup>5</sup></code></li>
	<li><code>0 &lt;= prices[i] &lt;= 10<sup>5</sup></code></li>
</ul>

auto speedUP = [](){
    ios::sync_with_stdio(false);
    cin.tie(0);
    cout.tie(0);
    return 'c';
}();
class Solution {
public:
    //Recursive
    //Important thing to learn, When I passed limit++ or ++limit than the answer was coming 
    //wrong but when I passed limit + 1 than the answer came correct.
    // int maxProfitReturn(vector<int>& prices, int index, int buy, int limit){
    //     //Base Case
    //     if(index >= prices.size() || limit == 2){
    //         return 0;
    //     }

    //     int profit = 0;
    //     if(buy){
    //         int buyStock = -prices[index] + maxProfitReturn(prices, index + 1, 0, limit);
    //         int skipStock = 0 + maxProfitReturn(prices, index + 1, 1, limit);
    //         profit = max(buyStock, skipStock);
    //     }
    //     else{
    //         int sellStock = prices[index] + maxProfitReturn(prices, index + 1, 1, limit + 1);
    //         int skipStock = maxProfitReturn(prices, index + 1, 0, limit);
    //         profit = max(sellStock, skipStock);
    //     }
    //     return profit;
    // }

    //Recursion + Memoization
    // int maxProfitUsingMemoization(vector<int>& prices, int index, int buy, int limit, vector<vector<vector<int>>> &dp){
    //     //Base Case
    //     if(index >= prices.size() || limit == 2){
    //         return 0;
    //     }

    //     if(dp[index][buy][limit] != -1){
    //         return dp[index][buy][limit];
    //     }
        
    //     int profit = 0;
    //     if(buy){
    //         int buyStock = -prices[index] + maxProfitUsingMemoization(prices, index + 1, 0, limit, dp);
    //         int skipStock = 0 + maxProfitUsingMemoization(prices, index + 1, 1, limit, dp);
    //         profit = max(buyStock, skipStock);
    //     }
    //     else{
    //         int sellStock = prices[index] + maxProfitUsingMemoization(prices, index + 1, 1, limit + 1, dp);
    //         int skipStock = maxProfitUsingMemoization(prices, index + 1, 0, limit, dp);
    //         profit = max(sellStock, skipStock);
    //     }
        
    //     dp[index][buy][limit] = profit;

    //     return dp[index][buy][limit];
    // }

    int maxProfitUsingTabulation(vector<int> &prices){
        int n = prices.size();
        vector<vector<vector<int>>> dp(n + 1, vector<vector<int>>(2, vector<int>(3, 0)));

        for(int index = n - 1 ; index >= 0 ; index--){
            for(int buy = 0 ; buy < 2 ; buy++){
                for(int limit = 1 ; limit >= 0 ; limit--){
                    int profit = 0;
                    if(buy){
                        int buyStock = -prices[index] + dp[index + 1][0][limit];
                        int skipStock = 0 + dp[index + 1][1][limit];
                        profit = max(buyStock, skipStock);
                    }
                    else{
                        int sellStock = prices[index] + dp[index + 1][1][limit + 1];
                        int skipStock = dp[index + 1][0][limit];
                        profit = max(sellStock, skipStock);
                    }
                    
                    dp[index][buy][limit] = profit;
                }
            }
        }
        return dp[0][1][0];
    }

    int maxProfit(vector<int>& prices) {
        // return maxProfitReturn(prices, 0, 1, 0);

        //3D DP
        // int n = prices.size();
        // vector<vector<vector<int>>> dp(n + 1, vector<vector<int>>(2, vector<int>(3, -1)));
        // return maxProfitUsingMemoization(prices, 0, 1, 0, dp);

        return maxProfitUsingTabulation(prices);
    }
};
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