# Best Time to Buy and Sell Stock III -- Two Transactions

Published: Jul 26, 2022

Hard Array

## Problem Description

You are given an array `prices` where `prices[i]` is the price of a given stock on the i-th 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).

Constraints:

• `1 <= prices.length <= 10**5`
• `0 <= prices[i] <= 10**5`

## Examples

``````example 1:
Input: prices = [3,3,5,0,0,3,1,4]
Explanation: 6 is the best profit which is from:
buy: index 3, sell: index 5, profit: 3
buy: index 6, sell: index 7, profit: 3
``````
``````example 2:
Input: prices = [1,2,3,4,5]
Explanation: 4 is the best profit which is from buying on the index 0 an selling on the index 4.
``````
``````example 3:
Input: prices = [7,6,4,3,1]
Explanation: the array value is ever descreasing. There is no profit available.
``````

## How to Solve

This is the third buy-sell-stock problem which has a slight variation to the basic one. In this problem, we can buy and sell stock at most twice. This is not an easy problem. However, if we focus on value differences in the array, we can find a couple of approaches to solve this.

Among those approaches, going over from left to right, from right to left, then, combine those might be an easy one. Like the basic buy and sell stock problem, calculate max profit and save it in a memoization array. Then, going over from right to left to calculate max profit and save it in another memoization array. When the max profit is calculated from right side, it sees how much decreased. Lastly, sum of left max profits and one index shifted right max profits are compared. The right profits came from one index shifted to the right, which means ranges are not overlapped. That’s how max profit from at most two transactions will be found.

## Solution

``````#include <vector>

using namespace std;

{
public:
int maxProfit(vector<int> &prices)
{
int n = prices.size();
vector<int> left(n, 0), right(n + 1, 0);
int left_min = prices, right_max = prices[n - 1];
for (int i = 1; i < n; ++i)
{
left[i] = max(left[i - 1], prices[i] - left_min);
left_min = min(left_min, prices[i]);
}
for (int i = n - 2; i >= 0; --i)
{
right[i] = max(right[i + 1], right_max - prices[i]);
right_max = max(right_max, prices[i]);
}
int max_profit = 0;
for (int i = 0; i < n; ++i)
{
max_profit = max(max_profit, left[i] + right[i + 1]);
}
return max_profit;
}
};
``````
``````class BestTimeToBuyAndSellStockThree {
public int maxProfit(int[] prices) {
int n = prices.length;
int[] left = new int[n], right = new int[n + 1];
int left_min = prices, right_max = prices[n - 1];
for (int i = 1; i < n; ++i) {
left[i] = Math.max(left[i - 1], prices[i] - left_min);
left_min = Math.min(left_min, prices[i]);
}
for (int i = n - 2; i >= 0; --i) {
right[i] = Math.max(right[i + 1], right_max - prices[i]);
right_max = Math.max(right_max, prices[i]);
}
int max_profit = 0;
for (int i = 0; i < n; ++i) {
max_profit = Math.max(max_profit, left[i] + right[i + 1]);
}
return max_profit;
}
}
``````
``````/**
* @param {number[]} prices
* @return {number}
*/
var maxProfit = function(prices) {
const n = prices.length;
let left = new Array(n).fill(0), right = new Array(n + 1).fill(0);
let left_min = prices, right_max = prices[n - 1];
for (let i = 1; i < n; i ++) {
left[i] = Math.max(left[i - 1], prices[i] - left_min);
left_min = Math.min(left_min, prices[i]);
}
for (let i = n - 2; i >= 0; i--) {
right[i] = Math.max(right[i + 1], right_max - prices[i]);
right_max = Math.max(right_max, prices[i]);
}
let max_profit = 0;
for (let i = 0; i < n; i++) {
max_profit = Math.max(max_profit, left[i] + right[i + 1]);
}
return max_profit;
};
``````
``````from typing import List

def maxProfit(self, prices: List[int]) -> int:
n = len(prices)
left, right = [0 for _ in range(n)], [0 for _ in range(n + 1)]
left_min, right_max = prices, prices[n - 1]
for i in range(1, n):
left[i] = max(left[i - 1], prices[i] - left_min)
left_min = min(left_min, prices[i])
for i in range(n - 2, -1, -1):
right[i] = max(right[i + 1], right_max - prices[i])
right_max = max(right_max, prices[i])
max_profit = 0
for i in range(n):
max_profit = max(max_profit, left[i] + right[i + 1])
return max_profit
``````
``````# @param {Integer[]} prices
# @return {Integer}
def max_profit(prices)
n = prices.length
left = Array.new(n, 0); right = Array.new(n + 1, 0)
left_min = prices; right_max = prices[n - 1]
(1...n).each do |i|
left[i] = [left[i - 1], prices[i] - left_min].max
left_min = [left_min, prices[i]].min
end
(n - 2).downto(0).each do |i|
right[i] = [right[i + 1], right_max - prices[i]].max
right_max = [right_max, prices[i]].max
end
max_profit = 0
(0...n).each do |i|
max_profit = [max_profit, left[i] + right[i + 1]].max
end
max_profit
end
``````

## Complexities

• Time: O(n) – n: length of the input array.
• Space: O(n)