Homework 9 | CS 88 Fall 2020

Instructions

Download hw09.zip. Inside the archive, you will find starter files for the questions in this homework, along with a copy of the OK autograder.

Submission: When you are done, submit with python3 ok --submit. You may submit more than once before the deadline; only the final submission will be scored. Check that you have successfully submitted your code on okpy.org. See this article for more instructions on okpy and submitting assignments.

Readings: This homework relies on following references:

Recall that the order of growth of a function expresses how long it takes for the function to run, and is defined in terms of the function's input sizes.

For example, let's say that we have the function get_x which is defined as follows:

def get_x(x):
    return x

get_x has one expression in it. That one expression takes the same amount of time to run, no matter what x is, or more importantly, how large x gets. This is called constant time, or O(1).

The main two ways that a function in your program will get a running time different than just constant time is through either iteration or recursion. Let's start with some iteration examples!

The (simple) way you figure out the running time of a particular while loop is to simply count the cost of each operation in the body of the while loop, and then multiply that cost by the number of times that the loop runs. For example, look at the following method with a loop in it:

def foo(n):
    i, sum = 1, 0
    while i <= n:
        sum,i = sum + i, i + 1
    return sum

This loop has one statement in it sum, i = sum + i, i + 1. This statement is considered to run in constant time, as none of its operations rely on the size of the input. Individually, sum = sum + 1 and i = i + 1 are both constant time operations. However, when we're looking at order of growth, we don't add the times together and get O(2), we take the maximum of those 2 values and use that as the running time. In 61A, we are not concerned with how long primitive functions, such as addition, multiplication, and variable assignment, take in order to run - we are mainly concerned with how many more times a loop is executed or how many more recursive calls occur as the input increases. In this example, we execute the loop n times, and for each iteration, we only execute constant time operations, so we get an order of growth of O(n).

Here are a couple of basic functions, along with their running times. Try to understand why they have the given running time.

O(n)

def bar(n):
    i, a, b = 1, 1, 0
    while i <= n:
        a, b, i = a + b, a, i + 1
     return a

O(n^2)

def bar(n):
    sum = 0
    a, b = 0, 0
    while a < n:
        while b < n:
            sum += (a*b)
            b += 1
        b = 0
        a += 1
    return sum

Efficiency

There is nothing to submit for this part. But doing these problems will be good practice. The solutions are given right below the question. Try covering the solution and see if you can solve the them!

For each question find the asymptotic runtime in big theta notation.

Question 1

What is the asymptotic run time of the baz function.

def baz(n):
    i, sum = 1, 0
    while i <= n:
        sum += bam(i)
        i += 1
    return sum

def bam(n):
    i, sum = 1, 0
    while i <= n:
        sum += i
        i += 1
    return sum

O(n²)

Question 2

def bonk(n):
    sum = 0
    while n >= 2:
        sum += n
        n = n / 2
    return sum

O(log(n))

Question 3

This question is very challenging. This is much beyond what we expect you to know for the exam. This is here merely to challenge you.

def boink(n):
    if n == 1:
        return 1
    sum = 0
    i = 1
    while i <= n:
        sum += boink(i)
        i += 1
    return sum

O(2ⁿ)

OOP

Question 4: Mint

Complete the Mint and Coin classes so that the coins created by a mint have the correct year and worth.

Each Mint instance has a year stamp. The update method sets the year stamp to the current_year class attribute of the Mint class.
The create method takes a subclass of Coin and returns an instance of that class stamped with the mint's year (which may be different from Mint.current_year if it has not been updated.)
A Coin's worth method returns the cents value of the coin plus one extra cent for each year of age beyond 50. A coin's age can be determined by subtracting the coin's year from the current_year class attribute of the Mint class.

Use OK to test your code:

python3 ok -q Mint

class Mint:
    """A mint creates coins by stamping on years.

    The update method sets the mint's stamp to Mint.current_year.

    >>> mint = Mint()
    >>> mint.year
    2020
    >>> dime = mint.create(Dime)
    >>> dime.year
    2020
    >>> Mint.current_year = 2100  # Time passes
    >>> nickel = mint.create(Nickel)
    >>> nickel.year     # The mint has not updated its stamp yet
    2020
    >>> nickel.worth()  # 5 cents + (80 - 50 years)
    35
    >>> mint.update()   # The mint's year is updated to 2100
    >>> Mint.current_year = 2175     # More time passes
    >>> mint.create(Dime).worth()    # 10 cents + (75 - 50 years)
    35
    >>> Mint().create(Dime).worth()  # A new mint has the current year
    10
    >>> dime.worth()     # 10 cents + (155 - 50 years)
    115
    >>> Dime.cents = 20  # Upgrade all dimes!
    >>> dime.worth()     # 20 cents + (155 - 50 years)
    125
    >>> m = Mint()
    >>> n = m.create(Nickel)
    >>> n.worth()
    5
    >>> n.year = 2015
    >>> n.worth()
    115
    """
    current_year = 2020

    def __init__(self):
        self.update()

    def create(self, kind):
        "*** YOUR CODE HERE ***"

    def update(self):
        "*** YOUR CODE HERE ***"

class Coin:
    def __init__(self, year):
        self.year = year

    def worth(self):
        "The worth is a coin's face value + 1 cent for each year over age 50."
        "*** YOUR CODE HERE ***"

class Nickel(Coin):
    cents = 5

class Dime(Coin):
    cents = 10

Question 5: Checking account

We'd like to be able to cash checks, so let's add a deposit_check method to our CheckingAccount class. It will take a Check object as an argument, and check to see if the payable_to attribute matches the CheckingAccount's holder. If so, it marks the Check as deposited, and adds the amount specified to the CheckingAccount's total.

Write an appropriate Check class, and add the deposit_check method to the CheckingAccount class. Make sure not to copy and paste code! Use inheritance whenever possible.

See the doctests for examples of how this code should work.

The Account class has been provided.

class CheckingAccount(Account):
    """A bank account that charges for withdrawals.

    >>> check = Check("Steven", 42)  # 42 dollars, payable to Steven
    >>> steven_account = CheckingAccount("Steven")
    >>> eric_account = CheckingAccount("Eric")
    >>> eric_account.deposit_check(check)  # trying to steal steven's money
    The police have been notified.
    >>> eric_account.balance
    0
    >>> check.deposited
    False
    >>> steven_account.balance
    0
    >>> steven_account.deposit_check(check)
    42
    >>> check.deposited
    True
    >>> steven_account.deposit_check(check)  # can't cash check twice
    The police have been notified.
    """
    withdraw_fee = 1
    interest = 0.01

    def withdraw(self, amount):
        return Account.withdraw(self, amount + self.withdraw_fee)



class Check(object):
    "*** YOUR CODE HERE ***"