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Best Time to Buy and Sell Stock III - Solution & Explanation

HardArrayDynamic Programming19 min readAsked at: Amazon, Microsoft, Apple +10
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Problem Statement

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

Approach Overview

Problem Overview: You are given an array where prices[i] represents the stock price on day i. You can complete at most two transactions (buy then sell). The goal is to compute the maximum profit possible without holding more than one stock at a time.

Approach 1: Dynamic Programming with 4 State Variables (O(n) time, O(1) space)

This approach models the trading process as four states: first buy, first sell, second buy, and second sell. Each state represents the best profit achievable after that operation. While iterating through the price array, update the states using simple max comparisons such as firstBuy = max(firstBuy, -price) and firstSell = max(firstSell, firstBuy + price). The key insight is that every transaction builds on the previous one, so the second buy depends on the profit from the first sell. This reduces the entire dynamic programming formulation into four running variables, giving an optimal O(n) solution with constant memory.

Approach 2: Two Arrays for Tracking Left and Right Profits (O(n) time, O(n) space)

This strategy splits the problem into two independent subproblems. First, traverse the array from left to right and compute leftProfit[i], the maximum profit achievable with one transaction in the range [0, i]. Then traverse from right to left to compute rightProfit[i], the best profit for one transaction in [i, n-1]. Combining these two arrays lets you evaluate every possible split point where the first transaction happens before the second. The final answer is the maximum value of leftProfit[i] + rightProfit[i]. This technique works well when reasoning about prefix and suffix profits in array problems.

Recommended for interviews: The 4‑state dynamic programming solution is usually expected in interviews because it demonstrates strong understanding of dynamic programming state transitions and optimization. The two‑array method is easier to reason about conceptually, but the constant‑space DP version shows deeper algorithmic skill and cleaner implementation.

Approach 1: Approach 1: Dynamic Programming with 4 State Variables

This approach utilizes dynamic programming with four state variables to track the profit in various stages of transactions: before the first buy, after the first buy (before the sell), after the first sell (before the second buy), and after the second buy (before the final sell). By iterating over the prices array, we update these states with the maximum profit achievable at each step.

This solution initializes four state variables representing the different transaction stages. We iterate over the price list to update these states, maximizing profit at each step. The final state, secondSell, holds the maximum profit obtainable.

Code

CC++JavaPythonC#JavaScript

C

C++

Java

Python

C#

JavaScript

Complexity

Time Complexity: O(n), where n is the number of days (prices length).
Space Complexity: O(1), constant space used regardless of input size.

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Approach 2: Approach 2: Two Arrays for Tracking Left and Right Profits

This approach involves creating two arrays that store the maximum profit achievable from the left side to each index and from each index to the right in the prices array. Combining these two arrays helps derive the total maximum profit achievable through two transactions.

This C solution uses two auxiliary arrays to track maximal profit potential from either direction along the price sequence. Profit amounts are compiled crosswise to determine peak profitability over the entire period.

Code

CC++JavaPythonC#JavaScript

C

C++

Java

Python

C#

JavaScript

Complexity

Time Complexity: O(n), as we iterate twice through prices.
Space Complexity: O(n), due to the use of two auxiliary arrays.

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Approach 3: Dynamic Programming

We define the following variables:

  • f1 represents the maximum profit after the first purchase of the stock;
  • f2 represents the maximum profit after the first sale of the stock;
  • f3 represents the maximum profit after the second purchase of the stock;
  • f4 represents the maximum profit after the second sale of the stock.

During the traversal, we directly calculate f1, f2, f3, f4. We consider that buying and selling on the same day will result in a profit of 0, which will not affect the answer.

Finally, return f4.

The time complexity is O(n), where n is the length of the prices array. The space complexity is O(1).

Code

PythonJavaC++GoTypeScriptRustC#

Python

Java

C++

Go

TypeScript

Rust

C#

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Complexity Comparison

ApproachComplexity
Approach 1: Dynamic Programming with 4 State Variables

Time Complexity: O(n), where n is the number of days (prices length).
Space Complexity: O(1), constant space used regardless of input size.

Approach 2: Two Arrays for Tracking Left and Right Profits

Time Complexity: O(n), as we iterate twice through prices.
Space Complexity: O(n), due to the use of two auxiliary arrays.

Dynamic Programming—

Detailed Complexity Analysis

ApproachTimeSpaceWhen to Use
Dynamic Programming with 4 State VariablesO(n)O(1)Best overall solution. Preferred in interviews due to optimal space usage and clean DP transitions.
Two Arrays for Left and Right ProfitsO(n)O(n)Useful for understanding the split of two transactions and easier to visualize prefix/suffix profits.

Video Solution

DP 37. Buy and Sell Stocks III | Recursion to Space Optimisation • take U forward • 317,022 views views

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Frequently Asked Questions

Is Best Time to Buy and Sell Stock III easy or hard?
LeetCode classifies this problem as Hard because it extends earlier stock problems with a two‑transaction constraint. The challenge is modeling the correct dynamic programming states and updating them in the right order.
Best Time to Buy and Sell Stock III Python/Java solution
Python and Java implementations typically follow the four‑state DP pattern. Maintain variables for firstBuy, firstSell, secondBuy, and secondSell and update them while iterating through the prices array. The logic stays identical across languages with O(n) time and O(1) space.
How to solve Best Time to Buy and Sell Stock III in O(n)?
Track four DP states while iterating through prices: firstBuy, firstSell, secondBuy, and secondSell. Update them sequentially using max comparisons based on the current price. This models two transactions directly and produces the maximum profit in a single pass with constant space.
What is the best approach for Best Time to Buy and Sell Stock III?
The most efficient approach uses dynamic programming with four state variables representing first buy, first sell, second buy, and second sell. Each state tracks the maximum profit achievable after that step. Iterating through the price array once updates these states, producing the optimal answer in O(n) time and O(1) space.
Is Best Time to Buy and Sell Stock III asked at Google/Amazon/Meta?
Best Time to Buy and Sell Stock variations frequently appear in interviews at companies like Amazon, Google, and Meta. Interviewers use this problem to test dynamic programming skills, state transitions, and understanding of transaction constraints.
What data structure is used in Best Time to Buy and Sell Stock III?
The main data structure is a simple array of stock prices. The optimal solution uses dynamic programming variables to track state transitions, while an alternative solution uses two auxiliary arrays to store prefix and suffix profits.
What is the time complexity of Best Time to Buy and Sell Stock III?
The optimal algorithms run in O(n) time where n is the number of days in the price array. Both the 4‑state dynamic programming method and the left/right profit array technique require a single linear pass (or two linear passes) through the array.

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