You are given an array prices where prices[i] is the price of a given stock on the ith day.
Find the maximum profit you can achieve. You may complete at most two transactions.
Note: You may not engage in multiple transactions simultaneously (i.e., you must sell the stock before you buy again).
Example 1:
Input: prices = [3,3,5,0,0,3,1,4]
Output: 6
Explanation: 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.
Example 2:
Input: prices = [1,2,3,4,5]
Output: 4
Explanation: 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.
Example 3:
Input: prices = [7,6,4,3,1]
Output: 0
Explanation: In this case, no transaction is done, i.e. max profit = 0.
Constraints:
1 <= prices.length <= 105
0 <= prices[i] <= 105
Intuition
Just introduce another state number of tries (i.e is 2)
Becomes 3D Dp and solve as same as Earlier question
classSolution {public:intmaxProfit(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<=1; buy++){for(int tries=1; tries<=2; tries++){ // Starting tries from 1 not 0 because it is base case condition //At tries = 0 we return 0 as cannot buy int profit;if(buy){ profit =max( -prices[index] +dp[index+1][0][tries],dp[index+1][1][tries]); }else{ profit =max( +prices[index] +dp[index+1][1][tries-1],dp[index+1][0][tries]); }dp[index][buy][tries] = profit; } } }returndp[0][1][2]; }};
Approach 3:
Space Optimized
C++
classSolution {public:intmaxProfit(vector<int>& prices) {int n =prices.size(); vector<vector<int>>cur(2,vector<int>(3,0)); vector<vector<int>>prev(2,vector<int>(3,0));for(int index=n-1; index>=0; index--){for(int buy=0; buy<=1; buy++){for(int tries=1; tries<=2; tries++){ // Starting tries from 1 not 0 because it is base case condition //At tries = 0 we return 0 as cannot buy int profit;if(buy){ profit =max( -prices[index] +prev[0][tries],prev[1][tries]); }else{ profit =max( +prices[index] +prev[1][tries-1],prev[0][tries]); }cur[buy][tries] = profit; } } prev = cur; }returnprev[1][2]; }};
Approach 4:
C++
Can maintain a N*4(Transaction number) StateInstead of N*2(Buy)*3(Transaction count)In this0123B S B SOn even tries you buy and odd you sellBase Cases will be At trasaction number ==4 we return0