Best Time to Buy and Sell Stock II

“Best Time to Buy and Sell Stock II” involves making as many transactions as you like (i.e., buy one and sell one share of the stock multiple times) to maximize profit, given a list of stock prices for different days. The task is to find a strategy that maximizes the total profit.

A problem that can be considered isomorphic, in terms of problem-solving strategy and not the actual problem context, is “Maximum Subarray” (LeetCode #53). In this problem, you are given an integer array nums, and you need to find a contiguous subarray (containing at least one number) which has the largest sum.

Here’s the reasoning:

  1. In “Best Time to Buy and Sell Stock II”, you want to maximize the total profit by buying and selling the stock. Essentially, you add all positive differences between the price of the stock for two consecutive days (i.e., sell it today if the price is higher than yesterday). This effectively translates to adding all positive integers in a subarray in “Maximum Subarray”.

  2. In “Maximum Subarray”, you want to find a contiguous subarray that has the largest sum. This problem requires a similar strategy—add all positive integers in a subarray—to solve it.

So, while the context of the problems is quite different—one deals with stock prices and the other deals with subarray sums—the problem-solving strategy applied to them is quite similar. Therefore, they are isomorphic in terms of problem-solving strategy.

“Best Time to Buy and Sell Stock II” does not have constraints on the number of transactions, whereas the subarray problem might have constraints on the length of the subarray, which could slightly modify the approach you would take to solve it.

“Best Time to Buy and Sell Stock II” is a problem that involves understanding the stock market transactions with the ability to execute as many transactions as desired (buy one and sell one share of the stock multiple times). Here are 10 problems to prepare for it:

  1. 121. Best Time to Buy and Sell Stock: This problem is the simpler version where you are only allowed to complete at most one transaction.

  2. 53. Maximum Subarray: Understanding this problem is key to realizing that the problem of buying and selling stock can be transformed into a maximum subarray problem.

  3. 122. Best Time to Buy and Sell Stock III: This is a slightly more complex version where you are only allowed to complete at most two transactions.

  4. 198. House Robber: This problem involves choosing the maximum amount of money, similar to choosing the maximum profit in stock transactions.

  5. 134. Gas Station: This problem also requires an understanding of circular sequences and choosing the right starting point, much like choosing when to buy and sell stocks.

  6. 189. Rotate Array: This problem is about array manipulation, which is helpful to understand for solving stock problems.

  7. 217. Contains Duplicate: Understanding this problem can help with grasping the concept of handling and processing arrays.

  8. 283. Move Zeroes: This problem is about array manipulation, which can help in solving stock problems.

  9. 628. Maximum Product of Three Numbers: This problem helps you practice array manipulation and finding maximum values.

  10. 714. Best Time to Buy and Sell Stock with Transaction Fee: This problem involves a similar scenario as buying and selling stocks but includes a transaction fee.

These involve array manipulation and dynamic programming, which are beneficial for tackling “Best Time to Buy and Sell Stock II”.

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class Solution:
    def maxProfit(self, prices: List[int]) -> int:
        @cache
        def trade(day_d):
            if day_d == 0:
                # Hold on day_#0 = buy stock at the price of day_#0
                # Not-hold on day_#0 = doing nothing on day_#0
                return -prices[day_d], 0
            
            prev_hold, prev_not_hold = trade(day_d-1)
            
            hold = max(prev_hold, prev_not_hold - prices[day_d] )
            not_hold = max(prev_not_hold, prev_hold + prices[day_d] )
            
            return hold, not_hold
        
        last_day= len(prices)-1
        
        # Max profit must come from not_hold state (i.e., no stock position) on last day
        return trade(last_day)[1]

Clarification Questions

What are the clarification questions we can ask about this problem?

Identifying Problem Isomorphism

Can you help me with finding the isomorphism for this problem?

Which problem does this problem map to the corresponding isomorphic problem on Leetcode ?

Language Agnostic Coding Drills

Here are the learning units that the code can be broken down into, arranged from simpler to more complex concepts:

  1. Variable Assignments and Arithmetic Operations: Basic knowledge of how to assign values to variables and how to perform simple arithmetic operations.

  2. Lists: Understanding how to define lists and access their elements, and understanding the length of a list.

  3. Conditional Statements (if): Learning how to use if statements to control program flow based on certain conditions.

  4. Defining Functions: Understanding how to define a function and how to call it, along with the concept of function recursion.

  5. Decorators: Learning what decorators are and how to use them. In this case, the @cache decorator is used to improve the efficiency of the recursive function.

  6. Tuples: Understanding how to use tuples to store multiple values and how to access the individual elements in a tuple.

  7. Classes and Methods: Learning how to define a class and a method within the class.

  8. Type Hinting: Understanding how to use type hinting to specify the expected type of function arguments and the return type.

  9. Dynamic Programming: An understanding of the dynamic programming approach to problem solving, where sub-problems are solved and their solutions stored for use in larger problems. This concept is key to understanding how the recursive function trade works in the provided code.

  10. Memoization: Understanding the concept of memoization, a technique often used in dynamic programming to store the results of expensive function calls and reusing them when the same inputs occur. This is being used here with the @cache decorator.

  11. Problem Solving and Algorithmic Thinking: An understanding of how to approach a problem in a systematic way, break it down into smaller parts, and translate the solution into a set of step-by-step instructions (an algorithm). This includes being able to understand the problem that the code is solving (maximizing profit from buying and selling stocks) and how it is solving it.

Targeted Drills in Python

Here are Python drills targeting each of the learning units identified:

  1. Variable Assignments and Arithmetic Operations:
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# Create variables a and b, assign them the values 5 and 2 respectively
# Then create a variable c that is the sum of a and b
# Finally, print the value of c
a = 5
b = 2
c = a + b
print(c)
  1. Lists:
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# Create a list called 'nums' with the elements 1, 2, 3, 4, 5
# Print the first element of 'nums'
# Print the length of 'nums'
nums = [1, 2, 3, 4, 5]
print(nums[0])
print(len(nums))
  1. Conditional Statements (if):
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# Create a variable 'x' with value 10
# Use an if statement to check if 'x' is greater than 5
# If 'x' is greater than 5, print 'x is greater than 5'
x = 10
if x > 5:
    print('x is greater than 5')
  1. Defining Functions:
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# Define a function 'add' that takes two parameters and returns their sum
# Call this function with the arguments 3 and 4, and print the result
def add(a, b):
    return a + b
print(add(3, 4))
  1. Decorators:
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# Define a simple decorator function that prints a message before and after a function is called
def my_decorator(func):
    def wrapper():
        print("Something is happening before the function is called.")
        func()
        print("Something is happening after the function is called.")
    return wrapper

# Apply this decorator to a function that just prints 'Hello'
@my_decorator
def say_hello():
    print("Hello!")

say_hello()
  1. Tuples:
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# Create a tuple 't' with the elements 1, 2, 3
# Print the first element of 't'
t = (1, 2, 3)
print(t[0])
  1. Classes and Methods:
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# Define a class 'Person' with an attribute 'name'
# And a method 'greet' that prints a greeting that includes the person's name
class Person:
    def __init__(self, name):
        self.name = name
    def greet(self):
        print(f"Hello, my name is {self.name}")

# Create an instance of Person with the name 'Alice' and call the greet method
p = Person('Alice')
p.greet()
  1. Type Hinting:
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# Define a function 'add' that takes two integers and returns their sum
# Use type hinting to indicate the type of the parameters and the return value
def add(a: int, b: int) -> int:
    return a + b
print(add(3, 4))
  1. Dynamic Programming:
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# Implement a dynamic programming solution to calculate the nth Fibonacci number
def fibonacci(n: int, dp: list) -> int:
    if n <= 1:
        return n
    if dp[n] != -1:
        return dp[n]
    dp[n] = fibonacci(n - 1, dp) + fibonacci(n - 2, dp)
    return dp[n]

n = 10
dp = [-1 for _ in range(n+1)]
print(fibonacci(n

, dp))
  1. Memoization:
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# Use the functools.lru_cache decorator to implement a memoized recursive function
# that calculates the nth Fibonacci number
from functools import lru_cache

@lru_cache(maxsize=None)
def fibonacci(n):
    if n < 2:
        return n
    return fibonacci(n - 1) + fibonacci(n - 2)

print(fibonacci(10))
  1. Problem Solving and Algorithmic Thinking:
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# Implement a function that takes a list of numbers and a target number
# The function should return two numbers from the list that add up to the target, or None if no such pair exists
def two_sum(nums, target):
    seen = {}
    for num in nums:
        complement = target - num
        if complement in seen:
            return complement, num
        seen[num] = True
    return None

print(two_sum([2, 7, 11, 15], 9))